Categorical, Homological and Combinatorial Methods in Algebra 9781470443689, 1470443686

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751

Categorical, Homological and Combinatorial Methods in Algebra AMS Special Session in honor of S.K. Jain’s 80th Birthday Categorical, Homological and Combinatorial Methods in Algebra March 16–18, 2018 Ohio State University, Columbus, Ohio

Ashish K. Srivastava André Leroy Ivo Herzog Pedro A. Guil Asensio Editors

Categorical, Homological and Combinatorial Methods in Algebra AMS Special Session in honor of S.K. Jain’s 80th Birthday Categorical, Homological and Combinatorial Methods in Algebra March 16–18, 2018 Ohio State University, Columbus, Ohio

Ashish K. Srivastava André Leroy Ivo Herzog Pedro A. Guil Asensio Editors

751

Categorical, Homological and Combinatorial Methods in Algebra AMS Special Session in honor of S.K. Jain’s 80th Birthday Categorical, Homological and Combinatorial Methods in Algebra March 16–18, 2018 Ohio State University, Columbus, Ohio

Ashish K. Srivastava André Leroy Ivo Herzog Pedro A. Guil Asensio Editors

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

2010 Mathematics Subject Classification. Primary 16-06, 18-06.

Library of Congress Cataloging-in-Publication Data Names: American Mathematical Society. Sectional meeting (2018: Columbus, Ohio) | Jain, S. K. (Surender Kumar), 1938– honoree. | Srivastava, Ashish K., editor. | Leroy, Andre (Andre Gerard), 1955– editor. | Herzog, Ivo, 1961– editor. | Guil Asensio, Pedro A., 1964– editor. Title: Categorical, homological and combinatorial methods in algebra: AMS sectional meeting in honor of S.K. Jain’s 80th birthday: Categorical, homological and combinatorial methods in algebra, March 16–18, 2018, Ohio State University, Columbus, Ohio / Ashish K. Srivastava, Andre Leroy, Ivo Herzog, Pedro A. Guil Asensio, editors. Description: Providence, Rhode Island: American Mathematical Society, [2020] | Series: Contemporary mathematics, 0271-4132; volume 751 | Includes bibliographical references. Identifiers: LCCN 2019056297 | ISBN 9781470443689 (paperback) | ISBN 9781470456085 (ebook) Subjects: LCSH: Jain, S. K. (Surender Kumar), 1938—Congresses. | Categories (Mathematics)– Congresses. | Algebra, Homological–Congresses. | Combinatorial analysis–Congresses. | AMS: Associative rings and algebras – Proceedings, conferences, collections, etc. | Category theory; homological algebra Classification: LCC QA169 .A474 2020 | DDC 512/.6–dc23 LC record available at https://lccn.loc.gov/2019056297 DOI: https://doi.org/10.1090/conm/751

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

25 24 23 22 21 20

Dedicated to Prof. S. K. Jain on his 80th Birthday

Contents

Preface

ix

On the Morita equivalence class of a finitely presented algebra Adel Alahmadi, Hamed Alsulami, and Efim Zelmanov

1

Quillen–Suslin theory for classical groups: Revisited over graded rings Rabeya Basu and Manish Kumar Singh

5

Natural elements of center of generalized quantum groups Punita Batra and Hiroyuki Yamane

19

The global dimension of the generalized Weyl algebras S −1 K[H, C][X, Y ; σ, a] V. V. Bavula and K. Alnefaie 33 Quasi-quantum groups obtained from the Tannaka-Krein reconstruction theorem D. Bulacu and B. Torrecillas

61

Homotopy categories of totally acyclic complexes with applications to the flat–cotorsion theory Lars Winther Christensen, Sergio Estrada, and Peder Thompson

99

Tor-pairs: Products and approximations Manuel Cort´ es-Izurdiaga

119

Model structures and relative Gorenstein flat modules and chain complexes Sergio Estrada, Alina Iacob, and Marco A. P´ erez

135

Strongly flat modules and right noetherian chain domains Alberto Facchini and Zahra Nazemian

177

Structure theory of graded regular graded self-injective rings and applications Roozbeh Hazrat, Kulumani M. Rangaswamy, and Ashish K. Srivastava 191 On the bijectivity of the antipode and serial quantum groups Miodrag Cristian Iovanov

205

Categories with negation Jaiung Jun and Louis Rowen

221

Quasi-Baer module hulls and their examples Jae Keol Park and S. Tariq Rizvi

271

vii

viii

CONTENTS

Strict Mittag-Leffler modules and purely generated classes Philipp Rothmaler

303

Rings over which cyclic modules are almost self-injective Surjeet Singh

329

On isoclasses of maximal subalgebras determined by automorphisms Alex Sistko

341

Preface This volume contains the proceedings of the Special Session on Categorical, Homological and Combinatorial Methods in Algebra in the Sectional Meeting of the American Mathematical Society held on March 16-18, 2018, at the Ohio State University, Columbus, Ohio. This Special Session was dedicated to the 80th birthday celebrations of Professor S. K. Jain. The speakers at the Special Session included Fields medalist Professor Efim Zelmanov. Just like the session itself, the volume is dedicated to Professor Jain. S. K. Jain is a prolific researcher and has contributed immensely to the study of noncommutative rings and modules over them. We feel honored to do our small part to celebrate his life and accomplishments. The articles of this volume aim to showcase the current state of art in categorical, homological and combinatorial aspects of algebra. For the most part, contributors to this volume delivered related talks at the Special Session. All papers were subject to a rigorous process of refereeing. We are thankful to all the participants and, in particular, to every contributor to this volume. We would like to thank all the anonymous referees for their careful reports submitted in a timely manner. Finally, we would like to express our gratitude to Christine Thivierge for her help and support in the preparation of this volume.

Ashish K. Srivastava Andr´e Leroy Ivo Herzog Pedro A. Guil Asensio

ix

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15118

On the Morita equivalence class of a finitely presented algebra Adel Alahmadi, Hamed Alsulami, and Efim Zelmanov To our teacher and friend S. K. Jain

Let F be a field and let A, B be F -algebras. The algebras A, B are Morita equivalent if their categories of (left) modules are equivalent. Throughout the paper we consider algebras with 1 and unital modules over them. Then A is Morita equivalent to B if and only if A ∼ = eMn (B)e, where Mn (B) is the algebra of n × n matrices over B and e = e2 ∈ Mn (B) is a full idempotent, i.e. Mn (B)eMn (B) = Mn (B). In [2], [3], [4] the authors described Morita equivalence classes of some important finitely presented algebras and, in particular, showed that these classes are countable. In this note we discuss Morita equivalence classes of arbitrary finitely presented algebras. Let σ be an automorphism of the field F . The mapping ϕ : A → B is called a σ-semilinear isomorphism if ϕ is a ring isomorphism and ϕ(αa) = σ(α)ϕ(a) for all α ∈ F , a ∈ A. Let V be a vector space over the field F . On the abelian group V we define a new multiplication by scalars: α · v = σ(α)v. We will denote the new vector space as V (σ) . For an F -algebra A define a new F -algebra structure A(σ) on A by keeping the ring structure and switching to the vector space A(σ) instead of A. −1 It is easy to see that the identical mapping A → A(σ ) is a σ-semilinear isomorphism. Suppose that an F -algebra A is generated by a finite collection of elements a1 , . . . , am . Consider the free associative algebra F x1 , . . . , xm  and the homomorphism ϕ F x1 , . . . , xm  −→ A, xi → ai , 1 ≤ i ≤ m. Let R be a subset of the ideal I = ker ϕ that generates I as an ideal. We say that the algebra A has presentation A = x1 , . . . , xm | R = 0. If the set R is finite then the algebra A is said to be finitely presented. This property does not depend on a choice of a generating system as long as the system is finite. 2010 Mathematics Subject Classification. Primary 20E18, 20F50, 20F40, 16R99. Key words and phrases. Morita equivalence, finitely presented algebra. The third author gratefully acknowledges the support from the NSF grant 1601920. c 2020 American Mathematical Society

1

2

ADEL ALAHMADI, HAMED ALSULAMI, AND EFIM ZELMANOV

It is easy to see that semilinearly isomorphic algebras are Morita equivalent. Hence the group Aut F acts on the class of Morita equivalence of the algebra A. Theorem 1. Let F be an algebraically closed field and let A be a finitely presented F -algebra. Then the action of Aut F on the class of Morita equivalence of A has countably many orbits. In other words, the Morita equivalence class is countable up to semilinear isomorphisms. The following observation is straightforward. Lemma 1. Let A =x1 , . . . , xm |

 j

αij wij = 0, 1 ≤ i ≤ n, αij ∈ F,

wij are words in x1 , . . . , xm . Then A(σ) = x1 , . . . , xm |



σ −1 (αij )wij = 0, 1 ≤ i ≤ n.

j

In [1] it was shown that finite presentation is a Morita invariant property. In other words, the Morita equivalence class of a finitely presented F -algebra A consists of finitely presented F -algebras. If the field F is countable then it immediately follows that the Morita equivalence class of A is countable up to isomorphisms. Assume therefore that the field F is not countable. Proof of Theorem 1. Let F0 be the prime subfield of F . Let X ⊂ F be a maximal subset of F that is algebraically independent over F0 . Then F is the algebraic closure of the purely transcendental extension F0 (X). Since the field F is uncountable it follows that the set X is uncountable as well. Let X0 be a countable subset of X. Let F0 (X0 ) be the purely transcendental F0 -extension generated by X0 and let F 0 (X0 ) be the algebraic closure of F0 (X0 ) in F. We claim that for any finite collection of elements α1 , . . . , αn ∈ F there exists an automorphism σ ∈ Aut F that maps α1 , . . . , αn to F 0 (X0 ). Indeed, there exists a   finite subset X ⊂ X such that α1 , . . . , αn ∈ F 0 (X ). Let p : X → X be a bijection such that p(X  ) ⊂ X0 . The bijection p extends to an automorphism σ ∈ Aut F . Clearly, σ(αi ) ∈ F 0 (X0 ). We say that an algebra is finitely presented over a subfield K ⊂ F if it has a presentation x1 , . . . , xm | R = 0, |R| < ∞, R ⊂ Kx1 , . . . , xm . If B is a finitely presented F -algebra then there exists an automorphism τ ∈ Aut F such that B (τ ) is finitely presented over F 0 (X0 ). Indeed, let  B =x1 , . . . , xm | αij wij = 0, 1 ≤ i ≤ n, αij ∈ F, j

wij are words in x1 , . . . , xm . There exists an automorphism σ ∈ Aut F such that σ(αij ) ∈ F 0 (X0 ) for all i, j. By (σ −1 ) is finitely presented over F Lemma 1 it implies that B 0 (X0 ).

MORITA EQUIVALENCE CLASS OF FINITELY PRESENTED ALGEBRA

3

An arbitrary algebra B that is Morita equivalent to A is finitely presented [1]. Hence there exists τ ∈ Aut F such that B (τ ) is finitely presented over F 0 (X0 ). Since the field F 0 (X0 ) is countable it completes the proof of Theorem 1.



Now we will show that, generally speaking, the class of Morita equivalence of a finitely presented algebra may be uncountable up to isomorphisms. Lemma 2. Let α ∈ F . Consider the algebra Aα = x1 , x2 | x21 + x22 + αx1 x2 = 0. Then Aα ∼ = Aβ if and only if β = ±α. Proof. Clearly, Aα ∼ = A−α . Suppose now that β = ±α. We will show that the algebra Aα does not contain generators y1 , y2 such that y12 + y22 + βy1 y2 = 0. Suppose the contrary and let y1 = α10 + α11 x1 + α12 x2 + y1 y2 = α20 + α21 x1 + α22 x2 + y2 be such generating elements; αij ∈ F ; y1 , y2 are linear combinations of monomials of length ≥ 2. The matrix   α11 α12 Q= α21 α22 is nonsingular, otherwise the elements y1 , y2 do not generate Aα . The linear part of y12 + y22 + βy1 y2 is equal to 2α10 (α11 x1 + α12 x2 ) + 2α20 (α21 x1 + α22 x2 ) + βα10 (α21 x1 + α22 x2 ) + βα20 (α11 x1 + α12 x2 ) = (2α10 α11 + 2α20 α21 + βα10 α21 + βα20 α11 )x1 + (2α10 α12 + 2α20 α22 + βα10 α22 + βα20 α21 )x2 = 0. Hence α10 (2α11 + βα21 ) + α20 (2α21 + βα11 ) = 0, α10 (2α12 + βα22 ) + α20 (2α22 + βα12 ) = 0. We have

 2α11 + βα21 2α12 + βα22

2α21 + βα11 2α22 + βα12



 =

2 β

β 2



α11 α21

 α12 . α22

If β = ±2 then this matrix is nonsingular, which implies α10 = α20 = 0. Suppose that this is the case, i.e. β = ±2, α10 = α20 = 0. The homogeneous component of degree 2 of y12 + y22 + βy1 y2 is equal to (α11 x1 + α12 x2 )2 + (α21 x1 + α22 x2 )2 + β(α11 x1 + α12 x2 )(α21 x1 + α22 x2 ) = 0.   1 β Let x = (x1 , x2 ). Then the left hand side is equal to xQT QxT . Since this 0 1 expression is equal to zero in the algebra x1 , x2 | x21 + x22 + αx1 x2 = 0

4

ADEL ALAHMADI, HAMED ALSULAMI, AND EFIM ZELMANOV

it follows that in the free algebra it is equal to γ(x21 + x22 + αx1 x2 ), γ ∈ F . Hence,     1 β 1 α T Q Q=γ . 0 1 0 1 It is well-known in the theory of quadratic forms that this equality implies β = ±α. We proved that β = ±2 or α = ±β. Similarly, α = ±2 or β = ±α. Thus, if α = ±β then α = ±2 and β = ±2 which again implies that α = ±β. This completes the proof of the lemma.  Corollary. If a complex number α ∈ C is not algebraic over Q then the Morita equivalence class of the algebra Aα is uncountable. Indeed, if α is not algebraic over Q then the orbit O = {σ(α), α ∈ Aut C} is uncountable. The system {Aβ , β ∈ O} lies in the Morita equivalence class of Aα and is uncountable by Lemma 1. Question. Let A be a C-algebra that is finitely presented over Q. Is it true that the Morita equivalence class of A is countable? Acknowledgement The authors are grateful to F. Eshmatov for helpful discussions. References [1] A. Alahmadi, H. Alsulami, Finite presentation is a Morita invariant property, Proc. Amer. Math. Soc., to appear. [2] Yuri Berest and George Wilson, Automorphisms and ideals of the Weyl algebra, Math. Ann. 318 (2000), no. 1, 127–147, DOI 10.1007/s002080000115. MR1785579 [3] X. Chen, A. Eshmtov, F. Eshmatov, V. Futorny, Automorphisms and ideals of noncommutatie deformations of C2 /Z2 , Arxiv preprint, 2016. [4] S. P. Smith, A class of algebras similar to the enveloping algebra of sl(2), Trans. Amer. Math. Soc. 322 (1990), no. 1, 285–314, DOI 10.2307/2001532. MR972706 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia Email address: [email protected] Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia Email address: [email protected] Department of Mathematics, University of California, San Diego, USA Email address: [email protected]

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15113

Quillen–Suslin theory for classical groups: Revisited over graded rings Rabeya Basu and Manish Kumar Singh Abstract. In this paper we deduce a graded version of Quillen–Suslin’s LocalGlobal Principle for the traditional classical groups, viz. general linear, symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This generalizes previous result of Basu–Rao–Khanna. Then, as an application, we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups. Finally, by using Swan–Weibel’s homotopy trick, we establish graded analogue of the Local-Global Principle for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes the previous result of Bak–Basu–Rao.

Introduction In [RRK], the first author with Ravi A. Rao and Reema Khanna revisited Quillen-Suslin’s Local-Global Principle for the linear, symplectic and orthogonal groups to show the equivalence of the Local-Global Principle and Normality property of their Elementary subgroups. A “relative version” is also established recently by the same authors; cf. [RRK2]. In [RRK], the authors have also deduced a Local-Global Principle for the commutator subgroups of the special linear groups, and symplectic groups. In this article we aim to revisit those results over commutative N-graded rings with identity. We follow the line of proof described in [RRK]. For the linear case, the graded version of the Local-Global Principle was studied by Chouinard in [C], and by J. Gubeladze in [G1], [G2]. Though the analogue results are expected for the symplectic and orthogonal groups, according to our best knowledge, that is not written explicitly in any existing literatures. By deducing the equivalence, we establish the graded version of the Local-Global Principle for the above two types of classical groups. To generalize the existing results, from the polynomial rings to the graded rings, one has to use the line of proof of Quillen–Suslin’s patching argument (cf. [Q], [S]), and Swan–Weibel’s homotopy trick. For a nice exposition we refer to [G1] by J. Gubeladze. 2010 Mathematics Subject Classification. Primary 11E57, 13A02, 13B25, 15B99; Secondary 13B99, 13C10, 13C99. Key words and phrases. Bilinear forms, symplectic and orthogonal forms, graded rings. Research by the first author was supported by SERB-MATRICS grant (File No. MTR/2017/000886) for the financial year 2018–2019. 5

6

RABEYA BASU AND MANISH KUMAR SINGH

Though, we are writing this article for commutative graded rings, one may also consider standard graded algebras which are finite over its center. For a commutative N-graded ring with 1, we establish: (1) (Theorem 2.8) Normality of the elementary subgroups is equivalent to the graded Local-Global Principle for the linear, symplectic and orthogonal groups. (2) (Theorem 3.8) Analogue of Quillen–Suslin’s Local-Global Principle for the commutator subgroups of the special linear and symplectic groups over graded rings. In [BBR], the following was established by the first author with A. Bak and R.A. Rao: Local-Global Principle for the Transvection Subgroups: An analogue of Quillen–Suslin’s Local-Global Principle for the transvection subgroup of the automorphism group of projective, symplectic and orthogonal modules of global rank at least 1 and local rank at least 3, under the assumption that the projective module has constant local rank and that the symplectic and orthogonal modules are locally an orthogonal sum of a constant number of hyperbolic planes. In this article, we observe that by using Swan–Weibel’s homotopy trick, one gets an analogue statement for the graded case. More precisely, we deduce the following fact: ∞ (3) (Theorem 4.4) Let A = i=0 Ai be a graded ring and Q  P ⊕ A0 be a projective A0 -module (Q  P ⊕ H(A0 ) for symplectic and orthogonal modules). If an automorphism of Q ⊗A0 A is locally in the transvection subgroup, then it is globally in the transvection subgroup. 1. Definitions and Notations Let us start by recalling the following well-known fact: Given a ring R and a subring R0 , one can express R as a direct limit of subrings which are finitely generated over R0 as rings. Considering R0 to be the minimal subring (i.e. the image of Z), it follows that every ring is a direct limit of Noetherian rings. Hence we may consider R to be Noetherian (cf. Pg 271 [M]). Throughout this paper we assume A to be a Noetherian, commutative graded ring with identity 1. We shall write A = A0 ⊕ A1 ⊕ A2 ⊕ · · · . As we know the multiplication in a graded ring satisfies the following property: For all i, j, Ai Aj ⊂ Ai+j . An element a ∈ A will be denoted by a = a0 + a1 + a2 + · · · , where ai ∈ Ai for each i, and and all but finitely many ai are zero. Let A+ = A1 ⊕A2 ⊕· · · . Graded structure of A induces a graded structure on Mn (A) (ring of n×n matrices). Let S be a multiplicatively closed subset of A0 . Then for a non-zero divisor s ∈ S we shall denote the localization of a matrix α ∈ Mn (A) to Mn (As ) as αs . Otherwise, αi , i ∈ Z will represent the i-th component of α. We shall use standard notations GLn (A) and SLn (A) to denote the group of invertible matrices and its subgroup of all invertible matrices with determinant 1 respectively. We recall the well-known “Swan–Weibel’s homotopy trick”, which is the main ingredient to handle the graded case. Definition 1.1. Let a ∈ A0 be a fixed element. We fix an element b = b0 + b1 + · · · in A and define a ring homomorphism  : A → A[X] given by (b) = (b0 + b1 + · · · ) = b0 + b1 X + b2 X 2 + · · · + bi X i + · · · .

QUILLEN–SUSLIN THEORY FOR CLASSICAL GROUPS

7

Then we evaluate the polynomial (b)(X) at X = a and denote the image by  + b (a) i.e. b+ (a) = (b)(a). Note that b+ (x) (y) = b+ (xy). Observe, b0 = b+ (0). We shall use this fact frequently. +

The above ring homomorphism  induces a group homomorphism at the GLn (A) level for every n ≥ 1, i.e. for α ∈ GLn (A) we get a map  : GLn (A) → GLn (A[X]) defined by α = α0 ⊕ α1 ⊕ α2 ⊕ · · · → α0 ⊕ α1 X ⊕ α2 X 2 · · · , where αi ∈ Mn (Ai ). As above for a ∈ A0 , we define α+ (a) as α+ (a) = (α)(a). Now we are going to recall the definitions of the traditional classical groups, viz. the general linear groups, the symplectic and orthogonal groups (of even size) and their type subgroups, viz. elementary (symplectic and orthogonal elementary resp.) subgroups. Let eij be the matrix with 1 in the ij-position and 0’s elsewhere. The matrices of the form {Eij (λ) : λ ∈ A | i = j}, where Eij (λ) = In + λeij are called the elementary generators. Definition 1.2. The subgroup generated by the set {Eij (λ) : λ ∈ A | i = j}, is called the elementary subgroup of the general linear group and is denoted by En (A). Observe that En (A) ⊆ SLn (A) ⊆ GLn (A). Let σ be a permutation defined by: For i ∈ {1, . . . , 2m} σ(2i) = 2i − 1 and σ(2i − 1) = 2i. With respect to this permutation, we define two 2m × 2m forms (viz.) ψm and ψm as follows: For m > 1, let



0 1 ψ 0 and for m > 1, ψm = m−1 ψ1 = , −1 0 0 I2



0 1 0 ψ and for m > 1, ψm = m−1 . ψ1 = 1 0 0 I2 Using above two forms, we define the following traditional classical groups: Definition 1.3. A matrix α ∈ GL2m (A) is called symplectic if it fixes ψm under the action of conjugation, i.e. αt ψm α = ψm . The group generated by all symplectic matrices is called the symplectic group and is denoted by Sp2m (A) = Spn (A), where n = 2m. Definition 1.4. A matrix α ∈ GL2m (A) is called orthogonal if it fixes ψm under the action of conjugation, i.e. αt ψm α = ψm . The group generated by all orthogonal matrices is called orthogonal group and is denoted by O2m (A) = On (A), where n = 2m.

8

RABEYA BASU AND MANISH KUMAR SINGH

Definition 1.5. The matrices of the form {seij (z) ∈ GL2m (A) : z ∈ A | i = j}, where seij (z) = I2m + zeij if i = σ(j) or seij (z) = I2m + zeij − (−1)i+j zeσ(j)σ(i) if i = σ(j) and i < j. are called symplectic elementary generators. The subgroup generated by symplectic elementary generators is called symplectic elementary group. Definition 1.6. The matrices of the form {oeij (z) : z ∈ A | i = j}, where oeij (z) = I2m + zeij − zeσ(j)σ(i) , if i = σ(j) and i < j. are called orthogonal elementary generators. The subgroup generated by orthogonal elementary generators is called orthogonal elementary group. Remark 1.7. It is a well known fact that the elementary subgroups are normal subgroups of the respective classical groups; for n ≥ 3 in the linear case, n = 2m ≥ 4 in the symplectic case, and n = 2m ≥ 6 in the orthogonal case. The linear case was due to A. Suslin (cf. [T]), symplectic was proved by V. Kopeiko (cf. [T]). Finally the orthogonal case was proved by Suslin–Kopeiko (cf. [SK]). We use the following notations to treat above three groups uniformly: G(n, A) will denote GLn (A) or Sp2m (A) or O2m (A). E(n, A) will denote En (A) or EO2m (A) or ESp2m (A). S(n, A) will denote SLn (A) or SO2m (A), and geij will denote the elementary generators of En (A) or ESpn (A)or EOn (A). Here SOn (A) is the subgroup of On (A) with determinant 1. Throughout the paper we will assume n = 2m to treat the non-linear cases. We shall assume n ≥ 3 while treating the linear case, and n ≥ 6; i.e. m ≥ 3 while treating the symplectic and orthogonal cases. Let  vψm in the symplectic case, and v = v ψm in the orthogonal case. Definition 1.8. Let v, w ∈ An be vectors of length n, then we define the inner product v, w as follows: (1) v, w = v t w in the linear case, (2) v, w = vw in the symplectic and orthogonal cases. Definition 1.9. Let v, w ∈ A, we define the map M : An × An → Mn (A) as follows: (1) M (v, w) = vwt in the linear case, (2) M (v, w) = v w  + w v in the symplectic case, (3) M (v, w) = v w  − w v in the orthogonal case. By [G, G], we denote the commutator subgroup of G i.e. group generated by elements of the form ghg −1 h−1 , for g, h ∈ G.

QUILLEN–SUSLIN THEORY FOR CLASSICAL GROUPS

9

Definition 1.10. A row (v1 , . . . , vn ) ∈ An is called unimodular row of length n if there exist b1 , . . . , bn ∈ A such that Σni=1 vi bi = 1; i.e. If I is the ideal generated by v1 , . . . , vn , then (v1 , . . . , vn ) is unimodular if and only if I = A. In the above case we say A has n-cover. 2. Local-Global Principle for classical groups Before discussing the main theorem we recall following standard facts. Lemma 2.1 (Splitting Lemma). The elementary generators of n × n matrices satisfy the following property: geij (x + y) = geij (x) geij (y) for all x, y ∈ A and for all i, j = 1, . . . , n with i = j. Proof. Standard. (cf. [RRK], §3, Lemma 3.2).

 k

Lemma 2.2. Let G be a group, and ai , bi ∈ G for i = 1, . . . , r. Let Jk = Π aj . j=1

Then

r

r

r

i=1

i=1

i=1

Π ai bi = Π Ji bi Ji−1 Π ai .

Proof. cf. [RRK], §3, Lemma 3.4.



Following structural lemma pays a key role in the proof of main theorem. It is well-known for the polynomial rings; cf. [RRK] (§3, Lemma 3.6). Here, we deduce the analogue for the graded rings. Lemma 2.3. Let G(n, A, A+ ) denote the subgroup of G(n, A) which is equal to In modulo A+ . Then the group G(n, A, A+ ) ∩ E(n, A) is generated by the elements of the type  geij (A+ ) −1 for some  ∈ E(n, A0 ). Proof. Let α ∈ E(n, A) ∩ G(n, A, A+ ). Then we can write r

α = Π geik jk (ak ) k=1

for some elements ak ∈ A, k = 1, . . . , r. As ak = (a0 )k + (a+ )k for some (a0 )k ∈ A0 and (a+ )k ∈ A+ . Using the splitting lemma (Lemma 2.1) we can rewrite the expression as: r    α = Π geik jk (a0 )k geik jk (a+ )k . k=1   We write t = Πtk=1 geik jk (a0 )k for t ∈ {1, 2, . . . , r}. Observe that r = In , as α ∈ G(n, A, A+ ). Then r   r   Π geik jk (a0 )k = AB (say) , α = Π k geik jk (a+ )k −1 k k=1 k=1 r r   −1   where A = Π k geik ,jk (a+ )k k and B = Π geik jk (a0 )k . Now go modulo k=1

k=1

A+ . Let bar (–) denote the quotient ring modulo A+ . Then, α = In = AB = In B = ¯In =⇒ B = In , as α ∈ G(n, A, A+ ). Since entries of B are in A0 , it follows that B = In . r    α = Π k geik jk (a+ )k −1 k ; k=1

as desired.



10

RABEYA BASU AND MANISH KUMAR SINGH

Now we prove a variant of “Dilation Lemma” mentioned in the statement of (4) of Theorem 2.8. Lemma 2.4. Assume the “Dilation Lemma” (Theorem 2.8 - (4)) to be true. Let αs ∈ E(n, As ), with α+ (0) = In . Then one gets α+ (b + d)α+ (d)−1 ∈ E(n, A) for some s, d ∈ A0 and b = sl , l  0. Proof. We have αs ∈ E(n, As ). Hence αs+ (X) ∈ E(n, As [X]). Let β + (X) = −1 α (X + d)α+ (d) , where d ∈ A0 . Then βs+ (X) ∈ E(n, As [X]) and β + (0) = In .  Hence by Theorem 2.8 - (4) their exists β(X) ∈ E(n, A[X]) such that βs (X) = +  βs (bX). Putting X = 1, we get the required result. +

Following is a very crucial result we need for our method of proof. There are many places we use this fact in a very subtle way. In particular, we mainly use this lemma for the step (4) ⇒ (3) of 2.8. Lemma 2.5. (cf.[HV], Lemma 5.1) Let R be a Noetherian ring and s ∈ R. Then there exists a natural number k such that the homomorphism G(n, sk R) → G(n, Rs ) (induced by localization homomorphism R → Rs ) is injective. Moreover, it follows that the induced map E(n, R, sk R) → E(n, Rs ) is injective. The next two lemmas will be used in intermediaries to prove the equivalent conditions mentioned in Theorem 2.8. We state it without proof. Lemma 2.6. Let v = (v1 , v2 , . . . , vn ) be a unimodular row over over a commutative semilocal ring R. Then the row (v1 , v2 , . . . , vn ) is completable; i.e. their exists  ∈ En (R) such that (v1 , v2 , . . . , vn ) = (1, 0, . . . , 0). Proof. cf. [RB1], Lemma 1.2.21.



Lemma 2.7. Let A be a ring and v ∈ E(n, A)e1 . Let w ∈ An be a column vector such that v, w = 0. Then In + M (v, w) ∈ E(n, A). Proof. cf. [RB1], Lemma 2.2.4.



Theorem 2.8. The followings are equivalent for any graded ring A = ⊕∞ i=0 Ai for n ≥ 3 in the linear case and n ≥ 6 otherwise. (1) (Normality): E(n, A) is a normal subgroup of G(n, A). (2) If v ∈ Umn (A) and v, w = 0, then In + M (v, w) ∈ E(n, A). (3) (Local-Global Principle): Let α ∈ G(n, A) with α+ (0) = In . If for every maximal ideal m of A0 αm ∈ E(n, Am ), then α ∈ E(n, A). (4) (Dilation Lemma): Let α ∈ G(n, A) with α+ (0) = In and αs ∈ E(n, As ) for some non-zero divisor s ∈ A0 . Then there exists β ∈ E(n, A) such that βs+ (b) = αs+ (b) for some b = sl ; l  0. i.e. αs+ (sl ) will be defined over A for l  0. (5) If α ∈ E(n, A), then α+ (a) ∈ E(n, A), for every a ∈ A0 . (6) If v ∈ E(n, A)e1 , and v, w = 0, then In + M (v, w) ∈ E(n, A).

QUILLEN–SUSLIN THEORY FOR CLASSICAL GROUPS

11

t   Proof. (6) ⇒ (5): Let α = Π In + a M (eik , ejk ) where a ∈ A, and t ≥ 1, a k=1

positive integer. Then t   α+ (b) = Π In + a+ (b) M (eik , ejk ) , k=1

where b ∈ A0 . Take v = ei and w = a+ (b)ej , Then α = In +M (v, w) and v, w = 0. Indeed, Linear case: M(v, w) = vwt = ei (a+ (b)ej )t = a+ (b)ei etj = a+ (b) M(ei , ej ), v, w = v t w = eti (a+ (b)ej )t = a+ (b) eti ej = a+ (b)ei , ej  = 0. Symplectic case: M(v, w) = vwt ψm + wv t ψm = ei (a+ (b)ej )t ψm + a+ (b)ej eti ψm = a+ (b)ei etj ψm + a+ (b)ej eti ψm = a+ (b) M(ei , ej ), ei , ej  = eti ψm ej = (eti ψm ei )(eti ej ) = ψm (eti ej ) = 0 hence, v, w = v t ψm w = eti ψm a+ (b)ej = a+ (b)eti ej = a+ (b)ei , ej  = 0. Orthogonal case: M(v, w) = v.wt ψm − w.v t ψm = ei .(a+ (b)ej )t ψm − a+ (b)ej .eti ψm = a+ (b)ei .etj ψm − a+ (b)ej .eti ψm = a+ (b) M(ei , ej ), ei , ej  = eti ψm ej = (eti ψm ei )(eti ej ) = ψm (eti ej ) = 0 hence, v, w = v t ψm w = eti ψm a+ (b)ej = a+ (b)eti ej = a+ (b)ei , ej  = 0. Hence applying (6) over ring A, we have α+ (b) ∈ E(n, A). Therefore α+ (b) ∈ E(n, A) for b ∈ A0 . (5) ⇒ (4): Since αs ∈ E(n, As ) with (α0s ) = In , the diagonal entries of α are of the form 1 + gii , where gii ∈ (A+ )s and off-diagonal entries are of the form gij , where i = j and gij ∈ (A+ )s . We choose l to be large enough such that sl is greater than the common denominator of all gii and gij . Then using (5), we get αs+ (sl ) ∈ E(n, As ). Since that α+ (sl ) permits a natural pullback (as denominators are cleared), we have α+ (sl ) ∈ E(n, A). (4) ⇒ (3): Since αm ∈ E(n, Am ), we have an element s ∈ A0 − m such that αs ∈ E(n, As ). Let s1 , s2 , . . . , sr ∈ A0 be non-zero divisors with si ∈ A0 − mi such that s1 , s2 , . . . , sr  = A. From (4) we have α (bi ) ∈ E(n, A), for some bi = si li with b1 + · · · + br = 1. Now consider αs1 s2 ...sr , which is the image of α in As1 ···sr . Due to Lemma 2.5, +

12

RABEYA BASU AND MANISH KUMAR SINGH

α → αs1 s2 ···sr is injective and hence we can perform our calculation in As1 ···sr and then pull it back to A. αs1 s2 ...sr = αs+1 s2 ...sr (b1 + b2 · · · + br ) + +   = (αs1 )s2 s3 ... (b1 + · · · + br ) (αs1 )s2 s3 ... (b2 + · · · + br )−1 · · ·  + +  (αsi )s ...sˆ ...s (bi + · · · + br ) (αsi )s ...sˆ ...s (bi+1 + · · · + br )−1 1 i r 1 i r + +   · · · (αsr )s1 s2 ...sr−1 (br ) (αsr )s1 s2 ...sr−1 (0)−1 Observing that each + +   (αsi )s1 s2 ...sˆi ...sr (bi + · · · + br ) (αsi )s1 ...sˆi ...sr (bi+1 + · · · + br )−1 ∈ E(n, A) due to Lemma 2.4 (here sˆi means we omit si in the product s1 · · · sˆi · · · sr ), we have αs1 ···sr ∈ E(n, As1 ···sr ) and hence α ∈ E(n, A). Remark 2.9. Following is a commutative diagram (here we are assuming si , sj  = A): A

/ As i

 A sj

 / As s i j

Let θi = αs+i (bi + · · · + br )αs+i (bi−1 + · · · + br ) ∈ Asi and θij be the image of θi in Asi sj and similarly πj = αs+j (bj + · · · + br )αs+j (bj−1 + · · · + br ) ∈ Asj and πsi ,sj    be its image in Asi sj . Then due to Lemma 2.5 the product θsi sj πsi sj can be identified with the product θi πj . (3) ⇒ (2): Since polynomial rings are special case of graded rings, the result follows by using (3) ⇒ (2) in §3, [RRK]. (2) ⇒ (1) ⇒ (6): The proof goes as in [RRK]. (§3, 2 ⇒ 1 ⇒ 7 ⇒ 6).



3. Local-Global Principle for commutator subgroup In this section we deduce an analogue Local-Global Principle for the commutator subgroup of the linear and the symplectic group over graded rings. Unless mentioned otherwise, we assume n ≥ 3 for the linear case and n ≥ 6 for the symplectic case. Let us begin with the following well-known fact for semilocal rings. Lemma 3.1. Let A be a semilocal commutative ring with identity. Then for n ≥ 2 in the linear case and n ≥ 4 in the symplectic case, one gets S(n, A) = E(n, A). Proof. cf. Lemma 1.2.25 in [RB1].



Remark 3.2. If α = (αij /1) ∈ G(n, As ), then α has a natural pullback β = (αij ) ∈ G(n, A) such that βs = α. If α ∈ S(n, As ) such that it admits a natural pullback β ∈ G(n, A), then β ∈ S(n, A).

QUILLEN–SUSLIN THEORY FOR CLASSICAL GROUPS

13

The next lemma deduces an analogue result of “Dilation Lemma” (Theorem 2.8-(4)) for S(n, A); the special linear (resp. symplectic) group. Lemma 3.3. Let α = (αij ) ∈ S(n, A). Then α+ (a) ∈ S(n, A), where a ∈ A0 . Hence if α ∈ G(n, A) with α+ (0) = In and αs ∈ S(n, As ) then αs+ (b) ∈ S(n, A) (after identifying αs+ (b) with its pullback by using Lemma 2.5), where s ∈ A0 is a non-zero divisor and b = sl with l  0, for n ≥ 1 in linear case and n ≥ 2 in symplectic case. Proof. Since det : A → A∗ (units of A) ⊂ A0 , and Ai Aj ⊂ Ai+j , the nonzero component of α does not contribute for the value of the determinant and hence det α = det α+ (0). Therefore, if β = α+ (a), then  + det β = det β + (0) = det α+ (a) (0) = det α+ (a · 0) = det α+ (0) = 1. Hence α+ (a) ∈ S(n, A). Since αs ∈ S(n, As ) with α+ (0) = In , the diagonal entries are of the form 1+gii , and off diagonal entries are gij , where gij ∈ (A+ )s for all i, j. We choose b ∈ (s) such that b = sl with l  0, so that b can dilute the denominator of each entries. Then αs+ (b) ∈ S(n, A).  The next lemma gives some structural information about commutators. Lemma 3.4. Let α, β ∈ S(n, A) and A0 be a commutative semilocal ring. Then the commutator subgroup [α, β] ∈ [αα0−1 , ββ0−1 ] E(n, A). Proof. Since A0 is semilocal, we have S(n, A0 ) = E(n, A0 ) by Lemma 3.1. −1 −1 Hence α+ (0), β + (0) ∈ E(n, A0 ). Let a = αα+ (0) and b = ββ + (0) . Then −1 +

[α, β] = [αα+ (0)

−1 +

α (0), ββ + (0)

β (0)]

−1 −1 +

−1

= aα+ (0)bβ + (0)α+ (0) a β (0) b−1      −1 = aba−1 b−1 bab−1 α+ (0)ba−1 b−1 baβ + (0)α+ (0)−1 a−1 b−1 bβ + (0) b−1 . Since E(n, A) is a normal subgroup of S(n, A), the elements bab−1 α+ (0)ba−1 b−1 , −1 −1 baβ + (0)α+ (0) a−1 b−1 and bβ + (0) b−1 are in E(n, A).  Corollary 3.5. Let α ∈ [S(n, A), S(n, A)] with α+ (0) = In , and let A0 be a semilocal commutative ring. Then using the normality property of the elementary (resp. elementary symplectic) subgroup α can be written as t

Π [βk , γk ] ,

k=1

for some t ≥ 1, and βk , γk ∈ S(n, A), with βk+ (0) = γk+ (0) = In , and  ∈ E(n, A) with + (0) = In .

14

RABEYA BASU AND MANISH KUMAR SINGH t

Proof. Since α ∈ [S(n, A), S(n, A)], α = Π [ak , bk ] for some t ≥ 1. Using −1

Lemma 3.4 we identify βk with ak ak + (0)

k=1

−1

and γk with bk bk + (0)

. This gives

t

α = Π [βk , γk ] . k=1

Then it follows that + (0) = In , as α+ (0) = In .



The next lemma gives a variant of “Dilation Lemma” (Lemma 3.3) for S(n, A); the special linear (resp. symplectic) group. Lemma 3.6. If α ∈ S(n, As ) with α+ (0) = In , then α+ (b + d)α+ (d)

−1

∈ S(n, A),

where s is a non-zero divisor and b, d ∈ A0 with b = si for some i  0. Proof. Let α+ (X) ∈ G(n, A[X]), then αs+ (X) ∈ E(n, As [X]). Let β + (X) = −1 α (X + d)α+ (d) . Then βs+ (X) ∈ S(n, As [X]). Hence by Lemma 3.3, β + (bX) ∈ S(n, A). Putting X = 1, we get result.  +

The following lemma makes use of Lemma 3.3 to deduce “Dilation Lemma” for the commutator subgroup [S(n, A), S(n, A)]. t

Lemma 3.7. Let αs = Π [βi s , γi s ]s for some t ≥ 1, such that βi s and γi s ∈ i=1

S(n, As ) and s ∈ E(n, As ) with γi+s (0) = βi+s (0) = s + (0) = In . Then (identifying with its pullback by using Lemma 2.5) αs+ (b + d)αs+ (d)−1 ∈ [S(n, A), S(n, A)], where b, d ∈ A0 with b = sl for some l  0. Proof. Without loss of generality (as we would conclude by the end of the proof), we can assume αs = [βs , γs ]s = βs γs βs −1 γs −1 s . Since βs+ (0) = In , by Lemma 3.3 βs + (b), γs + (b) ∈ S(n, A) for some b = sl with l  0. Also, s + (b) ∈ E(n, A) by (5) of Theorem 2.8. Hence −1

αs+ (b + d)αs+ (d)

−1 + −1 γs (b + d) −1 + −1 −1 d)+ γs (d)βs+ (d)γs+ (d) βs+ (d) . s (d)

= βs+ (b + d)γs+ (b + d)βs+ (b + d) + s (b +

Since E(n, As ) and S(n, As ) are normal subgroups in G(n, As ), by rearranging −1 −1 + as intermediary) we can consider γs+ (b + d)γs+ (d) and (using + s (b + d)s (d) −1 −1 βs+ (b + d)βs+ (d) together. Now by using Lemma 3.6 we get γs+ (b + d)γs+ (d) and −1 βs+ (b + d)βs+ (d) ∈ S(n, A). Hence (identifying with its pullback by using Lemma 2.5) it follows that αs+ (b + d)αs+ (d)

−1

∈ [S(n, A), S(n, A)]. 

Now we deduce the graded version of the Local-Global Principle for the commutators subgroups.

QUILLEN–SUSLIN THEORY FOR CLASSICAL GROUPS

15

Theorem 3.8. Let α ∈ S(n, A) with α+ (0) = In . If αp ∈ [S(n, Ap ), S(n, Ap )] for all p ∈ Spec(A0 ), then α ∈ [S(n, A), S(n, A)]. Proof. Since αp ∈ [S(n, Ap ), S(n, Ap )] ⊂ [S(n, Am ), S(n, Am )], we have for a maximal ideal m ⊇ p, s ∈ A0 − m such that αs ∈ [S(n, As ), S(n, As )]. Hence by Corollary 3.5, αs can be decomposed as t

αs = Π [βis , γis ]s i=1

where βis , γis ∈ S(n, As ) and s ∈ E(n, As ). Now using the “Dilation Lemma” (Lemma 3.3 for S(n, As ) and theorem 2.8 (4) for E(n, As )) on each of these elements, we have for b = si , i  0 t

αs+ (b) = Π [βi+s (b), γi+s (b)]+ s (b) ∈ [S(n, A), S(n, A)]E(n, A). i=1

Since E(n, As ) ⊆ [S(n, As ), S(n, As )], we have αs ∈ [S(n, As ), S(n, As )]. Let s1 , . . . , sr ∈ A0 non-zero divisors such that si ∈ A0 − mi and s1 , . . . , sr  = A0 . Then it follows that b1 + · · · + br = 1 for suitable (as before) bi ∈ (si ); and i = 1, . . . r. Now αs1 s2 ...sr = αs+1 s2 ...sr (1) = αs+1 s2 ...sr (b1 + b2 + · · · + br ) + +   = (αs1 )s2 s3 s4 ... (b1 + · · · + br ) (αs1 )s2 s3 s4 ... (b2 + · · · + br )−1· · ·  + +  (αsi )s ...sˆ ...s (bi + · · · + br ) (αsi )s ...sˆ ...s (bi+1 + · · · + br )−1 1 i r 1 i r +  +  · · · (αsr )s1 s2 ...sr−1 (br ) (αsr )s1 s2 ...sr−1 (0)−1 . Using Lemma 2.5 the product is well defined (see Remark 2.9) and using Lemma 3.7, we conclude that α ∈ [S(n, A), S(n, A)].  4. Auxiliary Result for Transvection subgroup Let R be a commutative ring with 1. We recall that a finitely generated projective R-module Q has a unimodular element if their exists q ∈ Q such that qR  R and Q  qR ⊕ P for some projective R-module P . In this section, we shall consider three types of (finitely generated) classical modules;  viz. projective, symplectic and orthogonal modules over graded rings. ∞ Let A = i=0 Ai be a commutative graded Noetherian ring with identity. For definitions and related facts we refer [BBR]; §1, §2. In that paper, the first author with A. Bak and R.A. Rao has established an analogous Local-Global Principle for the elementary transvection subgroup of the automorphism group of projective, symplectic and orthogonal modules of global rank at least 1 and local rank at least 3. In this article we deduce an analogous statement for the above classical groups over graded rings. We shall assume for every maximal ideal m of A0 , the symplectic and orthogonal module Qm  (A2m 0 )m with the standard form (for suitable integer m).

16

RABEYA BASU AND MANISH KUMAR SINGH

Remark 4.1. By definition the global rank or simply rank of a finitely generated projective R-module (resp. symplectic or orthogonal R-module) is the largest integer k such that ⊕k R (resp. ⊥k H(R)) is a direct summand (resp. orthogonal summand) of the module. Here H(R) denotes the hyperbolic plane. Let Q denote a projective, symplectic or orthogonal A0 -module of global rank ≥ 1, and total (or local) rank r + 1 in the linear case and 2r + 2 otherwise and let Q1 = Q ⊗A0 A. We use these notations to deal with the above three classical modules uniformly: (1) G(Q1 ) := the full automorphism group of Q1 . (2) T(Q1 ) := the subgroup generated by transvections of Q1 . (3) ET(Q1 ) := the subgroup generated by elementary-transvections of Q1 .  A) := E(r + 1, A) for linear case and E(2r + 2, A) otherwise. (4) E(r, In [BBR], the first author with Ravi A. Rao and A. Bak established the “Dilation Lemma” and the “Local-Global Principle” for the transvection subgroups of the automorphism group over polynomial rings. In this section, we generalize their results over graded rings. For the statements over polynomial rings we request the reader to look at Proposition 3.1 and Theorem 3.6 in [BBR]. Following is the graded version of the “Dilation Lemma”. Proposition 4.2 (Dilation Lemma). Let Q be a projective A0 -module, and Q1 = Q ⊗A0 A. Let s be a non-nilpotent element in A0 . Let α ∈ G(Q1 ) with α+ (0) = In . Suppose    E r + 1, (A0 )s   αs ∈ E 2r + 2, (A0 )s

in the linear case, otherwise.

Then there exists α  ∈ ET(Q1 ) and l  0 such that α  localizes at α+ (b) for some + (0) = In . b = (sl ) and α  As ), Proof. Let α ∈ G(Q1 ) and α(X) := α+ (X) ∈ G(Q1 [X]). Since αs ∈ E(r, +  αs (X) ∈ E(r, As [X]). Since α(0) := α (0) = In , we can apply “Dilation Lemma” for the ring A[X] (cf.[BBR], Proposition 3.1) and hence there exists α (X) ∈ s (X) = αs+ (bX) for some b ∈ (sl ), l  0. Substituting ET(Q1 [X]) such that α s := α s (1) = αs (b) = αs+ (b). This proves the X = 1, we get α  ∈ ET(Q1 ) and α proposition.  Lemma 4.3. If α ∈ G(Q1 ) with α+ (0) = In such that αs ∈ E(n, As ), then α (b + d)α+ (d)−1 ∈ ET(Q1 ) for some b ∈ (sl ) with l  0, and d ∈ A0 . +

−1

Proof. Let β = αs+ (b + d)αs+ (d) . Since αs ∈ E(n, As ) implies α+ (a) ∈ E(n, As ) for a ∈ As and hence β ∈ E(n, As ). Hence by Theorem 4.2, their exists a β ∈ ET(Q1 ) such that βs+ (b) = β. Hence the lemma follows. 

QUILLEN–SUSLIN THEORY FOR CLASSICAL GROUPS

17

Theorem 4.4 (Local-Global Principle). Let Q be a projective A0 -module, and Q1 = Q ⊗A0 A. Suppose α ∈ G(Q1 ) with α+ (0) = In . If    in the linear case, E r + 1, (A0 )m   αm ∈ otherwise. E 2r + 2, (A0 )m     for all m ∈ max (A0 ), then α ∈ ET Q1 ⊆ T Q1 .  Am ). Hence their Proof. Let α ∈ G(Q1 ) with α+ (0) = In . Since αm ∈ E(r,  exists a non-nilpotent s ∈ A0 − m, such that αs ∈ E(r, As ). Hence by the above s = αs+ (b), for some b = sl “Dilation Lemma” there exists α  ∈ ET(Q1 ) such that α with l  0. For each maximal ideal mi we can find a suitable bi . Since A is Noetherian, it follows that b1 + · · · + br = 1 for some positive integer r. Now we −1 observe that α+ (bi +· · ·+br )α+ (bi+1 + · · · + br ) ∈ ET(Q1 ) and hence calculating in the similar manner as we did it in Remark 2.9, we get α = α+ (1) = α+ (b1 + · · · + br )α+ (b2 + · · · + br ) −1

α+ (bi+1 + · · · + br )

−1

· · · α+ (bi + · · · + br )

· · · α+ (0)−1 ∈ ET(Q1 ) ⊆ T(Q1 ); 

as desired. Acknowledgments

The authors thank the referee for reading the manuscript carefully. The second author expresses his sincere thanks to IISER Pune for allowing him to work on this project and providing him their infrastructural facilities. References [BA] [BBR]

[C] [G1]

[G2]

[HV] [M]

[Q] [RB1] [RRK]

Hyman Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR0249491 A. Bak, Rabeya Basu, and Ravi A. Rao, Local-global principle for transvection groups, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1191–1204, DOI 10.1090/S0002-9939-09-101983. MR2578513 Leo G. Chouinard II, Projective modules over Krull semigroup rings, Michigan Math. J. 29 (1982), no. 2, 143–148. MR654475 Joseph Gubeladze, Classical algebraic K-theory of monoid algebras, K-theory and homological algebra (Tbilisi, 1987), Lecture Notes in Math., vol. 1437, Springer, Berlin, 1990, pp. 36–94, DOI 10.1007/BFb0086718. MR1079964 I. Dzh. Gubeladze, The Anderson conjecture and a maximal class of monoids over which projective modules are free (Russian), Mat. Sb. (N.S.) 135(177) (1988), no. 2, 169– 185, 271, DOI 10.1070/SM1989v063n01ABEH003266; English transl., Math. USSR-Sb. 63 (1989), no. 1, 165–180. MR937805 Roozbeh Hazrat and Nikolai Vavilov, K1 of Chevalley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), no. 1-2, 99–116, DOI 10.1016/S0022-4049(02)00292-X. MR1958377 Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR1011461 Daniel Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167– 171, DOI 10.1007/BF01390008. MR427303 Rabeya Basu; Results in Classical Algebraic K-theory, Ph.D. thesis, Tata Institute of Fundamental Research (2007), pp 70. Rabeya Basu, Ravi. A. Rao, and Reema Khanna, On Quillen’s local global principle, Commutative algebra and algebraic geometry, Contemp. Math., vol. 390, Amer. Math. Soc., Providence, RI, 2005, pp. 17–30, DOI 10.1090/conm/390/07291. MR2187322

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[RRK2] Rabeya Basu, Ravi. A. Rao, and Reema Khanna, On Quillen’s local global principle, Commutative algebra and algebraic geometry, Contemp. Math., vol. 390, Amer. Math. Soc., Providence, RI, 2005, pp. 17–30, DOI 10.1090/conm/390/07291. MR2187322 [S] A. A. Suslin, The structure of the special linear group over rings of polynomials (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477. MR0472792 [SK] A. A. Suslin and V. I. Kope˘ıko, Quadratic modules and the orthogonal group over polynomial rings (Russian), Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 71 (1977), 216–250, 287. Modules and representations. MR0469914 [T] M. S. Tulenbaev, The Schur multiplier of the group of elementary matrices of finite order: Algebraic numbers abd finite groups (Russian), Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 86 (1979), 162–169, 191–192. MR535488 Department of Mathematics, Indian Institute of Science Education and Research (IISER) Pune, India 411008 Current address: IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune, Maharashtra, India 411008 Email address: [email protected], [email protected] Department of Mathematics, Indian Institute of Science Education and Research (IISER) Pune, India 411008 Email address: [email protected]

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15114

Natural elements of center of generalized quantum groups Punita Batra and Hiroyuki Yamane Abstract. This paper gives elements in the (skew) center of the generalized quantum group corresponding to its irreducible finite dimensional modules. Finally we give a conjecture stating that those must form a basis of the center.

1. Introduction This paper is a continuation of [4], [5]. Let V be a finite-dimensional real linear space. Let l := dimR V , and assume l ≥ 1. Let {α1 , . . . , αl } be an R-basis of V . So V = ⊕li=1 Rαi . Let VZ := ⊕li=1 Zαi . Then VZ is a free Z-module with rankZ VZ = l. Let K be a field. Let K× := K \ {0}. Let χ : VZ × VZ → K× be a map such that χ(λ + μ, ν) = χ(λ, ν)χ(μ, ν) and χ(λ, μ + ν) = χ(λ, μ)χ(λ, ν). For the χ, in the same way as the Lusztig way [17, 3.1.1] to define the quantum groups Uq (g), we can define the Hopf algebra U = U (χ, π) over K, which we call the generalized quantum group. (U is the one of Subsection 2.2 with assuming VZ to be A. In Introduction, we may also assume VZ = A = Aπ . The notation π means the map from {1, . . . , l} to A defined by π(i) := αi .) For some χ, U can be the quantum groups, the small quantum groups at root of 1, the quantum superalgebras, or the one associated with the Nichols algebras in the Heckenberger’s list [10]. Further studies concerning U have been achieved by [1], [2], [3], [6], [11], [12], [13], etc. Let U = U − ⊗ U 0 ⊗ U + be the triangular decomposition (see (U 5)). Let U = ⊕λ∈VZ Uλ be the VZ -grading (see (U 4)). Let Rχπ,+ (⊂ VZ ) be the Kharchenko’s positive root system defined for U (see Theorem 2.2, which was originally given by [15]). As in the assumption of Theorem 2.4, assume that Rχπ,+ is a finite set and assume that χ(α, α) = 1 for all α ∈ Rχπ,+ . Let ω : VZ → K× be the map such that ω(λ + μ) = ω(λ)ω(μ). Let Zω (χ, π) := { Z ∈ U0 | ∀λ ∈ VZ , ∀X ∈ Uλ , ZX = ω(λ)XZ } (see (2.9)). In [5, Theorem 10.4] (see Theorem 2.4), we have the Harishχ,π : Zω (χ, π) → Bχ,π Chandra type isomorphism HCχ,π ω ω , where Bω is the K-subspace 0 π,+ of U defined by the equrations (e1)β -(e4)β for all β ∈ Rχ . As in (3.11), for λ, μ ∈ VZ , define the K-algebra homomorphism Λχλ,μ;ω : U 0 → K by Λχλ,μ;ω (Kλ Lμ ) := χ(λ, μ )χ(λ , μ)ω(λ ) for λ , μ ∈ VZ , where {Kλ Lμ |λ , μ ∈ VZ } is a K-basis of U 0 (see (U 3)). As in (3.12), let Finχω := {(λ, μ) ∈ VZ × VZ | dim L(Λχλ,μ;ω ) < ∞}, 2010 Mathematics Subject Classification. Primary 17B37, 17B10; Secondary 81R50. The second author is partially supported by JSPS Grand-in-Aid for Scientific Research (C), 16K05095. c 2020 American Mathematical Society

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where L(Λχλ,μ;ω ) is the finite-dimensional simple U -modules defined in the way that Λχλ,μ;ω is regarded as its highest weight. By our main result Theorem 3.7, for each χ χ (λ, μ) ∈ Finχω , we have Zλ,μ;ω ∈ Zω (χ, π) so that HCχ,π ω (Zλ,μ;ω ) can be viewed as the χ character of L(Λλ,μ;ω ). As a final stage of this paper, we state Conjecture (3.13), χ which states that { Zλ,μ;ω | (λ, μ) ∈ Finχω } is a K-basis of Zω (χ, π). 2. Generalized quantum groups 2.1. Preliminary. For x, y ∈ R, let Jx,y := {z ∈ Z|x ≤ z ≤ y}.Let K be n × r−1 a field. Let K n := K \ {0}. For n ∈ Z≥0 and x ∈ K, let (n)x := n  r=1 x , and (n)x ! := r=1 (r)x . For n ∈ Z≥0 , m ∈ J0,n and x ∈ K, define m x ∈ K by     n n n n−1  n−1  n−m n−1 m n−1 0 x := n x := 1, and m x := m x +x m−1 x = x m x + m−1 x (if n x! m ∈ J1,n−1 ). If (m)x !(n − m)x ! = 0, then m = (m)x(n) !(n−m)x ! . For x, y, z ∈ K, x   n−1  m(m−1) n n n−m m and n ∈ N, we have t=0 (y + xt z) = m=0 x 2 z . m xy For x ∈ K× , define oˆ(x) ∈ Z≥0 \ {1} by  min{ r  ∈ J2,∞ | (r  )x ! = 0 } if (r  )x ! = 0 for some r  ∈ J2,∞ , (2.1) oˆ(x) := 0 otherwise. Let A be an abelian group. Then A is a Z-module. Let χ : A × A → K× be a map such that (2.2) ∀λ, ∀μ, ∀ν ∈ A, χ(λ + μ, ν) = χ(λ, ν)χ(μ, ν), χ(λ, μ + ν) = χ(λ, μ)χ(λ, ν). Let l ∈ N and I := J1,l . Assume that there exists an injection π : I → A such that π(I) is a Z-basis of SpanZ (π(I)). Let Aπ := SpanZ (π(I)). Let αi := π(i) (i ∈ I). Then Aπ := ⊕i∈I Zαi . Let A+ π := ⊕i∈I Z≥0 αi . 2.2. Definition of U = U (χ, π). The facts mentioned in this subsection is well-known and can be proved in a standard way introduced by Drinfeld [7] and Lusztig [17, 3.1.1]. There exists a unique associative K-algebra (with 1) U = U (χ, π) satisfying the following conditions (U 1)-(U 6). (U 1) As a K-algebra, U is generated by the elements: (2.3)

Kλ , Lλ (λ ∈ A),

Ei , Fi (i ∈ I).

(U 2) The elements of (2.3) satisfy the following relations.

(2.4)

K0 = L0 = 1, Kλ Kμ = Kλ+μ , Lλ Lμ = Lλ+μ , Kλ Lμ = Lμ Kλ , Kλ Ei = χ(λ, αi )Ei Kλ , Lλ Ei = χ(−αi , λ)Ei Lλ , Kλ Fi = χ(λ, −αi )Fi Kλ , Lλ Fi = χ(αi , λ)Fi Lλ , [Ei , Fj ] = δij (−Kαi + Lαi ).

(U 3) Define the map ς1 : A × A → U by ς1 (λ, μ) := Kλ Lμ . Define the K-subalgebra U 0 = U 0 (χ, π) of U by U 0 := SpanK (ς1 (A × A)). Then ς1 is injective, and ς1 (A × A) is a K-basis of U 0 . (U 4) There exist K-subspaces Uλ = U (χ, π)λ of U for λ ∈ Aπ satisfying the following conditions (U 4 − 1)-(U 4 − 3).

ELEMENTS OF CENTER OF GENERALIZED QUANTUM GROUPS

21

(U 4 − 1) We have U 0 ⊂ U0 and Ei ∈ Uαi , Fi ∈ U−αi (i ∈ I). (U 4 − 2) We have Uλ Uμ ⊂ Uλ+μ (λ, μ ∈ Aπ ). (U 4 − 3) We have U = ⊕λ∈Aπ Uλ as a K-linear spaces. (U 5) Let U + = U + (χ, π) (resp. U − = U − (χ, π)) be the K-subalgebra (with 1) of U generated by Ei (resp. Fi ) (i ∈ I). Define the K-linear homomorphism ς2 : U − ⊗K U 0 ⊗K U + → U (χ, π) by ς2 (Y ⊗ Z ⊗ X) := Y ZX. Then ς2 is a K-linear isomorphism. (U 6) For λ ∈ Aπ , define Uλ+ = U + (χ, π)λ (resp. Uλ− = U − (χ, π)λ ) by Uλ+ := + U + ∩ Uλ (resp. Uλ− = U − ∩ Uλ ). Then for λ ∈ A+ π \ {0}, we have {X ∈ Uλ |∀i ∈ − I, [X, Fi ] = 0} = {0} and {Y ∈ U−λ |∀i ∈ I, [Ei , Y ] = 0} = {0}. − Uλ+ , U − = ⊕λ∈A+ U−λ , U0+ = U0− = K · 1U and Uα+i = Notice that U + = ⊕λ∈A+ π π − = K · Fi (i ∈ I). K · Ei , U−α i

We also regard U = U (χ, π) as a Hopf algebra (U, Δ, S, ε) by

(2.5)

Δ(Kλ ) = Kλ ⊗ Kλ , Δ(Lλ ) = Lλ ⊗ Lλ , Δ(Ei ) = Ei ⊗ 1 + Kαi ⊗ Ei , Δ(Fi ) = Fi ⊗ Lαi + 1 ⊗ Fi , S(Kλ ) = K−λ , S(Lλ ) = L−λ , S(Ei ) = −K−αi Ei , S(Fi ) = −Fi L−αi , ε(Kλ ) = ε(Lλ ) = 1, ε(Ei ) = ε(Fi ) = 0.

Let U +, := ⊕λ∈A U + Kλ , and U −, := ⊕λ∈A U − Lλ . Then U = SpanK (U −, U +, ) = SpanK (U +, U −, ). As in a standard way (see [7]), we have a bilinear form ϑ = ϑχ,π : U +, ×U −, → K having the following properties, see also Subsection 3.1 below.

(2.6)

ϑ(Kλ , Lμ ) = χ(λ, μ), ϑ(Ei , Fj ) = δij , ϑ(Kλ , Fj ) = ϑ(Ei , Lλ ) = 0,  (2) (1) ϑ(X + Y + , X − ) = k− ϑ(X + , (X − )k− )ϑ(Y + , (X − )k− ),  (1) (2) ϑ(X + , X − Y − ) = k+ ϑ((X + )k+ , X − )ϑ((X + )k+ , Y − ), ˜ − )), ϑ(S(X + ), X − ) = ϑ(X + , S −1 (X ϑ(X + , 1) = ε(X + ), ϑ(1, X − ) = ε(X − ),  ,(1) ,(1) ,(3) ,(3) X − X + = r+ ,r− ϑ((X + )r+ , S((X − )r− ))ϑ((X + )r+ , (X − )r− ) ,(2) ,(2) ·(X + )r+ (X − )r− ,  ,(3) ,(3) ,(1) ,(1) + − X X = r+ ,r− ϑ((X + )r+ , S((X − )r− ))ϑ((X + )r+ , (X − )r− ) ,(2) ,(2) ·(X − )r− (X + )r+

for λ, μ ∈ A, i, j ∈ I, and X + , Y + ∈ U +, , X − , Y − ∈ U −, , where (X + )k+ (x) ,(y) ,(y) and (X − )k− with x ∈ J1,2 (resp. (X + )r+ and (X − )r− with y ∈ J1,3 ) are any  (1) (2) elements of U +, and U −, respectively satisfying Δ(X ± ) = k± (X ± )k± ⊗(X ± )k± ,  ,(1) ,(2) ,(3) (resp. ((idU ⊗ Δ) ◦ Δ)(X ± ) = r± (X ± )r± ⊗ (X ± )r± ⊗ (X ± )r± ). We have (2.7) ϑ(X + Kλ , X − Lμ ) = χ(λ, μ)ϑ(X + , X − ) (λ, μ ∈ A, X + ∈ U +, , X − ∈ U −, ), (x)

and (2.8)

− ) = {0} if λ = μ. ϑ(Uλ+ , U−μ

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By (2.7)-(2.8), we can easily see: Lemma 2.1. Let χ : A × A → K× and π : I → A be as above. Let A be a Zsubmodule of A such that π(I) ⊂ A . Let K be a subfield of K such that χ(A , A ) ⊂ (K )× . Let η : A → A be the inclusion map. Define the map χ : A × A → (K )× by η ◦χ = χ|A ×A . Define the map π  : I → A by η ◦π  = π. Then there exists a Kalgebra monomorphism f : U (χ , π  ) ⊗K K → U (χ, π) such that f (Kλ Lμ ) = Kλ Lμ (λ, μ ∈ A ) and f (Ei ) = Ei , f (Fi ) = Fi (i ∈ I). Define the K-linear map Shχ,π : U (χ, π) → U 0 (χ, π) by Shχ,π (Y Kλ Lμ X) = ε(Y )ε(X)Kλ Lμ (X ∈ U + , Y ∈ U − , λ, μ ∈ A). 2.3. Kharchenko’s PBW theorem. Theorem 2.2. (Kharchenko’s PBW theorem [15, Theorem 2], [16, Theorem 2.2], see also [9, Theorem 3.14] and Section 4.) Keep the notation as above. π,+ of A+ Then there exists a unique pair of (Rχπ,+ , ϕπ,+ χ ) of a subset Rχ π \ {0} and π,+ π,+ a map ϕχ : Rχ → N satisfying the following. Let X := {(α, t) ∈ Rχπ,+ × N|t ∈ π,+ by z(α, t) := α. Let Y be the set of maps J1,ϕπ,+ (α) }. Define the map z : X → Rχ χ y : X → Z≥0 such that |{x ∈ X|y(x) ≥ 1}| < ∞ and (y(x))χ(z(x),z(x)) ! = 0 for all x ∈ X. Then  + y(x)z(x) = λ}|. ∀λ ∈ A+ π , dim U (χ, π)λ = |{y ∈ Y | x∈X

Theorem 2.3. ([8, Proposition 1], [12, Theorem 4.9]) If |Rχπ,+ | < ∞, then π,+ ϕπ,+ χ (Rχ ) = {1}. 2.4. Skew centers. Let ω : Aπ → K× be a Z-module homomorphism. (2.9)

Zω (χ, π) := { Z ∈ U (χ, π)0 | ∀λ ∈ Aπ , ∀X ∈ U (χ, π)λ , ZX = ω(λ)XZ }.

Define the Z-module homomorphism ρˆχ,π : Aπ → K× by ρˆχ,π (αj ) := χ(αj , αj )

(j ∈ I),

where αj := π(j), as above. 0 For each β ∈ Rχπ,+ , let Bχ,π ω (β) be the K-linear subspace of U (χ, π) formed by the elements  a(λ,μ) Kλ Lμ (λ,μ)∈A2

with a(λ,μ) ∈ K satisfying the following equations (e1)β -(e4)β . In (e1)β -(e4)β , let χ := ω(β) · χ(β,μ) qβ := χ(β, β), cβ := oˆ(qβ ) and ωλ,μ;β χ(λ,β) . χ = qβt , then (e1)β For (λ, μ) ∈ A2 and t ∈ Z \ {0}, if qβ = 1, cβ = 0 and ωλ,μ;β t χ,π the equation a(λ+tβ,μ−tβ) = ρˆ (β) · a(λ,μ) holds. χ = qβt for all t ∈ Z, then the equation (e2)β For (λ, μ) ∈ A2 , if cβ = 0 and ωλ,μ;β a(λ,μ) = 0 holds.

ELEMENTS OF CENTER OF GENERALIZED QUANTUM GROUPS

23

χ (e3)β For (λ, μ) ∈ A2 , if qβ = 1, cβ ≥ 2 and ωλ,μ;β = qβt for some t ∈ J1,cβ −1 , the equation ∞  a(λ+(cβ x+t)β,μ−(cβ x+t)β) ρˆχ,π (β)−(cβ x+t) x=−∞

=

∞ 

−cβ y

a(λ+cβ yβ,μ−cβ yβ) ρˆχ,π (β)

y=−∞

holds. χ = qβm for all m ∈ Z, then the cβ − 1 (e4)β For (λ, μ) ∈ A2 , if cβ ≥ 2 and ωλ,μ;β equations ∞  −(c x+t) a(λ+(cβ x+t)β,μ−(cβ x+t)β) ρˆχ,π (β) β x=−∞

=

∞ 

a(λ+cβ yβ,μ−cβ yβ) ρˆχ,π (β)−cβ y

y=−∞

(t ∈ J1,cβ −1 ) hold. Let := Bχ,π ω



Bχ,π ω (β).

π,+ β∈Rχ

Theorem 2.4. ([5, Theorem 10.4]) Assume A = Aπ . Assume |Rχπ,+ | < ∞. Assume that χ(α, α) = 1 for all α ∈ Rχπ,+ . Then we have the K-linear isomorphism χ,π HCχ,π : Zω (χ, π) → Bχ,π defined by HCχ,π (X). ω ω (X) := Sh ω The statement of [5, Theorem 10.4] has claimed that the above theorem holds if K is an algebraically closed field. However, by the argument in [5], we can easily see that it really holds for any field. 3. Elements of Zω (χ, π) via finite dimensional representations In Subsection 3, we use argument similar to that in [19, Sections 2 and 3]. 3.1. Hopf pairing. In Subsection 3.1, let A = (A, ΔA , SA , εA ) and B = (B, ΔB , SB , εB ) be Hopf algebras over K with SA and SB being bijective, and assume that there exists a Hopf pairing p : A × B → K, that is, p(a1 a2 , b) = p(a1 ⊗ a2 , ΔB (b)), p(a, b1 b2 ) = p(ΔA (a), b1 ⊗ b2 ), p(SA (a), b) = p(a, SB (b)), p(a, 1B ) = εA (a), and p(1A , b) = εB (b) hold for all a, a1 , a2 ∈ A and all b, b1 , b2 ∈ B, where p(a1 ⊗ a2 , b1 ⊗ b2 ) := p(a1 , b1 )p(a2 , b2 ). Note that from B, we obtain the Hopf algebra Bop = (B, y ◦ ΔB , SB−1 , εB ), where we define the K-linear isomorphism y : B ⊗ B → B ⊗ B by y(b1 ⊗ b2 ) := b2 ⊗ b1 , i.e., Bop is the opposite Hopf algebra of B. It is well-known that we have the Hopf algebra D = D(p, A, B) = (D, ΔD , SD , εD ) satisfying the conditions below. (D1) D include A and Bop as Hopf subalgebras. The K-linear homomorphism A ⊗ B → D, a ⊗ b → ab, is bijective. (D2) As a K-algebra, D has the multiplication such that for a ∈ A and b ∈  (3) (3) (1) (2) (2) −1 (1) B, ba = i,j p(SA (ai ), bj )p(ai , bj )ai bj , where ((ΔA ⊗ id) ◦ ΔA )(a) =

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P. BATRA AND H. YAMANE



(1)

(2)

(3)

ai ⊗ ai ⊗ ai and ((ΔB ⊗ id) ◦ ΔB )(b) =  (3) (1) (1) (3) (2) (2) ab = i,j p(SA−1 (ai ), bj )p(ai , bj )bj ai . i



(1) j bj

(2)

(3)

⊗ bj ⊗ bj ; we also have

We can easily see that the K-linear homomorphism B ⊗ A → D, b ⊗ a → ba, is bijective. Define the K-bilinear map P : D × D → K by P(a1 b1 , b2 a2 ) := p(a1 , b2 ) · p(a2 , b1 )

(a1 , a2 ∈ A, b1 , b2 ∈ B).

Define the left (resp. right) K-algebra action •l (resp. •r ) of D on D by  (1)  (2) (1) (2) x •l y := xi ySD (xi ) (resp. y •r x := SD (xi )yxi ), 

i

i

(1) (2) where ΔD (x) = i xi ⊗xi . We have y •r x1 x2 ) for x1 , x2 , y ∈ D. We have

x1 •l (x2 •l y) = x1 x2 •l y (resp. (y•r x1 )•r x2 =

Lemma 3.1. It follows that P(x •l y1 , y2 ) = P(y1 , y2 •r x)

(3.1)

(x, y1 , y2 ∈ D).

Proof. (cf. [19, Proposition 2.2.1]) Letting X be A or B, for c ∈ X and r ∈ N with r ≥ 2, let  (1) (r−1) (r) ⊗ ci i ci ⊗ · · · ⊗ ci := ((ΔX ⊗ idX ⊗ · · · ⊗ idX ) ◦ · · · ◦ (ΔX ⊗ idX ) ◦ ΔX )(c).    r−2



 (2) (1) (2) (1) (Note that ΔD (a) = i ai ⊗ ai (a ∈ A), and ΔD (b) = j bi ⊗ bj (b ∈ B).) Let a, a ´, a ` ∈ A and b, ´b, `b ∈ B. We have  (4) (3) (2) (1) (1) (3) (2) a •l a ´´b = i,j p(ai , ´bj )p(SA (ai ), ´bj )ai a ´SA (ai )´bj ,  (1) (1) (3) (3) (2) (2) (4) `b` a •r a = i,j p(ai , `bj )p(SA (ai ), `bj )`bj SA (ai )` aai ,  (1) (4) (3) (2) (2) (3) (1) b •l a ´´b = i,j p(´ ai , SB−1 (bj ))p(´ ai , bj )´ ai bj ´bSB−1 (bj ),  (3) (1) (1) (3) (4) (2) (2) `b` a •r b = i,j p(` ai , SB−1 (bj ))p(` ai , bj )SB−1 (bj )`bbj a `i . Hence we have P(a •l a ´´b, `b` a)  (4) (3) (3) (1) (1) (3) (2) = i,j p(ai , ´bj )p(SA (ai ), ´bj )p(ai a ´SA (ai ), `b)p(` a, ´bj )  (3) (4) (1) (3) = i p(SA (ai )` aai , ´b)p(ai a ´SA (ai ), `b)  (3) (4) ´ (1) `(1) (3) (3) (2) = i,j p(SA (ai )` aai , b)p(ai , bj )p(SA (ai ), `bj )p(´ a, `bj ) = P(´ a´b, `b` a •r a), and P(b •l a ´´b, `b` a)  (1) (4) (3) (2) (2) (3) (1) ai , SB−1 (bj ))p(´ ai , bj )p(´ ai , `b)p(` a, bj ´bSB−1 (bj )) = i,j p(´  (4) (2) (3) (1) = j p(´ a, SB−1 (bj )`bbj )p(` a, bj ´bSB−1 (bj ))  (1) (3) (2) (3) (1) −1 (4) ` (2) = i,j p(´ a, SB (bj )bbj )p(` ai , bj )p(` ai , ´b)p(` ai , SB−1 (bj )) = P(´ a´b, `b` a •r b). This completes the proof.

2

ELEMENTS OF CENTER OF GENERALIZED QUANTUM GROUPS

25

Let D∗ be the K-linear space formed by all K-linear homomorphisms from D to K. Define the right K-algebra action · of D on D∗ by (f · x)(y) := f (x •l y)

(f ∈ D∗ , x, y ∈ D).

Define the K-linear homomorphism ΩD : D → D∗ by (ΩD (x))(y) := P(y, x) (x, y ∈ D). By (3.1), we have (3.2)

ΩD (y) · x = ΩD (y •r x) (x, y ∈ D).

Lemma 3.2. Assume that p is non-degenerate. Let x, y ∈ D. Then (3.3)

εD (y)x = x •r y

⇐⇒

ΩD (x) · y = εD (y)ΩD (x).

Proof. Since p is non-degenerate, P is so. Hence ΩD is injective. Then (3.3) follows from (3.2). 2 3.2. Extension of U = U (χ, π). Keep the notation in Subsection 2.2. Recall I = J1,l and αi = π(i) (i ∈ I). Let I˘ := J1,2l+2 , i.e., I˘ = I ∪ Jl+1,2l+2 . Let αj ∈ A (j ∈ Jl+1,2l+2 ). Let ω : Aπ → K× be a Z-module homomorphism.

(3.4)

In Subsections 3.2 and 3.3, we assume the following (ass1)-(ass4). (ass1) There exists ξ ∈ K× \ {1} with oˆ(ξ) = 0. (ass2) A is a Z-module with rankZ A = 2l + 2 and A = ⊕i∈I˘Zαi . (ass3) One has χ(αi , αj+l+1 ) = χ(αj+l+1 , αi ) = ξ δij and χ(αi+l+1 , αj+l+1 ) = 1 for i, j ∈ J1,l+1 . (ass4) One has ω(αi ) = χ(αi , αl+1 ) and χ(αl+1 , αi ) = 1 for i ∈ I.

Let X and Y be a K-linear space, and let f : X × Y → K be a a K-bilinear map. We call K-linear space {x ∈ X|∀y ∈ Y, f (x, y) = 0} (resp. {y ∈ Y |∀x ∈ X, f (x, y) = 0}) of X (resp. Y ) the left kernel (resp. the right kernel). We call f non-degenerate if the right and left kernels are zero-dimensional. Lemma 3.3. Let U +,,0 := ⊕λ∈A KKλ (= U 0 ∩U +, ) and U −,,0 := ⊕λ∈A KLλ (= U ∩ U −, ). Then ϑ|U +,,0 ×U −,,0 is non-degenerate. In particular, ϑ is nondegenerate. 0

Proof. Let ϑ := ϑ|U +,,0 ×U −,,0 . Let A := ⊕li=1 Zαi and A := ⊕lj=1 Zαj+l . So A = A ⊕ A . Let (U −,,0 ) := ⊕λ∈A KLλ and (U +,,0 ) := ⊕μ∈A KK μ . Let z R (resp. R ) be the left (resp. right) kernel of ϑ . Let r ∈ R. Write r = t=1 rt  +,,0  for some z ∈ N and λt ∈ A (t ∈ J1,z ) with rt ∈ Kλt (U ) (t ∈ J1,z ). For l (t) (t)  y1 t ∈ J1,z , write λt = i=1 xi αi with xi ∈ Z. Since ϑ (r, hLαl+1 · · · Lyαl2l ) = 0 and (t)

(t)

ϑ (rt , hLyα1l+1 · · · Lyαl2l ) = ξ x1 y1 +···+xl yl ϑ (rt , h) for all h ∈ (U −,,0 ) and all yi ∈ Z (i ∈ J1,l ), we have rt ∈ R for all t ∈ J1,z . Since ϑ (rt , (U −,,0 ) ) = {0}, we have 2 rt = 0. Hence r = 0. Hence R = {0}. Similarly we have R = {0}. We have the Hopf algebra isomorphism D(ϑ, U + , (U − )op ) → U, X ⊗ Y → XY. From now on until the end of Section 3, we identify U with D, let the actions •r , •l , · of U be the ones defined for U by identifying ϑ with p.

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3.3. Central elements from finite dimensional modules under the assumption (3.4). Keep the notation as in Subsection 3.2. We have assumed (3.4). From now on until the end of this subsection fix λ, μ ∈ A, and fix a non-zero U -module V˘ such that there exists v˘ ∈ V˘ with v (λ , μ ∈ A) and Ei v˘ := 0 (i ∈ I). Kλ Lμ v˘ := χ(λ, μ )χ(λ , μ)˘ ˘(X)(h) := Xh (X ∈ U , Define the K-algebra homomorphism a ˘ : U → EndK (V˘ ) by a ˘ ˘ h ∈ V ), i.e., (˘ a, V ) is the K-algebra representation of U associated with V˘ . By − U−ν v˘, as a K-linear space, and Lemma 3.3, we have V˘ = ⊕ν∈A+ π − U−ν v˘ = { v ∈ V˘ | Kλ Lμ v = χ(λ + ν, μ )χ(λ , μ − ν)v (λ , μ ∈ A) }. − For ν ∈ A+ ˘ ν := dim U−ν v˘. π , let m

Assume dimK V˘ < ∞.

 Then dimK V˘ = ν∈A+ m ˘ ν . For k ∈ EndK (V˘ ), let TrV˘ (k) ∈ K denote the trace of π − k. Define ρ˘ ∈ EndK (V˘ ) by ρ˘(Y v˘) := ρˆχ,π (ν)Y v˘ for ν ∈ A+ π and Y ∈ U−ν . Since 2 χ,π   S (X) = ρˆ (−λ )X (λ ∈ A, X ∈ Uλ ), we have S 2 (Y1 )˘ ρ(Y2 v˘) = ρ˘(Y1 Y2 v˘) (Y1 , Y2 ∈ U ). a(X) ◦ ρ˘) (X ∈ U ). Define f˘ ∈ U ∗ by f˘(X) := TrV˘ (˘ Lemma 3.4. We have f˘ · X = ε(X)f˘ (X ∈ U ).  (1) (2) Proof. Let X, Y ∈ U . Write Δ(X) = r Xr ⊗ Xr . Then

(3.5)

(f˘ · X)(Y ) = = = = = = = = =

f˘(X •l Y )  ˘ (1) (2) f (Xr Y S(Xr )) r (1) (2) Tr ˘ (˘ a(Xr Y S(Xr )) ◦ ρ˘) r V (2) (1) Tr ˘ (˘ a(Y S(Xr )) ◦ ρ˘ ◦ a ˘(Xr )) r V (2) (1) Tr ˘ (˘ a(Y S(Xr )) ◦ a ˘(S 2 (Xr )) ◦ ρ˘) r V (2) (1) Tr ˘ (˘ a(Y S(Xr )S 2 (Xr )) ◦ ρ˘) r V (1) (2) a(Y S(S(Xr )Xr )) ◦ ρ˘) r TrV˘ (˘ ε(X)TrV˘ (˘ a(Y ) ◦ ρ˘) ε(X)f˘(Y ). 2

This completes the proof.

˘ 0 (Aπ ) := {1}. Then Define the Z-module homomorphism ω ˘ 0 : Aπ → K× by ω we can easily see the equations below. (3.6)

Zω˘ 0 (χ, π) = { X ∈ U | XY = Y X (Y ∈ U ) } . = { X ∈ U | ε(Y )X = X •r Y (Y ∈ U ) }.

Proposition 3.5. We have (3.7)

f˘ ∈ ΩU (Zω˘ 0 (χ, π))

ELEMENTS OF CENTER OF GENERALIZED QUANTUM GROUPS

and (3.8)

−1 ˘ HCχ,π ω ˘ 0 (ΩU (f )) =



27

ρˆχ,π (ν)m ˘ ν Kλ+ν Lμ−ν .

ν∈A+ π + Proof. For ν ∈ A+ π , let mν := dim Uν , let { Xν,x | x ∈ J1,mν } and { Y−ν,y | y ∈ − + ˘ ν,x , Y−λ ,y ) = δν,λ δx,y J1,mν } be K-base of Uν and U−ν respectively such that ϑ(X −  + ˘ ˘ (λ ∈ Aπ ), and let Y−ν,z ∈ U−ν (z ∈ J1,m ˘ | z ∈ J1,m ˘ ν ) be such that { Y−ν,z v ˘ ν } is a K−  + ∗ ˘ ˘ ˘ ˘   . Define t ∈ V by t ˘) = basis of U−ν v˘. For λ ∈ Aπ and z ∈ J1,m ˘ λ −λ ,z −λ ,z (Y−ν,y v δλ ,ν δz,y .   For λ , μ ∈ A+ π , x ∈ J1,mλ , y ∈ J1,mμ and λ , μ ∈ A, we have

f˘(Xλ ,x Kλ Y−μ ,y Lμ ) mν  = ν∈A+ t˘−ν,z (Xλ ,x Kλ Y−μ ,y Lμ ρ˘(Y˘−ν,z v˘)) π  z=1 mν = ν∈A+ ρˆχ,π (ν)t˘−ν,z (Xλ ,x Kλ Y−μ ,y Lμ Y˘−ν,z v˘) π z=1  mν = ν∈A+ ˆχ,π (ν)χ(λ , −μ − ν + μ)χ(ν + λ, μ ) z=1 ρ π ·t˘−ν,z (Xλ ,x Y−μ ,y Y˘−ν,z v˘)   mν = δλ ,μ ν∈A+ ˆχ,π (ν)χ(λ , −λ − ν + μ)χ(ν + λ, μ ) z=1 ρ π ·t˘ (Xλ ,x Y−λ ,y Y˘−ν,z v˘)  −ν,z mν (3.9) ˆχ,π (ν)t˘−ν,z (Xλ ,x Y−λ ,y Y˘−ν,z v˘) = δλ ,μ ν∈A+ z=1 ρ π ·ΩU (Y−λ ,x L−λ −ν+μ Xμ ,y Kν+λ )(Xλ ,x Kλ Y−μ ,y Lμ )  mν χ,π ˆ (ν)t˘−ν,z (Xλ ,x Y−λ ,y Y˘−ν,z v˘) = ν∈A+ z=1 ρ π ·ΩU (Y−λ ,x L−λ −ν+μ Xλ ,y Kν+λ )(Xλ ,x Kλ Y−μ ,y Lμ )  mν  mν χ,π  ˆ (ν)t˘−ν,z (Xν  ,x Y−ν  ,y Y˘−ν,z v˘) = ν,ν  ∈A+ x ,y  =1 z=1 ρ π ·ΩU (Y−ν  ,x L−ν  −ν+μ Xν  ,y Kν+λ )(Xλ ,x Kλ Y−μ ,y Lμ )  mν  mν χ,π ˆ (ν)t˘−ν,z (Xν  ,x Y−ν  ,y Y˘−ν,z v˘) = ΩU ( ν,ν  ∈A+ x ,y  =1 z=1 ρ π ·Y−ν  ,x L−ν  −ν+μ Xν  ,y Kν+λ )(Xλ ,x Kλ Y−μ ,y Lμ ). Hence f˘ ∈ ImΩU and  mν  mν χ,π  (f˘) = ˆ (ν)t˘−ν,z (Xν  ,x Y−ν  ,y Y˘−ν,z v˘) Ω−1  ,y  =1  ∈A+ U x z=1 ρ ν,ν π (3.10) ·Y−ν  ,x L−ν  −ν+μ Xν  ,y Kν+λ . ˘ By (3.3), (3.5) and (3.6), we have Ω−1 ˘ 0 (χ, π), which implies (3.7). By U ( f ) ∈ Zω (3.10), we have (3.8). This completes the proof. 2 Let Uπ0 = Uπ0 (χ, π) := ⊕ν,ν  ∈Aπ KKν Lν  and Uπ = Uπ (χ, π) := SpanK (U − Uπ0 U + ). Then Uπ (resp. Uπ0 ) is a K-subalgebra of U (resp. U 0 ). ˘ Lemma 3.6. Assume λ, μ − αl+1 ∈ Aπ . Let Z := Ω−1 U (f )L−αl+1 . Then Z ∈ Uπ ∩ Zω (χ, π). Proof. This easily follows from (3.10).

2

3.4. Conjecture on a basis of Zω (χ, π). In this subsection, assume A = Aπ . Then U = Uπ (χ, π). Let ω : A → K× be a Z-module homomorphism. For a Kalgebra homomorphism Λ : U 0 → K, there exists a simple U -module L(Λ) such − that there exists vΛ ∈ L(Λ) \ {0} such that L(Λ) = ⊕ν∈A+ U−ν vΛ , ZvΛ = Λ(Z)vΛ π and Ei vΛ = 0 (i ∈ I). For λ, μ ∈ A, define the K-algebra homomorphism Λχλ,μ;ω : U 0 → K by (3.11)

Λχλ,μ;ω (Kλ Lμ ) := χ(λ, μ )χ(λ , μ)ω(λ ) (λ , μ ∈ A).

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Let Finχω := {(λ, μ) ∈ A × A| dim L(Λχλ,μ;ω ) < ∞}.

(3.12)

χ ∈ Zω (χ, π) such Theorem 3.7. Let (λ, μ) ∈ Finχω . Then there exists a Zλ,μ;ω  χ χ,π − χ,π that HCω (Zλ,μ;ω ) = ν∈A+ ρˆ (ν)m ˘ ν Kλ+ν Lμ−ν , where m ˘ ν := dim U−ν vΛχλ,μ;ω . π

Proof. The claim follows from Proposition 3.5 and Lemmas 2.1 and 3.6.

2

Now we state the conjecture below. (3.13) χ Conjecture. The set { Zλ,μ;ω | (λ, μ) ∈ Finχω } must be a K-basis of Zω (χ, π). Remark 3.8. The above conjecture fits into [18]. 3.5. Z/3Z-quantum group. Assume that the characteristic of K is not 2 or 3. Assume l = 2. Then I = J1,2 . Assume A = Aπ . Let V be a two dimensional R-linear space such that A ⊂ V and {α1 , α2 } is an R-basis of V . Namely A = Aπ = Zα1 ⊕ Zα2 ⊂ V = Rα1 ⊕ Rα2 . Let ζ ∈ K be such that ζ 2 + ζ + 1 = 0. Let q ∈ K \ {0, 1, ζ}. (3.14)

Let χ(α1 , α1 ) = ζ, χ(α2 , α2 ) = q and χ(α1 , α2 )χ(α2 , α1 ) = q −1 .

Let β1 := α1 , β2 := 2α1 + α2 , β3 := α1 + α2 , and β4 := α2 . Then Rχπ,+ = {βt |t ∈ J1,4 }. Let γ1 := −β4 , γ2 := −β3 , γ3 := −β2 , γ4 := −β1 , γ5 = β4 , γ6 = β3 , γ7 = β2 , and γ8 = β1 . For t ∈ J1,8 , let γt := γs with s − t − 2 ∈ 8Z. For t (h) (h) h = (h1 , . . . , h8 ) ∈ R8≥0 and t ∈ J1,8 , let γt := s=1 ht γt . Let γ0 := 0 ∈ V . Let Bh := ∪8t=1 {γt−1 + uγt |0 ≤ u ≤ ht }, and Ch := ∩8t=1 (γt−1 + Rγt + R>0 γt ). Let Λ : U 0 (χ, π) → K be a K-algebra homomorphism. Let li := Λ(Kαi L−αi ) − (i ∈ I). Assume dim L(Λ) < ∞. Let m ˘ ν := dim U−ν vΛ . (ν ∈ A+ π ). By [3, Λ Lemma 6.6] and its proof, we can see that there exists a unique hΛ = (hΛ 1 , . . . , h8 ) ∈ (h)

(h)

(hΛ )

Z8≥0 such that γ8 = 0, {ν ∈ A+ ˘ ν > 0} ⊂ (BhΛ ∪ ChΛ ) ∩ A, and BhΛ ∩ A+ π |m π ⊂ + Λ Λ Λ {ν ∈ Aπ |m ˘ ν = 1}. By [3, (4.15)], we have hΛ , h , h , h ∈ J and, if o ˆ (q) = 0, 0,2 2 4 6 8 Λ Λ Λ , h , h , h ∈ J . Assume o ˆ (q) = 0. Then L(Λ) = {0} or there exists hΛ 0,ˆ o(q)−1 1 3 5 7 (m, n) ∈ Z≥0 × (2 + Z≥0 ) such that l2 = q m , l12 l2 = (ζq −1 )n , hΛ 2t = 2 (t ∈ J1,4 ), Λ Λ Λ hΛ 1 = h5 = m and h3 = h7 = n. (See also [20, Theorem 4.1], [3, Theorem 7.8. (cK4)].) Let H1 := {q s |s ∈ Z} and H2 := {(ζq −1 )t |t ∈ Z}. Assume oˆ(q) = 0. If Λ Λ Λ 2 Λ Λ Λ Λ l2 ∈ H1 , then hΛ 8 = h2 and h6 = h4 . If l1 l2 ∈ H2 , then h8 = h6 and h2 = h4 . 2πi For example, letting K = C, ξ := exp( 15 ) (∈ C) (i means the imaginary unit), Λ Λ Λ Λ ζ := ξ 5 and q := ξ 2 , if l1 = 1 and l2 = q, then hΛ 1 = 1, h2 = h8 = 0, h3 = h7 = 4 Λ Λ Λ h4 = h6 = 1, and h5 = 0. By the above argument and Theorems 2.4 and 3.7, we can easily convince ourself that the conjecture (3.13) for almost all χ of (3.14) must be true. 4. Appendix In this section, we mention a background of Theorem 2.2. Let I = J1,l be that of Subsections 2.1 and 2.2. Let X(I) be the set of all maps from N to I. Let the equivalence relation ∼ on the set X(I) × Z≥0 be such that (f, t) ∼ (f  , t ) if and only if t = t and f (k) = f  (k) for all k ∈ J1,t . Let X∼ (I) be the quotient set (X(I) × Z≥0 )/∼ of X(I) × Z≥0 by ∼. Let xf,t be the element of X∼ (I) such that

ELEMENTS OF CENTER OF GENERALIZED QUANTUM GROUPS

29

(f, t) is a representative of it. Regard X∼ (I) as the totally ordered set (X∼ (I), ≤) such that xf,t > xf  ,t if t ≥ 1, t ≥ 1, f (s) = f  (s) (s ∈ J1,t −1 and f (t ) < f  (t ) for some t ∈ J1,t ∩ J1,t or if t < t and f (s) = f  (s) (s ∈ J1,t ). (This is the lexicographical order in the sense of [16, Subsection 1.2].) We call xf,t a word. We call xf,0 an empty word. For (f, t) ∈ X(I) × Z≥0 , define f +,t ∈ X(I) by f +,t (k) := f (t + k) (k ∈ N). We also regard X∼ (I) as the monoid such that its unit is xf,0 and the multiplication is defined in the way that xf,t xf +,t ,k = xf,t+k . For i ∈ I, let xi := xf,1 for which f is such that f (1) = i. Then X∼ (I) can also be regarded as the free monoid generated by xi (i ∈ I). Let X∼ (I) := {xf,t ∈ X∼ (I)|t ≥ 1} and X∼ (I) := {xf,t ∈ X∼ (I)|t ≥ 2}. We say that u = xf,t ∈ X∼ (I) is a standard word (or a Lyndon-Shishov word)   if t ∈ J0,1 or if u ∈ X∼ (I) and u > wv for all v, w ∈ X∼ (I) with vw = u. Let  LS∼ (I) be the set of all standard words. Let LS∼ (I) := LS∼ (I) ∩ X∼ (I) . and LS∼ (I) := LS∼ (I) ∩ X∼ (I) . It is well-known [16, Corollary1.1] that 



∀u ∈ LS∼ (I) , ∃v, ∃w ∈ LS∼ (I) s.t. u = vw, v > w. 



For u ∈ LS∼ (I) , let u, ˙ u ¨ ∈ LS∼ (I) be such that u = u¨ ˙ u and there exists y ∈ X∼ (I)  ∼ with v = uy ˙ for all v, w ∈ LS (I) satisfying u = vw ([16, Theorem 1.1] implies u˙ > u ¨ and v > w). Let KX∼ (I) be the K-linear space such that X∼ (I) is its Kbase. Regard KX∼ (I) as the associative K-algebra such that X∼ (I) is a submonoid of KX∼ (I). Then KX∼ (I) can also be regarded as the free K-algebra generated by xi (i ∈ I). Let χ : A × A → K× , αi (i ∈ I) and A+ π be as in Subsections 2.1 and 2.2.  ∼ For xf,t ∈ X∼ (I), let θ(xf,t ) := tk=1 αf (k) (∈ A+ π ). For u = xf,t ∈ LS (I), define ∼ [u] ∈ KX (I) by  u if t ∈ J0,1 , [u] := [u][¨ ˙ u] − χ(θ(¨ u), θ(u)) ˙ −1 [¨ u][u] ˙ if t ≥ 2. We call [u] a super-letter. Theorem 4.1. (See [16, Theorems 1.1, 2.6 and Lemma 2.6]) (1) The set (4.1) {1} ∪ {[u1 ][u2 ] · · · [uk ]|k ∈ N, ur ∈ LS∼ (I) (r ∈ J1,k ), ur ≤ ur+1 (r ∈ J1,k−1 )} 

is a K-basis of KX∼ (I). Moreover, for v1 , v2 ∈ LS∼ (I) with v1 > v2 , [v1 ][v2 ] is a linear combination of elements [u1 ]n1 [u2 ]n2 · · · [uk ]nk in (4.1) with v2 ≤ u1 , v1 ≤ uk , and θ(un1 1 un2 2 · · · unk k ) = θ(v1 ) + θ(v2 ). (2) The same claim as that of (1) with the monomials u1 u2 · · · uk in place of [u1 ][u2 ] · · · [uk ] is true. ´ λ (λ ∈ A) and E ´i (i ∈ I) satisfying ´ +, be a Hopf K-algebra generated by K Let U ´ λ and E ´i in place of Kλ and Ei the same equations as those of (2.4) and (2.5) with K +,,0 + ´ ´ ´ +, generated by K ´λ respectively. Let U (resp. U ) be the K-subalgebra of U +, + ´ +,,0 ´ ´ ´ ´ ´ (λ ∈ A) (resp. Ei (i ∈ I)). Note U = SpanK (U U ). Let G := {Kλ |λ ∈ A}. ´ is a K-basis of U ´ +,,0 , and that K ´ λ = K ´ μ for λ = μ. Assume Assume that G +, ´ = {g ∈ U ´ |Δ(g) = g ⊗ g}. (Such Hopf algebra is called a Character that G Hopf algebra, see [16, Defnnition 1.11].) Let Γ = (Γ, ) be a well-ordered additive (commutative) monoid, see [14] for this terminology.

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Example 4.2. Let n ∈ N. Define the well-order  on Zn≥0 in the way that for n k = (k , . . . , kn ), r = (r1 , . . . , rn ) ∈ Zn≥0 , we have k ≺ r if and only if t=1 kt < n 1 n n t=1 rt or t=1 kt = t=1 rt and there exists b ∈ J1,n with kb < rb and kb = rb for all b ∈ J1,b−1 . Then (Zn≥0 , ) is a well-ordered additive monoid. Note that A+ π is a well-ordered additive monoid in a natural sense. Let D : X∼ (I) → Γ be a monoid homomorphism such that 0 ∈ / D(X∼ (I) ). Also, let D([u1 ][u2 ] · · · [uk ]) mean D(u1 u2 · · · uk ). Define the K-algebra epimor´ + by Φ(xi ) := E ´i (i ∈ I). For the element X = phism Φ : KX∼ (I) → U [u1 ][u2 ] · · · [uk ] in (4.1), let M (X) := uk . Let B be the set in (4.1). For u ∈ LS∼ (I) , we call [u] hard if Φ([u]) ∈ / SpanK ({Φ(X)|X ∈ B, D(X) = D(u), M (X) < ´ Y ∈ B, D(Y ) < D(u)}), see [16, Definition 2.6 (see also u} ∪ {gΦ(Y )|g ∈ G, Lemma 2.6)]. For a hard super-letter [u], let H([u]) := {k ∈ N|Φ([u]k ) ∈ SpanK ({Φ(X)|X ∈ B, D(X) = D(uk ), M (X) < u} ´ Y ∈ B, D(Y ) < D(uk )})}. ∪ {gΦ(Y )|g ∈ G, If H([u]) = ∅, let h([u]) := min H([u]). Otherwise, let h([u]) := ∞. Clearly h([u]) ≥ 2. Theorem 4.3. ([16, Defintion 2.7, Theorems 2.2 and 2.4 (see also Lemma 2.6)]) (1) Let h := h([u]). If h < ∞, then χ(θ(u), θ(u)) is a primitive t-th root of unity for some t ∈ N, and h = tpr for some r ∈ Z≥0 , where p := Char(K)(∈ {0} ∪ (N \ {1}). (2) Let H be the set of hard super-letters. Then the elements ´ [ui ] ∈ H, ni < h([ui ]), u1 < · · · < uk ) gΦ([u1 ]n1 [u2 ]n2 · · · [uk ]nk ) (g ∈ G, ´ +, . form a K-base of U (3) The same claim as that of (2) with the monomials un1 1 un2 2 · · · unk k in place of [u1 ]n1 [u2 ]n2 · · · [uk ]nk is true. ´ +,,0 → U ´ +, defined ´+ ⊗U Assume that we have the K-linear isomorphism ς´ : U by ς´(X ⊗ Z) := XZ. By Theorem 4.3, we can easily see that (4.2)

{Φ([u1 ]n1 [u2 ]n2 · · · [uk ]nk )|[ui ] ∈ H, ni < h([ui ]), u1 < · · · < uk } ´ +. is a K-basis of U

´ := (The claim of [u1 ]n1 [u2 ]n2 · · · [uk ]nk can be replaced by un1 1 un2 2 · · · unk k .) Let H o ˆ(χ(θ(u),θ(u)))ps s H ∪ {[u] |[u] ∈ H, s ∈ Z≥0 , 0 < oˆ(χ(θ(u), θ(u)))p < h([u])}, where ´ in the way that [u]x > [v]y if and p := Char(K). Define the total order < on H ´ : H ´ → (N \ {1}) ∪ {∞} only if u > v or u = v and x < y. Define the map h ´ as follows. If oˆ(χ(θ(u), θ(u))) = 0, let h([u]) := ∞ (= h([u])). Otherwise, let x ´ in place ´ ´ and h ) := oˆ(χ(θ(ux ), θ(ux ))). Then the same claim as (4.2) with H h([u] of H and h is true. ´ + as a K-linear space with ´+ = ⊕ U Assume that we have a direct sum U λ λ∈A+ π ´ ´μ+ ⊂ U ´ +, E ´i ∈ U ´α+ and U ´ +U ´ + . Let R ´ χ+ := {θ(ux )|[u]x ∈ H}. 1∈U Define the 0 λ λ+μ i + + + x x ´ ´ χ → N by ϕ´χ (β) := |{[u] ∈ H|θ(u ´ χ+ , ϕ´+ ) = β}|. Then the pair (R map ϕ´χ : R χ) +, is independent from choice of D (and Γ). Recall U from the sentence below ´ +, with U +, in a natural way. Then (R ´ χ+ , ϕ´+ (2.5). Identify U χ ) can be identified π,+ π,+ x x ´ and with (Rχ , ϕχ ) of Theorem 2.2. Let [u] ∈ H, α := θ([u] ), t ∈ J1,ϕ´+ χ (α) d := (α, t). Then we have y(d) ∈ J0,h([u] x )−1 , where y is the one of Theorem 2.2. ´

ELEMENTS OF CENTER OF GENERALIZED QUANTUM GROUPS

31

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I. Angiono and H. Yamane, The R-matrix of quantum doubles of Nichols algebras of diagonal type, J. Math. Phys. 56 (2015), no. 2, 021702, 19, DOI 10.1063/1.4907379. MR3390862 I. Angiono and H. Yamane, Bruhat order and nil-Hecke algebras for Weyl groupoids, J. Algebra Appl. 17 (2018), no. 9, 1850166, S. Azam, H. Yamane, and M. Yousofzadeh, Classification of finite-dimensional irreducible representations of generalized quantum groups via Weyl groupoids, Publ. Res. Inst. Math. Sci. 51 (2015), no. 1, 59–130, DOI 10.4171/PRIMS/149. MR3367089 P. Batra and H. Yamane, Skew centers of rank-one generalized quantum groups, Toyama Math. J. 37 (2015), 189–202. MR3468997 P. Batra and H. Yamane, Centers of generalized quantum groups, J. Pure Appl. Algebra 222 (2018), no. 5, 1203–1241, DOI 10.1016/j.jpaa.2017.06.015. MR3742226 M. Cuntz and I. Heckenberger, Finite Weyl groupoids, J. Reine Angew. Math. 702 (2015), 77–108, DOI 10.1515/crelle-2013-0033. MR3341467 V. G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR934283 I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), no. 1, 175–188, DOI 10.1007/s00222-005-0474-8. MR2207786 I. Heckenberger, Nichols algebras, ECNU Shanghai, July 2008 http://www.mathematik.unimarburg.de/ heckenberger/na.pdf I. Heckenberger, Classification of arithmetic root systems, Adv. Math. 220 (2009), no. 1, 59–124, DOI 10.1016/j.aim.2008.08.005. MR2462836 I. Heckenberger and H. Yamane, A generalization of Coxeter groups, root systems, and Matsumoto’s theorem, Math. Z. 259 (2008), no. 2, 255–276, DOI 10.1007/s00209-007-0223-3. MR2390080 I. Heckenberger and H. Yamane, Drinfel’d doubles and Shapovalov determinants, Rev. Un. Mat. Argentina 51 (2010), no. 2, 107–146. MR2840165 N. Jing, K. Misra, and H. Yamane, Kostant-Lusztig A-bases of multiparameter quantum groups, Representations of Lie algebras, quantum groups and related topics, Contemp. Math., vol. 713, Amer. Math. Soc., Providence, RI, 2018, pp. 149–164, DOI 10.1090/conm/713/14316. MR3845913 Y. Kobayashi, Well-ordered monoids -two numerical functions-, Surikaiseki Kenkyusho Kokyuroku, RIMS, vol. 1366 (2004), 111-120. https://repository.kulib.kyotou.ac.jp/dspace/bitstream/2433/25368/1/1366-13.pdf V. K. Kharchenko, A quantum analogue of the Poincar´ e-Birkhoff-Witt theorem (Russian, with Russian summary), Algebra Log. 38 (1999), no. 4, 476–507, 509, DOI 10.1007/BF02671731; English transl., Algebra and Logic 38 (1999), no. 4, 259–276. MR1763385 V. Kharchenko, Quantum Lie theory: A multilinear approach, Lecture Notes in Mathematics, vol. 2150, Springer, Cham, 2015. MR3445175 G. Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkh¨ auser Boston, Inc., Boston, MA, 1993. MR1227098 A. Sergeev and A. Veselov, Grothendieck rings of basic classical Lie superalgebras, Ann. T. Tanisaki, Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras, Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 941–961, DOI 10.1142/s0217751x92004117. MR1187582 H. Yamane, Representations of a Z/3Z-quantum group, Publ. Res. Inst. Math. Sci. 43 (2007), no. 1, 75–93. MR2317113

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India Email address: [email protected] Graduate School of Science and Engineering of Research, University of Toyama, 3190 Gofuku, Toyama-shi, Toyama 930-8555, Japan Email address: [email protected]

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15081

The global dimension of the generalized Weyl algebras S −1 K[H, C][X, Y ; σ, a] V. V. Bavula and K. Alnefaie This paper is dedicated to Professor S. K. Jain Abstract. The aim of the paper is to compute the global dimension of the algebras A in the title of the paper, their tensor products, some of their factor algebras. Some class of diskew polynomial rings belong to this class of algebras. Many classical algebras are examples of the algebras A (eg, U (sl2 ), Uq (sl2 ), the Heisenberg algebra and its quantum analogues, the Witten’s deformations, the Woronowicz’s deformation, Oq (SL2 (K)) and others). As a consequence, the exact value of the global dimension is computed for well-known quantum algebras, their localizations and tensor products.

1. Introduction Generalized Weyl algebras D(σ, a) with central element a. Definition, [2]-[9]. Let D be a ring, σ be a ring automorphism of D, a is a central element of D. The generalized Weyl algebra of rank 1 (GWA, for short) D(σ, a) = D[X, Y ; σ, a] is a ring generated by the ring D and two elements X and Y that are subject to the defining relations: (1.1) Xd = σ(d)X and Y d = σ −1 (d)Y for all d ∈ D, Y X = a and XY = σ(a). The ring D is called the base ring of the GWA. The automorphism σ and the element a are called the defining automorphism and the defining element of the GWA, respectively. Many popular algebras of small Gelfand-Kirillov dimension are GWAs (the first Weyl algebra A1 and its quantum analogue, the quantum plane, the quantum sphere and others). In particular, the following classical algebras are GWAs where either D = K[H, C] or D = K[H ±1 , C] (see Section 2): U (sl2 ), Uq (sl2 ), the Heizenberg algebra and its quantum analogues, the Witten’s and Woronowic’s deformations, Oq (SL2 (K)), Noetherian down-up algebras, etc. More examples of GWAs can be found in [16–20]. A more general construction of GWAs can be found in [13, 14].

2010 Mathematics Subject Classification. Primary 16E10, 16E05, 16P40, 16S35. Key words and phrases. The generalized Weyl algebra, the global dimension, simple module, the projective dimension, orbit, the projective resolution. c 2020 American Mathematical Society

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The global dimension of GWAs. The global dimension of GWAs were studied in a series of papers [5, 6, 11, 12]. For a ring D, we denote by lgd (D) its left global dimension and by gld (D) its left and right global dimension if they coincide. For each Noetherian ring, its left and right global dimension coincide. Theorem 1.1. ([11, Theorem 2.7].) lgd (A) = lgd (D), lgd (D) + 1 or ∞.

Let A = D(σ, a) be a GWA. Then

Let D be a ring and σ be its automorphism. An ideal I of D is called semistable (or σ-semistable) if σ i (I) = I for some i ≥ 1. An element a of a ring is called a regular element if it is not a zero divisor. When the ring D is a commutative Noetherian ring the theorem below gives more accurate estimates for the global dimension of the generalized Weyl algebra A = D(σ, a). Theorem 1.2. ([11, Theorem 3.7].) Let D be a commutative Noetherian ring of global dimension n < ∞, A = D(σ, a) be a GWA and the element a be a regular element of D. Suppose that gld A < ∞. Then gld A = n + 1 if and only if either there is a semistable maximal ideal of D of height n or there are maximal ideals p, q of D of height n such that σ i (p) = q for some i = 0 ∈ Z and a ∈ p, q. A weak (homological) dimension of a ring R is denoted by wd (R). Proposition 1.3. ([12, Corollary 4.4.(2)].) Let A = D(σ, a) be a GWA. Then (1) wd (A) ≥ wd (D). (2) If D is a Noetherian ring then gld (A) ≥ gld (D). gld (D) = ∞ then gld (A) = ∞.

In particular, if

The next result shows that the global dimension of a GWA A = D(σ, a) can be infinite but gld (D) = 1. Recall that the global dimension of a commutative Dedekind domain is 1. Theorem 1.4. ([12, Theorem 1.6].) Let A = D(σ, a) be a GWA where D is a commutative Dedekind domain and Da = pn1 1 · · · pns s is a product of distinct maximal ideals of D. Then ⎧ ∞ if a = 0 or ni ≥ 2 for some i, ⎪ ⎪ ⎪ ⎪ ⎪ 2 if a = 0, n1 = · · · = ns = 1, s ≥ 1 ⎪ ⎪ ⎪ ⎨ or a is invertible, and there exists an integer k ≥ 1 gld (A) = ⎪ such that either σ k (pi ) = pj for some i, j ⎪ ⎪ ⎪ ⎪ ⎪ or σ k (q) = q for some maximal ideal q of D, ⎪ ⎪ ⎩ 1 otherwise. Theorem 1.2 is a very effective tool in finding the global dimension of GWAs A provided gld (A) < ∞. The goal of the paper is to give an explicit criterion (Theorem 1.6) for gld (A) < ∞ for some GWAs A. In general, not much technique is available to verify whether an algebra has infinite global dimension or not. One of the promising directions is to select a relatively small class of modules that is ‘responsible’ for the algebra to have infinite global dimension. For GWAs with commutative Noetherian base ring which is a domain, there is such a class of modules (Theorem 1.5).

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

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Theorem 1.5. ([11, Theorem 3.5].) Let A = D(σ, a) be a GWA where D is a commutative Noetherian domain of finite global dimension and a = 0. Then gld (A) < ∞ if and only if pdA (A/A(X, p)) < ∞ for all prime ideals p of D such that a ∈ p. Assumption. In this paper, if it is not stated otherwise, the ring D = S −1 K[H, C] is a localization of the polynomial algebra K[H, C] in two variables H and C over a field K at a multiplicative set S ⊆ K[H, C] such that the Krull dimension of D is 2. Remark. In general, there are three options for the algebra D = S −1 K[H, C]: a field, a Dedekind domain or to have Krull dimension 2. If D is a field then either A  D[X, X −1 ; σ] or A = D[X, Y ; σ, 0], and so either gld (A) = 1 or gld (A) = ∞ (Proposition 3.1). If D is a Dedekind domain then Theorem 1.4 gives a formula for the global dimension of A. So, the only interesting case is the last one which is considered in the present paper. • The goal of the paper is to prove Theorem 1.6 and as a result to compute the global dimension of many popular algebras, see Section 2 for details. Theorem 1.6. Let K be an algebraically closed field, the algebra D = S −1 K[H, C] has Krull dimension 2, D∗ be its group of units and A = D[X, Y ; σ, a] be a GWA. Then gld (A) < ∞ if and only if either a ∈ D∗ or a ∈ D\{D∗ , 0} and ∂a ∂a grad (a) := ( ∂H , ∂C ) ≡ 0 mod m (equivalently, a ∈ / m2 ) for all maximal ideals m of D such that a ∈ m. As an application of Theorem 1.2 and Theorem 1.6 we compute the global dimension of many classical algebras that are examples of GWAs in Theorem 1.6, see Section 2 for details. Corollary 1.7. Let K be an algebraically closed field. (1) (2) (3) (4) (5) (6) (7)

gld (U sl(2)) = 3. gld (Uq sl(2)) = 3. The global dimension of Woronowicz’s deformation V is 3. The global dimension of Witten’s first deformation E is 3. The global dimension of Witten’s second deformation W is 3. The global dimension of the algebra Oq2 (so(K, 3)) is 3. The global dimension of the universal enveloping algebra H = U (N ) of the Heisenberg Lie algebra is 3. (8) The global dimension of the quantum Heisenberg algebra Hq is 3. (9) Let Λ(b) be the deformation of U sl(2) (see [2], [23], [6]) and char (K) = 0. Then gld (Λ(0)) =  3 and for b = 0 3 if α(μ) = α(μ + i) for some μ ∈ K and i ∈ N\{0}, gld (Λ(b)) = 2 otherwise, where α(H) is a solution of the equation α(H) − α(H − 1) = b. In particular, gld (U sl(2)) = 3 since U sl(2) = Λ(2H). (10) gld (Oq (SL2 (K))) = 3. The proof of Corollary 1.7 is given in Section 2.

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The global dimension of a class of diskew polynomial ring. The algebras in Corollary 1.7 are examples of the following construction of rings that first appeared in full generality in [11] and is a particular case of the construction of diskew polynomial ring [13, 14]: Let D be a ring, σ be its automorpmism and ρ, b are central elements of the ring D such that ρ is a unit of D and σ(ρ) = ρ. The ring E = DX, Y ; σ, b, ρ is generated by D, X and Y subject to the defining relations: For all elements d ∈ D, (1.2)

Xd = σ(d)X, Y d = σ −1 (d)Y and XY − ρY X = b.

The ring E is a very special case of a GWA. Lemma 1.8. ([11, Lemma 1.2].) The ring E = DX, Y ; σ, b, ρ is a GWA E = D[H][X, Y ; σ, a = H] where D[H] is as polynomial ring in a variable H with coefficients in D and σ is an extension of the automorphism σ of D to D[H] by the rule σ(H) = ρH + b. The next theorem is a relatively easy corollary of Theorem 1.4, Theorem 1.6 and Lemma 1.8. Its proof is given in Section 3. Theorem 1.9. Let D = S −1 K[C] be a localization of a polynomial algebra K[C] in a variable C at a multiplicative subset S of K[C] and E = DX, Y ; σ, b, ρ. Then gld (E) = 1, 2 or 3. Futhermore, (1) if D is a field then gld (E) = 1 or 2 and the exact value of gld (E) is given in Theorem 1.4 (E = D[H][X, Y ; σ, a = H] is a GWA where D[H] is a Dedekind domain, by Lemma 1.8). (2) If D is not a field and K is an algebraically closed field then gld (E) = 2 or 3, and gld (E) = 3 iff there are natural number n ≥ 1 and an element  i n−i−1 b). β ∈ K such that Dξn +D(σ n (C)−β) = D where ξn := n−1 i=0 σ (ρ The global dimension of tensor products of GWAs. For all left Noetherian algebras A and B, lgd(A ⊗ B) ≥ lgd A + lgd B, [1]. Definition, [12]. An algebra A is called the tensor homological minimal algebra (THM) with respect to a class of algebras Ω if lgd (A ⊗ B) = lgd A + lgd B for all B ∈ Ω. Let N be the class of countable Noetherian algebras (i.e., algebras have countable basis). Theorem 1.10. Let K be an algebraically uncountable closed field, Λ = ⊗ni=1 Λi be a tensor product of finitely generated GWAs in Theorem 1.6 (eg, all algebras in Corollary 1.7). Then thealgebra Λ is a THM algebra with respect to the class N . n In particular, lgd (Λ) = i=1 gld (Λi ). The paper is organized as follows. In Section 2, applications are given of Theorem 1.6 for computing the global dimension of many quantum algebras including the ones in Corollary 1.7. Entire Section 3 is about the proof of Theorem 1.6. The GWAs K[H, C][X, Y ; σ, a] and S −1 K[H, C][X, Y ; σ, a] where σ is an affine automorphism of the polynomial algebra D = K[H, C] are considered in Section 4. Up to change of variables, there are 4 types of affine automorphisms provided the ground field K is algebraically closed. In each of the four cases, an explicit value of the global dimension is given (Theorem 4.2, Theorem 4.3, Theorem 4.4 and Theorem 4.5).

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

37

2. The global dimension of Witten’s, Woronowic’z deformations and quantum algebras The aim of this section is to apply Theorem 1.6 for computing the global dimension of many classical algebras given in Corollary 1.7. In this section, K is an algebraically closed field. Examples of generalized Weyl algebras where a is central, [2]-[9]. 1. The (first) Weyl algebra A1 = Kx, ∂ | ∂x − x∂ = 1 over a field K is the GWA K[h][x, y := ∂; σ, a = h] with base ring K[h] and its K-automorphism σ defined by the rule σ(h) = h − 1. 2. The quantum plane Λ = Kx, y | xy = qyx where q is a central unit of K is the GWA K[h][x, y; σ, a = h] where σ(h) = qh. 3. For q, h = q − q −1 ∈ K = C such that q = ±1, the algebra Uq = Uq sl(2) is generated by X, Y, H− and H+ subject to the defining relations: H+ H− = H− H+ = 1, XH± = q ±1 H± X, Y H± = q ∓1 H± Y, [X, Y ] =

2 2 − H− H+ . h

It follows that the algebra Uq is a GWA,

(2.1) Uq  K[C, H, H −1 ](σ, a = C + H 2 /(q 2 − 1) − H −2 /(q −2 − 1) /2h) where σ(H) = qH, σ(C) = C. Corollary 2.1. gld (Uq sl(2)) = 3. Proof. Let U = Uq sl(2). ∂a (i) gld (U ) < ∞: Since ∂C = 1 (see (2.1)). Hence, gld (U ) < ∞, by Theorem 1.6. (ii) If q is a root of unity then gld (U ) = 3: If q n = 1 for some n ≥ 1 then σ = 1. Since gld (U ) < ∞, we must have gld (U ) = 3, by Theorem 1.2. (iii) If q is not a root of unity then gld (U ) = 3: Since q is not a root of unity, every σ-orbit of a maximal ideal of the algebra D = K[C, H, H −1 ] is infinite. Furthermore, for all integers i ≥ 1, n

Da + Dσ i (a) = Da + D(σ i (a) − a) = Da + D( Therefore, gld (U ) = 3, by Theorem 1.2.

q 2i − 1 2 q −2i − 1 −2 H − −2 H ) = D. q2 − 1 q −1 

4. Woronowicz’s deformation V is an algebra generated by elements V0 , V− and V+ subject to the defining relations, [25] where s ∈ K and s4 = 0, 1: [V0 , V+ ]s2 := s2 V0 V+ − s−2 V+ V0 = V+ , [V− , V0 ]s2 = V− , [V+ , V− ]1/s := s−1 V+ V− − sV− V+ = V0 . The algebra V is a GWA, V  K[u, v](σ, a = v), V+ ↔ x, V− ↔ y, V0 ↔ u, V− V+ ↔ v where σ : u → s2 (s2 u − 1), v → s2 v + su, is the automorphism of the polynomial ring K[u, v] in two variables u and v. Let H = u + s2 /(1 − s4 ) and Z = v + u/s(1 − s2 ) + s3 /(1 − s2 )(1 − s4 ).

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Then σ(H) = s4 H, σ(Z) = s2 Z and K[u, v] = K[H, Z]. So, (2.2)

V  K[H, Z](σ, a = Z +αH +β), V+ ↔ x, V− ↔ y, V0 ↔ H −s2 /(1−s4 )

where σ : H → s4 H, Z → s2 Z; α = −1/s(1 − s2 ) and β = s/(1 − s4 ). Corollary 2.2. The global dimension of Woronowicz’s deformation is 3. ∂a Proof. By (2.2), ∂Z = 1. Hence, gld (V ) < ∞, by Theorem 1.6. The maximal ideal (H, Z) of the polynomial algebra K[H, Z] is σ-invariant. Hence, gld (V ) = 3, by Theorem 1.2. 

5. Witten’s first deformation E is an algebra generated by elements E0 , E− and E+ subject to the defining relations, [25]: [E0 , E+ ]p := pE0 E+ − p−1 E+ E0 = E+ , [E− , E0 ]p = E− , [E+ , E− ] = E0 − (p − 1/p)E02 where p ∈ K and p4 = 0, 1. The element C = E− E+ + Witten’s first deformation is a GWA,

E0 (E0 +p) p(p2 +1)

is central in E.

E  K[C, H](σ, a = C − H(H + 1)/(p + p−1 )), E+ ↔ x, E− ↔ y, E0 ↔ pH where σ : C → C, H → p2 (H − 1). Let λ = p2 /(p2 − 1) and H  = H − λ. Then K[C, H] = K[C, H  ] and (2.3)

E  K[C, H  ](σ, a = C −

(H + λ)(H + λ + 1) ) p + p−1

where σ(C) = C and σ(H  ) = p2 H  . Corollary 2.3. The global dimension of Witten’s first deformation is 3. ∂a Proof. By (2.3), ∂C = 1. Hence, gld (E) < ∞, by Theorem 1.6. The maxi mal ideal (C, H ) of K[C, H  ] is σ-invariant. Hence, gld (E) = 3, by Theorem 1.2. 

6. Witten’s second deformation W is an algebra generated by W0 , W− and W+ subject to the defining relations, [25]: [W0 , W+ ]r = W+ , [W− , W0 ]r = W− , [W+ , W− ]1/r2 = W0 where r ∈ K and r 4 = 0, 1. The algebra W is a GWA (see p.99, [11]): (2.4)

W  K[H, C](σ, a = C − α), W+ ↔ X, W− ↔ Y, W0 ↔ H −

where σ(C) = r 4 C, σ(H) = r 2 H and α = (H −

r 1−r 2 )(H

−

r3 2 1−r 2 )/r (r

r 1 − r2

+ r −1 ).

Corollary 2.4. The global dimension of Witten’s second deformation is 3. ∂a Proof. By (2.4), ∂C = 1, and so gld (W ) < ∞, by Theorem 1.6. The maximal ideal (H, C) of K[H, C] is σ-invariant. Hence, gld (W ) = 3, by Theorem 1.2. 

7. The quantum group Oq2 (so(K, 3)) = K[H]σ; b = (q − q −1 )H, ρ = 1, σ(H) = q 2 H, q ∈ K, [24], is isomorphic to the GWA of degree 1: Oq2 (so(K, 3))  K[H, C](σ, a = C + H 2 /q(1 + q 2 )), σ(H) = q 2 H, σ(C) = C. Corollary 2.5. The global dimension of the quantum group Oq2 (so(K, 3)) is 3.

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

39

∂a Proof. Since ∂C = 1, gld (Oq2 (so(K, 3))) < ∞, by Theorem 1.6. The maximal ideal (H, C) of K[H, C] is σ-invariant. Hence, gld (Oq2 (so(K, 3))) = 3, by Theorem 1.2. 

8. Let N = KX ⊕KY ⊕KZ be the 3-dimensional Heisenberg Lie algebra where [X, Y ] = Z is a central element of N and H = U (N ) be its universal enveloping algebra. The algebra H is a GWA. H  K[H, Z][X, Y ; σ, H]

(2.5)

where σ(H) = H + Z. (Since XH = XY X = (Y X + [X, Y ])X = (H + Z)X = σ(H)X). Corollary 2.6. gld (H) = 3. ∂a Proof. By (2.5), ∂H = 1, and so gld (H) < ∞, by Theorem 1.6. The maximal ideal (H, Z) of K[H, Z] is σ-invariant. Hence, gld (H) = 3, by Theorem 1.2. 

9. The quantum Heisenberg algebra (see [20, 21, 26]) Hq = KX, Y, H | XH = q 2 HX, Y H = q −2 HY, XY − q −2 Y X = q −1 H, q ∈ K, q 4 = 0, 1 is a GWA (see p.302, [12]): Hq  K[H, C](σ, a = q 2 (C −

(2.6)

H )) q(1 − q 4 )

where σ(H) = q 2 H and σ(C) = q −2 C. Corollary 2.7. gld (Hq ) = 3. ∂a = q 2 = 0, and so gld (Hq ) < ∞, by Theorem 1.6. The Proof. By (2.6), ∂C maximal ideal (H, C) of K[H, C] is σ-invariant. Hence, gld (Hq ) = 3, by Theorem 1.2. 

10. Recall that the algebra U sl(2) = KX, Y, H | [H, X] = X, [H, Y ] = −Y, [X, Y ] = 2H is the universal enveloping algebra of the Lie algebra sl(2) over a field of characteristic zero K. Let us consider its deformation ([2, 6, 23]): (2.7)

Λ(b) = KX, Y, H | [H, X] = X, [H, Y ] = −Y, [X, Y ] = H

where b ∈ K[H]. The algebra Λ(b) is a GWA (see ([12, Example 3])): Λ(b)  K[H, C][X, Y ; σ, a = C − α]

(2.8)

where σ(H) = H − 1, σ(C) = C and α ∈ K[H] is a solution of the equation α − σ(α) = b (which exists as the map 1 − σ : K[H] → K[H] is a locally nilpotent map that is K[H] = ∪i≥1 ker(1 − σ)i ). Corollary 2.8.  gld (Λ(b)) =

(1) gld (Λ(0)) = 3 and if b = 0 then 3 2

if α(μ) = α(μ + i) for some μ ∈ K and i ∈ N\{0}, otherwise.

(2) gld (U sl(2)) = 3.

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V. V. BAVULA AND K. ALNEFAIE

∂a Proof. (1). By Eq. (2.8), ∂C = 1, and so gld (Λ(b)) < ∞, by Theorem 1.6. If b = 0 then a = C (since 0 − σ(0) = 0). Then the maximal ideals (H, C) and σ(H, C) = (H − 1, C) both contain the element a and are distinct. By Theorem 1.2, gld (Λ(0)) = 3. Suppose that b = 0. Since char (K) = 0, for every maximal ideal its σ-orbit is infinite. Since a = C − α, gld (A) = 3 if and only if there is a maximal ideal (C − λ, H − μ) of K[H, C] that contains a (i.e., λ = α(μ)) such that the maximal ideal

σ i (C − λ, H − μ) = (C − λ, H − i − μ) also contains the element a (i.e., λ = α(μ + i)) for some natural number i ≥ 1 if and only if α(μ) = α(μ + i) for some μ ∈ K and i ∈ N\{0}. Now, statement (1) follows from Theorem 1.2. (2). Since b = 2H, the element α can be chosen as α = H(H + 1) (since α − σ(α) = H(H + 1) − (H − 1)H = 2H). Hence, gld (U sl(2)) = 3, by statement (1), since α(−1) = α(0) = 0.  11. In [15], it was shown that the the algebra Oq (SL2 (K)) (the coordinate ring of the quantum SL2 (K) where q ∈ K is such that q 4 = 0, 1) is the GWA, Oq (SL2 (K))  K[H, C][X, Y ; σ, a = 1 + q −2 HC] where σ(H) = q 2 H and σ(C) = q 2 C. Corollary 2.9. gld (Oq (SL2 (K))) = 3. Proof. Since a = 1 + q −2 HC and grad(a) = q −2 (C, H), we have that gld (Oq (SL2 (K))) < ∞, by Theorem 1.6. The maximal ideal (H, C) of K[H, C] is σ-invariant. Hence,  gld (Oq (SL2 (K))) = 3, by Theorem 1.2. 3. Proof of Theorem 1.6 The aim of this section is to prove Theorem 1.6. The strategy of proving Theorem 1.6 is roughly as follows. A smal class of left ideals of the GWA A is considered that has property that gld (A) < ∞ iff their projective dimensions are finite (Theorem 1.5). For every such left ideal an explicit projective resolution is produced (Theorem 3.9). A criterion (Theorem 3.10) is given for all such left ideals to have finite projective dimension from which Theorem 1.6 follows (using some other results). In order to produce the projective resolution in Theorem 3.9, the 4 × 4 matrices e and d are used, see (3.6). We study their properties first (Proposition 3.7 and Lemma 3.8). Proposition 3.1. Let D be an arbitrary ring and A = D[X, Y ; σ, 0] be a GWA. Then l.gld (A) = r.gld (A) = ∞. Proof. Let us show that l.gld (A) = ∞. Using the Z-grading of the GWA A, we obtain the following short exact sequences of left A-modules: α

ϕ

β

ψ

(3.1)

0 → AX − →A− → AY → 0, α(u) = u, ϕ(v) = vY,

(3.2)

0 → AY − →A− → AX → 0, β(u) = u, ψ(v) = vX.

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

41

Notice that AX = ⊕i≥1 DX i and AY = ⊕i≥1 DY i (since a = 0). The short exact sequence (3.1) does not split since otherwise we would have a splitting homomorphism χ : A → AX, χα = id where id is the identity map on AX. Then X = χα(X) = χ(X) = Xχ(1) ∈ X · AX = ⊕i≥2 DX i , a contradiction. By a similar reason, the short exact sequence (3.2) does not split. So, the left A-modules AX and AY are not projective. Therefore, l.gld (A) = ∞. Since the opposite algebra Aop to the GWA A is the GWA Dop [Y, X; σ, 0], we have that  r.gld (A) = l.gld (Aop ) = l.gld (Dop [Y, X; σ, 0]) = ∞. Lemma 3.2. ([12, Lemma 7.3].) Let A = D[X, Y ; σ, a] be a GWA where D is a commutative ring, a is a regular element which is not a unit and p be an ideal of D containing a. (1) The following sequences of A-modules are exact: (3.3)

ϕ

ψ

→ Ap ⊕ A − → A(p, X) → 0 0 → A(σ(p), Y ) − where ϕ(ω) = (ωX, ω) and ψ(u, v) = u − vX,

(3.4)

ϕ

ψ

0 → A(p, X) − → Aσ(p) ⊕ A − → A(σ(p), Y ) → 0 where ϕ(ω) = (ωY, ω) and ψ(u, v) = u − vY . (2) The sequence (3.3) (or (3.4)) splits if and only if p(1 − p0 ) ⊆ Da for some element p0 ∈ p. (3) If, in addition, D is a Dedekind domain and p is a maximal ideal of D then the sequence (3.3) (or (3.4)) does not split if and only if a ∈ p2 .

Till the end of the section let D = S −1 K[H, C] be a localization of the polynomial algebra K[H, C] at a multiplicative set S. If S = {1} then D = K[H, C]. Let D∗ be the group of units of D. The polynomial algebra K[H, C] is a unique factorization domain (UFD). Hence, so is the algebra D. A set of ideals {ai | i ∈ I} of a ring R is called a co-maximal set of ideals if ai + aj = R for all i = j. Definition. We say that an element a ∈ D\D∗ is co-maximal if the ideal Da = p1 · · · ps is a product of co-maximal height 1 ideals p1 , . . . , ps , i.e., either s = 1 or otherwise, pi + pj = D for all i = j. Proposition 3.3. Let an algebra D be a localization of the polynomial algebra K[H, C] of Krull dimension 2. Suppose that a ∈ / D∗ . Then Da = pn1 1 · · · pns s (s ≥ 1) is a unique product of height 1 ideals pi of the algebra D (where ni is the multiplicity of pi ) and the following statements are equivalent. (1) pdA A(pi , X) < ∞. (2) The left ideal A(pi , X) of the algebra A is a projective left A-module. (3) The short exact sequence (3.3) splits where p = pi . (4) ni = 1 and pi + pj = D for all j = i. Proof. (1) ⇔ (2) ⇔ (3) Let p = pi . The polynomial algebra K[H, C] is a unique factorization domain, hence so is the algebra D. So, p = Dp for an element p ∈ p. Therefore, Ap  A  Aσ(p), as left A-modules. In particular, the middle terms of the short exact sequences (3.3) and (3.4) are free/projective A-modules. Now (1) ⇔ (2) ⇔ (3), by Lemma 3.2.(1).

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V. V. BAVULA AND K. ALNEFAIE

(3) ⇔ (4) By Lemma 3.2.(2), the short exact sequence (3.3) splits for p = pi = Dp if and only if 1 − p0 ∈ Dap−1 for some element p0 ∈ p if and only if the element ap−1 + p of the factor ring D/p is a unit if and only if ni = 1 and all the elements pj + p (j = i) are units of D/p (where pj = Dpj for some element pj ∈ pj ) if and  only if ni = 1 and pi + pj = D for all j = i. The next corollary is a ’dual’ version of Proposition 3.3. Corollary 3.4. Suppose that D and a be as in Proposition 3.3. Then Da = pn1 1 · · · pns s (s ≥ 1) is a unique product of height 1 ideals pi of the algebra D and the following statements are equivalent. (1) pdA A(σ(pi ), Y ) < ∞. (2) The left ideal A(σ(pi ), Y ) of the algebra A is a projective left A-module. (3) The short exact sequence (3.4) splits where p = pi . (4) ni = 1 and pi + pj = D for all j = i. Proof. (1) ⇔ (2) ⇔ (3) Let p = pi . The algebra D is a unique factorization domain. So, p = Dp for an element p ∈ p. Therefore, Ap  A  Aσ(p), as left A-modules. In particular, the middle terms of the short exact sequences (3.3) and (3.4) are free/projective A-modules. Now (1) ⇔ (2) ⇔ (3), by Lemma 3.2.(1). (3) ⇔ (4) By Lemma 3.2.(1), the short exact sequence (3.4) splits if and only if the short exact sequence (3.3) does. Now, the equivalence (3) ⇔ (4) follows from the equivalence (3) ⇔ (4) of Proposition 3.3.  Corollary 3.5. Suppose that the element a ∈ D\D∗ is co-maximal and an element d ∈ D\D∗ is a divisor of a. Let p = Dd. Then the left ideals A(p, X) and A(σ(p), Y ) of the GWA A = D(σ, a) are projective A-modules. Proof. Since the element a is co-maximal, the ring D is a unique factorization domain and d|a, the ideal p is a product of co-maximal height 1 prime ideals, say p = p1 · · · pt . Then D/p = D/p1 · · · pt = D/∩ti=1 pi 

t 

D/pi

i=1

since the ideals p1 , . . . , pt are co-maximal. Let p1 = Dp1 , . . . , pt = Dpt for some elements p1 ∈ p1 , . . . , pt ∈ pt . Since a is co-maximal and d|a, we have that the image of the element b = ad−1 ∈ D in the factor ring D/p is a unit. Therefore, cb = 1 − p0 for some elements c ∈ D and p0 ∈ p, i.e., d(1 − p0 ) = ca ∈ Da, and so p(1−p0 ) ⊆ Da. By Lemma 3.2.(2), the short exact sequence (3.3) splits. Therefore, the left ideals A(p, X) and A(σ(p), Y ) are projective A-modules.  Lemma 3.6. Let a ∈ D\D∗ be a co-maximal element and m = (H−α, C−β) be a maximal ideal of D such that a ∈ m where α, β ∈ K. Then a = a1 (H −α)+a2 (C−β) for some elements a1 , a2 ∈ D and for arbitrary choice of the elements a1 and a2 either (C − β)  a1 or (H − α)  a2 . Proof. Since m = (H − α, C − β) is a maximal ideal the elements H − α and C − β are not units. Suppose for some choice of the elements a1 and a2 , (C − β)|a1 and (H − α)|a2 , we seek a contradiction. Then a = (H − α)(C − β)b for some element b. The element a is co-maximal. So, D = D(H − α) + D(C − β) = m = D, a contradiction. 

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

43

The matrices e and d. Suppose that a ∈ D\D∗ , m = (H − α, C − β) is a maximal ideal of D such that a ∈ m where α, β ∈ K, i.e., a = a1 (H −α)+a2 (C −β) for some elements a1 , a2 ∈ D. We denote by Mn (D) the algebra of all n×n matrices with entries in D. The automorphism σ of the algebra D can be extended to an automorphism of the algebra Mn (D) by the rule: For a matrix (aij ) ∈ Mn (D), σ((aij )) = (σ(aij )). Let   a2 a1 ∈ M2 (D). A= C − β −(H − α)   1 −(H − α) −a2 −1 Then det(A) = −(a1 (H − α) + a2 (C − β)) = −a and A = − a −(C − β) a1 −1 i = {a | i ≥ 0}.  ∈ M2 (Da ) where Da = Sa D and Sa  H − α a2 −1 ˜ The matrix A = −det(A) A = satisfies the property that C − β −a1   1 0 ˜ ˜ (3.5) AA = AA = a where a = a . 0 1 Let us consider the following matrices of M4 (A): ⎛ ⎞ a2 −Y 0 a1 ⎜C − β −(H − α) 0 −Y ⎟ ⎟ and e=⎜ (3.6) ⎝ X 0 −σ(H − α) −σ(a2 )⎠ 0 X −σ(C − β) σ(a1 ) ⎛ ⎞ −Y 0 H − α a2 ⎜ C − β −a1 ⎟ 0 −Y ⎟. d=⎜ ⎝ X − (a2 ) ⎠ 0 −σ(a1 ) 0 X −σ(C − β) σ(H − α) The matrices e and d can be written as 2 × 2 matrices with entries in 2 × 2 matrices as follows     A −Y A˜ −Y e= and d = ˜ X −σ(A) X −σ(A)     X 0 Y 0 where X = and Y = . Let A4 = {(u1 , u2 , u3 , u4 ) | ui ∈ A} be 0 X 0 Y a free left A-module of rank 4 and the set {e1 = (1, 0, 0, 0), . . . , e4 = (0, 0, 0, 1)} be its canonical basis. The matrices e and d determine A-homomorphisms of A4 by the rule e : A4 → A4 , u = (u1 , . . . , u4 ) → ue, d : A4 → A4 , u = (u1 , . . . , u4 ) → ud. In particular, ker(e) = {u ∈ A4 | ue = 0}, im(e) = A4 e, ker(d) = {u ∈ A4 | ud = 0} and im(d) = A4 d. Proposition 3.7. (1) ed = 0 and de = 0. (2) ker(e) = im(d) and ker(d) = im(e).

44

V. V. BAVULA AND K. ALNEFAIE

Proof. (1).   A −Y A˜ ed = ˜ X −σ(A) X

 de =

A˜ X

 A −Y −σ(A) X

 −Y −σ(A)

 −Y ˜ −σ(A)









AA˜ − Y X −AY + Y σ(A) ˜ ˜ X A˜ − σ(A)X −XY + σ(AA)   a−a 0 = = 0, 0 −σ(a) + σ(a) =

˜ −YX ˜ + Y σ(A) ˜ AA −AY = ˜ XA − σ(A)X −XY + σ(AA)   a−a 0 = = 0. 0 −σ(a) + σ(a)

(2). (i) ker(e) = im(d): ker(e) = {(w1 , w2 ) ∈ A2 × A2 | (w1 , w2 )e = 0}. Notice that 0 = (w1 , w2 )e  A = (w1 , w2 ) X

−Y ˜ −σ(A)



 ⇔

w1 A + w2 X = 0 ˜ =0 w1 Y + w2 σ(A)

⇔ w1 A + w2 X = 0,

2 → since the second equation is redundant. Indeed, the map · σ(A) : A2 → A , w 1 0 ˜ ˜ wσ(A) is a monomorphism since σ(A) σ(A) = σ(AA) = σ(a)E where E = . 0 1 So, by applying the map · σ(A) to the second equation we have that ˜ = w1 AY + w2 σ(a)E = w1 AY + w2 XY 0 = w1 Y σ(A) + w2 σ(AA)

= (w1 A + w2 X)Y ⇔ w1 A + w2 X = 0 since the map ·Y : A → A , w → wY is a monomorphism. Therefore, 2

2

ker(e) = {(w1 , w2 ) ∈ A2 × A2 | w1 A + w2 X = 0}. The algebra A contains subalgebras A− = ⊕i≥0 DY i and A+ = ⊕i≥0 DX i . Clearly, A = A− ⊕ A+ X = A− Y ⊕ A+ . So, the elements w1 , w2 ∈ A2 can be uniquely written as follow: w1 = w1,+ X +w1,− and w2 = w2,+ + w2,− Y for some elements w1,+ , w2,+ ∈ A+ and w1,− , w2,− ∈ A− . Now, 0 = =

w1 A + w2 X = (w1,+ X + w1,− )A + (w2,+ + w2,− Y )X ˜ (w1,+ σ(A) + w2,+ )X + (w1,− + w2,− A)A.

Hence, using the Z-grading of the GWA A we see that (w1,+ σ(A)+w2,+ )X = 0 and ˜ = 0. Equivalently, w1,+ σ(A) + w2,+ = 0 and w1,− + w2,− A˜ = 0 (w1,− + w2,− A)A (since the maps ·X and ·A are injections). Finally, ˜ −w1,+ σ(A) + w2,− Y ) (w1 , w2 ) = (w1,+ X − w2,− A, ˜ w2,− (−Y )). = (w1,+ X, −w1,+ σ(A)) − (w2,− A, This means that the element (w1 , w2 ) of ker(e) is an  element ofthe A-submodule A˜ −Y = d. Therefore, of A4 generated by the 4 rows of the 4 × 4 matrix X −σ(A) (w1 , w2 ) ∈ im(d), i.e., ker(e) ⊆ im(d). By statement (1), ed = 0, and so we have im(d) ⊆ ker(e). Therefore, ker(e) = im(d).

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

45

(ii) ker(d) = im(e): The statement (ii) follows from the statement (i) in view ˜ of (A, A)-symmetry: The matrix d is obtained from the matrix e by replacing A ˜ by A, and vice versa. The facts used in the proof of statement (i), which are the ˜ equality (3.5) and statement (1), are also (A, A)-symmetrical.  Lemma 3.8.  We  keep the notations of Proposition 3.7. In the equalities below 1 0 1 stands for and the three matrices in the middle in statements 1 and 2 0 1 are equal to the three matrices on the RHS, respectively.     A 0 1 0 1 −A−1 Y (1) e = e− e0 e+ = . X X 0  0 0  Y  Y Y 0 0 X 0 (2) d = d− d0 d+ = . 0 σ(A) 0 1  σ(A−1 )X −1  XA 0 Y 0 (3) d+ e− = and e+ d− = . 0 −X 0 Y σ(A) Proof. Straightforward.  Let R be a ring and M be an R-module. A projective resolution of the Rmodule M of the type f0

fn−1

f1

fn−1

f0

f1

f0

· · · −→ Pn−1 −−−→ · · · → P1 −→ P0 −→ Pn−1 −−−→ Pn−2 → · · · → P1 −→ P0 −→ M →0 is called an n-periodic projective resolution or a projective resolution of period n. A projective resolution of the R-module M is called an eventually n-periodic projective resolution if it becomes n-periodic at certain point. An A-module M admits an nperiodic projective resolution if and only if there is an exact sequence of the type 0 → M → Pn−1 → · · · → P0 → M → 0 where P0 , . . . , Pn−1 are projective A-modules. A projective resolution for the left ideal A(σ(m), Y ). The next theorem gives such a resolution. Theorem 3.9. Let an algebra D be a localization of the polynomial algebra K[H, C] of Krull dimension 2. Suppose that the element a ∈ D\D∗ is co-maximal, m = (H − α, C − β) is a maximal ideal of D such that a ∈ m where α,β ∈ K, i.e., a = a1 (H − α) + a2 (C − β) for some elements a1 , a2 ∈ D. Then (3.7)

d

e

d

e

f

g

→ A4 − → A4 − → A4 − → A4 − → A3 − → A(σ(m), Y ) → 0 · · · → A4 −

is an eventually 2-periodic projective resolution of the left ideal A(σ(m), Y ) of the GWA A where the maps/matrices e and d are as in Proposition 3.7, f (u1 , u2 , u3 , u4 ) = (u1 Y + u3 σ(a1 ) + u4 σ(C − β), u2 Y + u3 σ(a2 ) − u4 σ(H − α), u1 (H −α)+u2 (C −β)+u3 X) and g(v1 , v2 , v3 ) = v1 σ(H −α)+v2 σ(C −β)−v3 Y . Proof. (i) g is an epimorphism (since A(σ(m), Y ) = A σ(H − α) + A σ(C − β) + AY ).

46

V. V. BAVULA AND K. ALNEFAIE

(ii) gf = 0: Let the set of elements e1 = (1, 0, 0, 0), . . . , e4 = (0, 0, 0, 1) be the canonical basis of the free A-module A4 = ⊕4i=1 Aei . Then gf (e1 ) gf (e2 ) gf (e3 )

gf (e4 )

= = = = = =

Y σ(H − α) − (H − α)Y = 0, Y σ(C − β) − (C − β)Y = 0, σ(a1 ) σ(H − α) + σ(a2 ) σ(C − β) − XY σ(a1 (H − α) + a2 (C − β)) − σ(a) σ(a) − σ(a) = 0, σ(C − β) σ(H − α) − σ(H − α) σ(C − β) = 0.

(iii) ker(g) ⊆ im(f ): Recall that the elements e1 , . . . , e4 are the canonical free generators of the left free A-module A4 . Consider their images under the homomorphism f , e1 e2 e3 e4

= = = =

f (e1 ) = (Y, 0, H − α), f (e2 ) = (0, Y, C − β), f (e3 ) = (σ(a1 ), σ(a2 ), X), f (e4 ) = (σ(C − β), −σ(H − α), 0).

By Lemma 3.6, either (C −β)  a1 or (H −α)  a2 . Up to interchanging the elements H and C, we may assume that (C − β)  a1 . In this case, a1 = 0. Furthermore, we can choose the element a1 from the set D∗ K[H]\{0} := {dp | d ∈ D∗ , p ∈ K[H]\{0}}. Let an element v = (v1 , v2 , v3 ) ∈ A3 belong to ker(g), g(v) = v1 σ(H − α) + v2 σ(C − β) − v3 Y = 0.  Notice that A = AY + i≥0 X i D. Then the element v1 (respectively, v2 ) up to  i addition of an element of Ae1 (respectively, Ae2 ) is a sum v1 = i≥0 X σ(αi )  i (respectively, v2 = i≥0 X σ(βi )) for some elements αi ∈ D (respectively, βi ∈ D). Furthermore, up to addition of an element of the left ideal Ae3 +Ae4 we can assume that all αi ∈ K[H] satisfy degH (αi ) < d1 where d1 is the dimension of the factor algebra

(3.8)

1 −1 KH i ) D/(D σ(a1 ) + D σ(C − β)) = σ(D/(Da1 + D(C − β)) = σ(⊕di=0  and σ(αi ) is defined up to D σ(a1 ) + D σ(C − β). By (3.8) and v1 , v2 ∈ i≥0 X i D,   we have v3 ∈ i≥1 X i D, i.e., v3 = i≥0 X i+1 γi for some elements γi ∈ D, and

(3.9)

σ(αi ) σ(H − α) + σ(βi ) σ(C − β) = σ(a) σ(γi ) for i ≥ 0.

Replacing the element a in (3.9) by the sum a = a1 (H − α) + a2 (C − β) and then taking the result modulo Dσ(C − β) we obtain the equality (σ(αi ) − σ(a1 ) σ(γi )) σ(H − α) ≡ 0 mod Dσ(C − β) in the domain D/Dσ(C − β). Hence, the first factor must be zero. This means that σ(αi ) ∈ D σ(a1 ) + D σ(C − β). Ae3

Ae4

+ ∈ im(f ). So, we can assume that v1 = 0 (all αi = 0). Then Hence, v1 ∈ the equalities (3.9) take the form after applying σ −1 : (3.10)

βi (C − β) = aγi for i ≥ 0.

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

47

Since the element a is co-maximal and a1 ∈ D∗ K[H]\{0}, (C − β)  a (since otherwise we would have (C − β)|a1 (H − α), a contradiction as (C − β)  a1 and (C − β)  (H − α)). By (3.10), (C − β)|γi for i ≥ 0, and so βi = γi a = γi Y X where γi = (C − β)−1 γi ∈ D. Now,   v = (0, v2 , v3 ) = (0, X i σ(γi ) σ(a), X i+1 γi (C − β)) ! =

i≥0



"

i≥0

X i σ(γi ) (0, σ(a), σ(C − β)X) ∈ im(f )

i≥0

since (0, σ(a), σ(C − β)X) = σ(C − β) e3 − (σ(C − β) σ(a1 ), σ(C − β) σ(a2 ) − σ(a), 0) = σ(C − β) e3 − σ(a1 ) e4 ∈ im(f ) as σ(C−β) σ(a2 )−σ(a) = −σ(H−β) σ(a1 ). The proof of statement (iii) is complete. (iv) ker(f ) = Aθ1 + Aθ2 + Aη1 + Aη2 = im(e) where θ1 = (a1 , a2 , −Y, 0), θ2 = (C − β, −(H − α), 0, −Y ), η1 = (X, 0, −σ(H − α), −σ(a2 )) and η2 = (0, X, −σ(C − β), σ(a1 )) are the rows of the matrix e: It can be easily verified that the elements θ1 , θ2 , η1 and η2 belong to ker(f ). An element u = (u1 , u2 , u3 , u4 ) ∈ A4 belongs to ker(f ) if and only if ⎧ ⎪ ⎨u1 Y = −u3 σ(a1 ) − u4 σ(C − β), u2 Y = −u3 σ(a2 ) + u4 σ(H − α), ⎪ ⎩ u1 (H − α) + u2 (C − β) + u3 X = 0. Then u1 a = u2 a =

u1 Y X = −u3 σ(a1 )X − u4 σ(C − β)X = −u3 Xa1 − u4 X(C − β), u2 Y X = −u3 σ(a2 )X + u4 σ(H − α)X = −u3 Xa2 + u4 X(H − α).

Hence, u1 = −u3 Xa1 a−1 − u4 X(C − β) a−1 and u2 = −u3 Xa2 a−1 + u4 X(H − α) a−1 . Now, the third equation of the system is redundant: u1 (H − α) + u2 (C − β) + u3 X

= −u3 X((a1 (H − α) + a2 (C − β)) a−1 − 1) −u4 X((C − β)(H − α) − (H − α)(C − β)) a−1 = −u3 X(aa−1 − 1) = 0.

So, thethird equationcan be dropped. Theelements u3 and  u4 are unique sums u3 = i≥1 γ−i Y i + i≥0 X i σ(γi ), u4 = i≥1 δ−i Y i + i≥0 X i σ(δi ) where γi , δi ∈ D and i ∈ Z. Now, u1

= −u3 Xa1 a−1 − u4 X(C − β) a−1   = − (γ−i Y i−1 a1 + δ−i Y i−1 (C − β)) − X i+1 (γi a1 + δi (C − β)) a−1 , i≥1

u2

i≥0 −1

−1

= −u3 Xa2 a + u4 X(H − α) a   = − (γ−i Y i−1 a2 − δ−i Y i−1 (H − α)) − X i+1 (γi a2 − δi (H − α)) a−1 . i≥1

i≥0

The conditions that u1 ∈ A and u2 ∈ A are equivalent to the following conditions.

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V. V. BAVULA AND K. ALNEFAIE

For all i ≥ 0, (3.11)

γi a1 + δi (C − β) ∈ Da,

(3.12)

γi a2 − δi (H − α) ∈ Da.

By multiplying the first inclusion by H − α and the second one by C − β and taking their sum we obtain that am " γi (a1 (H − α) + a2 (C − β)) = γi a, and so γi ∈ m. So, γi = γi (H − α) + γi (C − β) for some elements γi , γi ∈ D.

(3.13)

(iv-1) For all i ≥ 0, δi = γi a2 − γi a1 : Let Δi := δi − (γi a2 − γi a1 ). Suppose that Δi = 0, we seek a contradiction. Notice that (C − β) a2 = a − a1 (H − α) ≡ −a1 (H − α) mod Da. By taking (3.12) modulo Da and using (3.13), we obtain the equality 0 ≡ (γi (H − α) + γi (C − β)) a2 − δi (H − α) = −Δi (H − α). Notice that (H − α) a1 = a − a2 (C − β) ≡ −a2 (C − β) mod Da. Similarly, by taking (3.11) modulo Da and using (3.13), we obtain the equality 0 ≡ (γi (H − α) + γi (C − β)) a1 + δi (C − β) = Δi (C − β). So, we have Δi (H − α) = pi a, Δi (C − β) = qi a for some nonzero elements pi and qi (since Δi = 0). Since a = a1 (H − α) + a2 (C − β), the first (respectively, second) equality yields (H − α)|a2 (respectively, (C − β)|a1 ). This means that a = (H − α)(C − β) a for some a ∈ D\{0}. The element a is co-maximal. So, D = D(H −α)+D(C −β) = m, a contradiction. Therefore, Δi = 0, the proof of statement (iv-1) is complete. (iv-2) For all i ≥ 1, (γi a2 −δi (H −α)) a−1 = γi and (γi a1 +δi (C −β)) a−1 = γi : γi a2 − δi (H − α)

= = = γi a1 + δi (C − β) =

(γi (H − α) + γi (C − β))a2 − (γi a2 − γi a1 )(H − α) γi (a1 (H − α) + a2 (C − β)) γi a, (γi (H − α) + γi (C − β))a1 + (γi a2 − γi a1 )(C − β)

= γi (a1 (H − α) + a2 (C − β)) = γi a. By statements (iv-1) and (iv-2), u = (u1 , u2 , u3 , u4 )     γ−i Y i−1 ) θ1 − ( δ−i Y i−1 ) θ2 − ( X i σ(γi )) η1 − ( X i σ(γi )) η2 . = −( i≥1

i≥1

i≥0

i≥0

This finishes the proof of statement (iv). Now, the fact that (3.7) is an eventually 2-periodic resolution follows from Proposition 3.7.  Criterion for the left ideals A(m, X) and A(σ(m), Y ) to have finite projective dimension. Below such a criterion is given. Theorem 3.10. Let D be as in Theorem 3.9. Suppose that the element a ∈ D\D∗ is co-maximal, m = (H − α, C − β) is a maximal ideal of D such that a ∈ m where α, β ∈ K, i.e., a = a1 (H − α) + a2 (C − β) for some elements a1 , a2 ∈ D. The following statements are equivalent. (1) pdA A(σ(m), Y ) < ∞.

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

49

(2) pdA A(σ(m), Y ) = 1. (3) a ∈ / m2 . (4) The A-submodule A4 e of A4 is a projective A-module where e is as in Theorem 3.9. (5) The A-submodule A4 d of A4 is a projective A-module where d is as in Theorem 3.9. (6) pdA A(m, X) < ∞. (7) pdA A(m, X) = 1. Proof. (2) ⇒ (1), (7) ⇒ (6) The implications are obvious. (1) ⇔ (4) ⇔ (5) The implications follow at once from the exact sequence (3.7). (2) ⇐ (1) ⇔ (6) ⇒ (7) The ring D is a Noetherian ring of global dimension 2 and pdD m = 1. Then pdA (Am) = pdA (A ⊗D m) = pdD (m) = 1, by [11, Corollary 2.3]. Therefore, pdA (Am ⊕ A) = 1. Similarly, pdA (Aσ(m) ⊕ A) = 1. Now, by the short exact sequence (3.3) and (3.4) either the left ideals A(m, X) and A(σ(m), Y ) have both infinite projective dimension as left A-modules or otherwise they both have the same projective dimension as the left A-modules Am ⊕ A and Aσ(m) ⊕ A, i.e., 1. (5) ⇒ (3) In view of the exact sequence (3.7), there is a short exact sequence e

d

→ A4 − → A4 d → 0. 0 → A4 /A4 d −

(3.14)

By the assumption the submodule A4 d of A4 is projective. Hence, the short exact e sequence (3.14) splits. In particular, the monomorphism A4 /A4 d − → A4 admits  splitting. This means that there is a matrix e ∈ M4 (A) such that ee + d d = 1

(3.15)

for some matrix d ∈ M4 (A) where 1 is the identity element of the algebra M4 (A). The matrix algebra M4 (A) = M4 (K) ⊗K A = M4 (D)[X, Y ; σ, a]

(3.16)

is a GWA where the new automorphism σ ∈ AutK (M4 (D)) is a unique extension of the old automorphism σ ∈ AutK (D) that acts as the identity map on the subalgebra of scalar matrices M4 (K) in M4 (D) = M4 (K) ⊗K D. Using the Z-grading of the GWA M4 (A) and, in particular, the fact that the identity matrix 1 of M4 (K) belongs to the zero component M4 (D) of the GWA M4 (A), we see that the equality (3.15) yields the equality    d ∗ + 11 ∗ ∗

   1 0 −Y = 0 1 −σ(A)   1 0 for some matrices e11 , e21 , d11 , d12 ∈ M2 (D) where 1 = ∈ M2 (D). By 0 1 equating the (1, 1)-elements of the matrices on both sides of the equality we obtain the equality in the matrix algebra M2 (D):   1 0     ˜ . (3.17) A e11 − a e21 + d11 A + d12 a = 0 1 

A X

−Y ˜ −σ(A)



e11 Xe21

d12 Y ∗



A˜ X

50

V. V. BAVULA AND K. ALNEFAIE

The algebra M2 (D) contains the ideal M2 (m). Bearing in mind that     a1 H − α a2 a2 a ∈ m, A = and A˜ = C − β −(H − α) C − β −a1 and taking the equality (3.17) modulo the ideal M2 (m) we obtain the equality in the matrix algebra M2 (D)/M2 (m)  M2 (D/m)  M2 (K):       0 a2 1 0 a1 a2   e11 + d11 ≡ mod M2 (m). 0 0 0 −a1 0 1 By equating the (1, 1)-elements of the matrices on both sides of the equality we see that 1 ∈ Da1 + Da2 + m.

(3.18)

Equivalently, Da1 + Da2  m. Since a = a1 (H − α) + a2 (C − β) ∈ m, we conclude that a ∈ / m2 (since a = a1 (H − α) + a2 (C − β) ∈ m/m2 = K(H − α) ⊕ K(C − β) and either a1 ∈ K\{0} or a2 ∈ K\{0}, hence a = 0 where the bar means modulo m2 ). (3) ⇒ (5) By the assumption, a ∈ m\m2 which is equivalent to (3.18). This means that there are elements a1 , . . . , a4 ∈ D such that a1 a1 + a2 a2 + a3 (H − α) + a4 (C − β) = 1.

(3.19)

We have to show that the short exact sequence (3.14) splits, i.e., the equality (3.15)   holds for some elements    e , d ∈ M4 (A).     e −e22 a1 a4 Y σ(A) AY    11 CLAIM: Let e = ,d = , e11 =  a2 −a3 XA σ(e22 ) σ(A)X −σ(e11 )     −a3 −a4 and e22 = . Then ee + d d = 1 in M4 (A). −a2 a1 The equality ee +d d = 1 follows by direct computation and   using the equalities 1 0   ˜ ˜ ˜ ˜ 22 = and e11 A − Ae Y X = a, XY = σ(a), AA = AA, Ae11 − e22 A = 0 1   1 0 : 0 1        A −Y e11 Y σ(A) −e22 AY A˜ −Y   + ee + d d = ˜ XA σ(e22 ) σ(A)X −σ(e11 ) X −σ(A) X −σ(A)     ˜ 2   2 (A − e22 + e22 − A )Y Ae11 − aA − e22 A + Aa = ˜ + AA˜ − e ) σ(aA − Ae ˜  − Aa + e A) X(e11 − AA 11 22 11   1 0 = = 1. 0 σ(1) In more detail, Ae11

−

e22 A˜



a1 a2 = C − β −(H − α)     L 0 1 0 = = 0 L 0 1



a1 a2

a4 −a3



  −a3 − −a2

−a4 a1

 H −α C −β

a2 −a1



GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

51

where L := a1 a1 + a2 a2 + a3 (H − α) + a4 (C − β) = 1, by (3.19). Similarly,         a1 a4 a2 H − α a2 a1 −a3 −a4   ˜ e11 A − Ae22 = − a2 −a3 C − β −(H − α) C − β −a1 −a2 a1     L 0 1 0 = = . 0 L 0 1 The proof of the theorem is complete.  For a commutative ring D, we denote by Max (D) its maximal spectrum, the set of maximal ideals of D. For an element d ∈ D, let V(d) := {m ∈ Max (D) | d ∈ m}. Lemma 3.11. Let an algebra D be a localization of the polynomial algebra K[H, C] of Krull dimension 2, a ∈ D\D∗ and K be an algebraically closed field. Then the following two statements are equivalent. (1) For all m ∈ V(a), a ∈ / m2 . ∂a +D (2) For all m ∈ V(a), D ∂H

∂a ∂C

 m.

Proof. Fix a unit u ∈ D∗ ∩ K[H, C] such that a := ua ∈ K[H, C]. Then V(a) = V(a ) and for every m ∈ V(a) = V(a ), a ∈ m (respectively, a ∈ m2 ) if and only if a ∈ m (respectively, a ∈ m2 ). Since the field K is algebraically closed, every maximal ideal m is equal to m = (H − α, C − β) for some elements α, β ∈ K. Let m = (H − α, C − β) ∈ V(a). Now, the lemma follows from a ≡

∂a ∂a ∂a ∂a (H − α) + (C − β) ≡ u ( (H − α) + (C − β)) mod m.  ∂H ∂C ∂H ∂C

Every element a ∈ D\D∗ determines an algebraic curve V(a) = {m ∈ Max (D) | a ∈ m}. It is given by the equation a = 0 (in Max (D) ⊆ K 2 via m = (H−α, C−β) ↔ (α, β)). Definition. The algebraic curve a = 0 (in Max (D) ⊆ K 2 where a ∈ D\D∗ ) is called a smooth algebraic curve if it satisfies the second condition of Lemma 3.11 which means that the dimension of the tangent space at every point of the algebraic curve is a 1-dimensional vector space over the field K. Lemma 3.12. Let D and a be as in Lemma 3.11. If the algebraic curve a = 0 is smooth then the element a is co-maximal. Proof. The element a = p1 · · · ps is a product of irreducible elements of D. If s = 1 there is nothing to prove. Suppose that s ≥ 2. We have to show that Dpi + Dpj = D for all i = j. Suppose this is not the case for some i = j. Then there is a maximal ∂a ∂a ∈ m and ∂C ∈ m. This ideal m of D such that pi ∈ m and pj ∈ m. Clearly, ∂H contradicts to the assumption that the algebraic curve a = 0 is smooth.  Proof of Theorem 1.9. By Lemma 1.8, the algebra E = D[X, Y ; σ, a = H] is a GWA where D = D[H] and σ(H) = ρH + b. n−1 (i) For all natural numbers n ≥ 1 σ n (H) = ρn H+ξn where ξn = i=0 σ i (ρn−i−1 b): The statement is proven by induction on n. The initial case n = 1 is obvious

52

V. V. BAVULA AND K. ALNEFAIE

(σ(H) = ρH + b). So, let n > 1 and we assume that the equality holds for all n < n. Now, σ n (H) = σ(ρn−1 H + ξn−1 ) = ρn H + ρn−1 b + σ(ξn−1 ) = ρn H + ξn , as required. (ii) If D is a field then the algebra D = D[H] is a Dedekind domain and statement (1) follows from Theorem 1.4. Suppose that the algebra D is not a field. Then the algebra D = D[H] = S −1 K[C, H] has Krull dimension 2. (iii) gld (E) = 2 or 3: Since grad(a) = grad(H) = (1, 0), gld (E) < ∞, by Theorem 1.6, and so gld (E) = 2 or 3, by Theorem 1.1. (iv) gld (E) = 3 iff there are natural number n ≥ 1 and an element β ∈ K such that Dξn + D(σ n (C) − β) = D (where K is an algebraically closed field): By Theorem 1.2, gld (E) = 3 iff there is a maximal ideal m = (H, C −β) (where β ∈ K) such that H ∈ σ n (m) for some natural number n ≥ 1. By the statement (i), σ n (m) = (ρn H + ξn , σ n (C) − β). Hence, H ∈ σ n (m) iff Dξn + D(σ n (C) − β) = D. The proof of Theorem 1.9 is complete.  Proof of Theorem 1.6. If a = 0 then gld (A) = ∞, by Proposition 3.1. If a ∈ D∗ then A  D[X, X −1 ; σ] and gld (A) ≤ gld (D) + 1 < ∞, by [22, Theorem 7.5.3]. If a ∈ D\D∗ and a = 0 then by [11, Theorem 3.5], gld (A) < ∞ if and only if pdA A(p, X) < ∞ for all prime ideals p of the algebra D that contain the element a if and only if the element a is co-maximal (Proposition 3.3) and a ∈ / m2 for all maximal ideals m of D that contain a (Theorem 3.10) if and only if a is co-maximal and the algebraic curve a = 0 is smooth (Lemma 3.11) if and only if the algebraic curve a = 0 is smooth (Lemma 3.12).  Proof of Theorem 1.10. By [11, Corollary 4.5], the algebras Λi are THM with respect to the class of algebras N and the result follows.  4. The global dimension of S −1 K[H, C](σ, a) with affine automorphism σ In this section, A := K[H, C][X, Y ; σ, a] is a generalized Weyl algebra where D := K[H, C] is a polynomial algebra in two variables over an algebraically closed field K and σ is an affine automorphism of D, that is σ(H) = αH + βC + b and σ(C) = γH + δC + c   α β for some scalars α, β, . . . , c ∈ K such that det = 0. As an abstract algebra, γ δ the algebra A is generated by the elements H, C, X and Y subject to the defining relations: XH YC

= (αH + βC + b)X, = (γH + δC + c)Y,

Y X = a, XY = σ(a).

Many classical algebras are special cases of the algebra A: U (sl2 ), the Heisenberg algebra and its quantum analogues, Uq (so(K, 3)), the 2 × 2 quantum matrices, the

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

53

Witten’s and Woronowic’s deformations, the algebras similar to U (sl2 ) [2, 23], see Section 2. Since the field K is an algebraically closed field then up to an affine change of variables in the polynomial algebra D = K[H, C] (H  = α H + β  C + b , C  = γ  H + δ  C + c ) there are only 4 types of affine automorphisms [11, Lemma 4.1] where λ, μ ∈ K ∗ : (1) (2) (3) (4)

σ(H) = H − 1, σ(C) = λC, σ(H) = λH, σ(C) = μC, σ(H) = H − 1, σ(C) = C + H, σ(H) = λH + C, σ(C) = λC.

Let S be a multiplicative subset of K[H, C]\{0} such that σ(S) = S. Then S is a (left and right) Ore subset of the GWA A = D(σ, a) that consists of regular elements (i.e., non-zero-divisors) in A. The algebra A = S −1 A = D(σ, a) is a GWA where D = S −1 D and the automorphism σ is defined by the rule σ(s−1 d) = σ(s)−1 σ(d). As we already mentioned in the Introduction, we will assume that the Krull dimension of D is 2. The aim of the section is to calculate an explicit value of the global dimension of A (resp., A = S −1 D(σ, a)) in terms of the roots of the polynomial a ∈ K[H, C] (resp., a ∈ S −1 D) accordingly to four cases for the affine automorphism σ. Proposition 4.1. Let K be an algebraically closed field and A = K[H, C](σ, a) be a GWA. Then gld (A) < ∞ if and only if either a ∈ K ∗ or a ∈ / K and a = p1 · · · ps is a product of irreducible polynomials such that (pi ) + (pj ) = (1) for all i = j and the algebraic curves pi = 0 are smooth (i.e., the algebraic curve a = 0 is a disjoint union of smooth algebraic curves pi = 0). Proof. The proposition is a particular case of Theorem 1.6.  In the following theorems we assume that gld (A) < ∞ and we found the values of gld (A) for the four types of automorphism σ of D. Theorem 4.2. Let A = D(σ, a) be a GWA such that D = K[H, C], K is an algebraically closed field, σ(H) = H − 1 and σ(C) = λC where λ ∈ K ∗ . Suppose that gld (A) < ∞. Then gld (A) = 2, 3 and (1) gld (A) = 3 if and only if either char(K) = 0 or char(K) = 0 and there exist elements α, β ∈ K and i ∈ N\{0} such that a(α, β) = 0 and a(α + i, λ−i β) = 0. (2) gld (A) = 2 if and only if char(K) = 0 and if a(α, β) = 0 for some α, β ∈ K then a(α + i, λ−i β) = 0 for all i ∈ N\{0}. (3) Suppose that S is a multiplicative subset of D such that σ(S) = S and the algebra D = S −1 D has Krull dimension 2. Let A = D(σ, a) where a ∈ D. Suppose that gld (A) < ∞. Then gld (A) = 2, 3. Furthermore, gld (A) = 3 iff either char(K) = 0 and λ is a root of unity or char(K) = 0, λ is not a root of unity and (H − α, C) = D for some α ∈ K or there is a maximal ideal m = (H − α, C − β) of D (where α, β ∈ K) such that a ∈ m and a ∈ σ i (m) = (H − α − i, C − λ−i β) for some i ≥ 1. Proof. By Theorem 1.1, gld (A) = 2, 3 (since we assume that gld (A) < ∞). (1). If char(K) = p then the maximal ideal (C, H) of the algebra D is σ p invariant. Hence, gld (A) = 3, by Theorem 1.2, since gld (A) < ∞. If char(K) = 0

54

V. V. BAVULA AND K. ALNEFAIE

then for every maximal ideal m of D the orbit O(m) = {σ i (m) | i ∈ Z} is infinite. Recall that we assume that gld (A) < ∞. Then, by Theorem 1.2, gld (A) = 3 if and only if there exist elements α, β ∈ K and i ∈ N\{0} such that a(α, β) = 0 and a(α + i, λ−i β) = 0. (2). By Theorem 1.1, gld (A) = 2, 3 (since we assume that gld (A) < ∞). Now, statement (2) follows from statement (1). (3). Recall that the Krull dimension of the ring D is 2. Notice that there is a maximal ideal m = (H − α, C − β) of D (where α, β ∈ K) such that its σ-orbit is finite iff either char(K) = 0 and λ is a root of unity or char(K) = 0, λ is not a root of unity and (H − α, C) = D. By Theorem 1.1, gld (A) = 2, 3 (since gld (A) < ∞). Now, statement (3) follows from Theorem 1.2.  Theorem 4.3. Let A = D(σ, a) be a GWA such that D = K[H, C], K is an algebraically closed field, σ(H) = λH and σ(C) = μC where λ, μ ∈ K ∗ . (1) Suppose that gld (A) < ∞. Then gld (A) = 3. (2) Suppose that S is a multiplicative subset of D such that σ(S) = S and the algebra D = S −1 D has Krull dimension 2. Let A = D(σ, a) where a ∈ D. Suppose that gld (A) < ∞. Then gld (A) = 2, 3. Furthermore, gld (A) = 3 iff either λ, μ are roots of unity or λ is not a root of unity, μ is a root of unity and (H, C − β) = D for some β ∈ K or λ is a root of unity, μ is not a root of unity and (H −α, C) = D for some α ∈ K or λ, μ are not roots of unity and (H, C) = D or there is a maximal ideal m = (H −α, C −β) of D (where α, β ∈ K) such that a ∈ m and a ∈ σ i (m) = (H − λ−i α, C − μ−i β) for some i ≥ 1. Proof. (1). By Theorem 1.2, gld (A) = 3 since the maximal ideal (H, C) of the polynomial algebra D is σ-invariant. (2). Recall that the Krull dimension of the ring D is 2. By Theorem 1.1, gld (A) = 2, 3 (since we assume that gld (A) < ∞). In the last sentence of statement (2), the first four conditions are equivalent to the fact that there is a maximal ideal of D of height 2 such that its σ-orbit is finite. Now, the ‘iff’ statement follows from Theorem 1.2.  Suppose that σ is an automorphism of D = K[H, C] such that σ(H) = H − 1 and σ(C) = C + H. By induction on i, we have the equalities i(i − 1) for i ≥ 1. 2 Let p = char(K). Then the order or (σ) of the automorphism σ is a as follows ⎧ ⎪ ⎨4 if p = 2, (4.2) or (σ) = p if p > 2, ⎪ ⎩ ∞ if p = 0.

(4.1)

σ i (H) = H − i and σ i (C) = C + iH −

Theorem 4.4. Let A = D(σ, a) be a GWA such that D = K[H, C], K is an algebraically closed field, σ(H) = H − 1 and σ(C) = C + H. Suppose that gld (A) < ∞. Then gld (A) = 2, 3 and (1) gld (A) = 3 if and only if either char(K) = 0 or char(K) = 0 and there exist elements α, β ∈ K and i ∈ N\{0} such that a(α, β) = 0 and a(α + i, β − iα − i(i+1) 2 ) = 0.

GLOBAL DIMENSION OF THE ALGEBRAS S −1 K[H, C][X, Y ; σ, a]

55

(2) gld (A) = 2 if and only if char(K) = 0 and if a(α, β) = 0 for some α, β ∈ K then a(α + i, β − iα − i(i+1) 2 ) = 0 for all i ∈ N\{0}. (3) Suppose that S is a multiplicative subset of D such that σ(S) = S and the algebra D = S −1 D has Krull dimension 2. Let A = D(σ, a) where a ∈ D. Suppose that gld (A) < ∞. Then gld (A) = 2, 3. Furthermore, gld (A) = 3 iff either char(K) = 0 or char(K) = 0 and there is a maximal ideal m = (H − α, C − β) of D (where α, β ∈ K) such that a ∈ m and a ∈ σ i (m) = (H − α − i, C − β + iα + i(i+1) 2 ) for some i ≥ 1. Proof. By Theorem 1.1, gld (A) = 2, 3 (since we assume that gld (A) < ∞). (1). If char(K) = 0 then, by (4.2), the order of the automorphism σ is finite. Hence, gld (A) = 3, by Theorem 1.2, since gld (A) < ∞. If char(K) = 0 then for every maximal ideal m of D its orbit O(m) = {σ i (m) | i ∈ Z} is infinite. By Theorem 1.2, gld (A) = 3 if and only if there exists a maximal ideal m = (H − α, C − β) of D (where α, β ∈ K) such that a ∈ m and a ∈ σ i (m) for some i ≥ 1 iff a(α, β) = 0 and a(α + i, β − iα − i(i+1) 2 ) = 0 since, by (4.1), σ i (m) = (H − α − i, C + iH −

i(i + 1) i(i − 1) − β) = (H − (α + i), C − (β − iα − )). 2 2

(2). Since gld (A) = 2, 3. Now, statement (2) follows from statement (1). (3). Recall that the Krull dimension of D is 2. By Theorem 1.1, gld (A) = 2, 3 (since gld (A) < ∞). Now, repeat the proof of statement (1) by replacing the ring D by D.  Suppose that σ is an automorphism of D = K[H, C] such that σ(H) = λH + C and σ(C) = λC. By induction on i, we have the equalities (4.3)

σ i (H) = λi H + iλi−1 C and σ i (C) = λi C for i ≥ 1.

For each natural number n ≥ 1, let Pn = Pn (K) be the set of elements in K ∗ of order n. In general, the set Pn (K) can be an empty set: If F2 = {0, 1} is the field of characteristic 2 then Pn = ∅ for all n ≥ 2. For each natural number n ≥ 1, the set Mn := {λ ∈ K | λn = 1} is a subgroup of K ∗ . If n|m then Mn ⊆ Mm . Let us show that the order or (σ) of the automorphism σ is equal to  pm if λ ∈ Pm , p > 0, p  m, (4.4) or (σ) = ∞ otherwise. Proof of the equality in (4.4): Suppose that l := or (σ) < ∞. By (4.3), σ l = id if and only if λl = 1 and lλl−1 = 0 if and only if λ ∈ Ml , p > 0 and p|l. Let m = or (λ). Then p  m (otherwise, m = pj for some j ≥ 1, and so 0 = λm − 1 = (λj − 1)p implies λj = 1 and so j ≥ or(λ) = pj, a contradiction). Then l = im for some i ≥ 1 (since λl = 1). Notice that p|l if and only if p|im if and only if p|i (since p  m). Then, by (4.3), σ pm = id, hence l|pm (since l = or (σ)) and also pm|l (since m|l, p|l and p  m), and so l = pm. This proves that or(σ) < ∞ if and only if or(σ) = pm, λ ∈ Pm , p > 0 and p  m. Also, there is no automorphism σ such that λ ∈ Pm , m ≥ 1, p > 0 and p|m.  Theorem 4.5. Let A = D(σ, a) be a GWA such that D = K[H, C], K is an algebraically closed field, σ(H) = λH + C and σ(C) = λC where λ ∈ K ∗ . (1) Suppose that gld (A) < ∞. Then gld (A) = 3.

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(2) Suppose that S is a multiplicative subset of D such that σ(S) = S and the algebra D = S −1 D has Krull dimension 2. Let A = D(σ, a) where a ∈ D. Suppose that gld (A) < ∞. Then gld (A) = 2, 3. Furthermore, gld (A) = 3 iff either λ is a root of unity, char(K) = 0 and (H − α, C − β) = D for some α, β ∈ K such that β = 0 or (H, C) = D or λ is a root of unity and (H − α, C) = D for some α ∈ K or there is a maximal ideal m = (H − α, C − β) of D (where α, β ∈ K) such that a ∈ m and a ∈ σ i (m) = (H − (λ−i α − iλ−i−1 β), C − λ−i β) for some i ≥ 1. Proof. (1). The maximal ideal m = (H, C) is σ-invariant. Hence, gld (A) = 3, by Theorem 1.2, since gld (A) < ∞. (2). Recall that the Krull dimension of D is 2. By Theorem 1.1, gld (A) = 2, 3 (since gld (A) < ∞). Given a maximal ideal m = (H − α, C − β) of the algebra D (where α, β ∈ K). Then, by (4.3), for all i ∈ Z, (4.5)

σ i (m) = (H +iλ−1 C −λ−i α, C −λ−i β) = (H −(λ−i α−iλ−i−1 β), C −λ−i β).

Then σ i (m) = m for some i ≥ 1 iff (λ−i − 1)β = 0 and (λ−i − 1)α = iλ−i−1 β iff either β = 0, λi = 1 and char(K) = 0 or β = 0, α = 0 or β = 0, λi = 1 iff either λ is a root of unity, char(K) = 0 and (H − α, C − β) = D for some α, β ∈ K such that β = 0 or (H, C) = D or λ is a root of unity and (H − α, C) = D for some α ∈ K. Now, statement (2) follows from Theorem 1.2.  We finish the paper by considering some special cases of GWAs where the field K is not assumed to be algebraically closed. The computation of the global dimension in these cases is based on different ideas that can be useful in other situations. Proposition 4.6. Let A = D[X, Y ; σ, a] be a GWA of type 1 (σ(H) = H − 1, σ(C) = λC) such that a ∈ K[H]. Let K[H]a = pn1 1 · · · pns s be a product of maximal ideals of K[H] provided a ∈ K[H]\K. Then (1) The algebra A is a skew polynomial ring A [C, τ ] where A=K[H][X, Y ; σ, a] is a GWA with σ(H) = H − 1; τ ∈ AutK (A ) where τ (X) = λ−1 X, τ (Y ) = λY and τ (H) = H. (2) gld (A) = gld (A ) + 1 where ⎧ ∞ if a = 0 or ni ≥ 2 for some i, ⎪ ⎪ ⎪ ⎪ ⎪ 2 if a = 0, n1 = · · · = ns = 1, s ≥ 1 ⎪ ⎪ ⎪ ⎨ or a is invertible, and there exists an integer k ≥ 1 gld (A ) = ⎪ such that either σ k (pi ) = pj for some i, j ⎪ ⎪ ⎪ ⎪ ⎪ or σ k (q) = q for some maximal ideal q of K[H], ⎪ ⎪ ⎩ 1 otherwise. Proof. (1). Using the Z-grading of the GWA A, we see that as a vector space the algebra A is the tensor product of its subalgebras A ⊗ K[C]. Therefore, the algebra A is the skew polynomial algebra A [C, τ ] since CX = λ−1 XC, CY = λY C and CH = HC. (2). By [22, Theorem 7.5.(iii)] and statement (1), gld (A) = gld (A [C, τ ]) = gld (A ) + 1. Finally, by [12, Theorem 1.6], the expression for gld (A ) in statement (2) follows. 

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57

Let A = D[X, Y ; σ, a] be a GWA of type 1 (σ(H) = H − 1, σ(C) = λC) such that a ∈ K[C]. Then the algebra A is an Ore extension A [H; τ := idA , δ] of the GWA A = K[C][X, Y ; σ, a] with σ(C) = λC; and δ is a τ -derivative of algebra A given by the rule δ(X) = X, δ(Y ) = −Y and δ(C) = 0. Proposition 4.7 and Proposition 4.8 are special cases of generalized Weyl algebras of type 2 and the global dimension of these algebras can be found by applying some known results about GWAs and skew polynomial rings. Proposition 4.7. Let A = D[X, Y ; σ, a] be a GWA of type 2 (σ(H) = λH, σ(C) = μC) such that a ∈ K[H] and K[H]a = pn1 1 · · · pns s be a product of maximal ideals of K[H] provided a ∈ K[H]\K. Then (1) The algebra A is a skew polynomial ring A [C, τ ] where A=K[H][X, Y ; σ, a] is a GWA with σ(H) = λH; τ ∈ AutK (A ) where τ (X) = μ−1 X, τ (Y ) = μY and τ (H) = H. (2) gld (A) = gld (A ) + 1 where ⎧ ∞ if a = 0 or ni ≥ 2 for some i, ⎪ ⎪ ⎪ ⎪ ⎪ 2 if a = 0, n1 = · · · = ns = 1, s ≥ 1 ⎪ ⎪ ⎪ ⎨ or a is invertible, and there exists an integer k ≥ 1 gld (A ) = ⎪ such that either σ k (pi ) = pj for some i, j ⎪ ⎪ ⎪ ⎪ ⎪ or σ k (q) = q for some maximal ideal q of K[H], ⎪ ⎪ ⎩ 1 otherwise. Proof. (1). Using the Z-grading of the GWA A, we see that as a vector space the algebra A is the tensor product of its subalgebras A ⊗ K[C] where A is the GWA as in statement (1). Therefore, the algebra A is the skew polynomial algebra A [C, τ ] since CX = μ−1 XC, CY = μY C and CH = HC. (2). By [22, Theorem 7.5.(iii)] and statement (1), gld (A) = gld (A [C, τ ]) = gld (A ) + 1. Finally, by [12, Theorem 1.6], the expression for gld (A ) in statement (2) follows.  Proposition 4.8. Let A = D[X, Y ; σ, a] be a GWA of type 2 (σ(H) = λH, σ(C) = μC) such that a ∈ K[C] and K[C]a = pn1 1 · · · pns s be a product of maximal ideals of K[C] provided a ∈ K[C]\K. Then (1) The algebra A is a skew polynomial ring A [H, τ ] where A = K[C][X, Y ; σ, a] is a GWA with σ(C) = μC; τ ∈ AutK (A ) where τ (X) = λ−1 X, τ (Y ) = λY and τ (C) = C. (2) gld (A) = gld (A ) + 1 where ⎧ ∞ if a = 0 or ni ≥ 2 for some i, ⎪ ⎪ ⎪ ⎪ ⎪ 2 if a = 0, n1 = · · · = ns = 1, s ≥ 1 ⎪ ⎪ ⎪ ⎨ or a is invertible, and there exists an integer k ≥ 1 gld (A ) = ⎪ such that either σ k (pi ) = pj for some i, j ⎪ ⎪ ⎪ ⎪ ⎪ or σ k (q) = q for some maximal ideal q of K[C], ⎪ ⎪ ⎩ 1 otherwise.

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Proof. (1). Using the Z-grading of the GWA A, we see that as a vector space the algebra A is the tensor product of its subalgebras A ⊗ K[H] where A is the GWA as in statement (1). Therefore, the algebra A is the skew polynomial algebra A [H, τ ] since HX = λ−1 XH, HY = λY H and HC = CH. (2). By [22, Theorem 7.5.(iii)] and statement (1), gld (A) = gld (A [H, τ ]) = gld (A ) + 1. Finally, by [12, Theorem 1.6], the expression for gld (A ) in statement (2) follows.  Let A = D[X, Y ; σ, a] be a GWA of type 3 (σ(H) = H − 1, σ(C) = C + H) such that a ∈ K[H]. Then the algebra A is an Ore extension A [C; τ = idA , δ] of the GWA A = K[H][X, Y ; σ, a] with σ(H) = H − 1; and δ is a τ -derivative of algebra A given by the rule δ(X) = −XH, δ(Y ) = Y H and δ(H) = 0. Theorem 4.9. Let A = D[X, Y ; σ, a] be a GWA of type 4 (σ(H) = λH + C, σ(C) = λC) such that a ∈ K ∗ C. Then (1) gld (A) = 3. (2) The algebra A is a skew polynomial ring A [H; τ, δ] where A = K[C][X, Y ; σ, a] is a GWA with σ(C) = λC; τ ∈ AutK (A ) where τ (X) = λ−1 X, τ (Y ) = λY and τ (C) = C; and δ is a τ -derivative of the algebra A given by the rule δ(X) = −λ−1 CX, δ(Y ) = Y C and δ(C) = 0. (3) gld (A ) = 2. Proof. (3). By [12, Theorem 1.6], gld (A ) = gld (K[C]) + 1 = 2 since a is an irreducible element of K[C] and σ(K[C]a) = K[C]a. (2). Using the Z-grading of the GWA A, we see that as a vector space the algebra A is equal to the tensor product of its subalgebras A ⊗ K[H] where A is the GWA K[C][X, Y ; σ, a]. Therefore, the algebra A is the skew polynomial ring A [H; τ, δ] since XH = (λH + C)X, HY = Y (λH + C) and HC = CH (equivalently, HX − λ−1 XH = −λ−1 CX, HY − λY H = Y C and HC − CH = 0). (1). By [22, Theorem 7.5.(3).(i)] and statements (2), (3), we have gld (A) ≤ gld (A ) + 1 = 2 + 1 = 3 < ∞. Notice that the maximal ideal m = (C, H) of the algebra D = K[C, H] is σ-invariant of height 2. Therefore, gld (A) = gld (D) + 1 = 2 + 1 = 3, by Theorem 1.2.  References [1] Maurice Auslander, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9 (1955), 67–77. MR74406 [2] V. V. Bavula, Finite-dimensionality of Extn and Torn of simple modules over a class of algebras (Russian), Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 80–82, DOI 10.1007/BF01085496; English transl., Funct. Anal. Appl. 25 (1991), no. 3, 229–230 (1992). MR1139880 [3] V. V. Bavula, Classification of simple sl(2)-modules and the finite-dimensionality of the module of extensions of simple sl(2)-modules (Russian, with Ukrainian summary), Ukrain. Mat. Zh. 42 (1990), no. 9, 1174–1180, DOI 10.1007/BF01056594; English transl., Ukrainian Math. J. 42 (1990), no. 9, 1044–1049 (1991). MR1093625

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[4] V. V. Bavula, Simple D[X, Y ; σ, a]-modules (Russian, with Russian and Ukrainian summaries), Ukra¨ın. Mat. Zh. 44 (1992), no. 12, 1628–1644, DOI 10.1007/BF01061275; English transl., Ukrainian Math. J. 44 (1992), no. 12, 1500–1511 (1993). MR1215036 [5] V. V. Bavula, Generalized Weyl algebras and their representations (Russian), Algebra i Analiz 4 (1992), no. 1, 75–97; English transl., St. Petersburg Math. J. 4 (1993), no. 1, 71–92. MR1171955 [6] Vladimir Bavula, Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, Representations of algebras (Ottawa, ON, 1992), CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 83–107. MR1265277 [7] V. V. Bavula, Extreme modules over the Weyl algebra An (Russian, with English and Ukrainian summaries), Ukra¨ın. Mat. Zh. 45 (1993), no. 9, 1187–1197, DOI 10.1007/BF01058631; English transl., Ukrainian Math. J. 45 (1993), no. 9, 1327–1338 (1994). MR1300178 [8] V. V. Bavula, Description of two-sided ideals in a class of noncommutative rings. I (Russian, with Russian and Ukrainian summaries), Ukra¨ın. Mat. Zh. 45 (1993), no. 2, 209–220, DOI 10.1007/BF01060977; English transl., Ukrainian Math. J. 45 (1993), no. 2, 223–234. MR1232403 [9] V. V. Bavula, Description of two-sided ideals in a class of noncommutative rings. II (Russian, with Russian and Ukrainian summaries), Ukra¨ın. Mat. Zh. 45 (1993), no. 3, 307–312, DOI 10.1007/BF01061007; English transl., Ukrainian Math. J. 45 (1993), no. 3, 329–334. MR1238673 [10] Vladimir Bavula, Filter dimension of algebras and modules, a simplicity criterion of generalized Weyl algebras, Comm. Algebra 24 (1996), no. 6, 1971–1992, DOI 10.1080/00927879608825683. MR1386023 [11] Vladimir Bavula, Global dimension of generalized Weyl algebras, Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 81–107. MR1388045 [12] Vladimir Bavula, Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120 (1996), no. 3, 293–335. MR1399845 [13] V. V. Bavula, Generalized Weyl algebras and diskew polynomial rings, arXiv:1612.08941 (2016). [14] V. V. Bavula, Quiver generalized Weyl algebras, skew category algebras and diskew polynomial rings, Math. Comput. Sci. 11 (2017), no. 3-4, 253–268, DOI 10.1007/s11786-017-0313-5. MR3690043 [15] V. Bavula and F. van Oystaeyen, Krull dimension of generalized Weyl algebras and iterated skew polynomial rings: commutative coefficients, J. Algebra 208 (1998), no. 1, 1–34, DOI 10.1006/jabr.1998.7482. MR1643967 [16] Georgia Benkart and Tom Roby, Down-up algebras, J. Algebra 209 (1998), no. 1, 305–344, DOI 10.1006/jabr.1998.7511. MR1652138 [17] Thomas Cassidy, Homogenized down-up algebras, Comm. Algebra 31 (2003), no. 4, 1765– 1775, DOI 10.1081/AGB-120018507. MR1972891 [18] Ellen Kirkman and James Kuzmanovich, Fixed subrings of Noetherian graded regular rings, J. Algebra 288 (2005), no. 2, 463–484, DOI 10.1016/j.jalgebra.2005.01.024. MR2146140 [19] Ellen Kirkman, Ian M. Musson, and D. S. Passman, Noetherian down-up algebras, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3161–3167, DOI 10.1090/S0002-9939-99-04926-6. MR1610796 [20] Ellen E. Kirkman and Lance W. Small, q-analogs of harmonic oscillators and related rings, Israel J. Math. 81 (1993), no. 1-2, 111–127, DOI 10.1007/BF02761300. MR1231181 [21] Marie-Paule Malliavin, L’alg` ebre d’Heisenberg quantique (French, with English and French summaries), C. R. Acad. Sci. Paris S´er. I Math. 317 (1993), no. 12, 1099–1102. MR1257219 [22] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR934572 [23] S. P. Smith, A class of algebras similar to the enveloping algebra of sl(2), Trans. Amer. Math. Soc. 322 (1990), no. 1, 285–314, DOI 10.2307/2001532. MR972706

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[24] S. P. Smith, Quantum groups: an introduction and survey for ring theorists, Noncommutative rings (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., vol. 24, Springer, New York, 1992, pp. 131–178, DOI 10.1007/978-1-4613-9736-6 6. MR1230220 [25] Cosmas Zachos, Elementary paradigms of quantum algebras, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), Contemp. Math., vol. 134, Amer. Math. Soc., Providence, RI, 1992, pp. 351–377, DOI 10.1090/conm/134/1187297. MR1187297 [26] Alexander L. Rosenberg, The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra, Comm. Math. Phys. 142 (1991), no. 3, 567–588. MR1138051 School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom Email address: [email protected] School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom Email address: [email protected]

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15086

Quasi-quantum groups obtained from the Tannaka-Krein reconstruction theorem D. Bulacu and B. Torrecillas Abstract. We review the reconstruction theorem for quasi-quantum groups, also known as quasitriangular (QT for short) quasi-Hopf algebras. It allows to obtain for free the QT quasi-Hopf algebra structure of the quantum double of a quasi-Hopf algebra, as well as the biproduct quasi-Hopf algebra structure built on a smash product algebra. Our monoidal categorical approach can be used in order to obtain similar structures for different Hopf like algebras, too.

1. Introduction The Tannaka-Krein theory allows to reconstruct a compact group from its category of representations. It extends the Pontryagin duality which says that a locally compact abelian group identifies with its bidual. Note that the Pontryagin duality theorem applies to commutative groups and that for a noncommutative compact group its dual is not a group, it is a category of representations endowed with an additional structure. This additional structure is given by the tensor product of two representations, encoded in the multiplication of functions defined on the given group, and the dual of a representation. Later on the Tannaka-Krein theory was extended by Grothendieck to the case of algebraic groups. The interest for Tannaka-Krein theory has been renewed due to connections to quantum groups and rational conformal field theories. In this direction, by using a Tannaka-Krein type theory, Kazhdan and Lusztig [17] find a correspondence between a vertex-operator algebra (a conformal field theory) and a quantum group. Also, Majid gives in [19] reconstruction theorems for rational conform field theories and shows in [20] that to every topological quantum field theory one can associate a quasi-quantum group of internal symmetries. Meanwhile, the Tannaka-Krein philosophy was extended to several generalizations of quantum groups, see [15, 28] for nice surveys on this topic. The first goal of this note is to give the quasitriangular (QT for short) quasiHopf algebra structure of the quantum double D(H) of a finite dimensional quasiHopf algebra H by means of a Tannaka-Krein reconstruction theorem. This was the aim of [18] where it was explained how the construction should work: the centre of the monoidal category of H-representations (which identifies with the category of so called Yetter-Drinfeld modules over H) is a braided category which must be 2010 Mathematics Subject Classification. Primary 16W30; Secondaries 18D10; 16S34. Work supported by the project MTM2017-86987-P ”Anillos, modulos y algebra de Hopf”. c 2020 American Mathematical Society

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isomorphic to a category of representations over a certain algebra built on H ⊗ H ∗ ; then the general reconstruction principles should give us for free the all QT structure of D(H). We should point out that an explicit construction of the quantum double (as a so-called ”diagonal crossed product”) has been given afterwards by Hausser and Nill in [13, 14]. The Hausser-Nill description of the quantum double is based on direct computations and uses the generating matrix formalism instead of a Tannaka-Krein reconstruction principle; the latter fact was reformulated intrinsically in [3]. A different approach for defining D(H) was proposed by Schauenburg in [26]. Namely, he introduced the so called Yetter-Drinfeld modules of the second kind and defined an algebra structure on D(H) = H ⊗ H ∗ in such a way that the former is equivalent to the category of representations of the latter. Then he proved that the category of Yetter-Drinfeld modules of the second kind is isomorphic to the Yetter-Drinfeld category previously introduced by Majid and equivalent to the category of so called two-sided two-cosided Hopf modules over H; in either case, this gives a monoidal structure on the category of D(H)-representations, and so a quasi-bialgebra structure on D(H). The antipode of D(H) was computed by using the rigidity of the centre and a description of the braiding of the centre on two dual objects, say of V and W , in terms given by the strong monoidal structure of the dual functor and the transpose of the braiding of the centre on V and W . We refer to [26, & 9] for more details. In this paper we present a different method, entirely based on the TannakaKrein reconstruction principles, that provides a natural way for defining the algebra, the quasi-coalgebra, the antipode and the QT structure of D(H), as well as the isomorphism of categories between Yetter-Drinfeld modules and D(H)-modules. Furthermore, in our case, the axioms of a QT quasi-Hopf algebra are automatically guaranteed for D(H) due to the general results that we use, and this spares us for complicated computations. Actually, the most difficult part during the reconstruction procedure is to determine the algebra structure on D(H) = H ⊗ H ∗ and the isomorphism of categories between Yetter-Drinfeld modules, H YDH , and D(H)modules, D(H) M. To this end, we use the braided version of the reconstruction theorem, also due to Majid (see [22, & 9.4.2]). More generally, for (A, C) a YDdatum over a quasi-Hopf algebra H and A YD(H)C the category of (generalized) Yetter-Drinfeld modules defined by (A, C), we show that Nat(− ⊗ F, F ) is representable with representability object Homk (C, A), where F : A YD(H)C → k M is the forgetful functor to the category of k-vector spaces and Nat(− ⊗ F, F ) stands for natural transformations between the functors −⊗F and F . Thus, by the results proved in [22, & 9.4.2] it follows that B := Homk (C, A) has a k-algebra structure F and that F factors as A YD(H)C → B M → k M, with the former defined in terms of F and the latter being the forgetful functor. When C is finite dimensional we have Homk (C, A) ≡ A ⊗ C ∗ , where C ∗ is the linear dual of C, and it turns out that the algebra structure on A⊗C ∗ inherited from that of Homk (C, A) coming from the braided reconstruction theorem is just the one of A  C ∗ , the generalized diagonal crossed product algebra of A and C ∗ as it was defined in [10, 13]. Furthermore, in this situation, the functor F provides an isomorphism between the categories H C which A YD(H) and A  C ∗ M. For A = C = H we recover the category H YD is braided monoidal, and so the reconstruction theorems endow D(H) := H  H ∗ with a unique QT quasi-Hopf algebra structure such that the isomorphism of categories H YDH  D(H) M becomes braided monoidal, as Majid claimed in [18]. This is the topic of Section 4.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

63

In Section 2 we recall basic facts about rigid braided monoidal categories and functors related to them. For more details, as well as for concepts like algebra, coalgebra, bialgebra or Hopf algebra in a braided monoidal category and related topics we refer to [16, 22]. We also deal with quasi-monoidal functors (called multiplicative in [21]) in order to introduce quasi-bialgebras as the k-algebras H for which the forgetful functor H M → k M is quasi-monoidal, see Proposition 2.4. This result is known under the name of reconstruction theorem for quasi-bialgebras and generalizes the one obtained by Ulbrich [27] in the Hopf algebra case. A reconstruction theorem for quasi-quantum groups, i.e. for QT quasi-Hopf algebras, is presented in Section 3. Such kind of theorem appears for the first time in [21], with a reviewed version presented in [24, Lemma 4]. We should stress the fact that our version (Theorem 3.2) is different from those mentioned above, and this is mostly because of our interpretation for the forgetful functor to be left rigid quasi-monoidal and since we start from the beginning with a k-algebra and its category of representations instead of a certain functor F and Nat(F, F ), respectively. Therefore, our definition (3.3) for the reconstructed antipode is different (but perhaps equivalent) from the one given in [21, 24], and has a simpler form that allows to obtain, in an easy and elegant way, concrete definitions for the antipode in the case of a quantum double or of a biproduct quasi-Hopf algebra. Biproduct quasi-Hopf algebras B × H were introduced in [7], by generalizing a construction of Radford [25] given for Hopf algebras. In Section 5 we achieve the second goal of this paper; namely, to obtain the structure defining a biproduct quasi-Hopf algebra in a coceptual and less possible computational way. In fact, we will see that the biproduct structure can be reconstructed from the left rigid quasimonoidal structure of a certain forgetful functor; in other words, it is uncovered by the reconstruction theorems proved so far in this paper. More concretely, by using techniques involving bimonads it was proved in [2] that B C, the category of representations in C of a central bialgebra or Hopf algebra B (i.e. B is a bialgebra or a Hopf algebra in the weak centre of C), is monoidal; for a proof strictly connected to our context see Proposition 5.2. Furthermore, if C is left rigid then so is B C, cf. Proposition 5.3. If we take C = H M, a central bialgebra in C is nothing but a bialgebra in H H YD and B C =B#H M, where B#H is the smash product algebra constructed in [9]. So for B a bialgebra in the category of (left) YetterDrinfeld modules we have a quasi-monoidal functor B#H M → k M which turns to be left rigid when we restrict it to the finite dimensional objects, provided that B is a Hopf algebra in H H YD. Otherwise stated, we are in the position to apply our reconstruction theorem for quasi-Hopf algebras, and this leads to the structure of B × H. 2. Preliminaries 2.1. Braided monoidal categories and braided (quasi-)monoidal functors. A monoidal category is a category C together with a functor ⊗ : C × C → C, called the tensor product, an object 1 ∈ C, called the unit object, and natural isomorphisms a : ⊗ ◦ (⊗ × Id) → ⊗ ◦ (Id × ⊗) (the associativity constraint), l : ⊗ ◦ (1 × Id) → Id (the left unit constraint) and r : ⊗ ◦ (Id × 1) → Id (the right unit constraint) satisfying the so-called Pentagonal and Triangle Axioms, see for example [16, XI.2] for a detailed discussion. C is called strict if a, l and r are the identity natural transformations.

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Definition 2.1. Let C, D be monoidal categories and F : C → D a functor between them. We call F a quasi-monoidal functor if there exist a natural isomorphism ϕ2 = (ϕ2,X,Y : F (X)F (Y ) → F (X ⊗ Y ))X,Y ∈C and an isomorphism ϕ0 : F (1) → I in D (without any further conditions). Here ⊗ and 1 stand for the tensor product and the unit object of C, while  and I are the tensor product and the unit object of D, respectively. If, moreover, the couple (ϕ2 , ϕ0 ) behaves well with respect to the associativity and the left and right constraints of C and D we call F in Definition 2.1 a strong monoidal functor or a tensor functor; for a detailed definition of this concept we refer to [16, Definition XI.4.1]. It is immediate that any strong monoidal functor is quasi-monoidal. This justifies our terminology and also the concept of quasibialgebra that we will introduce soon. A monoidal category C is called left rigid or with left duality if any X ∈ C has a left dual, i.e. there exist X ∗ ∈ C and morphisms evX : X ∗ ⊗ X → 1 and coevX : 1 → X ⊗ X ∗ in C such that (2.1)

−1 rX ◦ (IdX ⊗evX ) ◦ aX,X ∗ ,X ◦ (coevX ⊗ IdX ) ◦ lX = IdX ,

(2.2)

−1 lX ∗ ◦ (evX ⊗ IdX ∗ ) ◦ a−1 X ∗ ,X,X ∗ ◦ (IdX ∗ ⊗coevX ) ◦ rX ∗ = IdX ∗ .

In this situation we also say that we have an adjunction (coevX , evX ) : X ∗ $ X, X∗ X

1

and denote evX = and coevX

 = . Hence, when C is strict monoidal the X X∗

1

following relations hold: X

 (2.3)

=

X

X∗

X

and X

 X ∗ = .

X∗

X∗

If C is a monoidal category with left duality then for any morphism f : X → Y define −1  rY IdY ∗ ⊗(f ⊗IdX ∗ ) ∗ Id ∗ ⊗coevX f∗ : Y ∗ → Y ∗ ⊗ 1 Y −→ Y ∗ ⊗ (X ⊗ X ∗ ) −→ Y ∗ ⊗ (Y ⊗ X ∗ ) a−1 Y ∗ ,Y,X ∗

−→

(Y ∗ ⊗ Y ) ⊗ X ∗

evY ⊗IdX ∗

−→

 lX ∗ 1 ⊗ X∗ → X∗ .

We call f ∗ the left transpose morphism of f in C. In a similar manner we can introduce the notion of a monoidal category with right duality or right rigid. A monoidal category is called rigid if it is left and right rigid. Definition 2.2. Let F : C → D be a quasi-monoidal functor between two left rigid monoidal √categories. We call F left rigid quasi-monoidal if there exist √ isomorphisms F (X ) ∼ = F (X)∗ , natural in X ∈ C. Here X is the left dual of X in C and F (X)∗ is the left dual of F (X) in D. A strong monoidal functor between two left rigid monoidal categories is left rigid quasi-monoidal, since it preserves dual objects and the left dual object is unique up to isomorphism.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

65

For an arbitrary category C, the switch functor τ : C × C → C × C is defined by τ (V, W ) = (W, V ), for any two objects V, W of C, and similarly on a pair of morphisms. A prebraiding on a monoidal category is a natural transformation c : ⊗ → ⊗ ◦ τ , satisfying the Hexagon Axioms (see for example [16, XIII.1]). A prebraiding c is called a braiding if it is a natural isomorphism. A (pre)braided monoidal category is a monoidal category with a (pre)braiding, in which case we denote it by (C, c). For the definition of a (co)algebra (resp. bialgebra, Hopf algebra) in a monoidal (resp. braided monoidal) category C and related topics we refer to [16, 22]. Let F : C → D be a quasi (resp. strong) monoidal functor between two (pre)braided categories (C, c) and (D, d). We say that F is (pre)braided quasi (resp. strong) monoidal functor if F (cX,Y ) ◦ ϕ2,X,Y = ϕ2,Y,X ◦ dF (X),F (Y ) , ∀ X, Y ∈ C. Similarly, if C and D are with left duality we then say that F is left rigid (pre)braided quasi (resp. strong) monoidal if it is left rigid quasi (resp. strong) monoidal and at the same time (pre)braided quasi (resp. strong) monoidal. Note that a braided category that is left (resp. right) rigid is right (resp. left) rigid as well, and therefore rigid monoidal. Let k be a field, and denote by k M the category of k-vector spaces and by fd M the full subcategory of k M consisting of finite dimensional vector spaces. It k is well-known that for any U ∈ k Mfd we have an adjuction (coevU , evU ) : U ∗ $ U , dual of U , evU : U ∗ ⊗ U " u∗ ⊗ u → u∗ (u) ∈ k and where U ∗ is the k-linear  coevV : k " κ → κ ui ⊗ ui ∈ U ⊗ U ∗ ; here {ui , ui }i are dual bases in U and U ∗ . i

For H a unital k-algebra with unit 1H we denote by H M the category of left H-representations. If M is a left H-module, unless otherwise specified, we will denote the action of H on M by H ⊗ M " h ⊗ m → h · m ∈ M . Let F : H M → k M be the functor that forgets the H-module structure. It is well-known that the monoidal structures on H M that make F strong monoidal (resp. quasi-monoidal) are in a one to one correspondence to the bialgebra (resp. quasi-bialgebra) structures on H; more details will be given in Subsection 2.2. When F is, moreover, left rigid will be discussed in Section 3. 2.2. Reconstruction theory for quasi-bialgebras. The statement of [16, Proposition XV.1.2] can be viewed as a first reconstruction type theorem for quasibialgebras. A slightly improved version of it is [1, Theorem 1], as the existence of the comultiplication and the counit morphisms is not required from the beginning; they can be “reconstructed” from the quasi-monoidal structure of the forgetful functor. For further use, we reformulate the result in [1, Theorem 1] in the language of quasi-monoidal functors and explain how the reconstruction theorem works. Throughout this Subsection k is a field and H is a unitary k-algebra with unit 1H . We next investigate the monoidal structures on H M for which the forgetful functor F : H M → k M is a quasi-monoidal functor. The case when F is, moreover, strong monoidal was treated in [1, Proposition 1]. Lemma 2.3. Let H be a unital k-algebra. Then giving a monoidal structure on M such that the forgetful functor F : H M →k M is a quasi-monoidal functor is H equivalent to giving a monoidal structure on H M that comes by a restriction of the

66

D. BULACU AND B. TORRECILLAS

monoidal structure on k M to H M. More exactly, this means that (a) for any two left H-modules X, Y the tensor product X ⊗ Y in k M admits a left H-module structure; (b) the tensor product in k M of two left H-module morphisms is a morphism in H M, and so ⊗ induces a functor from H M × H M to H M; (c) k, the unit object of k M, admits a left H-module structure; (d) there exist functorial isomorphisms a = (aX,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z))X,Y,Z∈H M , l = (lX : k ⊗ X → X)X∈H M and r = (rX : X ⊗ k → X)X∈H M in H M such that the Pentagon Axiom and the Triangle Axiom are satisfied. Proof. Everything follows from the definition of a quasi-monoidal functor, since in our case it acts as identity on objects and morphisms.  By Lemma 2.3 and [1, Theorem 1] we get the following result. Proposition 2.4. Let k be a field and H a k-algebra. Then there exists a one to one correspondence between (i) monoidal structures on H M such that the forgetful functor F : H M → k M is a quasi-monoidal functor; (ii) 5-tuples (Δ, ε, Φ, l, r) consisting of two k-algebra maps Δ : H → H ⊗ H and ε : H → k and invertible elements Φ ∈ H ⊗ H ⊗ H and l, r ∈ H such that, for all h ∈ H, the following relations hold: (2.4)



(IdH ⊗Δ)Δ(h) = Φ (Δ ⊗ IdH )Δ(h) Φ−1 ,

(2.5) (ε ⊗ IdH )Δ(h) = l−1 hl, (IdH ⊗ε)Δ(h) = r −1 hr, (2.6) (IdH ⊗ IdH ⊗Δ)(Φ)(Δ ⊗ IdH ⊗ IdH )(Φ) = (1H ⊗ Φ)(IdH ⊗Δ ⊗ IdH )(Φ)(Φ ⊗ 1H ), (2.7) (IdH ⊗ε ⊗ IdH )(Φ) = r ⊗ l−1 . Proof. If F : H M → k M is a quasi-monoidal functor, we have a left Hmodule structure · : H ⊗ (H ⊗ H) → H ⊗ H on H ⊗ H which defines Δ : H → H ⊗ H, Δ(h) = h · (1H ⊗ 1H ), ∀ h ∈ H. Now, giving a left H-module structure on k is equivalent to giving an algebra map ε : H → k. If we set Φ = aH,H,H (1H ⊗ 1H ⊗ 1H ) , l = lH (1k ⊗ 1H ) and r = rH (1H ⊗ 1k ) , then (H, Δ, ε, Φ, l, r) satisfies the conditions in (ii). Conversely, for (H, Δ, ε, Φ, l, r) as in (ii) we have a monoidal structure on H M. The tensor product is the tensor product over k: for two left H-modules V and W we have V ⊗ W a left H-module via the diagonal action induced by Δ, i.e. h · (v ⊗ w) = h1 · v ⊗ h2 · w, for all h ∈ H, where we denote Δ(h) = h1 ⊗ h2 (summation implicitly understood); this is the sigma or Sweedler notation. The unit is k, regarded as a left H-module via h · κ = ε(h)κ, for all h ∈ H and κ ∈ k.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

67

The associativity and the unit constraints are defined by aV,W,Z ((v ⊗ w) ⊗ z) := X 1 · v ⊗ (X 2 · w ⊗ X 3 · z), lV (1 ⊗ v) := l · v

and

rV (v ⊗ 1) := r · v,

for all V, W, Z ∈ H M and v ∈ V , w ∈ W , z ∈ Z, where, in what follows we denote the tensor components of Φ by capital letters, and the ones of Φ−1 by small letters, namely Φ = X1 ⊗ X2 ⊗ X3 = T 1 ⊗ T 2 ⊗ T 3 = V 1 ⊗ V 2 ⊗ V 3 = · · · Φ−1 = x1 ⊗ x2 ⊗ x3 = t1 ⊗ t2 ⊗ t3 = v 1 ⊗ v 2 ⊗ v 3 = · · · The remaining details can be found in the proof of [1, Theorem 1].



A unital k-algebra H endowed with a structure as in (ii) of Proposition 2.4 is called a quasi-bialgebra. Following [12] or the proof of [1, Proposition 1], any quasi-bialgebra is equivalent to one for which l = r = 1H (see [16, Section XV.3] for terminology). This is why all the quasi-bialgebras considered in the sequel will obey l = r = 1H . Thus, from now on, by a quasi-bialgebra H we mean a 4-tuple (H, Δ, ε, Φ), where H is an associative algebra with unit 1H , Φ is an invertible element in H ⊗ H ⊗ H (called the reassociator of H) and Δ : H → H ⊗ H and ε : H → k are unital algebra morphisms (called the comultiplication and the counit of H) satisfying the identities (2.4), (2.6) and (2.8) (2.9)

(ε ⊗ IdH )Δ(h) = h = (IdH ⊗ε)Δ(h), ∀ h ∈ H, (IdH ⊗ε ⊗ IdH )(Φ) = 1H ⊗ 1H .

One can easily see that (2.8), (2.6) and (2.9) also imply that (2.10)

(ε ⊗ IdH ⊗ IdH )(Φ) = (IdH ⊗ IdH ⊗ε)(Φ) = 1H ⊗ 1H .

2.3. Bimodule (co)algebras and bicomodule algebras over quasibialgebras. If H is a quasi-bialgebra, the category H MH of H-bimodules can be canonically identified with the category of left H ⊗ H op -modules. As H ⊗ H op is a quasi-bialgebra via the tensor product algebra and coalgebra structure, H MH is monoidal. A (co)algebra in H MH will be called an H-bimodule (co)algebra. As far as we are concerned, an H-bimodule coalgebra C is an H-bimodule (denote the actions by h · c and c · h) with a comultiplication Δ : C → C ⊗ C and a counit ε : C → k satisfying the following relations, for all c ∈ C and h ∈ H: (2.11) (2.12) (2.13)

Φ · (Δ ⊗ IdC )(Δ(c)) · Φ−1 = (IdC ⊗Δ)(Δ(c)), Δ(h · c) = h1 · c1 ⊗ h2 · c2 , Δ(c · h) = c1 · h1 ⊗ c2 · h2 , ε(h · c) = ε(h)ε(c), ε(c · h) = ε(c)ε(h),

where we used the notation Δ(c) = c1 ⊗c2 . An example of an H-bimodule coalgebra is H itself. The notion of H-bicomodule algebra cannot be introduced in a similar manner, because we cannot consider H-comodules or H-bicomodules. Nevertheless, by [13] we have the following. Definition 2.5. Let H be a quasi-bialgebra. By an H-bicomodule algebra we mean a 6-tuple (A, λ, ρ, Φλ , Φρ , Φλ,ρ ), where A is a unital associative algebra, λ : A → H ⊗ A and ρ : A → A ⊗ H are unital algebra morphisms, and where Φλ ∈ H ⊗ H ⊗ A, Φρ ∈ A ⊗ H ⊗ H and Φλ,ρ ∈ H ⊗ A ⊗ H are invertible elements, such that:

68

D. BULACU AND B. TORRECILLAS

- (A, λ, Φλ ) is a left H-comodule algebra, i.e., for all u ∈ A, (2.14)

(IdH ⊗λ)(λ(u))Φλ = Φλ (Δ ⊗ IdA )(λ(u)), (1H ⊗ Φλ )(IdH ⊗Δ ⊗ IdA )(Φλ )(Φ ⊗ 1A )

(2.15) (2.16) (2.17)

= (IdH ⊗ IdH ⊗λ)(Φλ )(Δ ⊗ IdH ⊗ IdA )(Φλ ), (ε ⊗ IdA ) ◦ λ = IdA , (IdH ⊗ε ⊗ IdA )(Φλ ) = (ε ⊗ IdH ⊗ IdA )(Φλ ) = 1H ⊗ 1A ; - (A, ρ, Φρ ) is a right H-comodule algebra, i.e., for all u ∈ A,

(2.18) (2.19) (2.20) (2.21)

Φρ (ρ ⊗ IdH )(ρ(u)) = (IdA ⊗Δ)(ρ(u))Φρ , (1A ⊗ Φ)(IdA ⊗Δ ⊗ IdH )(Φρ )(Φρ ⊗ 1H ) = (IdA ⊗ IdH ⊗Δ)(Φρ )(ρ ⊗ IdH ⊗ IdH )(Φρ ), (IdA ⊗ε) ◦ ρ = IdA , (IdA ⊗ε ⊗ IdH )(Φρ ) = (IdA ⊗ IdH ⊗ε)(Φρ ) = 1A ⊗ 1H ; - the following compatibility relations hold, for all u ∈ A,

(2.22) (2.23) (2.24)

Φλ,ρ (λ ⊗ IdH )(ρ(u)) = (IdH ⊗ρ)(λ(u))Φλ,ρ , (1H ⊗ Φλ,ρ )(IdH ⊗λ ⊗ IdH )(Φλ,ρ )(Φλ ⊗ 1H ) = (IdH ⊗ IdH ⊗ρ)(Φλ )(Δ ⊗ IdA ⊗ IdH )(Φλ,ρ ), (1H ⊗ Φρ )(IdH ⊗ρ ⊗ IdH )(Φλ,ρ )(Φλ,ρ ⊗ 1H ) = (IdH ⊗ IdA ⊗Δ)(Φλ,ρ )(λ ⊗ IdH ⊗ IdH )(Φρ ).

For an H-bicomodule algebra (A, λ, ρ, Φλ , Φρ , Φλ,ρ ) we denote, for all u ∈ A, ρ(u) = u0 ⊗ u1 , (ρ ⊗ IdH )(ρ(u)) = u0,0 ⊗ u0,1 ⊗ u1 etc. λ(u) = u[−1] ⊗ u[0] , (IdH ⊗λ)(λ(u)) = u[−1] ⊗ u[0,−1] ⊗ u[0,0] etc. Furthermore, in analogy with the notation for the reassociator Φ of H, we will write ˜ ρ1 ⊗ X ˜ ρ2 ⊗ X ˜ ρ3 = Y˜ρ1 ⊗ Y˜ρ2 ⊗ Y˜ρ3 = · · · Φρ = X ˜1ρ ⊗ x ˜2ρ ⊗ x ˜3ρ = y˜ρ1 ⊗ y˜ρ2 ⊗ y˜ρ3 = · · · Φ−1 ρ =x and similarly for the element Φλ ; when there is no danger of confusion we will omit the subscripts ρ or λ for the tensor components of the elements Φρ , Φλ or for the −1 tensor components of the elements Φ−1 ρ , Φλ . Also, for simplicity we denote 1

2

3

Φλ,ρ = Θ1 ⊗ Θ2 ⊗ Θ3 = Θ1 ⊗ Θ2 ⊗ Θ3 = Θ ⊗ Θ ⊗ Θ = · · · , 1 2 3 1 2 3 ˜1 ˜2 ˜3 Φ−1 λ,ρ = θ ⊗ θ ⊗ θ = θ ⊗ θ ⊗ θ = θ ⊗ θ ⊗ θ = · · · .

3. Reconstruction theory for quasi-quantum groups 3.1. The reconstruction theorem for quasi-Hopf algebras. We proved in Proposition 2.4 a reconstruction type theorem for quasi-bialgebras. The purpose of this Section is to provide a similar result for quasi-Hopf algebras. More precisely, we show that the antipode of a quasi-Hopf algebra H can be ”reconstructed” from a left rigid quasi-monoidal structure of the forgetful functor F : H Mfd → k Mfd . This is stated also in [11, Subsection 1.2.2] but without any further clue.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

69

Recall that a quasi-Hopf algebra is a quasi-bialgebra H together with an antialgebra morphism S : H → H and elements α, β ∈ H such that, (3.1)

S(h1 )αh2 = ε(h)α and S(h1 )βh2 = ε(h)β, ∀ h ∈ H,

(3.2)

X 1 βS(X 2 )αX 3 = 1 and S(x1 )αx2 βS(x3 ) = 1.

Let H be a quasi-bialgebra and denote by H Mfd the category of finite dimensional left representations over H. It can be seen easily that the forgetful functor F : H Mfd → k Mfd is a quasi-monoidal functor. We next see when F is a left rigid quasi-monoidal functor. Lemma 3.1. Let H be a quasi-bialgebra over k. Then the forgetful functor F : H Mfd → k Mfd is a left rigid quasi-monoidal functor if and only if: (LR1) For any V ∈ H Mfd the left dual of V in k Mfd , V ∗ , admits a left HH module structure with respect to which there exists an adjunction (coevH V , evV ) : ∗ fd ∗ V $ V in H M ; in other words, V with a suitable structure becomes a left dual for V in H Mfd . √ √ (LR2) For any morphism f : V → W in H Mfd we have f = f ∗ , where f is the left transpose morphism of f in H Mfd while f ∗ is the left transpose morphism of f in k Mfd . √

∼ Proof. The functor F acts as identity on objects and morphisms. So F (V ) = √ F (V )∗ reduces to the choosing of√V as being V ∗ , with additional structures as in (LR1) that come from those of V (recall that the left dual of an object is√unique up to an isomorphism). Furthermore, due to this choice we must require f =√f ∗ , for any f : V → W in H Mfd , and this is because the k-linear isomorphisms V ∼ =V∗ fd are natural in V ∈ H M . Hence our proof is complete.  We present now a revised version of the reconstruction theorem for quasi-Hopf algebras. Note that in the proof below we need H to be an object of H Mfd , so we have to assume from the beginning that H is finite dimensional. Theorem 3.2. Let H be a finite dimensional unital algebra over a field k. Then there exists a bijective correspondence between • quasi-Hopf algebra structures on H; • left rigid monoidal stuctures on H Mfd for which the functor F : H Mfd → fd that forgets the left H-action is a left rigid quasi-monoidal functor. kM Proof. It is well-konwn that for any quasi-Hopf algebra H (not necessarily finite dimensional) the forgetful functor F : H Mfd → k Mfd is a left rigid quasimonoidal functor. If V is a finite dimensional left H-module then V ∗ = Homk (V, k) with the left H-action (h · v ∗ )(v) = v ∗ (S(h) · v), for all v ∗ ∈ V ∗ , h ∈ H and v ∈ V , is a left H-module and the maps evV : V ∗ ⊗ V → k, evV (v ∗ ⊗ v) = v ∗ (α · v), ∀ v ∗ ∈ V ∗ , v ∈ V,  coevV : k → V ⊗ V ∗ , coevV (1) = β · vi ⊗ v i , i

are left H-linear, where {vi }i and {v i }i are dual bases in V and V ∗ . Furthermore, (coevV , evV ) : V ∗ $ V is an adjunction in H Mfd that makes F a left rigid quasimonoidal functor. By Proposition 2.4 and the reduction of the definition of a quasi-bialgebra to the case l = r = 1H , we have only to show that a finite dimensional quasi-bialgebra

70

D. BULACU AND B. TORRECILLAS

H is a quasi-Hopf algebra, provided that the forgetful functor F : H Mfd → k Mfd is a left rigid quasi-monoidal functor. For this, let H be a finite dimensional quasi-bialgebra and {hi }i a basis in H with dual basis {hi }i in H ∗ . Regard H ∈ H Mfd via its multiplication and denote by · : H ⊗ H ∗ → H ∗ the left H-module structure of H ∗ , and by (ι, ) : H ∗ $ H the adjunction in H Mfd as in (LR1). We claim that this data defines completely the left rigid monoidal structure on H Mfd as well as the quasi-Hopf algebra structure on H, provided that the forgetful functor F : H Mfd → k Mfd is left rigid quasi-monoidal. Indeed, consider S : H → H given by  (h · hi )(1H )hi , ∀ h ∈ H. (3.3) S(h) = i

Furthermore, for V ∈ H Mfd and v ∈ V , a fixed element, define ϕv √: H → V by ϕv (h) = h · v, for all h ∈ H, a left H-linear morphism. Then ϕ∗v = ϕv : V ∗ → H ∗ is a left H-linear morphism, i.e. ϕ∗v (h · v ∗ ) = h · ϕ∗v (v ∗ ), for all h ∈ H and v ∗ ∈ V ∗ . This is clearly equivalent to    ϕ∗v (v ∗ )(hi )hi = h · v ∗ (ϕv (hi ))hi = v ∗ (hi · v)h · hi , (h · v ∗ ) ◦ ϕv = h · ϕ∗v (v ∗ ) = h · i

i

i

∗

as elements in H , for all h ∈ H. Evaluating both sides of the above equality on 1H we get that  (3.4) (h · v ∗ )(v) = v ∗ (hi · v)(h · hi )(1H ) = v ∗ (S(h) · v), ∀ h ∈ H, i

and this describes completely the left H-module structure of V ∗ , for any V ∈ H Mfd . Since the definition of S depends only on H and the structure in (3.4) turns any V ∗ into a left H-module, it follows that S : H → H is an anti-algebra morphism. It remains to define the distinguished elements α, β ∈ H that together with S definedby (3.3) obey the relations (3.1) and (3.2). Towards this end, we write ι(1H ) = xj ⊗ xj ∈ H ⊗ H ∗ and define j

(3.5)

α=



(hi ⊗ 1H )hi and β =

i



xj (1H )xj .

j

We next see that α, β determine completely the adjunction (ιV , V ) : V ∗ $ V in (LR1) that behaves well with respect to the left H-module structures in (3.4). This claim follows from (LR2) since for any morphism in f : V → W in H Mfd we have W∗

(3.6)

 V

W



V∗

W∗ V

W∗ V

k

k

W V∗

W V∗

  ∗ h fh V W = and fh = , = f ∗ , and this implies ∗ fh V W



f

k

k

where, as the notation suggests, the evaluation and coevaluation morphisms with the letter V or W nearby is our diagrammatic notation for the evaluation and coevaluation morphisms corresponding to V , and respectively to W , in H Mfd . For V ∈ H Mfd and v ∈ V fixed arbitrary consider again ϕv : H → V the left H-linear morphism defined above. By the equality in (3.6) involving the evaluation morphisms, specialized for ϕv , we get that V (v ∗ ⊗ h · v) = (v ∗ ◦ ϕv ⊗ h), for all

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

71

v ∗ ∈ V ∗ and h ∈ H. By taking h = 1H we see that V (v ∗ ⊗ v)

(v ∗ ◦ ϕv ⊗ 1H )  v ∗ (ϕv (hi ))(hi ⊗ 1H )

= =

i

(3.5)

v ∗ (α · v),

=

for all v ∗ ∈ V ∗ and v ∈ V , so V is completely determined by α. Similarly, by taking f = ϕv in the equality of (3.6) that involves the coevalu⊗ IdH ∗ )ι = (IdV ⊗ϕ∗v )ιV , as morphisms from ation morphisms we deduce that (ϕv ∗ k to H ⊗ H . If we write ιV (1k ) := yl ⊗ y l ∈ V ⊗ V ∗ we then get l



ϕv (xi ) ⊗ xi =



i

yl ⊗ y l ◦ ϕv ∈ V ⊗ H ∗ ,

l

and therefore



y l (v)yl =



(3.5)

xi (1H )xi v = β · v,

i

l

for all v ∈ V . It follows that ιV (1k ) =



β · vs ⊗ v s ,

s

where {vs , v }s are dual bases in V and V ∗ . So β describes completely the coevaluation morphisms ιV , V ∈ H Mfd . It is clear at this moment that V and ιV are H-linear morphisms, for all V ∈ H Mfd , if and only if (3.1) are satisfied, and that (ιV , V ) obeys (2.3) if and only if (3.2) are fulfilled. Otherwise stated, the triple (S, α, β) defines an antipode for the quasi-bialgebra H, and therefore H is a quasi-Hopf algebra. This ends the proof of the theorem.  s

For further use, recall that the antipode S of a quasi-Hopf algebra H is anticoalgebra morphism in the following sense: there exists an invertible element f = f 1 ⊗ f 2 ∈ H ⊗ H, called the Drinfeld twist or gauge transformation, such that ε(f 1 )f 2 = ε(f 2 )f 1 = 1 and, for all h ∈ H, f Δ(S(h))f −1 = (S ⊗ S)(Δcop (h)),

(3.7)

where Δcop (h) = h2 ⊗h1 . f can be described explicitly: first we define γ, δ ∈ H ⊗H by (3.8)

γ = S(x1 X 2 )αx2 X13 ⊗ S(X 1 )αx3 X23 (2.4),(3.1)

=

(3.9)

S(X 2 x12 )αX 3 x2 ⊗ S(X 1 x11 )αx3 ,

δ = X11 x1 βS(X 3 ) ⊗ X21 x2 βS(X 2 x3 ) (2.4),(3.1) 1

=

x βS(x32 X 3 ) ⊗ x2 X 1 βS(x31 X 2 ).

With this notation f and f −1 are given by the formulas (3.10)

f

=

(S ⊗ S)(Δcop (x1 ))γΔ(x2 βS(x3 )),

(3.11)

f −1

=

Δ(S(x1 )αx2 )δ(S ⊗ S)(Δcop (x3 )).

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D. BULACU AND B. TORRECILLAS

3.2. The reconstruction theorem for QT quasi-Hopf algebras. By definition, a quasitriangular, QT for short, quasi-bialgebra (resp. quasi-Hopf algebra) is a quasi-bialgebra (resp. quasi-Hopf algebra) for which its category of left representations is braided monoidal. Explicitly, a quasi-bialgebra (resp. quasi-Hopf algebra) H is QT if and only if there exists an invertible element R ∈ H ⊗ H (called an R-matrix) such that (3.12)

(Δ ⊗ IdH )(R) = X 2 R1 x1 Y 1 ⊗ X 3 x3 r 1 Y 2 ⊗ X 1 R2 x2 r 2 Y 3 ,

(3.13) (3.14)

(IdH ⊗Δ)(R) = x3 R1 X 2 r 1 y 1 ⊗ x1 X 1 r 2 y 2 ⊗ x2 R2 X 3 y 3 , Δcop (h)R = RΔ(h), ∀ h ∈ H,

where, as before, Δcop (h) = h2 ⊗h1 , for all h ∈ H, is the coopposite comultiplication of H. For the sake of completeness and also for further use we briefly describe the inverse correspondences between the braided monoidal structures on H M and the family of R-matrices of H. We use the reconstruction formalism in the presentation of this result. Proposition 3.3. Let H be a unital k-algebra and F : H M → k M the functor forgetting the H-module structure. Then there exists a one to one correspondence between • QT quasi-bialgebra (resp. quasi-Hopf algebra) structures on H; • braided monoidal (resp. left rigid braided monoidal) structures on H M (resp. fd M ) for which F is quasi-monoidal (resp. left rigid quasi-monoidal as a functor H from H Mfd to k Mfd ). Proof. Let H be quasi-bialgebra, so that H M is monoidal. Suppose that is a braided monoidal category via a braiding c, and consider H ∈ H M via its multiplication. Then cH,H (1 ⊗ 1) := R2 ⊗ R1 ∈ H ⊗ H determines completely the braiding c, in the sense that

HM

(3.15)

cX,Y (x ⊗ y) = R2 · y ⊗ R1 · x , ∀ x ∈ X ∈ H M , y ∈ Y ∈ H M .

Now, it can be easily checked that cX,Y is a morphism in H M if and only if (3.14) holds, and that the commutativity of the two Hexagon Axioms involving c and a in the definition of a braided monoidal category is equivalent to (3.12) and (3.13). Moreover, c is a natural isomorphism if and only if R ∈ H ⊗ H is invertible. Conversely, if (H, R) is a QT quasi-bialgebra, i.e. there is an invertible R ∈ H ⊗ H such that (3.12)-(3.14) are satisfied, then c defined by (3.15) is a braiding for H M; for more details we refer to [16, Proposition XV.2.2]. Our one to one correspondence follows now from Proposition 2.4 (resp. Theorem 3.2).  Note that, in the quasi-Hopf algebra case the invertibility of an element R ∈ H ⊗ H satisfying (3.12)-(3.14) is equivalent to (3.16)

(ε ⊗ IdH )(R) = (IdH ⊗ε)(R) = 1H ,

a condition quite easy to verify. We refer to [8] for a proof of this fact.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

73

4. Application: The QT quasi-Hopf algebra structure of a quantum double The main goal in [18] was to give the quasi-Hopf algebra structure of the quantum double D(H) of a finite dimensional quasi-Hopf algebra H, by using a Tannaka-Krein reconstruction type theorem. It is well understood by now that this can be done in two steps. First we have to introduce Yetter-Drinfeld modules over H by computing the centre of the monoidal category of H-representations; this has been done by S. Majid in [18]. Secondly, we have to identify the category of Yetter-Drinfeld modules with a category of representations over an algebra that will turn out to be the desired quantum double. As we mentioned in the Introduction, this has been done for the first time by Hausser and Nill in [13, 14], and afterwards by Schauenburg [26]. But only the Schauenburg version of the quantum double is obtained through a Tannaka-Krein type theorem, and this with two exceptions: the algebra structure and the antipode of D(H). The purpose of this Section is to uncover a way, entirely based on the reconstruction principles, that allows to define the quantum double of H. Actually, we will use our version of the reconstruction theorems to obtain, in a transparent and easy way, the QT quasi-Hopf algebra structure of D(H). Towards this end, we have to recall the notion of Yetter-Drinfeld datum (YDdatum for short) and the definition of the category of Yetter-Drinfeld modules over a quasi-biagebra. By a YD-datum over a quasi-bialgebra H we mean a couple (A, C) consisting of an H-bicomodule algebra A and an H-bimodule coalgebra C. We next recall from [10] the definition of a left-right Yetter-Drinfeld module. For A = H = C we obtain the category of left-right Yetter-Drinfeld modules over H, as it was introduced in [18]. Definition 4.1. Let (A, C) be a YD-datum over the quasi-bialgebra H. A left-right Yetter-Drinfeld module is a k-vector space V with the following additional structure: - V is a left A-module; we write · for the left A-action; - we have a k-linear map ρV : V → V ⊗ C, ρV (v) = v(0) ⊗ v(1) , called the right C-coaction on V , such that, for all v ∈ V , ε(v(1) )v(0) = v and (θ 2 · v(0) )(0) ⊗ (θ 2 · v(0) )(1) · θ 1 ⊗ θ 3 · v(1) (4.1)

=x ˜1ρ · (˜ x3λ · v)(0) ⊗ x ˜2ρ · (˜ x3λ · v)(1)1 · x ˜1λ ⊗ x ˜3ρ · (˜ x3λ · v)(1)2 · x ˜2λ ;

- the following compatibility relation holds: (4.2)

u0 · v(0) ⊗ u1 · v(1) = (u[0] · v)(0) ⊗ (u[0] · v)(1) · u[−1] ,

for all u ∈ A, v ∈ V . A YD(H)C will be the category of left-right Yetter-Drinfeld modules and maps preserving the actions by A and the coactions by C. 4.1. The algebra structure of D(H). Let F : C → D be a functor such that Nat(− ⊗ F, F ) : C opp → Set is representable. Here C opp is the opposite category associated to C and Nat(− ⊗ F, F ) stands for the set of natural transformations ξ : − ⊗ F → F , where for any object M of D we denote by M ⊗ F : C → D the functor that sends N ∈ C to (M ⊗ F )(N ) = M ⊗ F (N ); if f : N → N  is a morphisms in C then (M ⊗ F )(f ) = IdM ⊗F (f ), a morphism in D. Otherwise

74

D. BULACU AND B. TORRECILLAS

stated, there exists an object B of D and functorial bijections

(θM : HomD (M, B) −→ Nat(M ⊗ F, F ))M ∈D .

(4.3)

In this case we say that F satisfies the representability assumption for modules. Recall from [20, 22] the following general result. Proposition 4.2. Let F : C → D be a functor satisfying the representability assumption for modules (4.3). Then there is an algebra B in D such that F factors U as F : C → B D → D, where U is the forgetful functor. Proof. Let B an object of D satisfying (4.3) and take μ = θB (IdB ), a natural transformation between B ⊗ F and F . μ determines completely θ, in the sense that θM (f ) = (μX (f ⊗ IdF (X) ))X∈C , for all f : M → B in D. Thus, if we denote B F (X)

μX =

μX

F (X)

(4.4)

, for any M ∈ D we have that ⎛

M F (X)

⎜ fh ⎜ μX θM (f ) = ⎜ ⎜ ⎝

F (X)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

, ∀ f ∈ HomD (M, B).

X∈C

Define ξ := (μX (IdB ⊗μX ))X∈C , a natural transformation from B ⊗B ⊗F to F . Then there exists a unique morphism mB : B ⊗ B → B such that θB⊗B (mB ) = ξ. Hence mB is the unique morphism in D fulfilling B B F (X)

B B F (X)

(4.5)

μX

F (X)

μX

, ∀ X ∈ C.

= μX

F (X)

A unit for mB is the unique morphism η B : 1 → D obeying θ1 (η B ) = 1F , the identity natural transformation of F . In other words, η B is the unique morphism from 1 to B in C satisfying (4.6)

μX (η B ⊗ IdF (X) ) = IdF (X) , ∀ X ∈ C.

Now, it is easy to see that (B, mB , η B ) is an algebra in D and that μX : B ⊗ F (X) → F (X) provides a left B-module structure on F (X) in D, for all U X ∈ D. Therefore F factors as F : C → B D → D, as stated. We refer to [22, & 9.4] for a detailed proof.  We next apply Proposition 4.2 for C = A YD(H)C , the category of left-right Yetter-Drinfeld module over the YD-datum (A, C) over the quasi-Hopf algebra H, D = k M and F : C → D the forgetful functor.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

75

If H is a quasi-Hopf algebra with bijective antipode and A is an H-bicomodule algebra, we define the elements p˜λ , q˜λ ∈ H ⊗ A by (4.7)

˜ λ2 S −1 (X ˜ λ1 β) ⊗ X ˜ λ3 , q˜λ = q˜1 ⊗ q˜2 = S(˜ p˜λ = p˜1 ⊗ p˜2 = X x1λ )α˜ x2λ ⊗ x ˜3λ .

Then, for all u ∈ A, the following relations hold (see [13, 14]): (4.8) (4.9)

pλ [S −1 (u[−1] ) ⊗ 1A ] = p˜λ [1H ⊗ u], λ(u[0] )˜ ˜ λ1 )1 p1 ⊗ X ˜ λ2 S −1 ((X ˜ λ1 )2 p2 ) ⊗ X ˜ λ3 , x3λ )[−1] p˜1 S −1 (˜ x2λ ) ⊗ (˜ x3λ )[0] p˜2 = (X x ˜1λ ⊗ (˜

(4.10) θ 1 p˜1 ⊗ θ 2 p˜20 ⊗ θ 3 p˜21 = Θ2[−1] p˜1 S −1 (Θ1 ) ⊗ Θ2[0] p˜2 ⊗ Θ3 , ˜ λ1 ⊗ q˜2 X ˜ 2 ˜2 X ˜3 (4.11) q˜1 X x1λ )q1 (˜ x2λ )1 ⊗ q2 (˜ x2λ )2 ⊗ x ˜3λ , [−1] λ ⊗ q [0] λ = S(˜ where qL = q1 ⊗ q2 ∈ H ⊗ H is qλ specialized for A = H and pR = p1 ⊗ p2 := x1 ⊗ x2 βS(x3 ) ∈ H ⊗ H. Recall from [6, Corollary 3.9] the following result. Proposition 4.3. Let H be a quasi-Hopf algebra with bijective antipode and (A, C) a YD-datum over H. We have a functor F = •⊗C : A M → A YD(H)C . For any left A-module M, F(M) = M ⊗ C is an object of A YD(H)C via the structure given, for all u ∈ A, m ∈ M and c ∈ C, by: (4.12)

u · (m ⊗ c) = u[0]0 · m ⊗ u[0]1 · c · S −1 (u[−1] ), 3

3

2 2 ˜ 2 2 ˜ X λ )0 · m ⊗ θ1 X λ )1 1 · c1 ρM⊗C (m ⊗ c) = θ0 x ˜1ρ (˜ q[0] x ˜2ρ (˜ q[0]

(4.13)

2

3

1

2 2 ˜ ˜ λ g2 ) ⊗ θ3 x ˜ λ g 1 ). X X λ )1 2 · c2 · S −1 (˜ · S −1 (θ 1 q˜[−1] ˜3ρ (˜ q[0] q1 X

If ϕ is a left A-linear map then F(ϕ) = ϕ ⊗ IdC is a morphism in A YD(H)C . In particular, A ⊗ C is a left-right Yetter-Drinfeld module over the YD-datum (A, C). Note also that, if pL = p1 ⊗ p2 ∈ H ⊗ H is the element pλ specialized for A = H then pL and qL , pR defined above satisfy the following equalities (see again [13, 14]): (4.14)

˜ λ2 )1 p1 S −1 (X ˜ λ1 ) ⊗ (X ˜ λ2 )2 p2 ⊗ X ˜ λ3 = x (X ˜1λ p˜1 ⊗ x ˜2λ p˜2[−1] ⊗ x ˜3λ p˜2[0] ,

(4.15)

S −1 (q1 h(2,1) )h1 ⊗ q2 h(2,2) = S −1 (q1 ) ⊗ hq2 , ∀ h ∈ H,

(4.16)

1 1 Y 2 p12 P 2 S(X 3 ) ⊗ X(2,2) Y 3 p2 S(X 2 ), p1 ⊗ p21 g 1 ⊗ p22 g 2 = X11 Y 1 p11 P 1 ⊗ X(2,1)

where P 1 ⊗ P 2 is a second copy of pR and f −1 = g 1 ⊗ g 2 is as in (3.11). We need to connect V ∈ A YD(H)C to F(V ) via a morphism in A YD(H)C . In the Hopf case this is just the C-coaction on V ; in the quasi-Hopf case this is induced by the C-coaction on V . Lemma 4.4. If V ∈ A YD(H)C via the A-action · : A ⊗ V " u ⊗ v → u · v ∈ V and C-coaction ρV : V " v → v(0) ⊗ v(1) ∈ V ⊗ C then ρV : V → V ⊗ C given by ρV (v) = (˜ p2 · v)(0) ⊗ (˜ p2 · v)(1) · p˜1 , ∀ v ∈ V, is a morphisms in A YD(H)C , where V ⊗ C = F(V ) as objects of A YD(H)C .

76

D. BULACU AND B. TORRECILLAS

Proof. For all u ∈ A and v ∈ V we have ρV (u · v)

=

(˜ p2 u · v)(0) ⊗ (˜ p2 u · v)(1) · p˜1

(4.8)

(u[0,0] p˜2 · v)(0) ⊗ (u[0,0] p˜2 · v)(1) · u[0,−1] p˜1 S −1 (u[−1] )

(4.2)

u[0]0 · (˜ p2 · v)(0) ⊗ u[0]1 · (˜ p2 · v)(1) · p˜1 S −1 (u[−1] )

= =

(4.12)

=

u · ρV (v),

V is left A-linear. Similarly, we compute that and this shows that λ ρV ⊗C ρV (v) (4.13)

2 2 ˜3 2 2 ˜3 Xλ )0 · (˜ Xλ )1 1 · (˜ θ0 x ˜1ρ (˜ q[0] p2 · v)(0) ⊗ θ1 x ˜2ρ (˜ q[0] p2 · v)(1)1

=

2 2 ˜3 ˜ λ2 g 2 ) ⊗ θ 3 x ˜ λ1 g 1 ) X Xλ )1 2 · (˜ ·˜ p11 S −1 (θ 1 q˜[−1] ˜3ρ (˜ q[0] p2 · v)(1)2 · p˜12 S −1 (˜ q1 X (4.11),(3.7)

=

(4.9)

2 ˜ λ3 · v)(0) θ0 x ˜1ρ · (X 2 ˜ λ3 · v)(1) · (X ˜ λ2 )1 S −1 (θ 1 q2 (X ˜ λ1 )(2,2) p22 g 2 ) ⊗θ1 x ˜2ρ · (X 1

(4.15),(4.16)

=

˜ λ3 · v)(1) · (X ˜ λ2 )2 S −1 (q1 (X ˜ λ1 )(2,1) p21 g 1 )(X ˜ λ1 )1 p1 ⊗θ 3 x ˜3ρ · (X 2 ˜ λ3 · v)(0) θ2 x ˜1ρ · (X 0

2 ˜ λ1 q2 X 1 Y 3 p2 ) ˜ λ3 · v)(1) · (X ˜ λ2 )1 X 2 S −1 (θ 1 X ⊗θ1 x ˜2ρ · (X (2,2) 1

˜ λ3 · v)(1) · (X ˜ λ2 )2 X 3 S −1 (q1 X 1 Y 2 p12 P 2 )X11 Y 1 p11 P 1 ⊗θ 3 x ˜2ρ · (X (2,1) 2 (4.15)

2 2 2 −1 1 ˜ 1 1 2 ˜ λ3 · v)(0) ⊗ θ 2 x ˜3 ˜2 θ0 x ˜1ρ · (X (θ Xλ X p ) 1 ˜ρ · (Xλ · v)(1)1 · (Xλ )1 X S

=

˜ λ3 · v)(1) · (X ˜ λ2 )2 X 3 S −1 (αp12 P 2 )p11 P 1 ⊗θ 3 x ˜2ρ · (X 2 (3.1),(3.2)

=

(4.14)

⊗θ 3 x ˜3ρ · (˜ x3λ p˜2[0] · v)(1)2 · x ˜2λ p˜2[−1]

(4.1),(4.10)

=

2 2 θ0 x ˜1ρ · (˜ x3λ p˜2[0] · v)(0) ⊗ θ1 x ˜2ρ · (˜ x3λ p˜2[0] · v)(1)1 · x ˜1λ p˜1 S −1 (θ 1 )

2 2 θ0 · (Θ2[0] p˜2 · v(0) )(0) ⊗ θ1 · (Θ2[0] p˜2 · v(0) )(1) · Θ2[−1] p˜1 S −1 (θ 1 Θ1 )

⊗θ 3 Θ3 · v(1) (4.2)

=

(˜ p2 · v(0) )(0) ⊗ (˜ p2 · v(0) )(1) · p˜1 ⊗ v(1)

=

(˜ ρV ⊗ IdC )ρV (v),

V is right C-colinear, finishing the proof. for all v ∈ V . This shows that λ



We can uncover now a representabiltiy object for Nat(− ⊗ F, F ). Proposition 4.5. Let H be a quasi-Hopf algebra and (A, C) a YD-datum over H, denote C := A YD(H)C and consider the forgetful functor F : C → k M. For any k-vector space M , we have a bijection between Nat(M ⊗F, F ) and Homk (M ⊗C, A). Consequently, F satisfies the representability assumption for modules. Proof. According to Proposition 4.3, A ⊗ C is an object of C via the structure given by (4.17)

u · (u ⊗ c) = u[0]0 u ⊗ u[0]1 · c · S −1 (u[−1] ), 2 2 ˜3 2 2 ˜3 X λ )0 u ⊗ θ1 X λ )1 1 · c1 ρA⊗C (u ⊗ c) = θ0 x ˜1ρ (˜ q[0] x ˜2ρ (˜ q[0]

(4.18)

2 2 ˜3 ˜ 2λ g 2 ) ⊗ θ 3 x ˜ 1λ g 1 ), X X λ )1 2 · c2 · S −1 (˜ ·S −1 (θ 1 q˜[−1] ˜3ρ (˜ q[0] q1 X

for all u, u ∈ A and c ∈ C.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

77

We show that the map χM : Homk (M ⊗ C, A) → Nat(M ⊗ F, F ) given by (4.19)   χM (g) = ξg := ξV : M ⊗ V " m ⊗ v → g(m ⊗ (˜ p2 · v)(1) · p˜1 ) · (˜ p2 · v)(0) ∈ V V ∈C , for all g ∈ Homk (M ⊗ C, A), is bijective with inverse χ−1 M : Nat(M ⊗ F, F ) → Homk (M ⊗ C, A) defined, for all ξ ∈ Nat(M ⊗ F, F ), by (4.20)

χ−1 M (ξ) := gξ : M ⊗ C " m ⊗ c → IdA ⊗ε, ξA⊗C (m ⊗ 1A ⊗ c) ∈ A.

Indeed, for all g ∈ Homk (M ⊗ C, A), we see that, for all m ∈ M and c ∈ C, gξg (m ⊗ c) = IdA ⊗ε, (ξg )A⊗C (m ⊗ 1A ⊗ c) = (4.17)

=

(4.18)

=

(4.17)

=

p2 · (1A ⊗ c))(1) · p˜1 ) · (˜ p2 · (1A ⊗ c))(0)  IdA ⊗ε, g(m ⊗ (˜ IdA ⊗ε, g(m ⊗ (˜ p2[0]0 ⊗ p˜2[0]1 · c · S −1 (˜ p2[−1] ))(1) · p˜1 ) ·(˜ p2[0]0 ⊗ p˜2[0]1 · c · S −1 (˜ p2[−1] ))(0) 

2 ˜3 ˜ 1λ g 1 )˜ X λ )1 2 ·(˜ IdA ⊗ε, g m⊗θ 3 x ˜3ρ (˜ q[0] p2[0]1 ·c·S −1 (˜ p2[−1] ))2 ·S −1 (˜ q1 X p1 2 2 ˜3 2 2 ˜3 X λ )0 p˜2[0]0 ⊗ θ1 X λ )1 1 · (˜ x ˜1ρ (˜ q[0] x ˜2ρ (˜ q[0] p2[0]1 · c · S −1 (˜ p2[−1] ))1 · θ0

2 ˜ 2 g2 )  X ·S −1 (θ 1 q˜[−1] λ 2 2 ˜3 2 2 ˜ 2λ g 2 ) X λ p˜[0] )1 · c · S −1 (˜ X ε, θ1 x ˜2ρ · ((˜ q[0] p2[−1] ))1 · S −1 (θ 1 q˜[−1]

2 ˜3 2 ˜ 1λ g 1 )˜ X λ p˜[0] )1 · c · S −1 (˜ ˜3ρ · ((˜ q[0] p2[−1] ))2 · S −1 (˜ q1 X p1 g m ⊗ θ3 x 3

2 2 ˜ X λ p˜2[0] )0 x ˜1ρ (˜ q[0] θ0

=

g(m ⊗ (˜ q 2 p˜2[0] )1 · c · S −1 (˜ q 1 p˜2[−1] )˜ p1 )(˜ q 2 p˜2[0] )0

=

g(m ⊗ c),

and this shows that χ−1 M χM = IdHomk (M ⊗C,A) . Note that in the last equality we used the relation (4.21)

q˜2 p˜2[0] ⊗ S −1 (˜ q 1 p˜2[−1] )˜ p1 = 1A ⊗ 1H ,

which follows easily from the definitions of pλ , qλ and the axioms of an H-bicomodule algebra and of a quasi-Hopf algebra. Now, let ξ : M ⊗ F → F be a natural transformation and for any v ∈ V ∈ C YD(H) define ϕv : A → V by ϕv (u) = u · a, for all u ∈ A. Clearly ϕv is left A A-linear, so F(ϕv ) = ϕv ⊗ IdC : A ⊗ C → V ⊗ C is a morphism in A YD(H)C . Hence, by the naturality of ξ we get that ξV ⊗C (IdM ⊗ϕv ⊗ IdC ) = (ϕv ⊗ IdC )ξA⊗C . Evaluating this equality on m ⊗ 1A ⊗ c we deduce that (4.22)

ξV ⊗C (m ⊗ v ⊗ c) = ϕv ⊗ IdC , ξA⊗C (m ⊗ 1A ⊗ c), ∀ m ∈ M, v ∈ V, c ∈ C.

Likewise, by the naturality of ξ applied to the morphism ρ˜V defined in Lemma ρV ). Since (IdV ⊗ε)˜ ρV = IdV it follows that 4.4, we have that ρ˜V ξV = ξV ⊗C (IdM ⊗˜ (4.23)

p2 · v)(0) ⊗ (˜ p2 · v)(1) · p˜1 ), ξV (m ⊗ v) = IdV ⊗ε, ξV ⊗C (m ⊗ (˜

78

D. BULACU AND B. TORRECILLAS

for all m ∈ M and v ∈ V . These relations allow to compute, for all m ∈ M and v ∈ V , that (ξgξ )V (m ⊗ v)

=

gξ (m ⊗ (˜ p2 · v)(1) · p˜1 ) · (˜ p2 · m)(0)

=

IdA ⊗ε, ξA⊗C (m ⊗ 1A ⊗ (˜ p2 · v)(1) · p˜1 ) · (˜ p2 · v)(0)

=

ϕ(p˜2 ·v)(0) ⊗ ε, ξA⊗C (m ⊗ 1A ⊗ (˜ p2 · v)(1) · p˜1 )

=

p2 · v)(1) · p˜1 ) (IdV ⊗ε)(ϕ(p˜2 ·v)(0) ⊗ IdC ), ξA⊗C (m ⊗ 1A ⊗ (˜

(4.22)

=

(4.23)

=

IdV ⊗ε, ξV ⊗C (m ⊗ (˜ p2 · v)(0) ⊗ (˜ p2 · v)(1) · p˜1 ) ξV (m ⊗ v).

Thus ξgξ = ξ, and from here we conclude that χM and χ−1 M are indeed bijective inverses. Finally, it is well known that Homk (M ⊗ C, A)  Homk (M, Homk (C, A)), and therefore, for any k-vector space M , B := Homk (C, A) is a representability object for Nat(M ⊗ F, F ). The functorial bijections θM : Homk (M, B) → Nat(M ⊗ F, F ) are given by (4.24)   θM (f ) = ξV : M ⊗ V " m ⊗ v → f (m), (˜ p2 · v)(1) · p˜1  · (˜ p2 · v)(0) ∈ V V ∈C , for all f ∈ Homk (M, B).



Corollary 4.6. With notation as above, B = Homk (C, A) admits a k-algebra structure such that the forgetful functor F : C = A YD(H)C → k M factors as U C F A YD(H) → B M → k M, where U is the functor that forgets the B-module structure. Proof. We apply Proposition 4.2 to our context. If we define ω  = ω 1 ⊗ · · · ⊗ ω ∈ H ⊗5 by (4.25) 1 ˜3 2 ˜ λ2 g 2 ) ⊗ θ 2 x ˜ λ1 g 1 ) ⊗ S −1 (θ 1 X ˜ λ3 )1 ⊗ θ 3 x ˜ λ3 )1 , ˜2ρ (X ˜3ρ (X ω  := S −1 (X 0 ˜ρ (Xλ )0 ⊗ θ1 x 1 2 5

then the multiplication on B given by (4.26)

(f % g)(c) = f, g(ω 5 · c2 · ω 1 )[0]1 ω 4 · c1 · ω 2 S −1 (g(ω 5 · c2 · ω 1 )[−1] ) g(ω 5 · c2 · ω 1 )[0]0 ω 3 ,

for all f, g ∈ B, is associative and unital with unit 1B (c) = ε(c)1A , for all c ∈ C. Then F is the functor that acts as identity on objects and morphisms; an object V of C becomes a left B-module via the action given, for all f ∈ B and v ∈ V , by (4.27)

f · v = f, (˜ p2 · v)(1) · p˜1  · (˜ p2 · v)(0) .

Indeed, we have that μ := θB (IdB ) is given in our case by   p2 · v)(1) · p˜1  · (˜ p2 · v)(0) ∈ V V ∈C . μ = μV : B ⊗ V " f ⊗ v → f, (˜ Thus, according to (4.5) the unique multiplication on B that turns each μV into a left B-module structure on V is given by −1 (f % g)(c) = θB⊗B (μA⊗C (IdB ⊗μA⊗C )), c

= IdA ⊗ε, μA⊗C (f ⊗ μA⊗C (g ⊗ 1A ⊗ c)),

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

79

for all f, g ∈ B. Now, for all f ∈ B, u ∈ A and c ∈ C we have that μA⊗C (f ⊗ u ⊗ c) p2 · (u ⊗ c))(0) f, (˜ p2 · (u ⊗ c))(1) · p˜1  · (˜

= (4.17)

=

f, (˜ p2[0]0 u ⊗ p˜2[0]1 · c · S −1 (˜ p2[−1] ))(1) · p˜1  ·(˜ p2[0]0 u ⊗ p˜2[0]1 · c · S −1 (˜ p2[−1] ))(0)

(4.18)

=

(3.7)

2 ˜3 2 1 1 ˜ λ1 p˜2 Xλ p˜[0] )1 2 · c2 · S −1 (˜ f, θ 3 x ˜3ρ (˜ q[0] q1 X p  [−1]1 g )˜ 2 2 ˜3 2 2 2 ˜3 2 Xλ p˜[0] )0 u ⊗ θ1 Xλ p˜[0] )1 1 ·(θ0 x ˜1ρ (˜ q[0] x ˜2ρ (˜ q[0] 2 2 ˜ λ2 p˜2 X ·c1 · S −1 (θ 1 q˜[−1] [−1]2 g ))

(2.14)

=

˜ λ3 )1 · c2 · S −1 (X ˜ λ1 g 1 ) f, θ 3 x ˜3ρ (X 2 2 2 ˜3 −1 1 ˜ 2 2 ˜ λ3 )0 u ⊗ θ 2 x x ˜1ρ (X (θ Xλ g )) ·(θ0 1 ˜ρ (Xλ )1 1 · c1 · S

=

f, ω 5 · c2 · ω 1  · (ω 3 u ⊗ ω 4 · c1 · ω 2 ),

(4.21)

where in the last equality we used the definition of ω  from (4.25). In particular, we get that IdA ⊗ε, μA⊗C (f ⊗ u ⊗ c) = f (c)u, for all f ∈ B, u ∈ A and c ∈ C, from where we see that  (f % g)(c) = IdA ⊗ε, μA⊗C f ⊗ g(ω 5 · c2 · ω 1 )[0]0 ω 3  ⊗g(ω 5 · c2 · ω 1 )[0]1 ω 4 · c1 · ω 2 S −1 (g(ω 5 · c2 · ω 1 )[−1] )  = f, g(ω 5 · c2 · ω 1 )[0]1 ω 4 ·c1 · ω 2 S −1 (g(ω 5 · c2 · ω 1 )[−1] )g(ω 5 · c2 · ω 1 )[0]0 ω 3 , for all f, g ∈ B and c ∈ C, as stated. Finally, % is unital with unit given by θk−1 (1F ). As χ−1 k (1F )(c) = IdA ⊗ε, 1A ⊗ c = ε(c)1A = 1B (c), for all c ∈ C, it follows that  1B defined above is a unit for %. Remark 4.7. Let (A, C) be a YD-datum over the quasi-Hopf algebra H, and consider C ∗ the linear dual of C. Since C is a coalgebra in H MH , C ∗ is an algebra in H MH via the following structure: - the multiplication is given by the convolution product, i.e. (c∗ d∗ )(c) = ∗ c (c1 )d∗ (c2 ), for all c∗ , d∗ ∈ C ∗ and c ∈ C, while the unit is ε; - the H-bimodule structure of C ∗ is determined by (h  c∗  h )(c) = c∗ (h · c · h), for all c∗ ∈ C ∗ , c ∈ C and h, h ∈ H. We have a linear map ϕ : A ⊗ C ∗ " u ⊗ c∗ → (c → c∗ (c)u) ∈ Homk (C, A), which is an isomorphism in the case when C is finite dimensional. If we consider }i dual bases in C and C ∗ then the inverse of ϕ is ϕ−1 : Homk (C, A) " {ci , ci  f (ci ) ⊗ ci ∈ A ⊗ C ∗ . Thus, when C is finite dimensional there exists a f → i

unique algebra structure on A ⊗ C ∗ that turns ϕ into an algebra morphism, where we consider Homk (C, A) as a k-algebra as in Corollary 4.6. More exactly, the multiplication is (4.28)

(u  c∗ )(u  d∗ ) = uu[0]0 ω 3  (ω 2 S −1 (u[−1] )  c∗  u[0]1 ω 4 )(ω 1  d∗  ω 5 ),

for all u, u ∈ A and c∗ , d∗ ∈ C ∗ , where ω  is as in (4.25) and we write u  c∗ in place of u ⊗ c∗ in order to distinguish this algebra structure on A ⊗ C ∗ . The unit of

80

D. BULACU AND B. TORRECILLAS

it is 1A ⊗ ε. In other words, A ⊗ C ∗ endowed with this algebra structure is nothing but A  C ∗ , the right generalized diagonal crossed product of A and C ∗ as it was introduced in [10]. This also justifies the notation for ω  . The multiplication defined in (4.28) makes sense even if C is not finite dimensional, in which case we have an algebra morphism ϕ : A  C ∗ → B = Homk (C, A), and therefore a (restriction of scalars) functor R : B M → A  C ∗ M. F Consequently, the functor F : C = A YD(H)C → k M factors also as C → A  C ∗ M U → k M, where U is the functor that forgets the A  C ∗ -module structure. Here F is the functor that acts as identity on objects and morphism, and associates to any V ∈ C the left A  C ∗ -module structure given, for all u ∈ A, c∗ ∈ C ∗ and v ∈ V , by p2 · v)(1) · p˜1 u · (˜ p2 · v)(0) . (u  c∗ ) → v = c∗ , (˜

(4.29)

When C is finite dimensional one can identify A YD(H)C with the category A  C ∗ M. Theorem 4.8. Let H be a quasi-Hopf algebra with bijective antipode and (A, C) a YD-datum over H. If C is finite dimensional then the categories A YD(H)C and A  C ∗ M are isomorphic. Proof. We claim that F defined at the end of Remark 4.7 is an isomorphism of categories. Towards this end, we define G : A  C ∗ M → A YD(H)C as follows: if V is a left A  C ∗ -module then G(V ) = V , regarded as an object in A YD(H)C via the structure u · v = (u  ε)v, ∀ u ∈ A, v ∈ V,  2 2 V " v → (˜ q0  ci )v ⊗ q˜1 · ci · S −1 (˜ qλ1 ) ∈ V ⊗ C, i

where by juxtaposition we denoted the left A  C ∗ -action on V , and {ci , ci }i are dual bases in C and C ∗ . G sends a morphism to itself. R The functor G is well defined because it equals the composition A  C ∗ M → G C ∗ C ∗ A M → A YD(H) , where C  A and R, G are defined as follows. ∗ • C  A is the generalized left diagonal crossed product of C ∗ and A, i.e. ∗ C  A is the k-vector space C ∗ ⊗ A equipped with the algebra structure given by (c∗  u)(d∗  u ) = (Ω1 · c∗ · Ω5 )(Ω2 u0 [−1] · d∗ · S −1 (u1 )Ω4 )  Ω3 u0 [0] u , for all c∗ , d∗ ∈ C ∗ ∈ C ∗ and u, u ∈ A, where we write c∗  u in place of c∗ ⊗ u to distinguish the new algebraic structure, and denoted by Ω = Ω1 ⊗ · · · ⊗ Ω5 ∈ H ⊗5 the element 1

1

1

2

3

2 ˜ ρ θ 3 ) ⊗ S −1 (f 2 X ˜ ρ ). ˜ ρ )[−1] x ˜ ρ )[0] x ˜ ρ )[−1] x ˜ 1 θ 1 ⊗ (X ˜ 2 θ 2 ⊗ (X ˜3λ θ[0] ⊗ S −1 (f 1 X Ω = (X 1 λ 2 λ [−1]

Then C ∗  A is a k-algebra with unit ε  1A , and it is isomorphic to A  C ∗ as an algebra. More exactly, the map q 2 u[0] )0 ϑ : C ∗  A " c∗  u → q 1 (˜  S −1 (˜ q 1 u[−1] )  c∗  q 2 (˜ q 2 u[0] )1 ∈ A  C ∗ is an algebra isomorphism, cf. [10, Corollary 4.7 & Proposition 4.9]. Here qρ = ˜ ρ1 ⊗ S −1 (αX ˜ ρ3 )X ˜ ρ2 ∈ A ⊗ H and qλ = q˜1 ⊗ q˜2 ∈ H ⊗ A is as in (4.7). q 1 ⊗ q 2 := X

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

81

The inverse of ϑ is ϑ−1 : A  C ∗ → C ∗  A determined by ϑ−1 (u  c∗ ) = (u0 p1 )[−1] p˜1  c∗  S −1 (u1 p2 )  (u0 p1 )[0] p˜2 , ˜1ρ ⊗ x ˜2ρ βS(˜ x3ρ ) ∈ A ⊗ H and for all u ∈ A and c∗ ∈ C ∗ , where pρ = p1 ⊗ p2 := x 1 2 pλ = p˜ ⊗ p˜ ∈ H ⊗ A is defined by (4.7). The functor R is the isomorphism of categories produced by ϑ. • G : C ∗ A M → A YD(H)C is the functor G defined in [10, Lemma 8.3]. Namely, G(V ) = V as k-module, with structure maps defined by (4.30)

u · v = (ε  u)v,

(4.31)

ρV : V → V ⊗ C, ρV (v) =



(ci  p1[0] )v ⊗ S −1 (p2 ) · ci · p1[−1] ,

i

for all v ∈ V and u ∈ A. G sends a morphism to itself. One can easily check that G = GR and F = LF, where L is the inverse of R and F : A YD(H)C → C ∗ A M is the functor defined in [10, Lemma 8.2]; namely, F(V ) = V with left C ∗  A-module structure given by (4.32)

(c∗  u)v := c∗ , q 2 · (u · v)(1) q1 · (u · v)(0) ,

for all c∗ ∈ C ∗ , u ∈ A and v ∈ V , and F(f ) = f , for any morphism f in A YD(H)C . By [10, Theorem 8.4] the functors F, G are inverses, and so the same are F and G.  Remark 4.9. Another proof for Theorem 4.8 can be obtained from [6, Corollary 3.13 i)]. We presented the one above in order to introduce the algebra C ∗  A and the algebra isomorphism ϑ : C ∗  A → A  C ∗ , which will be intensively used in what follows. Note also that both C ∗  A and A  C ∗ provide explicit realizations for D(H), the quantum double of a quasi-Hpf algebra H, by considering A = C = H. By analogy with the Hopf case, the quantum double of H is build on the algebra H ∗  H, although the reconstruction procedure gives H  H ∗ as a realization of D(H). Nevertheless, we will see that H ∗  H  H  H ∗ as QT quasi-Hopf algebras. 4.2. The quasi-Hopf algebra structure of D(H). In what follows, H is a finite dimensional quasi-Hopf algebra and {ei , ei }i are dual bases in H and H ∗ . We denote H YD(H)H by H YDH and H  H ∗ by D(H); the latter will be called the quantum double of H. The k-algebra structure of D(H) is given by (4.33) (h  ϕ)(h  ϕ ) = hh(2,1) ω 3  (ω 2 S −1 (h1 )  ϕ  h(2,2) ω 4 )(ω 1  ϕ  ω 5 ), for all h, h ∈ H and ϕ, ϕ ∈ H ∗ , where ω  = ω 1 ⊗ · · · ⊗ ω 5 ∈ H ⊗5 is given by (4.34)

3 3 ω  := S −1 (X 1 g 1 ) ⊗ S −1 (x1 X 2 g 2 ) ⊗ x21 y 1 X13 ⊗ x22 y 2 X(2,1) ⊗ x3 y 3 X(2,2) ,

and ,  are the regular left and right H-actions that turn H ∗ into an H-bimodule algebra, i.e. (h  ϕ  h )() = ϕ(h h), for all h, , h ∈ H and ϕ ∈ H ∗ .

82

D. BULACU AND B. TORRECILLAS

According to [5], the category H YDH is (braided) monoidal. The tensor product of M, N in H YDH is M ⊗ N with structure given by (4.35) h · (m⊗n) = h1 · m ⊗ h2 · n, (4.36) ρM ⊗N (m ⊗ n) = x1 X 1 · (y 2 · m)(0) ⊗ x2 · (X 3 y 3 · n)(0) ⊗ x3 (X 3 y 3 · n)(1) X 2 (y 2 · m)(1) y 1 , for all m ∈ M , n ∈ N . So the monoidal structure of H YDH is given in such a way that the forgetful functor H YDH → H M is strong monoidal, and since the forgetful functor H M → k M is quasi-monoidal it follows that F : H YDH → k M is quasi-monoidal, too. As H YDH identifies with D(H) M, we have that the forgetful functor (still denoted by F ) from D(H) M to k M is quasi-monoidal, and so by Proposition 2.4 we get that D(H) admits a quasi-bialgebra structure. In the first part of this Subsection we describe this structure of D(H) explicitly. Since we will use very often the categorical isomorphism between D(H) M and H H YD , for the convenience of the reader, record that D(H) is a left-right YetterDrinfeld module over H via the structure given by: h · (h  ϕ) = (h  ε)(h  ϕ) = hh  ϕ,   h  ϕ → (q21  ei )(h  ϕ) ⊗ q22 ei S −1 (q1 ) = q21 h(2,1) ω 3

(4.37)

i

(4.38)

i

 (ω 2 S −1 (q1 h1 )  ei  q22 h(2,2) ω 4 )(ω 1  ϕ  ω 5 ) ⊗ ei ,

for all ϕ ∈ H ∗ and h, h ∈ H. Furthermore, any left-right Yetter-Drinfeld module M is a left D(H)-module via the left D(H)-action given by (4.39) (h  ϕ)m = ϕ, (p2 · m)(1) p1 h · (p2 · m)(0) , ∀ ϕ ∈ H ∗ , h ∈ H, m ∈ M, where pL := p1 ⊗ p2 ∈ H ⊗ H is the element pλ defined in (4.7) specialized for A = H. Consequently, if M, N ∈ H YDH then M ⊗ N ∈ H YDH with structure as in (4.35) and (4.36), and therefore M ⊗ N is a left D(H)-module with structure given by (h  ϕ)(m ⊗ n) = ϕ, (p2 · (m ⊗ n))(1) · p1 h · (p2 · (m ⊗ n))(0) = ϕ, (p21 · m ⊗ p22 · n)(1) · p1 h · (p21 · m ⊗ p22 · n)(0) = ϕ, x3 (X 3 y 3 p22 · n)(1) X 2 (y 2 p21 · m)(1) y 1 p1  (4.40)

h1 x1 X 1 · (y 2 p21 · m)(0) ⊗ h2 x2 · (X 3 y 3 p22 · n)(0) .

By using the reconstruction theorem for quasi-bialgebras (Proposition 2.4) we find now the quasi-bialgebra structure of D(H). For the definition of (braided) monoidally isomorphic categories we refer to [16, 22]. Proposition 4.10. Let H be a finite dimensional quasi-Hopf algebra H. Then D(H) has a unique quasi-bialgebra structure with respect to which the isomorphism of categories in Theorem 4.8 becomes a monoidal isomorphism.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

83

Proof. By the proof of Proposition 2.4, the comultiplication of D(H), denoted in what follows by ΔD , is given by ΔD (h  ϕ) = (h  ϕ)(1D(H) ⊗ 1D(H) ) = (h  ϕ)((1H  ε) ⊗ (1H  ε)) (4.40)

= ϕ, x3 (X 3 y 3 p22 · (1H  ε))(1) X 2 (y 2 p21 · (1H  ε))(1) y 1 p1  h1 x1 X 1 · (y 2 p21 · (1H  ε))(0) ⊗ h2 x2 · (X 3 y 3 p22 · (1H  ε))(0)

(4.37)

= ϕ, x3 (X 3 y 3 p22  ε)(1) X 2 (y 2 p21  ε)(1) y 1 p1  h1 x1 X 1 · (y 2 p21  ε)(0) ⊗ h2 x2 · (X 3 y 3 p22  ε)(0) ,

for all ϕ ∈ H ∗ and h ∈ H. If we take ϕ = ε in (4.38) we get that (h  ε)(0) ⊗ (h  ε)(1)  q21 h(2,1)  S −1 (q1 h1 )  ei  q22 h(2,2) ⊗ ei , =

(4.41)

i

for all h ∈ H. Thus, if we denote by Q1 ⊗ Q2 a second copy of qL , we can compute: ΔD (h  ϕ)  ϕ, x3 ei X 2 ej y 1 p1 h1 x1 X 1 · (Q21 (y 2 p21 )(2,1)  S −1 (Q1 (y 2 p21 )1 )  ej = i,j

 Q22 (y 2 p21 )(2,2) ) ⊗ h2 x2 · (q21 (X 3 y 3 p22 )(2,1)  S −1 (q1 (X 3 y 3 p22 )1 )  ei  q22 (X 3 y 3 p22 )(2,2) ) (4.10)  2 2 = ϕ, x3 ei ej Y12 p1 S −1 (Y 1 )h1 x1 X 1 · ((Q2 Y(2,2) p22 )1  S −1 (Q1 Y(2,1) p21 ) i,j 2 p22 )2 )  ej  (Q2 Y(2,2) 3 ⊗ h2 x2 · ((q2 X23 )1 Y(2,1)  S −1 (S(X 2 )q1 X13 Y13 )  ei  (4.15) 3  (q2 X23 )2 Y(2,2) ) = ϕ, x3 ei ej S −1 (Y 1 )h1 x1 X 1 · (Y12  ej  Y22 ) (4.21)

i,j

3 3  S −1 (S(X 2 )q1 X13 Y13 )  ei  (q2 X23 )2 Y(2,2) ) ⊗ h2 x2 · ((q2 X23 )1 Y(2,1) (4.37)  = ϕ, x3 ei ej S −1 (Y 1 )h1 x1 q 1 (y 1 Y 2 )1  ej  Y22 i,j

⊗ h2 x2 (y 3 Y23 )1  S −1 (y 2 Y13 )q 2 y21  ei  (y 3 Y23 )2 = h1 x1 q 1 (y 1 Y 2 )1  S −1 (Y 1 )  ϕ2  Y22 ⊗ h2 x2 (y 3 Y23 )1  S −1 (y 2 Y13 )q 2 y21  ϕ1  x3 (y 3 Y23 )2 , for all h ∈ H and ϕ ∈ H ∗ , where (4.10) and (4.21) are applied to the case when A = H. Also, in the last but one equality we used the equation (4.42)

X 1 ⊗ S(X 2 )q1 X13 ⊗ q2 X23 = q 1 y11 ⊗ S(q 2 y21 )y 2 ⊗ y 3 ,

and in the last equality the coalgebra structure of H ∗ dual to the algebra structure of H, i.e. ΔH ∗ (ϕ) = ϕ1 ⊗ ϕ2 if and only if ϕ(hh ) = ϕ1 (h)ϕ2 (h ), for all h, h ∈ H. Note that (4.42) can be easily obtained from the definitions of qL and qR and the axioms of a quasi-Hopf algebra, where qR = q 1 ⊗ q 2 = X 1 ⊗ S −1 (αX 3 )X 2 ∈ H ⊗ H is qρ specialized for A = H.

84

D. BULACU AND B. TORRECILLAS

Thus we showed that the comultiplication ΔD of D(H) is given, for all h ∈ H and ϕ ∈ H ∗ , by ΔD (h  ϕ) = h1 x1 q 1 (y 1 Y 2 )1  S −1 (Y 1 )  ϕ2  Y22 ⊗h2 x2 (y 3 Y23 )1  S −1 (y 2 Y13 )q 2 y21  ϕ1  x3 (y 3 Y23 )2 .

(4.43)

Similar computations yield explicit formulas for the counit εD and the reassociator ΦD of D(H). Namely, by the proof of Proposition 2.4 we have εD (h  ϕ)

= (4.39)

(h  ϕ) · 1k

=

ϕ, (p2 · 1k )(1) p1 h · (p2 · 1k )(0)

=

ε(p2 )ϕ, p1 h · 1k = ε(h)ϕ(S −1 (β)),

for all ϕ ∈ H ∗ and h ∈ H, while ΦD

=

aD(H),D(H),D(H)(1D(H) ⊗ 1D(H) ⊗ 1D(H) )

=

X 1 · (1H  ε) ⊗ X 2 · (1H  ε) ⊗ X 3 · (1H  ε)

(4.37)

=

(X 1  ε) ⊗ (X 2  ε) ⊗ (X 3  ε).

Concluding, D(H) is a quasi-bialgebra with multiplication given in (4.33), unit 1D = 1H  ε, comultiplication ΔD as in (4.43), and counit and reassociator given by (4.44)

εD (h  ϕ) = ε(h)ϕ(S −1 (β)), ∀ h  ϕ ∈ D(H),

(4.45)

ΦD = (X 1  ε) ⊗ (X 2  ε) ⊗ (X 3  ε),

and this is the unique quasi-bialgebra structure on the diagonal crossed product algebra D(H) that turns the isomorphism in Theorem 4.8 into an isomorphism of monoidal categories.  Remark 4.11. If we apply the reconstruction theorem to the quasi-monoidal functor H ∗ H M  H YDH → k M, by computations similar to the ones performed in the proof of Proposition 4.10 we find that H ∗  H is a quasi-bialgebra with the algebra structure given by the diagonal crossed product of H ∗ and H, comultiplication Δ(ϕ  h) = (ε  X 1 Y 1 )(p11 x1  ϕ2  Y 2 S −1 (p2 )  p12 x2 h1 ) ⊗(X12  ϕ1  S −1 (X 3 )  X22 Y 3 x3 h2 ), and counit and reassociator given by ε(ϕ  h) = ε(h)ϕ(S −1 (α)), ∀ ϕ  h ∈ D(H), Φ = (ε  X 1 ) ⊗ (ε  X 2 ) ⊗ (ε  X 3 ). So, in a less computational way, we obtained the quasi-bialgebra structure of the quantum double of H built on H ∗ ⊗ H, see [3, 13, 14]. It is isomorphic as a quasibialgebra with D(H) since the map ϑ considered in the proof of Theorem 4.8, which in our situation (A = C = H) becomes ϑ : H ∗  H " ϕ  h → q 1 (q2 h2 )1  S −1 (q1 h1 )  ϕ  q 2 (q2 h2 )2 ∈ H  H ∗ , is a quasi-bialgebra isomorphism. We leave the verifications to the reader.

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

85

We end this Subsection by computing the quasi-Hopf algebra structure of D(H). For this, we will make use of the reconstruction theorem for quasi-Hopf algebras as it was stated in Theorem 3.2. In what follows, we denote by {θij }i,j the basis of D(H)∗ dual to the basis {ei  ej }i,j of D(H), i.e. θij (es  et ) = δi,s δj,t , where δi,j is the Kronecker’s delta. From the ”cop”-version of [5, Theorem 3.2] the category of finite dimensional left-right Yetter-Drinfeld modules over H, H YDHfd , is rigid monoidal. As far as we are concerned, the left dual of D(H) in H YDHfd is D(H)∗ considered as a left-right Yetter-Drinfeld module over H with the left H-action defined by h · θi,j

=



θi,j (S(h) · (es  et ))θs,t

s,t

(4.37)

(4.46)

=



θi,j (S(h)es  et )θs,t =

s,t



ei (S(h)es )θs,j ,

s

and the right H-action given by (4.47)  θi,j → θi,j , f 1 · (g 2 · (es  et ))(0) θs,t ⊗ S −1 (f 2 (g 2 · (es  et ))(1) g 1 ) s,t

=



θi,j , (f 1 q21  S −1 (q1 )  eu  q22 )(g 2 es  et )θs,t ⊗ S −1 (f 2 eu g 1 )

s,t,u

=



θi,j , (f 1 q21  S −1 (q1 )g 1  eu  f 2 q22 )(g 2 es  et )θs,t ⊗ S −1 (eu ),

s,t,u

for all i, j, extended by linearity. The evaluation evD (resp. the coevaluation coevD ) fd morphism of the left dual object of D(H) in H YDH is given by evD (ϕ ⊗ h, h  ϕ ) = ϕ ⊗ h, α · (h  ϕ ) = ϕ(αh )ϕ (h), for all ϕ ⊗ h ∈ D(H)∗ ≡ H ⊗ H ∗ and h  ϕ ∈ D(H) (resp. coevD (1k ) =  β · (ei  ej ) ⊗ θi,j ). i,j

Note that this (left) rigid monoidal structure on H YDHfd makes the forgetful functor H YDHfd → H Mfd (left) rigid monoidal, and therefore the forgetful functor fd  H YDHfd → k Mfd is (left) rigid quasi-monoidal. Thus Theorem 3.2 D(H) M applies, and so D(H) is a quasi-Hopf algebra. Theorem 4.12. Let H be a finite dimensional quasi-Hopf algebra. Then D(H), the quantum double of H, is a quasi-Hopf algebra that contains H as a quasi-Hopf subalgebra. Proof. By (3.3), specialized to our context, the antipode SD of D(H) is given by SD (h  ϕ)

=



(h  ϕ)θi,j , 1D ei  ej

i,j

(4.39)

=

 i,j

ϕ, (p2 · θi,j )(1) p1 h · (p2 · θi,j )(0) , 1H  εei  ej

86

D. BULACU AND B. TORRECILLAS

for all ϕ ∈ H ∗ and h ∈ H. As (h · θi,j )(1H  ε) = θi,j (S(h)  ε), for all h ∈ H, we get that SD (h  ε) (4.46)

=



ϕ, (θs,j )(1) p1 h · (θs,j )(0) , 1H  εS(p2 )es  ej

j,s (4.47)

=



θi,t (S(h)  ε)ϕ, S −1 (eu )p1 

i,j,s,t,u

=

θs,j , (f 1 q21  S −1 (q1 )g 1  eu  f 2 q22 )(g 2 ei  et )S(p2 )es  ej  ϕ, S −1 (eu )p1 (S(p2 )f 1 q21  S −1 (q1 )g 1  eu  f 2 q22 )(g 2 S(h)  ε)

=

(S(p2 )f 1 q21  S −1 (q1 )g 1  ϕ ◦ S −1  S(p1 )f 2 q22 )(g 2 S(h)  ε),

u

for all h ∈ H and ϕ ∈ H ∗ . Similarly, the general formulas in (3.5) give the elements αD and βD that together with SD define the antipode of D(H). More exactly,  (4.37)  evD (θi,j ⊗ 1D )ei  ej = θi,j (α  ε)ei  ej = α  ε, αD = i,j

and likewise, since coevD (1k ) = βD =





i,j

β · (ei  ej ) ⊗ θi,j , we get that

i,j

θi,j (1D )β · (ei  ej ) = β · (1H  ε) = β  ε.

i,j

Hence, the formulas (4.48) αD = α  ε, βD = β  α, (4.49) SD (h  ϕ) = (S(p2 )f 1 q21  S −1 (q1 )g 1  ϕ ◦ S −1  S(p1 )f 2 q22 )(g 2 S(h)  ε), together with the ones found in the proof of Proposition 4.10 define on D(H) a quasi-Hopf algebra structure. It is immediate that iD : H " h → h  ϕ ∈ D(H) is an injective quasi-Hopf algebra morphism, so H can be regarded as a quasi-Hopf subalgebra of D(H).  Remark 4.13. Continuing the ideas in Remark 4.11, the reconstructed antipode S for the quasi-bialgebra H ∗  H is given, for all ϕ ∈ H ∗ and h ∈ H, by S(ϕ  h) = (ε  S(h)f 1 )(p11 g 1 S(q 2 )  ϕ ◦ S −1  f 2 S −1 (p2 )  p12 g 2 S(q 1 )), while the distinguished elements α, β in D(H) are ε  α and ε  β. In this way the quasi-bialgebra isomorphism ϑ defined in Remark 4.11 becomes a quasi-Hopf algebra isomorphism. 4.3. The QT structure of D(H). If H is a quasi-Hopf algebra with bijective antipode the category H YDH can be identified with the right centre of the monoidal category H M, cf. [5,18]. Thus H YDH is a braided monoidal category, the braiding c being given on M, N ∈ H YDH by (4.50)

cM,N (m ⊗ n) = n(0) ⊗ n(1) · m,

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

87

for all m ∈ M and n ∈ N . Hence Proposition 3.3 implies that D(H) has a unique QT structure such that the monoidal category isomorphism H YDH ∼ = D(H) M becomes braided monoidal. We compute this QT structure of D(H) as follows. Theorem 4.14. Let H be a finite dimensional quasi-Hopf algebra. Then D(H), the quantum double of H, is a QT quasi-Hopf algebra with the R-matrix given by  (4.51) RD = (q22 ei S −1 (q1 )  ε) ⊗ (q21  ei ). i

This QT structure of D(H) turns the isomorphism sition 4.10 into a braided monoidal one.

H YD

H

∼ = D(H) M from Propo-

Proof. From the above comments we only have to transfer the braided structure of H YDH to the category D(H) M through the categorical isomorphism in Proposition 4.10, and then to apply Proposition 3.3 in order to get the explicit form of the R-matrix of D(H). By using these correspondences we get that the braided structure of D(H) M is given by the braiding c defined, for all M, N ∈ D(H) M, by cM,N (m ⊗ n)

= =

(4.50)

=

cM,N (m ⊗ n) n(0) ⊗ n(1) · m  (q21  ei )n ⊗ q22 ei S −1 (q1 ) · m i

=



(q21  ei )n ⊗ (q22 ei S −1 (q1 )  ε)m,

i

for all m ∈ M and n ∈ N . Now, by the proof of Proposition 3.3 we have that (RD )21 = cD(H),D(H) (1D ⊗ 1D ) and this leads to the formula of RD in (4.51).  Remark 4.15. Applying the same procedure to H ∗  H we find that  R := (ε  S −1 (p2 )ei p11 ) ⊗ (ei  p12 ) i

is the unique R-matrix on H ∗  H that turns the monoidal isomorphism H YDH ∼ = ∗ H ∗ H M into a braided monoidal one. Moreover, one can check that ϑ : (H  H, R) → (D(H), RD ) is a QT quasi-Hopf algebra isomorphism, i.e. (ϑ ⊗ ϑ)(R) = RD , where ϑ is as in Remark 4.11. 5. Application: biproduct quasi-Hopf algebras To a quasi-Hopf algebra H and a Hopf algebra B in the category of left YetterDrinfeld modules over H one can associate a new quasi-Hopf algebra B × H, called the biproduct of B and H. This construction was done in [7] and, similar to the Hopf case [25], characterizes the quasi-Hopf algebras with a projection. We should stress the fact that in [7] the proof of the fact that B × H is a quasi-Hopf algebra is based on technical and complicated computations. In what follows we present a less computational and more elegant way to introduced B × H; it has a categorical flavour, and so can be used in order to obtain similar results for various Hopf like algebras.

88

D. BULACU AND B. TORRECILLAS

5.1. Central Hopf algebras. In what follows C is a monoidal category and Wl (C) is the left weak centre of C as in [5] (see also [16, VIII.4] for the right version of the centre construction). For B an algebra in C we give sufficient conditions under which B C, the category of left B-modules in C, becomes a left rigid monoidal category. Definition 5.1. A central bialgebra (resp. Hopf algebra) in C is a bialgebra (resp. Hopf algebra) in the pre-braided monoidal category Wl (C). For the convenience of the reader, record that an object (B, cB,− : B ⊗ − → − ⊗ B) of Wl (C) is a central bialgebra in C if and only if • B has an algebra structure (B, mB , η B ) in C such that B BX



B BX X e X BX r r , ∀ X ∈ C, where cB,X = e ; (5.1) and = = e e e

X B X B X B X B X B • B has a coalgebra structure (B, ΔB , εB ) in C such that B X BX BX  e BX e e = r (5.2) and = , ∀ X ∈ C; r  e X X X B B X B B • the algebra and coalgebra structures on B are compatible in the following sense: 1 B B B B B B  r 1

B B

(5.3) , = r r , = r r and εB η B = Id1 , =  r

1 B B B B 1 B B B B B B where we denoted cB,B =

.

B B Furthermore, a central bialgebra (B, cB,− ) is a central Hopf algebra if there exists a morphism S : B → B in C such that B B BX BX   B e r Sh Sh . (5.4) , ∀ X ∈ C, and Sh = r = e = Sh



B X B X B B B A central bialgebra B turns the functor B ⊗ − : C → C into a bimonad on C, see [23] for the definition of a bimonad and [2, & 5.3] for a proof of this fact. On the other hand, for B an algebra in C we have a one to one correspondence between monoidal structures on B C making the forgetful functor B C → C strong monoidal and bimonad structures on B ⊗ −, cf. [23, Theorem 7.1]. Consequently, for any central bialgebra B the category B C is monoidal in such a way that the forgetful

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

functor result.

BC

89

→ C becomes strong monoidal. More exactly, we have the following

Proposition 5.2. Let C be a monoidal category and B a central bialgebra in C. Then B C, the category of left representations over B in C, is a monoidal category with structure given by: • for any X, Y ∈ B C the object X ⊗ Y of C is a left B-module as well, via the morphism B X Y  BX e (5.5) μX⊗Y = , with cB,X = e , P P P P X B X Y where, for simplicity, we assumed that C is strict. In this way the tensor product on C induces a tensor product on B C. • the unit object 1 of C is as well the unit object of B C, by considering it in B C via μ1 = εB ; • the associativity, left and right unit constraints coincide with those of C. With respect to this monoidal structure on B C the forgetful functor B C → C becomes strong monoidal. Proof. Although it follows from the above comments, for the sake of completeness we include a short direct proof. Note that by μX we denoted the left B-module structure morphism of X ∈ B C. By the unit conditions in (5.3) and (5.1) we can easily deduce that μX⊗Y (η B ⊗ IdX⊗Y ) = IdX⊗Y , for all X, Y ∈ B C. Moreover, since cB,− is a natural transformation and μX : B ⊗ X → X is a morphism in C we get that B BX P P e

B BX =

P P

e , where cB,X =

BX B B e and cB,B = , X B

B B

X B X B and this allows to compute B B X Y B B X Y B B X Y    B BX Y e e

 e P P (5.3) (5.1) P P P P = = = , e P P e e P P P P P P P P P P P P P P P P X Y X Y X Y X Y as required. Thus the tensor product on C induces a tensor product on B C. The last two conditions in (5.3) are equivalent to the fact that εB is an algebra morphism in C and this implies that 1 ∈ B C via εB . Furthermore, using the counit property in (5.2) and the fact that εB is a counit for ΔB , one can see easily that the left and right unit constraints of C produce left and right unit constraints for B C.

90

in

D. BULACU AND B. TORRECILLAS

For any X, Y, Z ∈ B C the morphism aX,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z) is B C, as the next computation shows: B X Y Z  e  

e

yright 2020. AMS. rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

P P

P P

B X Y Z   e e P P P P

=

P P

B X  e (5.2)

=



Y Z B Y



P P

P P

P P



, where cB,Y =

.



Y B P P

X Y Z X Y Z X Y Z Thus the associativity constraint of C induces an associativity constraint for B C. It is immediate that with this monoidal structure on B C one has that the forgetful  functor from B C to C is a strong monoidal functor, so we are done. BC

A situation when

is monoidal with left duality is the following.

Proposition 5.3. Let C be a monoidal category with left duality and (B, cB,− ) a central Hopf algebra in C. Then B C is with left duality and with respect to this left duality structure the forgetful functor B C → C becomes a left rigid strong monoidal functor. Proof. In view of the previous result, we only have to show that B C admits a left rigid structure in such a way that the forgetful functor maps left dual objects into left dual objects, of course provided that C is left rigid and that (B, cB,− ) is a central Hopf algebra in C. For this, consider X ∈ B C and take an adjunction (coevX , evX ) : X ∗ $ X. We show that X ∗ admits a left B-module structure in C with respect to which evX , coevX become morphisms in B C. Indeed, from the definition of an object in Wl (C) and since cB,− is a natural transformation, B X∗ B

B X∗ B 

B X∗ Sh

•

P P

B X∗

 =

•

•

, where cB,X ∗ =

P P

•

, and so μX ∗ :=

X∗ B

X B

X∗

X∗ B defines a left B-module structure on X ∗ in C since B B X∗ B B X∗ B B X∗ Sh Sh Sh Sh Sh Sh  • • • P P

•

=

 • P P

 P P



X

∗

• (2.3)

= 

B B X∗ Sh Sh



P P

=

(∗2 )

•

P P

X

=  P P

X

∗

EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 11/2/2023 9:50 AM via TEXAS TECH UNIVERSITY

Sh •

 . P P





∗

B B X∗





(∗1 )



P P





∗



X∗

X∗

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

91

Here (∗1 ) refers to the naturality of cB,− applied to the morphism mB cB,B : B ⊗ B → B, while (∗2 ) refers to the well known fact that the antipode of a braided Hopf algebra is an anti-algebra morphism. In addition, on can immediately check that μX ∗ (η B ⊗ IdX ∗ ) = IdX ∗ , and so indeed μX ∗ determines a left B-module structure on X ∗ in C. With the structure μX ∗ in mind, the morphism evX : X ∗ ⊗X → 1 in C becomes left B-linear: B X∗ X 

B X∗ X  Sh •

•

B X∗

• •

P P 

P P

1

(2.3)

=

(5.4)

Sh





X

•

P P

1

(5.2)

=

 Sh

B X∗ X = r .

(5.2) 1 (5.4)

P P

1

Similarly, coevX : 1 → X ⊗ X ∗ becomes a left B-linear morphism in C since the naturality of cB,− under the morphism coevX implies that B B B    B e  e  P P P Sh B P Sh  P  Sh • (5.4) (2.3) P  (5.4) r  • = = = = ,

P Sh P P P ∗ XX P P P P



X X∗

∗ X X X X∗ X X∗ as needed. This finishes our proof.



5.2. Biproduct quasi-Hopf algebras. We specialize the results in the previous Subsection to the case when H is a quasi-bialgebra or quasi-Hopf algebra, H cop C = H M and B is a Hopf algebra within Wl (H M) ≡ H , where H YD := H cop YD H cop is H with the opposite comultiplication. Consequently, H YD has a preH braided monoidal that makes the forgetful functor H YD → M strong monoidal. H H H Note that if (M, λM ), (N, λN ) ∈ H H YD then (M ⊗ N, λM ⊗N ) ∈ H YD, where M ⊗ N is a left H-module via the comultiplication Δ of H and, for all m ∈ M and n ∈ N , λM ⊗N (m ⊗ n) = X 1 (x1 Y 1 · m)(−1) x2 (Y 2 · n)(−1) Y 3 ⊗ X 2 · (x1 Y 1 · m)(0) ⊗ X 3 x3 · (Y 2 · n)(0) .

92

D. BULACU AND B. TORRECILLAS

Here we used for the structure morphisms of an object V in H H YD a notation similar H to the one for an object of H YD ; namely, the left H-action on V was denoted by ·, while the left H-coaction on V was denoted by λV : V " v → v(−1) ⊗ v(0) ∈ H ⊗ V . Also, the unit object is k viewed trivially as an object of H H YD, and the associativity, left and right unit constraints coincide with those of H M. Moreover, the catH egory H H YD is pre-braided with the pre-braiding determined, for all M, N ∈ H YD, by

(5.6)

cM,N (m ⊗ n) = m(−1) · n ⊗ m(0) , ∀ m ∈ M, n ∈ N.

We know from Proposition 5.2 that B (H M) has a monoidal structure with respect to which the forgetful functor F : B (H M) → H M is strong monoidal. Since the forgetful functor from H M to k M is a quasi-monoidal functor, we obtain that the forgetful functor from B (H M) to k M is quasi-monoidal. But B (H M) identifies with the category of left representations over the smash product k-algebra B#H, cf. [9], and so by Proposition 2.4 we conclude that B#H admits a quasibialgebra structure with respect to which the forgetful functor from B#H M to k M is quasi-monoidal. We next compute this quasi-bialgebra structure on B#H. Recall that B#H, the smash product algebra associated to B and H, is a k-algebra with multiplication given by (b#h)(b #h ) = (x1 · b)(x2 h1 · b )#x3 h2 h , ∀ b, b ∈ B and h, h ∈ H, and unit 1B #1H . Proposition 5.4. Let H be a quasi-bialgebra and B a bialgebra within the category of left Yetter-Drinfeld modules over H, H H YD. Then B#H is a quasibialgebra via the structure defined by Δ(b#h) = y 1 X 1 · b1 #y 2 Y 1 (x1 X 2 · b2 )(−1) x2 X13 h1 (5.7)

⊗y13 Y 2 · (x1 X 2 · b2 )(0) #y23 Y 3 x3 X23 h2 , Φ = 1#X 1 ⊗ 1#X 2 ⊗ 1#X 3 ,

and ε(b#h) = εB (b)ε(h), for all b ∈ B and h ∈ H. Here · is the left action of H on B, Δ(b) = b1 ⊗b2 and εB are the comultiplication and the counit of B in H H YD, and X 1 ⊗X 2 ⊗X 3 = · · · is the reassociator Φ of H with inverse Φ−1 = x1 ⊗x2 ⊗x3 = · · · . Proof. For X ∈ B (H M) denote by · the action of H on X, and by  the action of B on X. By (5.5) we know that for any X, Y ∈ B (H M) their tensor product over k, X ⊗ Y , is as well an object of B (H M) via the left H-action given by h · (x ⊗ y) = h1 · x ⊗ h2 · y, and the left B-action defined by the following

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

93

composition: b ⊗ (x ⊗ y)

→

(b1 ⊗ b2 ) ⊗ (x ⊗ y)

aB,B,X⊗Y

−→

X 1 · b1 ⊗ (X 2 · b2 ⊗ (X13 · x ⊗ X23 · y))

IdB ⊗a−1 B,X,Y

−→

IdB ⊗(cB,X ⊗IdY )

−→

IdB ⊗aX,B,Y

−→

X 1 · b1 ⊗ ((x1 X 2 · b2 ⊗ x2 X13 · x) ⊗ x3 X23 · y) X 1 · b1 ⊗ (((x1 X 2 · b2 )(−1) x2 X13 · x ⊗ (x1 X 2 · b2 )(0) ) ⊗x2 X23 · y)  X 1 · b1 ⊗ Y 1 (x1 X 2 · b2 )(−1) x2 X13 · x ⊗(Y 2 · (x1 X 2 · b2 )(0) ⊗ Y 3 x3 X23 · y)



 1 1  y X · b1 ⊗ y 2 Y 1 (x1 X 2 · b2 )(−1) x2 X13 · x   ⊗ y13 Y 2 ·(x1 X 2 ·b2 )(0) ⊗ y23 Y 3 x3 X23 ·y

a−1 B,X,B⊗Y

−→

(y 1 X 1 ·b1 )  (y 2 Y 1 (x1 X 2 ·b2 )(−1) x2 X13 ·x)

→

⊗(y13 Y 2 ·(x1 X 2 ·b2 )(0) )  (y23 Y 3 x3 X23 ·y). In other words, we have that b  (x ⊗ y) = (y 1 X 1 · b1 )  (y 2 Y 1 (x1 X 2 · b2 )(−1) x2 X13 · x) ⊗ (y13 Y 2 · (x1 X 2 · b2 )(0) )  (y23 Y 3 x3 X23 · y), for all b ∈ B, x ∈ X and y ∈ Y . Thus, by using the natural identification B (H M) ≡ we obtain that the tensor product on B#H M is given by

B#H M

(b#h) · (x ⊗ y)

= b  (h · (x ⊗ y)) = b  (h1 · x ⊗ h2 · y) = (y 1 X 1 · b1 )  (y 2 Y 1 (x1 X 2 · b2 )(−1) x2 X13 h1 · x) ⊗(y13 Y 2 · (x1 X 2 · b2 )(0) )  (y23 Y 3 x3 X23 h2 · y) = (y 1 X 1 · b1 #y 2 Y 1 (x1 X 2 · b2 )(−1) x2 X13 h1 ) · x ⊗(y13 Y 2 · (x1 X 2 · b2 )(0) #y23 Y 3 x3 X23 h2 ) · y,

for all b ∈ B, h ∈ H, x ∈ X ∈ B#H M and y ∈ Y ∈ B#H M. Consequently, by Proposition 2.4 we deduce that B#H is a quasi-bialgebra via the comultiplication defined by Δ(b#h)

=

(b#h) · (1B #1H ⊗ 1B #1H )

= y 1 X 1 · b1 #y 2 Y 1 (x1 X 2 · b2 )(−1) x2 X13 h1 ⊗ y13 Y 2 · (x1 X 2 · b2 )(0) #y23 Y 3 x3 X23 h2 , for all b ∈ B and h ∈ H, as stated. Furthermore, by the proof of Proposition 5.2 we know that k is the unit object of B#H M via the structure (b#h) · κ = b  (h · κ) = ε(h)b  κ = εB (b)ε(h)κ, for all κ ∈ k, b ∈ B and h ∈ H, and therefore by the proof of Proposition 2.4 we deduce that the counit of B#H is given by ε(b#h) = (b#h) · 1k = εB (b)ε(h), for all b ∈ B and h ∈ H, as needed. Finally, again by the proof of Proposition 2.4, the reassociator Φ of B#H is reconstructed as Φ = aB#H,B#H,B#H (1B #1H ⊗ 1B #1H ⊗ 1B #1H ),

94

D. BULACU AND B. TORRECILLAS

and since the associativity constraint of conclude that Φ

B#H M

is the same as that of

HM

we

= (1B #X 1 )(1B #1H ) ⊗ (1B #X 2 )(1B #1H ) ⊗ (1B #X 3 )(1B #1H ) = 1B #X 1 ⊗ 1B #X 2 ⊗ 1B #X 3 ,

where we identified H as a subalgebra of B#H via the natural inclusion i(h) =  1B #h, for all h ∈ H. So the proof is finished. In what follows, we denote by B × H the vector space B ⊗ H (elements b ⊗ h will be written b × h) equipped with the algebra structure given by the smash product and the quasi-bialgebra structure from Proposition 5.4. We will call it the biproduct of B and H. We next show that B × H is a quasi-Hopf algebra, provided that H is a quasiHopf algebra and B is a braided Hopf algebra in H H YD. First, we will do this in the finite dimensional case, and this is because H Mfd is a category with left duality and so Proposition 5.3 applies. Secondly, we will note that the obtained quasi-Hopf algebra structure on B × H is just the one considered in [7], so it can be also considered in the infinite dimensional case. Proposition 5.5. Let H be a finite dimensional quasi-Hopf algebra and B a finite dimensional Hopf algebra within H H YD. Then the biproduct quasi-bialgebra B × H is, moreover, a quasi-Hopf algebra with antipode given by (5.8)

s(b × h) = (1B × S(b(−1) h))(U 1 · S B (b(0) ) × U 2 ),

for all b ∈ B, h ∈ H, and distinguished elements αB×H = 1 × α and βB×H = 1 × β. Here U = U 1 ⊗ U 2 := g 1 S(q 2 ) ⊗ g 2 S(q 1 ) ∈ H ⊗ H. Proof. The category C = H Mfd is monoidal with left duality and B is a fd braided Hopf algebra in Wl (H Mfd ) = H H YD . Hence, by Proposition 5.3 we get fd fd that B (H M ) ≡ B×H M is a monoidal category with left duality, too. Furthermore, the forgetful functor from B×H Mfd to H Mfd is a left rigid strong monoidal functor, thus so is the forgetful functor from B×H Mfd to k Mfd . The latter corroborated with Theorem 3.2 guarantees a quasi-Hopf algebra structure on B × H. We next see that it is exactly the one in the statement. fd Denote by S B the antipode of the Hopf algebra B in H H YD . As a morphism fd in H H YD , S B is left H-colinear, that is, (5.9)

S B (b)(−1) ⊗ S B (b)(0) = b(−1) ⊗ S B (b(0) ),

for all b ∈ B. This allows to compute the structure of X ∗ in B×H Mfd , for all X ∈ fd ∗ B×H M , as follows. We know that X is a left H-module via the action given by ∗ ∗ (h · x )(x) = x (S(h) · x), for all h ∈ H, x∗ ∈ X ∗ and x ∈ X. In addition, keeping in mind the definitions of the evaluation and coevaluation morphisms of X ∗ in H Mfd , by the proof of Proposition 5.3 we get that, moreover, X ∗ ∈ B (H Mfd ) ≡ B×H Mfd

RECONSTRUCTION FOR QUASI-QUANTUM GROUPS

95

via the structure defined by the following composition: b ⊗ x∗ → (S B (b)(−1) · x∗ ⊗ S B (b)(0) ) ⊗ (β · xl ⊗ xl ) = (b(−1) · x∗ ⊗ S B (b)(0) ) ⊗ (β · xl ⊗ xl ) aX ∗ ,B,X⊗X ∗ −→ X 1 b(−1) · x∗ ⊗ (X 2 · S B (b(0) ) ⊗ (X13 β · xl ⊗ X23 · xl ))

IdX ∗ ⊗a−1 B,X,X ∗

−→ →

X 1 b(−1) · x∗ ⊗ ((x1 X 2 · S B (b(0) ) ⊗ x2 X13 β · xl ) ⊗ x3 X23 · xl ) X 1 b(−1) · x∗ ⊗ ((x1 X 2 · S B (b(0) ))  (x2 X13 β · xl ) ⊗ x3 X23 · xl )

−→

(y 1 X 1 b(−1) ·x∗ ⊗y 2 ·((x1 X 2 ·S B (b(0) ))  (x2 X13 β ·xl )))⊗y 3 x3 X23 ·xl

−→ = =

x∗ , S(y 1 X 1 b(−1) )αy 2 · ((x1 X 2 · S B (b(0) ))  (x2 X13 β · xl ))y 3 x3 X23 · xl x∗ , S(y 1 X 1 b(−1) )αY 2 · (x1 X 2 · S B (b(0) ) × x2 X13 βS(y 3 x3 X23 ) · xl xl x∗ , (1B × S(y 1 b(−1) )αy 2 )(x1 · S B (b(0) ) × x2 βS(y 3 x3 ) · xl xl ,

a−1 X ∗ ,X,X ∗ evX ⊗IdX ∗

where {xl , xl }l are dual bases in X and X ∗ . Otherwise stated, we have (b  x∗ )(x) = x∗ , (1B × S(y 1 b(−1) )αy 2 )(x1 · S B (b(0) ) × x2 βS(y 3 x3 )) · x, for all b ∈ B, x∗ ∈ X ∗ and x ∈ X. In conclusion, X ∗ is a left B × H-module via the action given by (b × h) · x∗ , x = x∗ , (1B × S(y 1 b(−1) h)αy 2 )(x1 · S B (b(0) ) × x2 βS(y 3 x3 )) · x, for all b ∈ B, h ∈ H, x∗ ∈ X ∗ and x ∈ X. Once more, we have considered H as a subalgebra of B × H via the natural inclusion H " h → 1B × h ∈ B × H. Let now {bi , bi }i and {ej , ej }j be dual bases in B and H, respectively, so that {bi ⊗ ej }i,j is a basis in B × H. If we denote by E i,j the dual basis of {bi ⊗ ej }i,j in (B × H)∗ then by the proof of Theorem 3.2 we conclude that B × H is a quasi-Hopf algebra with antipode given, for all b ∈ B and h ∈ H, by s(b × h)

=

(b × h) · E i,j , 1B × 1H bi × ej

=

E i,j , (1B × S(y 1 b(−1) h)αy 2 )(x1 · S B (b(0) ) × x2 βS(y 3 x3 ))bi ⊗ ej

=

(1B × S(y 1 b(−1) h)αy 2 )(x1 · S B (b(0) ) × x2 βS(y 3 x3 ))

=

(1B × S(b(−1) h)q1 )(p1 · S B (b(0) ) × p2 S(q2 ))

=

(1B × S(b(−1) h))(q11 p1 · S B (b(0) ) × q12 p2 S(q2 ))

(∗)

(1B × S(b(−1) h))(U 1 · S B (b(0) ) × U 2 ),

=

and distinguished elements αB×H = E i,j , α · (1B × 1H )bi ⊗ ej = E i,j (1B × α)bi ⊗ ej = 1B × α, βB×H = E i,j (1B × 1H )β · (bi ⊗ ej ) = β · (1B × 1H ) = 1B × β. Note that, in the last two computations we used the fact that the evaluation and coevaluation morphisms are the same in H Mfd and B×H Mfd (see the proof of Proposition 5.3), and (*) refers to the equality q11 p1 ⊗ q12 p2 S(q2 ) = U 1 ⊗ U 2 , which is just the first relation of (4.11) in [4].  Summing up, the finite dimensional case allows to find in a natural way the biproduct quasi-Hopf algebra structure on B ⊗ H. As long as it is independent of dual bases, is just a matter of computation to show that this structure remains valid in general; this was already done in [7].

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Acknowledgments. The first author thanks the University of Almeria (Spain) for its warm hospitality. The authors also thank Bodo Pareigis for sharing his ”diagrams” program. References [1] Alessandro Ardizzoni, Daniel Bulacu, and Claudia Menini, Quasi-bialgebra structures and torsion-free abelian groups, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104) (2013), no. 3, 247–265. MR3114473 [2] Alain Brugui`eres, Steve Lack, and Alexis Virelizier, Hopf monads on monoidal categories, Adv. Math. 227 (2011), no. 2, 745–800, DOI 10.1016/j.aim.2011.02.008. MR2793022 [3] D. Bulacu and S. Caenepeel, The quantum double for quasitriangular quasi-Hopf algebras, Comm. Algebra 31 (2003), no. 3, 1403–1425, DOI 10.1081/AGB-120017773. MR1971069 [4] Daniel Bulacu and Stefaan Caenepeel, On integrals and cointegrals for quasi-Hopf algebras, J. Algebra 351 (2012), 390–425, DOI 10.1016/j.jalgebra.2011.11.006. MR2862216 [5] D. Bulacu, S. Caenepeel, and F. Panaite, Yetter-Drinfeld categories for quasi-Hopf algebras, Comm. Algebra 34 (2006), no. 1, 1–35, DOI 10.1080/00927870500345869. MR2194347 [6] D. Bulacu, S. Caenepeel, and B. Torrecillas, Doi-Hopf modules and Yetter-Drinfeld modules for quasi-Hopf algebras, Comm. Algebra 34 (2006), no. 9, 3413–3449, DOI 10.1080/00927870600794123. MR2252681 [7] Daniel Bulacu and Erna Nauwelaerts, Radford’s biproduct for quasi-Hopf algebras and bosonization, J. Pure Appl. Algebra 174 (2002), no. 1, 1–42, DOI 10.1016/S00224049(02)00014-2. MR1924081 [8] Daniel Bulacu and Erna Nauwelaerts, Quasitriangular and ribbon quasi-Hopf algebras, Comm. Algebra 31 (2003), no. 2, 657–672, DOI 10.1081/AGB-120017337. MR1968919 [9] Daniel Bulacu, Florin Panaite, and Freddy Van Oystaeyen, Quasi-Hopf algebra actions and smash products, Comm. Algebra 28 (2000), no. 2, 631–651, DOI 10.1080/00927870008826849. MR1736752 [10] Daniel Bulacu, Florin Panaite, and Freddy Van Oystaeyen, Generalized diagonal crossed products and smash products for quasi-Hopf algebras. Applications, Comm. Math. Phys. 266 (2006), no. 2, 355–399, DOI 10.1007/s00220-006-0051-z. MR2238882 [11] Damien Calaque and Pavel Etingof, Lectures on tensor categories, Quantum groups, IRMA Lect. Math. Theor. Phys., vol. 12, Eur. Math. Soc., Z¨ urich, 2008, pp. 1–38, DOI 10.4171/0471/1. MR2432988 [12] V. G. Drinfeld, Quasi-Hopf algebras (Russian), Algebra i Analiz 1 (1989), no. 6, 114–148; English transl., Leningrad Math. J. 1 (1990), no. 6, 1419–1457. MR1047964 [13] Frank Hausser and Florian Nill, Diagonal crossed products by duals of quasi-quantum groups, Rev. Math. Phys. 11 (1999), no. 5, 553–629, DOI 10.1142/S0129055X99000210. MR1696105 [14] Frank Hausser and Florian Nill, Doubles of quasi-quantum groups, Comm. Math. Phys. 199 (1999), no. 3, 547–589, DOI 10.1007/s002200050512. MR1669685 [15] Andr´ e Joyal and Ross Street, An introduction to Tannaka duality and quantum groups, Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 413–492, DOI 10.1007/BFb0084235. MR1173027 [16] Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR1321145 [17] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. III, J. Amer. Math. Soc. 7 (1994), no. 2, 335–381, DOI 10.2307/2152762. MR1239506 [18] S. Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), no. 1, 1–9, DOI 10.1023/A:1007450123281. MR1631648 [19] Shahn Majid, Reconstruction theorems and rational conformal field theories, Internat. J. Modern Phys. A 6 (1991), no. 24, 4359–4374, DOI 10.1142/S0217751X91002100. MR1126612 [20] Shahn Majid, Quasi-quantum groups as internal symmetries of topological quantum field theories, Lett. Math. Phys. 22 (1991), no. 2, 83–90, DOI 10.1007/BF00405171. MR1122044 [21] Shahn Majid, Tannaka-Kre˘ın theorem for quasi-Hopf algebras and other results, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), Contemp. Math., vol. 134, Amer. Math. Soc., Providence, RI, 1992, pp. 219–232, DOI 10.1090/conm/134/1187289. MR1187289

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[22] Shahn Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995. MR1381692 [23] I. Moerdijk, Monads on tensor categories, J. Pure Appl. Algebra 168 (2002), no. 2-3, 189–208, DOI 10.1016/S0022-4049(01)00096-2. Category theory 1999 (Coimbra). MR1887157 [24] Reinhard H¨ aring-Oldenburg, Reconstruction of weak quasi Hopf algebras, J. Algebra 194 (1997), no. 1, 14–35, DOI 10.1006/jabr.1996.7006. MR1461480 [25] David E. Radford, The structure of Hopf algebras with a projection, J. Algebra 92 (1985), no. 2, 322–347, DOI 10.1016/0021-8693(85)90124-3. MR778452 [26] Peter Schauenburg, Hopf modules and the double of a quasi-Hopf algebra, Trans. Amer. Math. Soc. 354 (2002), no. 8, 3349–3378, DOI 10.1090/S0002-9947-02-02980-X. MR1897403 [27] K.-H. Ulbrich, On Hopf algebras and rigid monoidal categories, Israel J. Math. 72 (1990), no. 1-2, 252–256, DOI 10.1007/BF02764622. Hopf algebras. MR1098991 [28] J. Vercruysse, Hopf algebras–Variant notions and reconstruction theorems, in Compositional methods in quantum physics and linguistics; Oxford University Press, 2012, pp. 115–145. Faculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania Email address: [email protected] Department of Algebra and Analysis, Universidad de Almer´ıa, E-04071 Almer´ıa, Spain Email address: [email protected]

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15112

Homotopy categories of totally acyclic complexes with applications to the flat–cotorsion theory Lars Winther Christensen, Sergio Estrada, and Peder Thompson To S.K. Jain on the occasion of his eightieth birthday Abstract. We introduce a notion of total acyclicity associated to a subcategory of an abelian category and consider the Gorenstein objects they define. These Gorenstein objects form a Frobenius category, whose induced stable category is equivalent to the homotopy category of totally acyclic complexes. Applied to the flat–cotorsion theory over a coherent ring, this provides a new description of the category of cotorsion Gorenstein flat modules; one that puts it on equal footing with the category of Gorenstein projective modules.

Introduction Let A be an associative ring. It is classic that the stable category of Gorenstein projective A-modules is triangulated equivalent to the homotopy category of totally acyclic complexes of projective A-modules. Under extra assumptions on A this equivalence can be found already in Buchweitz’s 1986 manuscript [6]. In this paper we focus on a corresponding equivalence for Gorenstein flat modules. It could be pieced together from results in the literature, but we develop a framework that provides a direct proof while also exposing how closely the homotopical behavior of cotorsion Gorenstein flat modules parallels that of Gorenstein projective modules. The category of Gorenstein flat A-modules is rarely Frobenius, indeed we prove in Theorem 4.5 that it only happens when every module is cotorsion. This is evidence that one should restrict attention to the category of cotorsion Gorenstein flat A-modules; in fact, it is already known from work of Gillespie [15] that this category is Frobenius if A is coherent. The associated stable category is equivalent to the homotopy category of F-totally acyclic complexes of flat-cotorsion A-modules; this follows from a theorem by Estrada and Gillespie [12] combined with recent work of Bazzoni, Cort´es Izurdiaga, and Estrada [3]. The proof in [12] involves model structures on categories of complexes of projective modules, and one goal of 2010 Mathematics Subject Classification. Primary 16E05; Secondary 18G25, 18G35. Key words and phrases. Cotorsion pair; Gorenstein object; stable category; totally acyclic complex. The first author was partly supported by Simons Foundation collaboration grant 428308. The second author was partly supported by grant MTM2016-77445-P and FEDER funds and by grant 19880/GERM/15 from the Fundaci´ on S´ eneca-Agencia de Ciencia y Tecnolog´ıa de la Regi´ on de Murcia. c 2020 American Mathematical Society

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this paper—with a view towards extending the result to non-affine schemes [7]—is to give a proof that avoids projective modules; we achieve this with Corollary 5.3. The pure derived category of flat A-modules is the Verdier quotient of the homotopy category of complexes of flat A-modules by the subcategory of pure-acyclic complexes; its subcategory of F-totally acyclic complexes was studied by Murfet and Salarian [21]. We show in Theorem 5.6 that this subcategory is equivalent to the homotopy category of F-totally acyclic complexes of flat-cotorsion A-modules, and thus to the stable category of cotorsion Gorenstein flat modules. Combining this with results of Christensen and Kato [8] and Estrada and Gillespie [12], one can derive that under extra assumptions on A, made explicit in Corollary 5.9, the stable category of Gorenstein projective A-modules is equivalent to the stable category of cotorsion Gorenstein flat A-modules. Underpinning the results we have highlighted above are a framework, developed in Sections 1–3, and two results, Theorems 4.4 and 5.2, that show—as the semantics might suggest—that the cotorsion Gorenstein flat modules are, indeed, the Gorenstein modules naturally attached to the flat–cotorsion theory. ∗ ∗ ∗ Let A be an abelian category and U a subcategory of A. In 1.1 we define a right Utotally acyclic complex to be an acyclic homA (−, U ∩ U⊥ )-exact complex of objects from U with cycle objects in U⊥ . Left U-total acyclicity is defined dually, and in the case of a self-orthogonal subcategory, left and right total acyclicity is the same; see Proposition 1.5. These definitions recover the standard notions of totally acyclic complexes of projective or injective objects; see Example 1.7. In the context of a cotorsion pair (U, V) the natural complexes to consider are right U-totally acyclic complexes, left V-totally acyclic complexes, and (U ∩ V)-totally acyclic complexes for the self-orthogonal category U ∩ V. In Section 2 we define left and right U-Gorenstein objects to be cycles in left and right U-totally acyclic complexes. In the context of a cotorsion pair (U, V), we show that the categories of right U-Gorenstein objects and left V-Gorenstein objects are Frobenius categories whose projective-injective objects are those in U∩V; see Theorems 2.11 and 2.12. In Section 3 the stable categories induced by these Frobenius categories are shown to be equivalent to the corresponding homotopy categories of totally acyclic complexes. In particular, Corollary 3.9 recovers the classic results for Gorenstein projective objects and Gorenstein injective objects. The literature contains a variety of generalized notions of totally acyclic complexes and Gorenstein objects; see for example Sather-Wagstaff, Sharif, and White [23]. We make detailed comparisons in Remark 2.3; at this point it suffices to say that our notion of Gorensteinness differs from the existing generalizations by exhibiting periodicity: For a self-orthogonal category W, the category of (WGorenstein)-Gorenstein objects is simply W; see Proposition 2.8.

1. Total acyclicity and other terminology Throughout this paper, A denotes an abelian category; we write homA for the homsets and the induced functor from A to abelian groups. Tacitly, subcategories of A are assumed to be full and closed under isomorphisms. A subcategory of A is called additively closed if it is additive and closed under direct summands.

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A complex of objects from A is referred to as an A-complex. We use homological notation for complexes, so for a complex T the object in degree i is denoted Ti and Zi (T ) denotes the cycle subobject in degree i. Let U and V be subcategories of A. The right orthogonal of U is the subcategory U⊥ = {N ∈ A | Ext1A (U, N ) = 0 for all U ∈ U} ; the left orthogonal of V is the subcategory ⊥

V = {M ∈ A | Ext1A (M, V ) = 0 for all V ∈ V} .

In case U⊥ = V and ⊥ V = U hold, the pair (U, V) is referred to as a cotorsion pair. In this section and the next, we develop notions of total acyclicity, and corresponding notions of Gorenstein objects, associated to any subcategory of A. Our primary applications are in the context of a cotorsion pair. 1.1 Definition. Let U and V be subcategories of A. (r) An A-complex T is called right U-totally acyclic if the following hold: (1) T is acyclic. (2) For each i ∈ Z the object Ti belongs to U. (3) For each i ∈ Z the object Zi (T ) belongs to U⊥ . (4) For each W ∈ U ∩ U⊥ the complex homA (T, W ) is acyclic. (l) An A-complex T is called left V-totally acyclic if the following hold: (1) T is acyclic. (2) For each i ∈ Z the object Ti belongs to V. (3) For each i ∈ Z the object Zi (T ) belongs to ⊥ V. (4) For each W ∈ ⊥ V ∩ V the complex homA (W, T ) is acyclic. 1.2 Example. Let U and V be subcategories of A. For every W ∈ U ∩ U⊥ a = complex of the form 0 −→ W −−→ W −→ 0 is right U-totally acyclic; similarly, for ⊥ every W ∈ V ∩ V such a complex is left V-totally acyclic. 1.3 Proposition. Let U and V be subcategories of A. (r) An A-complex T is right U-totally acyclic if and only if the following hold: (1) T is acyclic. (2) For each i ∈ Z the object Ti belongs to U ∩ U⊥ . (3) For each U ∈ U the complex homA (U, T ) is acyclic. (4) For each W ∈ U ∩ U⊥ the complex homA (T, W ) is acyclic. (l) An A-complex T is left V-totally acyclic if and only if the following hold: (1) T is acyclic. (2) For each i ∈ Z the object Ti belongs to ⊥ V ∩ V. (3) For each V ∈ V the complex homA (T, V ) is acyclic. (4) For each W ∈ ⊥ V ∩ V the complex homA (W, T ) is acyclic. Proof. (r): A complex T that satisfies Definition 1.1(r) trivially satisfies conditions (1), (3), and (4), while (2) follows from 1.1(r.2) and 1.1(r.3) as U⊥ is closed under extensions. Conversely, a complex T that satisfies conditions (1)–(4) in the statement trivially satisfies conditions (1), (2), and (4) in Definition 1.1(r). Moreover it follows from (2) and (3) that also condition 1.1(r.3) is satisfied. The proof of (l) is similar. 

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1.4 Example. A right A-totally acyclic complex is a contractible complex of injective objects, and a left A-totally acyclic complex is a contractible complex of projective objects. In this paper we call a subcategory W of A self-orthogonal if Ext1A (W, W  ) = 0 holds for all W and W  in W. 1.5 Proposition. Let W be a subcategory of A. The following conditions are equivalent (i ) W is self-orthogonal. (ii ) Every object in W belongs to W⊥ . (iii ) Every object in W belongs to ⊥ W. (iv ) One has W ∩ W⊥ = W = ⊥ W ∩ W. Moreover, if W satisfies these conditions, then an A-complex is right W-totally acyclic if and only if it is left W-totally acyclic. Proof. Evidently, (i ) implies (iv ), and (iv ) implies both (ii ) and (iii ). Conditions (ii ) and (iii ) each precisely mean that Ext1A (W, W  ) = 0 holds for all W and W  in W, so either implies (i ). Now assume that W satisfies (i )–(iv ). Parts (1) are the same in Proposition 1.3(r) and 1.3(l), and so are parts (2) per the assumption W ∩ W⊥ = ⊥ W ∩ W. Part (3) in 1.3(r) coincides with part (4) in 1.3(l) by the assumption W = ⊥ W ∩W,  and 1.3(r.4) coincides with 1.3(l.3) per the assumption W ∩ W⊥ = W. 1.6 Definition. For a self-orthogonal subcategory W of A, a right, equivalently left, W-totally acyclic complex is simply called a W-totally acyclic complex. 1.7 Example. The subcategory Prj(A) of projective objects in A is self-orthogonal, and a Prj(A)-totally acyclic complex is called a totally acyclic complex of projective objects. In the special case where A is the category Mod(A) of modules over a ring A these were introduced by Auslander and Bridger [1]; see also Enochs and Jenda [10]. The terminology is due to Avramov and Martsinkovsky [2]. Dually, Inj(A) is the subcategory of injective objects in A, and an Inj(A)-totally acyclic complex is called a totally acyclic complex of injective objects; see Krause [19]. The case A = Mod(A) was first considered in [10]. 1.8 Remark. For a cotorsion pair (U, V) in A, the subcategory U ∩ V is selforthogonal. It follows from Proposition 1.3 that every right U-totally acyclic complex and every left V-totally acyclic complex is (U ∩ V)-totally acyclic. 2. Gorenstein objects In line with standard terminology, cycles in totally acyclic complexes are called Gorenstein objects. 2.1 Definition. Let U and V be subcategories of A. (r) An object M in A is called right U-Gorenstein if there is a right Utotally acyclic complex T with Z0 (T ) = M . Denote by RGorU (A) the full subcategory of right U-Gorenstein objects in A. (l) An object M in A is called left V-Gorenstein if there is a left V-totally acyclic complex T with Z0 (T ) = M . Denote by LGorV (A) the full subcategory of left V-Gorenstein objects in A.

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For a self-orthogonal subcategory W one has RGorW (A) = LGorW (A), see Proposition 1.5; this category is denoted GorW (A), and its objects are called W-Gorenstein. Notice that if U is an additive subcategory, then so is RGorU (A); similarly for V and LGorV (A). 2.2 Example. Let U and V be subcategories of A. Objects in U ∩ U⊥ are right U-Gorenstein and objects in ⊥ V ∩ V are left V-Gorenstein; see Example 1.2. 2.3 Remark. We compare our definitions of total acyclicity and Gorenstein objects with others that already appear in the literature. (1) For an additive category W, Iyengar and Krause [17] define a “totally acyclic complex over W.” For additive subcategories U and V of an abelian category, a right U-totally acyclic complex is totally acyclic over U ∩ U⊥ in the sense of [17, def. 5.2], and a left V-totally acyclic complex is totally acyclic over ⊥ V ∩ V. In particular, for a self-orthogonal additive subcategory W of an abelian category, a W-totally acyclic complex is the same as an acyclic complex that is totally acyclic over W in the sense of [17, def. 5.2]. (2) For an additive subcategory W of an abelian category, Sather-Wagstaff, Sharif, and White [23] define a “totally W-acyclic” complex. A right or left W-totally acyclic complex is totally W-acyclic in the sense of [23, def. 4.1]; the converse holds if W is self-orthogonal. For a self-orthogonal additively closed subcategory W of a module category, Geng and Ding [13] study the associated Gorenstein objects. (3) For subcategories U and V of Mod(A) with Prj(A) ⊆ U and Inj(A) ⊆ V, Pan and Cai [22] define “(U, V)-Gorenstein projective/injective” modules. In this setting, a right U-Gorenstein module is (U, U ∩ U⊥ )-Gorenstein projective in the sense of [22, def. 2.1], and a left V-Gorenstein module is (⊥ V ∩ V, V)-Gorenstein injective in the sense of [22, def. 2.2]. (4) For a complete hereditary cotorsion pair (U, V) in an abelian category, Yang and Chen [26] define a “complete U-resolution.” Every right Utotally acyclic complex is a complete U-resolution in the sense of [26, def. 3.1]. (5) For a pair of subcategories (U, V) in an abelian category, Becerril, Mendoza, and Santiago [4] define a “left complete (U, V)-resolution.” If (U, V) is a cotorsion pair, then a right U-totally acyclic complex is a left complete (U, U ∩ V)-resolution in the sense of [4, def. 3.2]. The key difference between Definition 1.1 and those cited above is that 1.1— motivated by 4.1—places restrictions on the cycle objects in a totally acyclic complex; the significance of this becomes apparent in Proposition 2.8. 2.4 Remark. Given a cotorsion pair (U, V) in A, it follows from Remark 1.8 that there are containments RGorU (A) ⊆ GorU∩V (A) ⊇ LGorV (A) . 2.5 Example. A right A-Gorenstein object is injective, and a left A-Gorenstein object is projective; see Example 1.4. The subcategory Prj(A) is self-orthogonal, and a Prj(A)-Gorenstein object is called Gorenstein projective; see [1, 10] for the special case A = Mod(A). Similarly,

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an Inj(A)-Gorenstein object is called Gorenstein injective; see [19] and see [10] for the case A = Mod(A). The next three results, especially Proposition 2.8, are motivated in part by [23, Theorem A]. We consider what happens when one iterates the process of constructing Gorenstein objects. Starting from a self-orthogonal additively closed subcategory, our construction iterated twice returns the original subcategory. The construction in [23] is, in contrast, “idempotent.” 2.6 Lemma. Let U and V be additively closed subcategories of A. One has ⊥

RGorU (A) ∩ RGorU (A) = U ∩ U⊥ = RGorU (A) ∩ RGorU (A)⊥

⊥

LGorV (A) ∩ LGorV (A) =

⊥

and

⊥

V ∩ V = LGorV (A) ∩ LGorV (A) .

In particular, RGorU (A) is self-orthogonal if and only if RGorU (A) = U ∩ U⊥ holds, and LGorV (A) is self-orthogonal if and only if LGorV (A) = ⊥ V ∩ V holds. For a self-orthogonal category W one has (2.6.1)

⊥

GorW (A) ∩ GorW (A) = W = GorW (A) ∩ GorW (A)⊥ .

Proof. Set W = U ∩ U⊥ and notice that W is self-orthogonal and additively closed. By Example 2.2 objects in W are right U-Gorenstein, and by Proposition 1.3 the subcategory W is contained in both ⊥ RGorU (A) and RGorU (A)⊥ . Let G be a right U-Gorenstein object. By Proposition 1.3 there are exact sequences η  = 0 → G → T  → G → 0

and

η  = 0 → G → T  → G → 0

where G and G are right U-Gorenstein, while T  and T  belong to W. If G belongs to ⊥ RGorU (A), then η  splits, so G is a summand of T  and hence in W. Similarly, if G is in RGorU (A)⊥ , then η  splits, and it follows that G is in W. This proves the first set of equalities, and the ones pertaining to LGorV (A) are proved similarly. The remaining assertions are immediate in view of Proposition 1.5.  2.7 Remark. Let U be an additively closed subcategory of A. It follows from Example 2.2 and Lemma 2.6 that objects in U ∩ U⊥ are both right U-Gorenstein and right RGorU (A)-Gorenstein. On the other hand, a right RGorU (A)-Gorenstein object belongs by Definition 1.1(r.3) to RGorU (A)⊥ , so any object that is both right U- and right RGorU (A)-Gorenstein belongs to U ∩ U⊥ . In symbols, RGorU (A) ∩ RGorRGorU (A) (A) = U ∩ U⊥ . For an additively closed subcategory V, a similar argument yields LGorV (A) ∩ LGorLGorV (A) (A) =

⊥

V∩V.

2.8 Proposition. Let W be a self-orthogonal additively closed subcategory of A. A right or left GorW (A)-totally acyclic complex is a contractible complex of objects from W. In particular, one has (2.8.1)

LGorGorW (A) (A) = W = RGorGorW (A) (A) .

Moreover, the following hold • If (GorW (A), GorW (A)⊥ ) is a cotorsion pair, then one has (2.8.2)

LGorGorW (A)⊥ (A) = GorW (A) .

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• If (⊥ GorW (A), GorW (A)) is a cotorsion pair, then one has (2.8.3)

RGor⊥ GorW (A) (A) = GorW (A) .

Proof. A right GorW (A)-totally acyclic complex T is by Proposition 1.3 and (2.6.1) an acyclic complex of objects from W, and by Definition 1.1 the cycles Zi (T ) belong to GorW (A)⊥ . As W is contained in GorW (A), it follows from Proposition 1.3(r.3) that the cycles Zi (T ) are contained in W⊥ . It now follows from Definition 1.1 that T is W-totally acyclic, whence the cycles Zi (T ) belong to GorW (A) and hence to W, see (2.6.1). Thus T is an acyclic complex of objects from W with cycles in W ⊂ W⊥ and, therefore, contractible. A parallel argument shows that a left GorW (A)-totally acyclic complex is contractible. Assume that (GorW (A), GorW (A)⊥ ) is a cotorsion pair; by (2.6.1) and Remark 2.4 one has LGorGorW (A)⊥ (A) ⊆ GorW (A). To prove the opposite containment, let T be a W-totally acyclic complex. By Definition 1.1 it is an acyclic complex of objects from W ⊆ GorW (A)⊥ , see (2.6.1), and homA (W, T ) is acyclic for every object W in GorW (A) ∩ GorW (A)⊥ . Moreover, the cycle objects Zi (T ) belong to GorW (A) by Definition 2.1, so T is per Definition 1.1 a left GorW (A)⊥ -totally acyclic complex. A parallel argument proves the last assertion.  ˇ ˇ ıˇcek [24, thm. 4.6] show that the 2.9 Example. Let A be a ring. Saroch and Stov´ subcategory GorInj (A) of Gorenstein injective A-modules is the right half of a cotorsion pair, so by Proposition 2.8 one has LGorGorInj (A) (A) = Inj(A) = RGorGorInj (A) (A) and RGor⊥ GorInj (A) (A) = GorInj (A). 2.10 Lemma. Let U and V be additive subcategories of A. (r) The subcategory RGorU (A) is closed under extensions. (l) The subcategory LGorV (A) is closed under extensions. Proof. Let 0 → M  → M → M  → 0 be an exact sequence where M  and M  are right U-Gorenstein objects. Let T  and T  be right U-totally acyclic complexes with Z0 (T  ) = M  and Z0 (T  ) = M  . Per Remark 2.3(1) it follows from [23, prop. 4.4] that there exists an A-complex T that satisfies conditions (1), (2), and (4) in Definition 1.1(r), has Z0 (T ) = M , and fits in an exact sequence 0 −→ T  −→ T −→ T  −→ 0 . The functor Z(−) is left exact, and since T  is acyclic a standard application of the Snake Lemma yields an exact sequence 0 −→ Zi (T  ) −→ Zi (T ) −→ Zi (T  ) −→ 0 for every i ∈ Z. As U⊥ is closed under extensions, it follows that Zi (T ) belongs to U⊥ for each i and thus T is right U-totally acyclic by Definition 1.1. This proves (r) and a similar argument proves (l).  2.11 Theorem. Let U be an additively closed subcategory of A. The category RGorU (A) is Frobenius and U∩U⊥ is the subcategory of projective-injective objects. Proof. Set W = U ∩ U⊥ and notice that W is additively closed. It follows from Lemma 2.10 that RGorU (A) is an exact category. It is immediate from Example 2.2 and Proposition 1.3 that objects in W are both projective and injective in RGorU (A). It is now immediate from Definition 2.1 that RGorU (A) has enough projectives and

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injectives. It remains to show that every projective and every injective object in RGorU (A) belongs to W. Let P be a projective object in RGorU (A). By Definition 2.1 and Proposition 1.3 there is an exact sequence 0 → P  → W → P → 0 in A with P  ∈ RGorU (A) and W ∈ W. As all three objects belong to RGorU (A) it follows by projectivity of P that the sequence splits, so P is a summand of W , and thus in W. A dual argument shows that every injective object in RGorU (A) belongs to W. Thus RGorU (A) is a Frobenius category and W is the subcategory of projective-injective objects.  2.12 Theorem. Let V be an additively closed subcategory of A. The category LGorV (A) is Frobenius and ⊥ V ∩ V is the subcategory of projective-injective objects. 

Proof. Parallel to the proof of Theorem 2.11. 3. An equivalence of triangulated categories

Generalizing the classic result, we prove here that the stable category of right/left Gorenstein objects is equivalent to the homotopy category of right/left totally acyclic complexes. 3.1 Lemma. Let U be a subcategory of A; let T and T  be right U-totally acyclic complexes. Every morphism ϕ : Z0 (T ) → Z0 (T  ) in A lifts to a morphism φ : T → T  of A-complexes. Proof. Let a morphism ϕ : Z0 (T ) → Z0 (T  ) be given; to see that it lifts to a morphism φ : T → T  of complexes it is sufficient to show that ϕ lifts to morphisms φ1 : T1 → T1 and φ0 : T0 → T0 . As T1 is in U and T  is right U-totally acyclic, one obtains per Proposition 1.3(r.3) an exact sequence 0 −→ homA (T1 , Z1 (T  )) −→ homA (T1 , T1 ) −→ homA (T1 , Z0 (T  )) −→ 0 . 

In particular, there is a φ1 ∈ homA (T1 , T1 ) with ∂1T φ1 = ϕ∂1T . As T0 is in U ∩ U⊥ and T is right U-totally acyclic, it follows that the sequence 0 −→ homA (Z−1 (T ), T0 ) −→ homA (T0 , T0 ) −→ homA (Z0 (T ), T0 ) −→ 0 is exact, whence there exists a φ0 ∈ homA (T0 , T0 ) that lifts ϕ.



3.2 Lemma. Let U be a subcategory of A and φ : T → T  be a morphism of right U-totally acyclic complexes. If the cycle subobject Z0 (T ) has a decomposition  ∈ U, then φ is null-homotopic. Z0 (T ) = Z ⊕ Z with Z ⊆ ker φ0 and Z  Proof. The goal is to construct a family of morphisms σi : Ti → Ti+1 such that  T T  = φ0 |Z . By Definition 1.1 each φi = ∂i+1 σi + σi−1 ∂i holds for all i ∈ Z. Set ϕ object Zi (T  ) is in U⊥ . It follows that there is an exact sequence,

 Z1 (T  )) −→ homA (Z,  T1 ) −→ homA (Z,  Z0 (T  )) −→ 0 . 0 −→ homA (Z,  T1 ) with ∂1T σ  = ϕ.  Set σ 0 = 0 ⊕ σ ; by In particular, there is a σ  ∈ homA (Z, exactness of the sequence 

0 −→ homA (Z−1 (T ), T1 ) −→ homA (T0 , T1 ) −→ homA (Z0 (T ), T1 ) −→ 0 , σ 0 lifts to a morphism σ0 : T0 → T1 .

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107

We proceed by induction to construct the morphisms σi for i ≥ 1. The image of the morphism φ1 − σ0 ∂1T is in Z1 (T  ) as one has 



∂1T (φ1 − σ0 ∂1T ) = φ0 ∂1T − ∂1T σ0 ∂1T 

= (0 ⊕ ϕ)∂  1T − ∂1T (0 ⊕ σ )∂1T 

= (0 ⊕ (ϕ  − ∂1T σ ))∂1T =0. As T1 is in U and Z2 (T  ) is in U⊥ per Definition 1.1, there is an exact sequence 0 −→ homA (T1 , Z2 (T  )) −→ homA (T1 , T2 ) −→ homA (T1 , Z1 (T  )) −→ 0 . 

In particular, there is a σ1 ∈ homA (T1 , T2 ) with ∂2T σ1 = φ1 − σ0 ∂1T . Now let i ≥ 1 and assume that σj has been constructed for 0 ≤ j ≤ i. The standard computation 



T T T T (φi+1 − σi ∂i+1 ) = (φi − ∂i+1 σi )∂i+1 ∂i+1 T = (σi−1 ∂iT )∂i+1 =0 T is in Zi+1 (T  ). As Ti+1 is in U and Zi+2 (T  ) shows that the image of φi+1 − σi ∂i+1 ⊥ is in U , the existence of the desired σi+1 follows as for i = 0. Finally, we prove the existence of the morphisms σi for i ≤ −1 by descending  induction. The morphism φ0 − ∂1T σ0 : T0 → T0 restricts to 0 on Z0 (T ); indeed one has    (φ0 − ∂1T σ0 )|Z0 (T ) = 0 ⊕ ϕ  − ∂1T (0 ⊕ σ  ) = 0 ⊕ (ϕ  − ∂1T σ ) = 0 . ∼ Thus it induces a morphism ζ−1 from T0 / Z0 (T ) = Z−1 (T ) to T  with ζ−1 ∂ T = 0



φ0 − ∂1T σ0 . As T0 is in U ∩ U⊥ it follows that the sequence

0

0 −→ homA (Z−2 (T ), T0 ) −→ homA (T−1 , T0 ) −→ homA (Z−1 (T ), T0 ) −→ 0 is exact. In particular, there is a σ−1 ∈ homA (T−1 , T0 ) with σ−1 |Z−1 (T ) = ζ−1 and,  therefore, σ−1 ∂0T = φ0 − ∂1T σ0 . Now let i ≤ −1 and assume that σj has been constructed for 0 ≥ j ≥ i. The standard computation 







T T T T T T σi )∂i+1 = ∂i+1 (φi+1 − σi ∂i+1 ) = ∂i+1 (∂i+2 σi+1 ) = 0 (φi − ∂i+1 

T σi restricts to 0 on Zi (T ). It follows that it shows that the morphism φi − ∂i+1 T σi . Since induces a morphism ζi−1 on Ti / Zi (T ) ∼ = Zi−1 (T ) with ζi−1 ∂iT = φi − ∂i+1 Ti is in U ∩ U⊥ , it follows as for i = 0 that the desired σi−1 exists. 

3.3 Proposition. Let U be a subcategory of A. Let T and T  be right U-totally acyclic complexes and ϕ : Z0 (T ) → Z0 (T  ) be a morphism in A. (a) If φ : T → T  and ψ : T → T  are morphisms that lift ϕ, then φ − ψ is null-homotopic. (b) If ϕ is an isomorphism and φ : T → T  is a morphism that lifts ϕ, then φ is a homotopy equivalence. Proof. (a): Immediate from Lemma 3.2 as (φ − ψ)|Z0 (T ) = ϕ − ϕ = 0. (b): Let φ : T  → T be a lift of ϕ−1 ; see Lemma 3.1. The restriction of 1T −φ φ to Z0 (T ) is 0, so it follows from part (a) that 1T − φ φ is null-homotopic. Similarly,  1T − φφ is null-homotopic; that is, φ is a homotopy equivalence. 

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⊥ 3.4 Definition. Let U and V be subcategories of A. Denote by KR U-tac (U ∩ U ) and L ⊥ KV-tac ( V ∩ V) the homotopy categories of right U-totally acyclic complexes and left V-totally acyclic complexes. ⊥ The subcategory U∩U⊥ is self-orthogonal, so the categories KR (U∩U⊥ )-tac (U∩U ) L ⊥ ⊥ and K(U∩U⊥ )-tac (U∩U ) coincide, see Proposition 1.5, and are denoted Ktac (U∩U ). The self-orthogonal subcategory ⊥ V ∩ V similarly gives a category Ktac (⊥ V ∩ V). For a cotorsion pair (U, V) all of these homotopy categories are Ktac (U ∩ V). ⊥ If U is an additive subcategory, then the homotopy category KR U-tac (U ∩ U ) is L ⊥ triangulated; similarly for V and KV-tac ( V ∩ V).

3.5 Lemma. Let U be an additively closed subcategory of A. Let T be a right Utotally acyclic complex; if Zi (T ) belongs to U for some i ∈ Z, then T is contractible. Proof. Set W = U ∩ U⊥ and notice that W is additively closed. To prove that T is contractible it is enough to show that Zi := Zi (T ) belongs to W for every i ∈ Z. There are exact sequences 0 −→ Zj+1 −→ Tj+1 −→ Zj −→ 0

(∗)

with Tj+1 in W and Zj+1 , Zj ∈ U⊥ ; see Definition 1.1 and Proposition 1.3. Without loss of generality, assume that Z0 is in U and hence in W. Let j ≥ 0 and assume that Zj is W. The sequence (∗) splits as Zj is in U and Zj+1 is in U⊥ . It follows that Zj+1 is in W, so by induction Zi is in W for all i ≥ 0. Now let j < 0 and assume that Zj+1 is in W. The sequence (∗) splits as homA (T, Zj+1 ) is acyclic by Definition 1.1. It follows that Zj belongs to W, so by  descending induction Zi is in W for all i ≤ 0. 3.6 Proposition. Let U be an additively closed subcategory of A. • For every right U-Gorenstein object M fix a right U-totally acyclic complex T with Z0 (T ) = M and set T˙ R (M ) = T . • For every morphism ϕ : M → M  of right U-Gorenstein objects fix by 3.1 a lift φ : T˙ R (M ) → T˙ R (M  ) of ϕ and set T˙ R (ϕ) = [φ]. This defines a functor ⊥ T˙ R : RGorU (A) −→ KR U-tac (U ∩ U ) . For every morphism ϕ in RGorU (A) that factors through an object in U ∩ U⊥ one has T˙ R (ϕ) = [0]. In particular, T˙ R (M ) is contractible for every M in U ∩ U⊥ . ˙ R (M ) → Proof. Let M be a right U-Gorenstein object. Denote by ιM : T ˙TR (M ) the fixed lift of 1M ; that is, [ιM ] = T˙ R (1M ). As the morphisms 1T˙ R (M ) ˙ R (M )) = M , it follows from Lemma 3.2 that the difference and ιM agree on Z0 (T ˙ R (M ) ˙ T M − ι is null-homotopic. That is, one has T˙ R (1M ) = [1TR (M ) ], which is the 1 ⊥ identity on T˙ R (M ) in KR U-tac (U ∩ U ). ϕ

ϕ

Let M  −−→ M −−→ M  be morphisms of right U-Gorenstein objects. The ˙ R (ϕ ) to Z0 (T ˙ R (M  )) are both ϕϕ . It now restrictions of T˙ R (ϕϕ ) and T˙ R (ϕ) T ˙ R (ϕ ) are follows from Lemma 3.2 that the homotopy classes T˙ R (ϕϕ ) and T˙ R (ϕ) T equal. Thus T˙ R is a functor. For an object M in the additively closed subcategory U ∩ U⊥ it follows from Lemma 3.5 that T˙ R (M ) is contractible. Finally, if a morphism ϕ : M  → M  in

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RGorU (A) factors as

ψ

109

ψ

M  −−→ M −−→ M  ˙ R (ψ) T ˙ R (ψ  ) factors through where M is in U ∩ U⊥ , then T˙ R (ϕ) = T˙ R (ψψ  ) = T  the contractible complex T˙ R (M ), so one has T˙ R (ϕ) = [0][0] = [0]. 3.7 Remark. Let U be an additively closed subcategory of A. Let M be a right U-Gorenstein object in A and T a right U-totally acyclic complex with Z0 (T ) ∼ = M . It follows from Proposition 3.3 that T and T˙ R (M ) are (U ∩ U⊥ ). isomorphic in KR U-tac  Let ϕ : M → M be a morphism of right U-Gorenstein objects in A. For every ˙ R (M  ) that lifts ϕ, Proposition 3.3 yields [φ] = T˙ R (ϕ). morphism φ : T˙ R (M ) → T Let U be an additively closed subcategory of A. Recall from Theorem 2.11 that RGorU (A) is a Frobenius category with U∩U⊥ the subcategory of projective-injective objects. Denote by StRGorU (A) the associated stable category. It is a triangulated category, see for example Krause [20, 7.4], and it is immediate from Proposition 3.6 ⊥ that T˙ R induces a triangulated functor TR : StRGorU (A) −→ KR U-tac (U ∩ U ). 3.8 Theorem. Let U be an additively closed subcategory of A. There is a biadjoint triangulated equivalence StRGorU (A) o

TR Z0

/

⊥ KR U-tac (U ∩ U ) .

Proof. Set W = U ∩ U⊥ and notice that W is additively closed. The functors TR and Z0 are triangulated. We prove that (TR , Z0 ) is an adjoint pair; a parallel argument shows that (Z0 , TR ) is an adjoint pair. Let M be a right U-Gorenstein object and T be a right U-totally acyclic complex. The assignment [φ] −→ [Z0 (φ)] defines a map ΦM,T : homKRU-tac (W) (TR (M ), T ) −→ homStRGorU (A) (M, Z0 (T )) . By Lemma 3.1 there is a morphism of A-complexes εT : TR (Z0 (T )) → T . The assignment [ϕ] −→ [εT ] TR (ϕ) defines a map ΨM,T in the opposite direction. Let [φ] ∈ homKRU-tac (W) (TR (M ), T ) be given. Let φM,T : TR (M ) → TR (Z0 (T )) be a representative of the homotopy class TR (Z0 (φ)), cf. Remark 3.7. The composite εT φM,T agrees with φ on M = Z0 (TR (M )), so Lemma 3.2 yields [φ] = [εT φM,T ] = [εT ] TR (Z0 (φ)) = ΨM,T ΦM,T ([φ]) . Now let [ϕ] ∈ homStRGorU (A) (M, Z0 (T )) be given. Let ϕM,T : TR (M ) → TR (Z0 (T )) be a lift of ϕ; that is, a representative of the homotopy class TR (ϕ). One now has ΦM,T ΨM,T ([ϕ]) = ΦM,T ([εT ] TR (ϕ)) = ΦM,T ([εT ϕM,T ]) = [Z0 (εT ϕM,T )] = [Z0 (εT ) Z0 (ϕM,T )] = [1Z0 (T ) ϕ] = [ϕ] . Thus Φ

M,T

is an isomorphism.

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The unit of the adjunction is the identity as one has Z0 (TR (−)) = 1StRGorU (A) , and it is straightforward to check that εT defined above determines the counit R ε : TR (Z0 (−)) → 1KU-tac (W) . To show that ε is an isomorphism, let T ∈ KR U-tac (W) be given and consider a lift of the identity Z0 (T ) → Z0 (TR (Z0 (T ))) to a morphism ιT : T → TR (Z0 (T )); see Lemma 3.1. The composite εT ιT agrees with 1T on Z0 (T ), so εT ιT is a homotopy equivalence by Lemma 3.2. Similarly, ιT εT is a homotopy equivalence. It follows that εT is a homotopy equivalence, i.e. [εT ] is an isomorphism  in KR U-tac (W). 3.9 Corollary. Let (U, V) be a cotorsion pair in A. There is a biadjoint triangulated equivalence /

TR

StGorU∩V (A) o

Ktac (U ∩ V) .

Z0

Proof. This is Theorem 3.8 applied to the self-orthogonal additively closed subcategory U ∩ V and written in the notation from Definitions 2.1 and 3.4.  3.10 Example. Applied to the cotorsion pair (A, Inj(A)), Corollary 3.9 recovers the well-known equivalence of the stable category of Gorenstein injective objects and the homotopy category of totally acyclic complexes of injective objects; see [19, prop. 7.2]. Applied to the cotorsion pair (Prj(A), A), the corollary yields the corresponding equivalence StGorPrj (A)  Ktac (Prj(A)). 3.11 Remark. Let U and V be additively closed subcategories of A. In 3.1–3.8 we have focused on right U-totally acyclic complexes and right U-Gorenstein objects. There are, of course, parallel results about left V-totally acyclic complexes and left V-Gorenstein objects. In particular, there is a biadjoint triangulated equivalence StLGorV (A) o

TL Z0

/

KLV-tac (⊥ V ∩ V) .

Notice that applied to a cotorsion pair (U, V) this also yields Corollary 3.9. 4. Gorenstein flat-cotorsion modules In this section and the next, A is an associative ring. We adopt the convention that an A-module is a left A-module; right A-modules are considered to be modules over the opposite ring A◦ . The category of A-modules is denoted Mod(A). Given a cotorsion pair (U, V) in Mod(A) the natural categories of Gorenstein objects to consider are RGorU (A), LGorV (A), and Gor(U∩V) (A); see Remark 1.8. For each of the absolute cotorsion pairs (Prj(A), Mod(A)) and (Mod(A), Inj(A)), two of these categories of Gorenstein objects coincide and the third is trivial. We start this section by recording the non-trivial fact that the cotorsion pair (Flat(A), Cot(A)) exhibits the same behavior. For brevity we denote the self-orthogonal subcategory Flat(A) ∩ Cot(A) of flat-cotorsion modules by FlatCot(A). Bazzoni, Cort´es Izurdiaga, and Estrada [3, thm. 1.3] prove: 4.1 Fact. An acyclic complex of cotorsion A-modules has cotorsion cycle modules.

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111

4.2 Proposition. A FlatCot(A)-totally acyclic complex is right Flat(A)-totally acyclic, and a left Cot(A)-totally acyclic complex is contractible. In particular, one has RGorFlat (A) = GorFlatCot (A)

and

LGorCot (A) = FlatCot(A) .

Proof. In a FlatCot(A)-totally acyclic complex, the cycle modules are cotorsion by 4.1, whence the complex is right Flat(A)-totally acyclic by Definition 1.1. By Remark 1.8 every right Flat(A)-totally acyclic complex is FlatCot(A)-totally acyclic, so the first equality of categories follows from Definition 2.1. In a left Cot(A)-totally acyclic complex, the cycle modules are flat-cotorsion, again by Definition 1.1 and 4.1, so such a complex is contractible, and the second equality follows.  We introduce a less symbol-heavy terminology. 4.3 Definition. A FlatCot(A)-totally acyclic complex is called a totally acyclic complex of flat-cotorsion modules. A cycle module in such a complex, that is, a FlatCot(A)-Gorenstein module, is called a Gorenstein flat-cotorsion module. Recall that a complex T of flat A-modules is called F-totally acyclic if it is acyclic and the complex I ⊗A T is acyclic for every injective A◦ -module I. 4.4 Theorem. Let A be right coherent. For an A-complex T the following conditions are equivalent (i ) T is a totally acyclic complex of flat-cotorsion modules. (ii ) T is a complex of flat-cotorsion modules and F-totally acyclic. (iii ) T is right Flat(A)-totally acyclic. Proof. Per Remark 1.8 condition (iii ) implies (i ). (i ) =⇒ (ii ): If T is a totally acyclic complex of flat-cotorsion modules, then by Proposition 1.3 it is an acyclic complex of flat-cotorsion modules. For every injective A◦ -module I the A-module HomZ (I, Q/Z) is flat-cotorsion, as A is right coherent. Now it follows by the isomorphism (∗)

HomA (T, HomZ (I, Q/Z)) ∼ = HomZ (I ⊗A T , Q/Z)

and faithful injectivity of Q/Z that I ⊗A T is acyclic. (ii ) =⇒ (iii ): If T is a complex of flat-cotorsion A-modules and F-totally acyclic, then T satisfies conditions (r.1) and (r.2) in Definition 1.1. By 4.1 the cycles modules of T are cotorsion, so T also satisfies condition (r.3). Further, as A is right coherent, every flat-cotorsion A-module is a direct summand of a module of the form HomZ (I, Q/Z), for some injective A◦ -module I; see e.g. Xu [25, lem. 3.2.3]. Now it follows from the isomorphism (∗) that HomA (T, W ) is acyclic for every W ∈ FlatCot(A). That is, T also satisfies condition 1.1(r.4).  Recall that an A-module M is called Gorenstein flat if there exists an F-totally acyclic complex F of flat A-modules with Z0 (F ) = M . The full subcategory of Mod(A) whose objects are the Gorenstein flat modules is denoted GFlat(A). Gillespie [15, cor. 3.4] proved that the category Cot(A) ∩ GFlat(A) is Frobenius if A is right coherent. That it remains true without the coherence assumption is an immediate consequence of [24, cor. 3.12] discussed ibid.; for convenience we include the statement as part of the next result.

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4.5 Theorem. The category Cot(A) ∩ GFlat(A) is Frobenius and FlatCot(A) is the subcategory of projective-injective objects. Moreover, the following conditions are equivalent. (i ) A is left perfect. (ii ) The category GFlat(A) is Frobenius. (iii ) One has GFlat(A) = Cot(A) ∩ GFlat(A). Furthermore, if A is right coherent then these conditions are equivalent to (iv ) An A-module is Gorenstein flat if and only if it is Gorenstein projective. Proof. By [24, cor. 3.12] the category GFlat(A) is closed under extensions, and GFlat(A) ∩ GFlat(A)⊥ is the subcategory FlatCot(A) of flat-cotorsion modules. It follows that Cot(A) ∩ GFlat(A) is closed under extensions, and that modules in FlatCot(A) are both projective and injective in Cot(A) ∩ GFlat(A). Let P be a projective object in Cot(A) ∩ GFlat(A); it fits in an exact sequence (∗)

0 −→ C −→ F −→ P −→ 0

where F is flat and C is cotorsion; see Bican, El Bashir, and Enochs [5]. As P is cotorsion it follows that F is flat-cotorsion. By [24, cor. 3.12] the category GFlat(A) is resolving, so C is Gorenstein flat. Thus, (∗) is an exact sequence in Cot(A) ∩ GFlat(A), whence it splits by the assumption on P . In particular, P is flat-cotorsion. Now let I be an injective object in Cot(A) ∩ GFlat(A). It fits by [24, cor. 3.12] in an exact sequence (†)

0 −→ I −→ F −→ G −→ 0

where F belongs to GFlat(A)⊥ and G is Gorenstein flat. It follows that F is Gorenstein flat and hence flat-cotorsion, still by [24, cor. 3.12]. Finally, G is cotorsion as both I and F are cotorsion. Thus, (†) is an exact sequence in Cot(A) ∩ GFlat(A), whence it splits by the assumption on I. In particular, I is flat-cotorsion. (i ) =⇒ (iii ): Assuming that A is left perfect, every flat A-module module is projective, whence every A-module is cotorsion. (iii ) =⇒ (ii ): Evident as Cot(A) ∩ GFlat(A) is Frobenius as shown above. (ii ) =⇒ (i ): Assume that GFlat(A) is Frobenius and denote by W its subcategory of projective-injective objects. To prove that A is left perfect it suffices by a result of Guil Asensio and Herzog [16, cor. 20] to show that the free module A(N) is cotorsion. As A(N) is flat, in particular Gorenstein flat, and as GFlat(A) by assumption has enough projectives, there is an exact sequence 0 → K → W → A(N) → 0 with W ∈ W. The sequence splits because A(N) is projective, so it suffices to show that modules in W are cotorsion. Fix W ∈ W, let F be a flat A-module, and consider an extension (‡)

0 −→ W −→ E −→ F −→ 0 .

As GFlat(A) by [24, cor. 3.12] is closed under extensions, the module E is Gorenstein flat. As W is injective in GFlat(A) it follows that the sequence (‡) splits, i.e. one has Ext1A (F, W ) = 0. That it, W is cotorsion. (iv ) =⇒ (ii ): By Theorem 2.11 the category of Gorenstein projective A-modules is Frobenius. (i ) =⇒ (iv ): If A is perfect and right coherent, then it follows from Theorem 4.4 that an A-module is Gorenstein flat if and only if it is Gorenstein projective. 

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By Theorem 4.5 the category GFlat(A) is only Frobenius when every A-module is cotorsion, and the take-away is that the appropriate Frobenius category to focus on is Cot(A) ∩ GFlat(A). If A is right coherent ring, then this category contains GorFlatCot (A), by Theorem 4.4 and 4.1, and one goal of the next section is to prove the reverse inclusion; that is Theorem 5.2. 5. The stable category of Gorenstein flat-cotorsion modules Recall that an A-complex P is called pure-acyclic if the complex N ⊗A P is acyclic for every A◦ -module N . In particular, an acyclic complex P of flat A-modules is pure-acyclic if and only if all cycle modules Zi (P ) are flat. 5.1 Fact. Let M be an A-complex. It follows1 from Gillespie [14, cor. 4.10] that there exists an exact sequence of A-complexes 0 −→ M −→ C −→ P −→ 0 where C is a complex of cotorsion modules and P is a pure-acyclic complex of flat modules. The first theorem of this section shows that if A is right coherent, then the cotorsion modules in GFlat(A) are precisely the non-trivial Gorenstein modules associated to the cotorsion pair (Flat(A), Cot(A)); namely the Gorenstein flat-cotorsion modules or, equivalently, the right Flat(A)-Gorenstein modules. 5.2 Theorem. Let A be right coherent. There are equalities Cot(A) ∩ GFlat(A) = GorFlatCot (A) = RGorFlat (A) . Proof. The second equality is by Proposition 4.2, and the containment Cot(A) ∩ GFlat(A) ⊇ GorFlatCot (A) follows from 4.1 and Theorem 4.4. It remains to show the reverse containment. Let M be a Gorenstein flat A-module that is also cotorsion. By definition, there is an F-totally acyclic complex F of flat A-modules with Z0 (F ) = M . Further, 5.1 yields an exact sequence of A-complexes (1)

ι

π

0 −→ F −−→ T −−→ P −→ 0

where T is a complex of cotorsion modules and P is a pure-acyclic complex of flat modules. It follows that T is a complex of flat modules; moreover, since P is trivially F-totally acyclic, so is T . As A is right coherent, it now follows from Theorem 4.4 that T is a totally acyclic complex of flat-cotorsion modules. The functor Z(−) is left exact, and since F is acyclic a standard application of the Snake Lemma yields the exact sequence (2)

ι

π

0 −→ M −→ Z0 (T ) −−→ Z0 (P ) −→ 0

where ι and π are the restrictions of the morphisms from (1). As M is cotorsion and Z0 (P ) is flat, (2) splits. Set Z = Z0 (P ) and denote by the section with π = 1Z . By 4.1 the module Z0 (T ) is cotorsion, so it follows that Z is a flat-cotorsion module. Now, as Z−1 (P ) is flat, the exact sequence εP

0 −→ Z −−0→ P0 −→ Z−1 (P ) −→ 0 1 Although [14, cor. 4.10] is stated for commutative rings, it is standard that the result remains valid without this assumption; see for example the discussion before [12, thm. 4.2].

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Z splits; denote by σ the section with σεP 0 = 1 . By commutativity of the diagram

/Z

π

Z0 (T ) εT 0

εP 0

 T0

 / P0

π0

Z  one has σπ0 εT0 = σεP 0 π = 1 . It follows that σπ0 : T0 → Z is a split surjection T with section ε0 . As Z is flat and Z1 (T ) is cotorsion, there is an exact sequence

0 −→ HomA (Z, Z1 (T )) −→ HomA (Z, T1 ) −→ HomA (Z, Z0 (T )) −→ 0 . It follows that there is a homomorphism ζ : Z → T1 with ∂1T ζ = and, therefore, ∂1T ζ = εT0 as homomorphisms from Z to T0 . As ∂0T εT0 = 0 trivially holds, the homomorphisms ζ and εT0 yield a morphism of complexes: D = ···

/0

/Z

ρ

 T = ···

 / T2

=

/Z

∂1T

 / T0

εT 0 

ζ

∂2T

 / T1

∂0T

/0

/ ···

 / T−1

/ ···

This is evidently a split embedding in the category of complexes whose section given by the homomorphisms σπ0 ∂1T : T1 → Z and σπ0 : T0 → Z. The restriction of the split exact sequence of complexes (3)

0 −→ D −→ T −→ T  −→ 0 

to cycles is isomorphic to the split exact sequence 0 −→ Z −→ Z0 (T ) −→ M −→ 0, see (2), so it follows that the complex T  has Z0 (T  ) ∼ = M. In (3) both D and T are complexes of flat-cotorsion modules and F-totally acyclic, so also T  is a complex of flat-cotorsion modules and F-totally acyclic. Now it follows from Theorem 4.4 that T  is a totally acyclic complex of flat-cotorsion  modules, whence the module M ∼ = Z0 (T  ) is Gorenstein flat-cotorsion. 5.3 Corollary. Let A be right coherent. There is a triangulated equivalence StGorFlatCot (A)  KF-tac (FlatCot(A)) . 5.7.

Proof. Immediate from Theorems 3.8, 4.4, and 5.2; see also the diagram in 

5.4 Corollary. Let A be right coherent. The category GorFlatCot (A) is closed under direct summands. Proof. Immediate from the theorem as both Cot(A) and GFlat(A) are closed under direct summands; for the latter see [24, cor. 3.12].  Gorenstein flat A-modules are, within the framework of Sections 1–2, not born out of a cotorsion pair, not even out of a self-orthogonal subcategory of Mod(A). However, they form the left half of a cotorsion pair, and also out of that pair comes the Gorenstein flat-cotorsion modules.

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5.5 Remark. Let A be right coherent. Enochs, Jenda, and Lopez-Ramos [11, thm. 2.11] show that GFlat(A) is the left half of a cotorsion pair, and Gillespie [15, prop. 3.2] shows that GFlat(A) ∩ GFlat(A)⊥ is FlatCot(A)2 A right GFlat(A)-totally acyclic complex as well as a left GFlat(A)⊥ -totally acyclic complex is by Remark 1.8 and Definition 4.3 a totally acyclic complex of flat-cotorsion modules. For a right GFlat(A)-totally acyclic complex T , it follows from Definition 1.1 that HomA (G, T ) is acyclic for every Gorenstein flat A-module G, in particular for every Gorenstein flat-cotorsion module. That is, such a complex is contractible and, therefore, a right GFlat(A)-Gorenstein module is flat-cotorsion. On the other hand, the cycles in a left GFlat(A)⊥ -totally acyclic complex are by Definition 1.1 Gorenstein flat and by 4.1 cotorsion, so a left GFlat(A)⊥ -Gorenstein module is by Theorem 5.2 Gorenstein flat-cotorsion. Let Kpac (Flat(A)) denote the full subcategory of K(Flat(A)) whose objects are pure-acyclic; notice that it is contained in KF-tac (Flat(A)). Via 4.1 and the dual of 5.1 one could obtain the next theorem as a consequence of a standard result [18, prop. 10.2.7]; we opt for a direct argument. 5.6 Theorem. The composite I : KF-tac (FlatCot(A)) −→ KF-tac (Flat(A)) −→

KF-tac (Flat(A)) Kpac (Flat(A))

of canonical functors is a triangulated equivalence of categories. Proof. Let I be the composite of the inclusion followed by Verdier localization; notice that I is the identity on objects. We argue that the functor I is essentially surjective, full, and faithful. Let F be an F-totally acyclic complex of flat modules. By 5.1 there is an exact sequence (∗)

0 −→ F −→ C F −→ P F −→ 0

where C F is a complex of cotorsion modules and P F is in Kpac (Flat(A)). As F and P F are F-totally acyclic complexes of flat A-modules so is C F ; that is, C F belongs to KF-tac (FlatCot(A)). It follows from (∗) that F and C F are isomorphic in the (Flat(A)) . Thus I is essentially surjective. Verdier quotient KKF-tac pac (Flat(A)) Let F and F  be F-totally acyclic complexes of flat-cotorsion modules. A (Flat(A)) is a diagram in KF-tac (Flat(A)) morphism F → F  in KKF-tac pac (Flat(A)) (∗)

[α]

[ϕ]

F −−−→ X ←−−− F 

such that the complex Cone ϕ belongs to Kpac (Flat(A)). Let ι be the embedding X → C X from 5.1. It is elementary to verify that the composite ιϕ : F  → C X has a pure-acyclic mapping cone; see [9, lem. 2.7]. Since F  and C X are complexes of flat-cotorsion modules, so is Cone ιϕ. It now follows by way of 4.1 that Cone ιϕ is contractible; that is, ιϕ is a homotopy equivalence. Thus [ιϕ] has an 2 Saroch ˇ ˇ and Stov´ ıˇ cek [24, cor. 3.12] show that all of this is true without assumptions on A, and we used that crucially in the proof of Theorem 4.5. The results from [11] and [15] suffice to prove 4.5 for a right coherent ring.

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inverse in KF-tac (Flat(A)), i.e. [ιϕ]−1 = [ψ] for some morphism ψ : C X → F  . The commutative diagram > X aB }} BBB } BB[ϕ] [α] }} }} [ι] BBB } BB }  }} [ια] [ιϕ] X o / FA F CO

AA | | AA | AA [ιϕ] ||| A |  [ψια] AA || [1F ] A }|| F [ψια]



[1F ]

now shows that the morphism (∗) is equivalent to F −−−−→ F  ←−−−− F  , which is I(ψια). This shows that I is full. Finally, let α : F → F  be a morphism of F-totally acyclic complexes of flatcotorsion modules, and assume that I([α]) is zero. It follows that there is a commutative diagram in KF-tac (Flat(A)), F ~> `AAA ~ AA [1F  ] [α] ~~~ ~ [ϕ] AAA ~ AA ~~  ~~ [ϕα] [ϕ] / o F@ XO F

@@ } } @@ } @@ [ϕ] }}} @ } [0] @@ } [1F  ] ~}} F where the mapping cone of ϕ is in Kpac (Flat(A)). The diagram yields [ϕα] = [0] and, therefore, [ιϕ][α] = [ιϕα] = [0] where ι is the embedding X → C X from 5.1. As above, [ιϕ] is invertible in KF-tac (Flat(A)), so one has [α] = [0] in KF-tac (Flat(A)). That is, α is null-homotopic, and hence [α] = 0 in KF-tac (FlatCot(A)).  5.7 Summary. Let A be right coherent. By Theorems 3.8 and 5.6 there are triangulated equivalences KF-tac (FlatCot(A)) StRGorFlat (A) StGorFlatCot (A)

TR

I

/ KF-tac (Flat(A)) Kpac (Flat(A))

/ KR Flat-tac (FlatCot(A)) Ktac (FlatCot(A))

where the equalities come from Proposition 4.2 and Theorems 4.4 and 5.2. 5.8 Corollary. Let A be right coherent. There is a triangulated equivalence StGorFlatCot (A)  Proof. See the diagram in 5.7.

KF-tac (Flat(A)) . Kpac (Flat(A)) 

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In the special case where A is commutative noetherian of finite Krull dimension, the next result is immediate from [21, lem. 4.22] and Corollary 5.8. 5.9 Corollary. Let A be right coherent ring such that all flat A-modules have finite projective dimension. There is a triangulated equivalence of categories StGorPrj (A)  StGorFlatCot (A) . Proof. Under the assumptions on A, a complex of projective A-modules is totally acyclic if and only if it F-totally acyclic; see [8, claims 2.4 and 2.5]. By [12, thm. 5.1] there is now a triangulated equivalence of categories Ktac (Prj(A)) 

KF-tac (Flat(A)) . Kpac (Flat(A))

Now apply the equivalences from Example 3.10 and Corollary 5.8.



Acknowledgments We thank Petter Andreas Bergh, Tsutomu Nakamura, and Mark Walker for conversations and comments on an early draft of this paper. References [1] Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR0269685 [2] Luchezar L. Avramov and Alex Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393– 440, DOI 10.1112/S0024611502013527. MR1912056 [3] Silvana Bazzoni, Manuel Cort´ es Izurdiaga, and Sergio Estrada, Periodic modules and acyclic complexes, preprint arXiv:1704.06672; 20 pp. [4] Victor Becerril, Octavio Mendoza, and Valente Santiago, Relative Gorenstein objects in abelian categories, preprint arXiv:1810.08524; 47 pp. [5] L. Bican, R. El Bashir, and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385–390, DOI 10.1017/S0024609301008104. MR1832549 [6] Ragnar-Olaf Buchweitz, Maximal Cohen–Macaulay modules and Tatecohomology over Gorenstein rings, University of Hannover, 1986, available at http://hdl.handle.net/1807/16682. [7] Lars Winther Christensen, Sergio Estrada, and Peder Thompson, The stable category of Gorenstein flat sheaves on a noetherian scheme, preprint arXiv:1904.07661; 12 pp. [8] Lars Winther Christensen and Kiriko Kato, Totally acyclic complexes and locally Gorenstein rings, J. Algebra Appl. 17 (2018), no. 3, 1850039, 6, DOI 10.1142/S0219498818500391. MR3760008 [9] Lars Winther Christensen and Peder Thompson, Pure-minimal chain complexes, Rend. Semin. Mat. Univ. Padova, to appear. Preprint arXiv:1801.00302; 18 pp. [10] Edgar E. Enochs and Overtoun M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611–633, DOI 10.1007/BF02572634. MR1363858 [11] Edgar E. Enochs, Overtoun M. G. Jenda, and J. A. Lopez-Ramos, The existence of Gorenstein flat covers, Math. Scand. 94 (2004), no. 1, 46–62, DOI 10.7146/math.scand.a-14429. MR2032335 [12] Sergio Estrada and James Gillespie, The projective stable category of a coherent scheme, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 1, 15–43, DOI 10.1017/S0308210517000385. MR3922806 [13] Yuxian Geng and Nanqing Ding, W-Gorenstein modules, J. Algebra 325 (2011), 132–146, DOI 10.1016/j.jalgebra.2010.09.040. MR2745532 [14] James Gillespie, The flat model structure on Ch(R), Trans. Amer. Math. Soc. 356 (2004), no. 8, 3369–3390, DOI 10.1090/S0002-9947-04-03416-6. MR2052954 [15] James Gillespie, The flat stable module category of a coherent ring, J. Pure Appl. Algebra 221 (2017), no. 8, 2025–2031, DOI 10.1016/j.jpaa.2016.10.012. MR3623182

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[16] Pedro A. Guil Asensio and Ivo Herzog, Sigma-cotorsion rings, Adv. Math. 191 (2005), no. 1, 11–28, DOI 10.1016/j.aim.2004.01.006. MR2102841 [17] Srikanth Iyengar and Henning Krause, Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math. 11 (2006), 207–240. MR2262932 [18] Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, SpringerVerlag, Berlin, 2006. MR2182076 [19] Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162, DOI 10.1112/S0010437X05001375. MR2157133 [20] Henning Krause, Derived categories, resolutions, and Brown representability, Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 101–139, DOI 10.1090/conm/436/08405. MR2355771 [21] Daniel Murfet and Shokrollah Salarian, Totally acyclic complexes over Noetherian schemes, Adv. Math. 226 (2011), no. 2, 1096–1133, DOI 10.1016/j.aim.2010.07.002. MR2737778 [22] Qunxing Pan and Faqun Cai, (X , Y)-Gorenstein projective and injective modules, Turkish J. Math. 39 (2015), no. 1, 81–90, DOI 10.3906/mat-1306-48. MR3310713 [23] Sean Sather-Wagstaff, Tirdad Sharif, and Diana White, Stability of Gorenstein categories, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 481–502, DOI 10.1112/jlms/jdm124. MR2400403 ˇ ˇ [24] Jan Saroch and Jan Stov´ ıˇ cek, Singular compactness and definability for Σ-cotorsion and Gorenstein modules, preprint arXiv:1804.09080; 34 pp. [25] Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, SpringerVerlag, Berlin, 1996. MR1438789 [26] Xiaoyan Yang and Wenjing Chen, Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs, Comm. Algebra 45 (2017), no. 7, 2875–2888, DOI 10.1080/00927872.2016.1233226. MR3594565 L.W.C. Texas Tech University, Lubbock, TX 79409, U.S.A. Email address: [email protected] URL: http://www.math.ttu.edu/~lchriste S.E. Universidad de Murcia, Murcia 30100, Spain Email address: [email protected] URL: http://webs.um.es/sestrada P.T. Norwegian University of Science and Technology, 7491 Trondheim, Norway Email address: [email protected] URL: https://folk.ntnu.no/pedertho

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15082

Tor-pairs: Products and approximations Manuel Cort´es-Izurdiaga Abstract. Recently the author has studied rings for which products of flat modules have finite flat dimension. In this paper we extend the theory to characterize when products of modules in T have finite T -projective dimension, where T is the left hand class of a Tor-pair (T , S), relating this property with the relative T -Mittag-Leffler dimension of modules in S. We apply these results to study the existence of approximations by modules in T . In order to do this, we give short proofs of the well known results that a deconstructible class is precovering and that a deconstructible class closed under products is preenveloping.

Introduction Let R be an associative ring with unit. A Tor-pair over R is a pair of classes (T , S) of right and left R-modules respectively, such that T is the left Tor-orthogonal class of S and S is the right Tor-orthogonal class of T (see Section 1 for details). The main objective of this work is to study two problems about Tor-pairs over R: when T is closed under products and when T provides for approximations. The study of when T is closed under products is related with right Gorenstein regular rings. The ring R is said to be right Gorenstein regular [10, Definition 2.1] if the category of right R-modules is a Gorenstein category in the sense of [8, Definition 2.18]. These rings may be considered as the natural one-sided extension of classical Iwanaga-Gorenstein rings to non-noetherian rings (recall that the ring R is Iwanaga-Gorenstein if it is two sided noetherian with finite left and right self-injective dimension). In [4, Corollary VII.2.6] it is proved that the ring R is right Gorenstein regular if and only if the class of all right R-modules with finite projective dimension coincides with the class of all right modules with finite injective dimension. If we look at the class Projω of all modules with finite projective dimension, this condition has two consequences: the right projective finitistic dimension of R is finite (that is, Projω = Projn for some natural number n, where Projn denotes the class of all modules with projective dimension less than or equal to n); and the class Projω is closed under products. As in the classical case of products of projective 2010 Mathematics Subject Classification. Primary 16D40, 16E10, 16E30. Partially supported by grants MTM2017-86987-P and MTM2016-77445-P of Ministerio de Economia, Industria y Competitividad and FEDER; and by grant UAL18-FQM-B008-A-E of Universidad de Almer´ıa, Consejer´ıa de Econom´ıa, Conocimiento, Empresas y Universidad and FEDER. c 2020 American Mathematical Society

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modules studied in [5], this last property implies that products of modules with finite flat dimension have finite flat dimension. Consequently, the first step in order to understand right Gorenstein regular rings is to study rings with this property. This study is developed in [6]. In the first part of this paper we extend the theory of [6] to characterize, for a fixed Tor-pair (T , S), when products of modules in T have finite T -projective dimension (see Definition 1.2 for the definition of relative dimensions). As in the case of the flat modules, this property is related with the ML(T )-projective dimension of modules in S, see Theorem 2.2 (where ML(T ) is the class of all Mittag-Leffler modules with respect to T , see Definition 2.1). In the second part of the paper we are interested in approximations by modules in T and in Tn (modules with T -projective dimension less than or equal to n). The relationship of these approximations with the first part of the paper comes from the fact that if a class of right R-modules is preenveloping then it is closed under products [14, Propostion 1.2]. Therefore, the following natural question arises: if Tn is closed under products, when is it preenveloping? One tool in order to construct approximations of modules is that of deconstruction of classes, because a deconstructible class is always precovering, [18, Theorem 2.14] and [9, Theorem 5.5], and a deconstructible class closed under direct products is preenveloping [18, Theorem 4.19]). The procedure of deconstruction of a class X consists on finding a set S such that each module in X is S-filtered, which means that for each X ∈ X there exists a continuous chain of submodules of X, ∈ S. {Xα : α < κ} (where κ is a cardinal), whose union is X and such that XXα+1 α In Section 3 we give easy proofs of [18, Theorem 2.14] and [9, Theorem 5.5] (in Theorem 3.3) and of [18, Theorem 4.19] (in Theorem 3.8), and we prove that Tm is deconstructible for each natural number m, so that it is always precovering and it is preenveloping precisely when it is closed under products (see Corollary 3.3). Throughout the paper R will be an associative ring with unit. We shall denote by Mod-R and R-Mod the categories of all right R-modules and left R-modules respectively. Given a class X of right R-modules, we shall denote by Prod(X ) the class consisting of all modules isomorphic to a direct products of modules in X . The classes of flat and projective modules will be denoted by FlatR and ProjR respectively. If there is no possible confussion, we shall omit the subscript R. The cardinal of a set X will be denoted by |X|. 1. Tor-pairs, relative dimensions and relative Mittag-Leffler modules Given a class X of right (resp. left) R-modules we shall denote by X  (resp X ) the class of all left (resp. right) R-modules M satisfying TorR 1 (X, M ) = 0 (resp TorR (M, X) = 0) for each X ∈ X . Recall that a Tor-pair is a pair of classes 1 (T , S) such that T =  S and S = T  . Given a class X of right R-modules (resp. left R-modules), the pair ( (X  ), X  ) (resp. ( X , ( X ) )) is a Tor-pair, which is called the Tor-pair generated by X . Given a class X of left modules, a short exact sequence of right modules 

0

A

f

B

g

C

0

is called X -pure if the sequence 0

A⊗X

B⊗X

C ⊗X

0

TOR-PAIRS: PRODUCTS AND APPROXIMATIONS

121

is exact for each X ∈ X . In such case, f is called an X -pure monomorphism and g an X -pure epimorphism. Note that each pure exact sequence is X -pure exact. Proposition 1.1. Let (T , S) be a Tor-pair. Then T is closed under direct limits, pure submodules and S-pure quotients. Proof. T is closed under direct limits since the Tor functor commutes with direct limits. T is closed under pure submodules by [2, Proposition 9.12]. In order to see that it is closed under S-pure quotients, take f : T → T  a pure epimorphism with T ∈ T and denote by ι the inclusion of Ker f into T . Given S ∈ S and applying − ⊗R S we get the exact sequence Tor1 (T  , S)

TorR 1 (T, S)

Ker f ⊗R S

ι⊗S

T ⊗R S

Since T ∈ T , the first term is zero and, since Ker f is a S-pure submodule of T ,   ι ⊗R S is monic. Then TorR 1 (T , S) = 0 and, as S is arbitrary, T belongs to T .  A class X of right R-modules is called resolving if it contains all projective modules and is closed under extensions and kernels of epimorphisms. A cotorsion pair (F, C) is hereditary if F is resolving. Similarly, we shall call a Tor-pair (T , S) hereditary if T is resolving. The following result is the Tor-pair version of the well known characterizations of hereditary cotorsion pairs [12, Theorem 1.2.10]. Proposition 1.2. Let (T , S) be a Tor-pair. The following assertions are equivalent: (1) The Tor pair is hereditary. (2) S is resolving. (3) T ornR (T, S) = 0 for each T ∈ T , S ∈ S and nonzero natural number n. Proof. (1) ⇒ (3). From (1) follows that all syzygies of any module in T belong to T . Then (3) is a consequence of [17, Corollary 6.23]. (3) ⇒ (1). Given a short exact sequence 0

K

T1

T2

0

with T1 , T2 ∈ T , the induced long exact sequence when tensoring with any S ∈ S R R ∼ gives an isomorphism TorR 2 (T2 , S) = Tor1 (K, S). Then (3) gives that Tor1 (K, S) = 0 and K ∈ S. (3) ⇔ (2). Follows from the previous proof, since (3) is left-right symmetric.  Proposition 1.3. Let (T , S) be a hereditary Tor-pair. Then T is closed under S-pure submodules. Proof. We can argue as in [2, Proposition 9.12]. Let T be a module in T and K a S-pure submodule of T . Let S be any module in S and take 0

S

F

S

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a projective presentation of S. We can construct the following commutative diagram with exact rows: K ⊗R S 

TorR 1 (K, S)

0

g

f

T ⊗R S 

TorR 1 (T, S)

0

K ⊗R F T ⊗R F

Now g is monic as the inclusion K → T is S-pure and S  ∈ S by Proposition 1.2. R Then f is monic and TorR 1 (K, S) = 0 since Tor1 (T, S) = 0. Because S is arbitrary, we conclude that K ∈ T .  Example 1.1. (1) The pair of classes (Flat, R-Mod) is a hereditary Torpair. (2) Recall that a left R-module C is cyclically presented provided that C ∼ = R R for some x ∈ R. A right module X satisfying Tor (X, C) = 0 for 1 Rx each cyclically presented left module C is called torsion-free. We shall denote by TFree the class consisting of all torsion-free right modules. Then (TFree, TFree ) is a Tor-pair. We shall use the homological notation for projective resolutions so that, for a given a right R-module M , a projective resolution of M will be denoted ···

P1

d1

P0

d0

M

0

Then the nth-syzygy of M will be Ker dn for each natural number n. Definition 1.2. Let X be a class of left R-modules containing all projective modules. (1) Given a nonzero natural number n and a left R-module M , we shall say that M has projective dimension relative to X (or X -projective dimension) less than or equal to n (written pdX (M ) ≤ n) if there exists a projective resolution of M such that its (n − 1)st syzygy belongs to X . We shall denote by Xn the class of all modules with X -projective dimension less than or # equal to n (if n = 0, X0 will be X ). Moreover, we shall denote Xω = n pdT (A) then pdT (C) = pdT (B). (3) If pdT (B) = pdT (A), then pdT (C) ≤ pdT (A) + 1. Proof. Given a nonzero natural number n and S ∈ S we have, by [17, Corollary 6.30], the exact sequence (1.1)

TorR n+1 (B, S)

TorR n+1 (C, S)

TorR n (A, S)

TorR n (B, S)

TorR n (C, S)

Set nA = pdT (A) and nB = pdT (B); take SA , SB ∈ S with TorR nA (A, SA ) = 0 and TornB (B, SB ) = 0.

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If nB < nA , then the sequence (1.1) for n = nA + 1 gives TornA +2 (C, S) = 0 for each S ∈ S. Moreover, for n = nA and S = SA , the sequence (1.1) gives TornA +1 (C, SA ) = 0. Consequently pdT (C) = nA + 1. If nA < nB , then the sequence (1.1) for n = nB gives TornB +1 (C, S) = 0 for all S ∈ S. The same sequence for n = nB and S = SB gives that TorR nB (C, SB ) = 0, so that pdT (C) = nB . Finally, if nA = nB , the sequence (1.1) for n = nA + 1 gives that TorR nA +2 (C, S)  = 0 for each S ∈ S, so that pdT (C) ≤ nA + 1. 2. Tor-pairs closed under products As we have mentioned before, in [6], rings for which direct products of flat modules have finite flat dimension are characterized. Let (T , S) be a hereditary Tor-pair. In this section we study rings for which direct products of modules in T have finite T -projective dimension. The main result relates this property with the projective dimension relative to some Mittag-Leffler modules of the modules belonging to S. Mittag-Leffler modules were introduced by Raynaud and Gruson in their seminal paper [15]. We shall work with the following relativization of the concept, introduced in [16]. Definition 2.1. Let X be a class of right R-modules and M a left R-module. We say that M is X -Mittag-Leffler  any familyof modules in X , {Xi : i ∈ I},  if for X the canonical morphism from i ⊗R M to i∈I i∈I (Xi ⊗R M ) is monic. We shall denote by ML(X ) the class consisting of all X -Mittag-Leffler left Rmodules. Theorem 2.2. The following assertions are equivalent for a hereditary Tor-pair (T , S) and a natural number n. (1) Each product of modules in T has T -projective dimension less than or equal to n. (2) Each module in S has finite ML(T )-projective dimension less than or equal to n + 1. Consequently: pdT (Prod(T )) = n ⇔ pdML(T ) (S) = n + 1 Proof. Fix {Ti : i ∈ I} a family of modules in T and S an object of S. Take a projective resolution of S, ···

d2

d1

P1

d0

P0

S

0

and consider the short exact sequence 0

dn

Pn

Kn

Kn−1

0

 where Kn−1 = ker dn−1 and K−1 = S if n = 0. Tensoring by i∈I Ti we can construct the following commutative diagram with exact rows:      f  0 i∈I Ti ⊗R Kn i∈I Ti ⊗ Pn i∈I Ti ⊗ Kn−1 0



g

i∈I Ti ⊗ Kn



h

i∈I Ti ⊗ Pn

 i∈I

Ti ⊗ Kn−1

0

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(note that the last row is exact because Kn−1 ∈ S by Proposition 1.2). Since Pn is Mittag-Leffler, h is monic and, consequently, f is monic if and only if g is monic. By [17, Corollary 6.23 and Corollary 6.27] there exists a exact sequence       f 0 TorR n+1 i∈I Ti , S i∈I Ti ⊗R Kn i∈I Ti ⊗ Pn   so that f is monic if and only if TorR n+1 i∈I Ti , S = 0. The conclusion is that for a fixed  family {T  i : i ∈ I} in T and module S ∈ S, g is monic if and only if TorR T , S = 0. n+1 i∈I i Now using that both {Ti : i ∈ I} and S are arbitrary we get, by Lemma 1.5, that all products of modules in T have T -projective dimension less than or equal to n if and only if each module in S has ML(T )-projective dimension less than or equal to n + 1.  As an inmediate consequence we get a characterization of when the left hand class of a Tor-pair is closed under products. Corollary 2.1. The following assertions are equivalent for a hereditary Torpair (T , S). (1) T is closed under products. (2) Each module in S has ML(T )-projective dimension less than or equal to 1. If we apply this result to the Tor-pair induced by the flat modules, we get the following well known results. Recall that the class of flat Mittag-Leffler modules is closed under extensions since, if 0

K

M

N

0

is a short exact sequence in R-Mod with K and N flat and Mittag-Leffler, then M is flat, the sequence is pure and, for each family of right R-modules {Xi : i ∈ I} there exists a commutative diagram       0 i∈I Xi ⊗R K i∈I Xi ⊗ M i∈I Xi ⊗ N g

f

0

 i∈I

Xi ⊗ K

 i∈I

Xi ⊗ M

h

 i∈I

Xi ⊗ N

0

from which follows that g is monic, as so are f and h. Corollary 2.2. (1) Pure submodules of flat Mittag-Leffler right modules are Mittag-Leffler. (2) R is right coherent if and only if each submodule of a projective right module is Mittag-Leffler with respect to the flat modules. Proof. (1) If we apply the previous result to the Tor-pair (FlatR , R-Mod) we get that each flat right module has ML(R-Mod)-projective dimension less than or equal to 1, as R-Mod is closed under products. Noting that FlatR consists of all pure quotients of projective modules and that ML(R-Mod) is the class of all Mittag-Leffler modules, this is equivalent to all pure submodules of projective right modules being (flat) Mittag-Leffler modules. Now let M be a flat Mittag-Leffler right module and K a pure submodule of M . Let f : P → M K be an epimorphism with P projective. Constructing the pullback

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of f along the projection M → exact rows and colummns:

M K

we get the following commutative diagram with 0

0

Ker f

Ker f

0

K

Q

P

0

0

K

M

M K

0

0

0

Since the first column is pure and P is projective, Ker f is flat Mittag-Leffler by the previous proof. Then, as the class of flat Mittag-Leffler modules is closed under extensions, Q is flat Mittag-Leffler as well. But the middle row splits, so that K is isomorphic to a direct summand of Q. Thus, K is flat Mittag-Leffler. (2) If we consider the Tor-pair (Mod-R, R Flat), we get that R is right coherent if and only if R Flat is closed under products if and only if (by the left version of Corollary 2.1) each right module has projective dimension relative to the MittagLeffler modules less than or equal to 1. But this is equivalent to each submodule of a projective module being Mittag-Leffler with respect to the flat modules.  Now, what about the class Tm where m is a nonzero natural number? When is it closed under products? The following result, which extends [6, Proposition 4.1], gives the answer. Proposition 2.3. The following assertions are equivalent for a Tor pair (T , S). (1) Each module in Prod(T ) has finite T -projective dimension. (2) pdT (Prod(T )) is finite. (3) There exists a natural number m such that each module in Prod(Tm ) has finite T -projective dimension. (4) There exists a natural number m such that pdT (Prod(Tm )) is finite. (5) For any natural number m each module in Prod(Tm ) has finite T -projective dimension. (6) For any natural number m, pdT (Prod(Tm )) is finite. Moreover, when all these conditions are satisfied then pdT (Prod(Tm )) ≤ pdT (Prod(Tm+1 )) ≤ pdT (Prod(T )) + m + 1 for each natural number m. If, in addition pdT (ProdT ) = 0 (that is, T is closed under products), then Tm is closed under products for each natural number m. Proof. (1) ⇔ (2), (3) ⇔ (4) and (5) ⇔ (6) follow from Lemma 1.4. (1) ⇔ (4) and (5) ⇒ (1) are trivial. (1) ⇒ (5) is proved by dimension shifting noting that, if the result is true for some natural number m and {Ti : i ∈ I} is a family of modules having T -projective

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dimension less than or equal to m + 1 then, for each i ∈ I there exists a short exact sequence 0

Ki

Pi

Ti

0

with Pi projective and Ki ∈ Tm . Since the direct product is an exact functor, these sequences give the exact sequence    0 0 (2.1) i∈I Ki i∈I Pi i∈I Ti in which both the first and second  term have finite T -projective dimension by the induction hyphotesis. Then so has i∈I Ti by Proposition 1.4. In order to prove the last inequality we shall proceed by induction on m. Suppose that we have proved the result for some natural number m. The first inequality is trivial, since Tm ⊆ Tm+1 . In order to prove the other one simply note that for any family of modules {Ti : i ∈ I} in Tm+1 we can construct, as above, a short exact sequence  0 0 K P i∈I Ti with P ∈ Prod(T ) and K ∈ Prod(Tm ). Using Proposition 1.4 and the induction hyphotesis we get the desired inequality. Finally, if pdT (Prod(T )) = 0 we induct on m. If Tm = Tm−1 , the result follows from the induction hyphotesis. If Tm = Tm−1 , then the preceeding inequality gives pdT (Prod(Tm )) ≤ m. In addition m ≤ pdT (Prod(Tm )) as well, so that Tm is closed under products.  As an application of this result we can characterize when the class Tω is closed under products: Corollary 2.4. The following assertions are equivalent for a hereditary Torpair (T , F). (1) Tω is closed under direct products. (2) pdT (Tω ) and pdML(T ) (S) are finite. That is, the right finitistic T -projective dimension is finite and each module in S has finite ML(T )-projective dimension. Proof. (1) ⇒ (2). If Tω is closed under direct products, we can apply Lemma 1.4 to get that pdT (Tω ) is finite. That is, Tω = Tn for some natural number n. Now pdML(T ) (S) is finite as a consequence of Theorem 2.2 and Proposition 2.3. (2) ⇒ (1). Since pdT (Tω ) is finite, there exists a natural number n such that Tω = Tn . Now, as pdML(T ) (S), apply Corollary 2.1 to get that each product of modules in T has finite T -projective dimension. By Proposition 2.3, Tn is closed under products as well.  Recall that a class X of right R-modules is definable if it is closed under direct products, direct limits and pure submodules. As a consequence of the results of this section we can characterize when, fixed a Tor-pair (T , S), the classes Tm are definable for each natural number m. The same proof of [6, Proposition 4.7] gives: Proposition 2.5. Let (T , S) be a hereditary Tor-pair and n a natural number. Then Tn is closed under direct limits and pure submodules.

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Proof. The closure under direct limits follows from Lemma 1.5 and the fact that the TorR n+1 functor commutes with direct limits. In order to see that Tn is closed under pure submodules take T ∈ Tn and T  a pure submodule of T . Arguing as in [6, Proposition 4.7] we get, for each S ∈ S, the exact sequence T   0 TorR TorR TorR 0 n+1 (T , S) n+1 (T, S) n+1 T  , S Now, if T ∈ Tn then so does T  by Lemma 1.5.



Putting all things together, we charactize when Tm is a definable class for each m ∈ N. This result extends [2, Proposition 9.12] Corollary 2.6. The following assertions are equivalent for a hereditary Torpair (T , S). (1) Each module in S has ML(T )-projective dimension less than or equal to 1. (2) T is closed under products. (3) Tm is a definable category for each natural number m. Proof. (1) ⇔ (2) is Corollary 2.1. (2) ⇔ (3) follows from propositions 2.3 and 2.5.  3. Approximations by modules in Tm In this section we study the existence of approximations by modules in Tm for each natural number m. Let X be a class of right R-modules and M a module. An X -precover of M is a morphism f : X → M with X ∈ X such that for each X  ∈ X , the induced morphism HomR (X  , X) → HomR (X  , M ) is epic. The X -precover f is said to be an X -cover if it is minimal in the sense that each endomorphism g of X satisfying f g = f is an isomorphism. The class X is called precovering or covering if each right module has an X -precover or an X -cover respectively. Dually are defined X -preenvelopes and X -envelopes, and the corresponding preenveloping and enveloping classes. Most of the known examples of classes providing for approximations are part of a “small” cotorsion pair (F, C) (in the sense that it is generated by a set, i. e., there exists a set of modules G such that C = G ⊥ ). This is due to the fact that a cotorsion pair generated by a set always provide for precovers and preenvelopes, [13, Theorem 3.2.1] and [13, Lemma 2.2.6]. Moreover, by [13, Theorem 4.2.1], the left hand class of a cotorsion pair generated by a set is deconstructible (the definition will be precised later) and it has recently proved that deconstructible classes are precovering (see [18, Theorem 2.14] for a proof in exact categories and [9, Theorem 5.5] for a proof in module categories), and that deconstructible classes closed under products are preenveloping [18, Theorem 4.19]. In this paper we are going to work with deconstructible classes. We are going to give easier proofs of the aforementioned results concerning deconstructible classes and approximations. Next we will use these results to prove that, if (T , S) is a Torpair and m a natural number, then Tm is always precovering, and is preenveloping povided it is closed under direct products, i. e., the conditions of Corollary 2.6 are satisfied. Given a class of right R-modules G, a G-filtration of a module M is a continuous chain of submodules of M , (Gα : α < κ), where κ is a cardinal, such that M =

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129

#

Gα , G0 = 0 and GGα+1 ∈ G for each α < κ. We shall denote by Filt −G α the class of all G-filtered modules. We shall say that a class of modules X is deconstructible if there exists a set of modules G such that X = Filt −G. We begin proving that a deconstructible class is precovering. Given a class of right modules X and a module M , the trace of X in M is the submodule  tX (M ) = Im f α i, and its projective cover P (Li ) has to be an injective E(Lp(i) ) for some p(i) < i, and has composition series Lp(i) , . . . , Li . Each projective is injective, so it is the projective cover of its top, and vice-versa. Thus, P (Li ) = E(Lp(i) ), and E(Li ) = P (Lq(i) ), which shows that p and q are inverse functions. Moreover, E(Li+1 ) has at least length q(i), since Li+1 has E(Li )/Li as an essential extension; but in fact we can not have E(Li+1 ) = E(Li )/Li since then as it is projective, E(Li )/Li would be a direct summand of E(Li ), which is not possible. Thus, q(i + 1) > q(i). Hence, q is an increasing invertible function, and thus there is k such that q(i) = i + k (in fact, to get this conclusion once we know q is bijective, it would have been enough to prove that q is only weakly increasing). This shows that all injective indecomposables have length k + 1.  Corollary 2.8. Let H be a coserial Hopf algebra. Then exactly one of the following two hold: (i) H is co-Frobenius and as coalgebra, H is isomorphic to a direct sum of serial coalgebras of bounded type Cn,k and Lk , for various n, k. (ii) H is not co-Frobenius and as coalgebra, H is isomorphic to a direct sum of serial coalgebras of unbounded (infinite) type C∞ and L∞ , in which case MH is a hereditary category. Proof. There are two mutually exclusive cases to consider: (1) If some finite dimensional injective comodule exists, then Rat(H ∗ ) = 0, all injective indecomposable comodules are finite dimensional (see [DNR]; see also [I5, I3]) and H is co-Frobenius. In this case, by Proposition 2.7, H is isomorphic as a coalgebra to a direct sum of connected serial coalgebras, each of which is coFrobenius of type Cn,k or Lk . (2) If there is no finite dimensional injective comodule, then H, as a coalgebra, is (n or a direct sum of connected serial coalgebras Di , each of which having either A ∞ A∞ as its Ext quiver. Again, the other types are excluded by the same reason as in Proposition 2.7 - a vertex which is either a left or right endpoint gives rise to a simple left or right injective, leading to a case (1) situation. Since all injective indecomposables are infinite dimensional, each of these coalgebras Di are of type  C∞ or L∞ . 3. Questions We close this paper with a number of open questions. The first two refer to our earlier criterion on the (strong) surjectivity and bijectivity of the antipode. Question 3.1. If the antipode of H is surjective, does it follow that every simple right H-comodule is isomorphic to the dual of a (necessarily simple) left H-comodule V ? Question 3.2. If every simple right H-comodule L is isomorphic to the dual of a (necessarily simple) right H-comodule V , does it follow that S is bijective (so 2 (−)S is dense)? Of course, it is not possible that the answer to both of the above questions is yes. The next two questions are related to co-serial Hopf algebras and other classes of Hopf algebras which are contained in the co-serial ones. In view of the classifications

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of pointed co-serial Hopf algebras [I7] and of Hopf algebras which are co-monomial [DIN], we mention the following: Question 3.3. Give a structure theorem or detailed classification of co-serial Hopf algebras. Another question not related to the bijectivity of the antipode but still in the world of serial algebras concerns algebras of finite type. Classically, algebras of finite representation type (i.e. finite dimensional algebras which have only finitely many indecomposable representations up to isomorphism) appear in the representation theory of groups in positive characteristic. It is well known that a group algebra KG of a finite group G has finite type either if it is semisimple or the characteristic p of K divides the order of the group and the p-Sylow subgroups of G are cyclic. Finite quantum groups of finite representation type were studied and characterized in the cocommutative case in [FV], and in the case where the Hopf algebra is dual pointed in [LL], where it is shown that such Hopf algebras of finite representation type are all monomial, and then by other results of [ChHYZ] and [I7] they are serial (see also [CR], [DIN]). When G is group with normal cyclic p-Sylow subgroup, then KG is serial, but in general it does not have to be, but it is a so called Brauer-tree algebra. Hence, we formulate the following: Question 3.4. Give a description of the representation theory of a finite dimensional Hopf algebra H of finite (co)representation type. When is such an H (co)serial, or a Brauer-tree algebra? The bijectivity of the antipode allows one to see that certain properties are left-right symmetric for Hopf algebras - such as co-serial, co-Frobenius, and in some situations, Noetherian. A particularly interesting bijectivity question motivated by noncommutative geometry is about Noetherian Hopf algebras. While co-Frobenius coalgebras are largely regarded as functions on locally compact quantum groups, Noetherian Hopf algebras, which include universal enveloping algebras of finite dimensional Lie algebras or coordinate rings of affine algebraic groups and their quantized versions, can be geometrically regarded as functions on quantum algebraic groups (see the surveys [Br, G]). In [Sk], it is proved that a Noetherian Hopf algebra has an injective antipode, and semiprime Noetherian or affine PI Hopf algebras have bijective antipodes. We end by reasserting here the following very interesting Conjecture of Skryabin from [Sk], reported also by [Br, G]. Conjecture 3.5 (Skryabin). Let H be a Noetherian Hopf algebra. Then the antipode of H is bijective. Acknowledgment The author acknowledges the careful work of the referee, whose remarks greatly improved the presentation. References [A]

[Ag]

Eiichi Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. MR594432 A. L. Agore, Monomorphisms of coalgebras, Colloq. Math. 120 (2010), no. 1, 149–155, DOI 10.4064/cm120-1-11. MR2652613

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Department of Mathematics, University of Iowa, McLean Hall, Iowa City, IA, USA; and Simion Stoilow Institute of the Romanian Academy, PO-Box 1-764, 014700 Bucharest, Romania Email address: [email protected]; [email protected]

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15120

Categories with negation Jaiung Jun and Louis Rowen In honor of our friend and colleague, S.K. Jain. Abstract. We continue the theory of T -systems from the work of the second author, describing both ground systems and systemic modules over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Prime ground systems are introduced as a way of developing geometry. The polynomial system over a prime system is prime, and there is a weak Nullstellensatz. Also, the polynomial A[λ1 , . . . , λn ] and Laurent polynomial systems A[[λ1 , . . . , λn ]] in n commuting indeterminates over a T -semiringgroup system have dimension n. For systemic modules, special attention also is paid to tensor products and Hom. Abelian categories are replaced by “semi-abelian” categories (where Hom(A, B) is not a group) with a negation morphism.

Contents 1. Introduction 2. Background 3. The structure theory of ground triples via congruences 4. The geometry of prime systems 5. Tensor products 6. The structure theory of systemic modules via congruences 7. Functors among semiring ground triples and systems 8. Appendix A: Interface between systems and tropical mathematics 9. Appendix B: The categorical underpinning Acknowledgment References

2010 Mathematics Subject Classification. Primary 08A05, 08A30, 14T05, 16Y60, 20N20; Secondary 06F05, 08A72, 12K10, 13C60. Key words and phrases. Category, pseudo-triple, matroid, pseudo-system, triple, system, semiring, semifield, congruence, module, negation, surpassing relation, symmetrization, congruence, tropical algebra, supertropical algebra, bipotent, meta-tangible, symmetrized, hypergroup, hyperfield, polynomial, prime, dimension, algebraic, tensor product. c 2020 American Mathematical Society

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1. Introduction This paper, based on [39], is the continuation of a project started in [55, 56] as summarized in [57], in which a generalization of classical algebraic theory was presented to provide applications in related algebraic theories. It is an attempt to understand why basic algebraic theorems are mirrored in supertropical algebra, spurred by the realization that some of the same results were obtained in parallel research on hypergroups and hyperfields [6, 24, 34–36, 61] and fuzzy rings [15, 17, 24], which lack negatives although each has an operation resembling negation 1 2 . The underlying idea is to take a set T that we want to study. In the situation considered here, T also has a partial additive algebraic structure which is not defined on all of T ; this is resolved by having T act on a set A with a fuller algebraic structure. Lorscheid [44, 45] developed this idea when T is a monoid which is a subset of a semiring A. Since semirings may lack negatives, we introduce a formal negation map (−) in Definition 2.3, resembling negation, often requiring that T generates A additively. In tropical algebra, (−) can be taken to be the identity map. Or it can be supplied via “symmetrization” (§2.2), motivated by [2, 21, 22, 53]. Together with the negation map, A and T comprise a pseudo-triple (A, T , (−)). This is rounded out to a pseudo-system (A, T , (−), ) with a surpassing relation , often a partial order (PO), replacing equality in the algebraic theory. Ironically, equality in classical mathematics is the only situation in which  is an equivalence. We set forth a systemic foundation for affine geometry (based on prime systems) and representation theory, as well as to lay out the groundwork for further research, cf. [3] for linear algebra, [20] for exterior algebra, [37] for projective modules, and [38] for homology, and other work in progress. Familiar concepts from classical algebra were applied in [56] to produce new triples and systems, e.g., direct powers [56, Definition 2.6], matrices [56, §6.5], involutions [56, §6.6], polynomials [56, §6.7], localization [56, §6.8], and tensor products [56, §8.6]. Recently tracts, a special case of pseudo-systems, were introduced in [7] in order to investigate matroids. At the end of this paper we also view systems categorically, to make them formally applicable to varied situations. Since the main motivation comes from tropical and supertropical algebra, in Appendix A (§8), we coordinate the systemic theory with the main approaches to tropical mathematics, demonstrating the parallels of some tropical notions such as bend congruences (introduced by J. Giansiracusa and N. Giansiracusa in [23], and tropical ideals (introduced by Maclagan and Rincon in [46]). For example, [57, Proposition 7.5] says that the bend relation implies ◦-equivalence, cf. Definition 8.8. Category theory involving explicit algebraic structure can be described in universal algebra in terms of operations and identities, reviewed briefly in §2.6. But the “surpassing relation”  also is required. A crucial issue here is the “correct” definition of morphism. One’s initial instinct would be to take “homomorphisms,” preserving all the structure from universal algebra. However, this does not tie into hyperfields, for which a more encompassing definition pertains in [34, 36]. Accordingly, we define the more general 1 In a hypergroup, for each element a there is a unique element called −a, such that 0 ∈ a  (−a). 2 A fuzzy ring A has an element ε such 1 + ε is in a distinguished ideal A , and we define 0 (−)a = εa.

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-morphism in Definition 2.37, satisfying f (a + a )  f (a) + f (a ). Another key point is that in the theory of semirings and their modules, homomorphisms are described in terms of congruences. The trivial congruences contain the diagonal, not just zero. Mor(A, B) denotes the semigroup of -morphisms from A to B, and has the sub-semigroup Hom(A, B) of homomorphisms. Since (Mor(A, B) , +) and (Hom(A, B), +) no longer are groups, one needs to weaken the notion of additive and abelian categories, respectively in §9.2.2 to semi-additive categories and semi-abelian categories, [26, §1.2.7]. The tensor product and its abstraction to monoidal categories, an important aspect of algebra, is exposed in [18] for monoidal abelian categories. But, to our dismay, the functoriality of the tensor product runs into stumbling blocks because of the asymmetry involved in . So we have a give and play between -morphisms and homomorphisms, which is treated in §5. 1.1. Objectives. Our objectives in this paper are as follows: (1) Lay out the notions in §2 of T -module, negation map, “ground” triples and T -systems, which should parallel the classical structure theory of algebras. In the process, we consider convolutions and the ensuing construction of polynomials, Laurent series, etc. (2) Introduce affine geometry in §3.2 in terms of polynomials, with special attention to the theory of prime ground triples, to lay out the groundwork for the spectrum of prime congruences. (3) Elevate congruences to their proper role in the theory, in §3 and §6, since the process of modding out ideals suffers from the lack of negation. Investigate which classical module-theoretic concepts (such as Hom and direct sums) have analogs for modules over ground triples, viewed in terms of their congruences. Special attention is given to the tensor product (Definition 5.5). In general -morphisms do not permit us to build tensor categories, as seen in Example 5.6, but we do have the usual theory using homomorphisms, in view of [43]. (4) Express these notions in categorical terms. This should parallel the theory of modules over semirings, which has been developed in the last few years by Katsov [40–42], Katsov and Nam [43], Patchkoria [51], Macpherson [48], Takahashi [59]. (5) Provide the functorial context for the main categories of this paper, as indicated in the diagram given in §7. (6) Relate this theory in Appendix A to other approaches in tropical algebra. (7) Define negation morphisms and negation functors, together with a surpassing relation, in the context of N -categories, in Appendix B. In the process, we generalize abelian categories to semi-abelian categories with negation, and lay out the role of functor categories. 1.2. Main results. Our results in geometry require prime congruences (Definition 3.7). Proposition A (Proposition 3.17). For every T -congruence Φ on a com√ mutative T -semiring system, Φ is an intersection of prime T -congruences.

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Theorem B (Theorem 4.6). Over a commutative prime triple A, any nonzero polynomial f ∈ T [λ] of degree n cannot have n + 1 distinct ◦-roots in T . Corollary C (Corollary 4.7). If (A, T , (−)) is a prime commutative triple with T infinite, then so is the polynomial triple (A[λ], TA[λ] , (−)). This corollary plays a key role in the geometry of systems, in terms of the prime spectrum. Theorem D (Theorem 4.26, Artin-Tate lemma, -version). Suppose A = A[a1 , a2 , . . . , an ] is a -affine system over A, and K a subsystem of A , with A having a -base v1 = 1, . . . , vd of A over K. Then K is -affine over A. Theorem E (Theorem 4.33). If (A, T , (−)) is a semiring-group system and (A = A[a1 , . . . , am ], T  , (−)) is a -affine semiring-group system, in which (f, g)(ai , 0) is invertible for every symmetrically functionally tangible pair (f, g) 0 has a of polynomials, then a1 , . . . , am are symmetrically algebraic over T , and A symmetric base over T . The next result is rather easy, but its statement is nice. Theorem F (Theorem 4.37). Both the polynomial A[λ1 , . . . , λn ] and Laurent polynomial systems A[[λ1 , . . . , λn ]] in n commuting indeterminates over a T semiring-group system have dimension n. Tensor products are described in detail in §5.1. From the categorical point of view: Proposition G (Proposition 9.16). The category of systemic modules over A is a monoidal semi-abelian category, with respect to negated T -tensor products. 2. Background Here is a review of what is needed to understand the main results. 2.1. Basic structures. A semiring† (A, +, ·, 1) is an additive abelian semigroup (A, +) and a multiplicative monoid (A, ·, 1) satisfying the usual distributive laws. A semiring is a semiring† which contains a 0 element. A semidomain is a semiring A such that A \ {0} is closed under multiplication, i.e., ab = 0 only when a = 0 or b = 0 for all a, b ∈ A. We do not assume commutativity of multiplication, since we want to consider matrices and other noncommutative structures more formally. But we often specialize to the commutative situation (for example in considering prime spectra) when the proofs are simpler. We stay mainly with the associative case, in contrast to [56]. Distributivity is a stickier issue, since we do not want to forego the applications to hypergroups. As customary, N denotes the nonnegative integers, Q the rational numbers, and R the real numbers, all ordered monoids under addition. Let us give a brief review of the theory of triples, from [56].

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2.1.1. T -modules. First we put T into the limelight, starting with the following fundamental basic structure. Definition 2.1. A (left) T -module† over a set T is an additive semigroup (A, +) with a scalar multiplication T × A → A satisfying a( uj=1 bj ) = uj=1 (abj ), ∀a ∈ T (distributivity with respect to T ). A T -module over T is a T -module† (A, +, 0A ) satisfying a0A = 0A for any a∈T. In other words, T acts on A. Often T is a multiplicative group. For example, T ∪ {0} might be a hyperfield generating A inside its power set. Or T might be an ordered group, and A its supertropical semiring (or symmetrized semiring). Or A might be a fuzzy ring, and T its subset of invertible elements, as described in [56, Appendix B]. Definition 2.2. A T -monoid module over a monoid (T , ·, 1) is a T -module (A, +) that also respects the monoid structure, i.e., (A, +) satisfies the following additional axioms, ∀ai ∈ T , b ∈ A: (i) 1T b = b. (ii) (a1 a2 )b = a1 (a2 b). We delete the prefix T - when it is unambiguous. When (T , ·) is a group, we call A a group module to emphasize this fact. In §4.2 we shall see how localization permits us to reduce from monoid modules to group modules. 2.1.2. Negation maps. We need some formalism to get around the lack of negation. Definition 2.3. A negation map on a T -module A is a semigroup automorphism (−) of (A, +) and a map T → T of order at most two, also denoted as (−), respecting the T -action in the sense that a((−)b) = ((−)a)b for a ∈ T , b ∈ A. When T is a subset of A, we require that (−) on A restricts to (−) on T . For monoid modules, it is enough to know (−)1T : Lemma 2.4. Let T be a monoid, and take ε in T with ε2 = 1. (i) There is a unique negation map on T and A for which (−)1T = ε, given by (−)a = εa and (−)b = εb for a ∈ T , b ∈ A. Furthermore, (2.1)

(−)(ab) = ((−)a)b = a((−)b). (ii) When A also is a semiring and T ⊆ A generates (A, +), then (2.1) holds for any a, b ∈ A.

Proof. (i) a((−)b) = a(εb) = a 1T εb = εa 1T b = ((−)a)b, and (−)(ab) = ε(ab) = (εa)b = ((−)a)b.  (ii) Write a = i ai for ai ∈ T . Then    (−)(ab) = ε ai b = ((−)ai )b = ai ((−)b) = a((−)b). i

i

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We write a(−)b for a + ((−)b), and a◦ for a(−)a, called a quasi-zero. Thus, in classical algebra, the only quasi-zero is 0 itself if the negation map is the usual negation. Of special interest are the sets A◦ := {a◦ : a ∈ A} and T ◦ := {a◦ : a ∈ T }, inferred from the “diodes” of [21], Izhakian’s thesis [28], and the “ghost ideal” of [31] and studied explicitly in the symmetrized case in [56, §3.5.1], and [12, §4] (over the Boolean semifield B). Example 2.5. Major instances of negation maps: (i) Equality (taking ε = 1). (ii) The switch map in symmetrization, in Definition 2.19 below (taking ε = (0, 1) under the twisted multiplication). (iii) The negation map in a hypergroup (taking ε = −1). (iv) The negation map (−)a → εa in a fuzzy ring. 2.1.3. Pseudo-triples and triples. Definition 2.6. (i) A pseudo-triple (A, T , (−)) is a T -module A with a negation map (−). (ii) A TA -pseudo-triple (A, TA , (−)) is a pseudo-triple (A, TA , (−)), with TA designated as a distinguished subset of A. The elements of TA are called tangible. (iii) A TA -triple is a TA -pseudo-triple (A, TA , (−)), in which TA ∩ A◦ = ∅ and TA generates (A \ {0}, +). (iv) A T -semiring pseudo-triple (A, TA , (−)) is a TA -pseudo-triple for which A is also a semiring, and the semiring multiplication on A restricts to the TA -module† multiplication TA × A → A. Remark 2.7. The condition that TA ∩ A◦ = ∅ in (iii) fails in the max-plus algebra, but holds in the other examples of interest to us. When 1A ∈ A, we can put TA = T 1A , and thus get a TA -pseudo-triple. The most straightforward way of ensuring that TA generates (A, +) is to restrict (A, +) case, we define the height of an to the T -submodule generated by TA . In this  t element c ∈ A as the minimal t such that c = i=1 ai with each ai ∈ TA . (0 has height 0.) The height of A is the maximal height of its elements (which is said to be ∞ if these heights are not bounded). The following properties play a basic role in the theory. Definition 2.8. (i) A T -triple (A, TA , (−)) has unique negation if a0 + a1 ∈ A◦ for ai ∈ TA implies a1 = (−)a0 . (ii) (A, TA , (−)) is meta-tangible if a0 + a1 ∈ TA for any a1 = (−)a0 in TA . (iii) (A very important special case) A T -pseudo-triple (A, TA , (−)) is (−)bipotent if a + b ∈ {a, b} whenever a, b ∈ TA with b = (−)a. In other words, a + b ∈ {a, b, a◦ } for all a, b ∈ TA . The unique negation property is needed to get started in the theory, and metatangibility is a rather pervasive property studied in [56].

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Definition 2.9. A homomorphism ϕ : (A, TA , (−)) → (A , TA , (−)) of pseudo-triples is a T -module homomorphism ϕ : A → A satisfying ϕ(a) ∈ TA ∪ {0}, 3 ∀a ∈ TA , and ϕ((−)a) = (−)ϕ(a), ∀a ∈ TA . 2.1.4. Surpassing relations and systems. Having generalized negation to the negation map, our next goal is a workable substitute for equality. Definition 2.10 ( [56, Definition 4.5]). A surpassing relation on a pseudotriple (A, T , (−)), denoted , is a partial pre-order on T and on A satisfying the following, for elements ai , a ∈ T and bi , b ∈ A: (i) b1  b whenever b1 + c◦ = b for some c ∈ A◦ . (ii) If b1  b2 then (−)b1  (−)b2 . (iii) If b1  b2 then ab1  ab2 . (iv) If b1  b2 and b1  b2 for i = 1, 2 then b1 + b1  b2 + b2 . (v) If a1  a2 then a1 = a2 . By a surpassing PO, we mean a surpassing relation that is a PO. A T -surpassing relation on a T -pseudo-triple A is a surpassing relation which also satisfies the following extra condition: b◦  a for any a ∈ T , b ∈ A. Lemma 2.11. Condition (i) is a formal consequence of the following weaker condition: 0  c◦ , ∀c ∈ A. Proof. Since 0  c◦ , it follows from (iv) that b1 = b1 + 0  b1 + c◦ = b.



Definition 2.12. Let (A, T , (−)) be a T -pseudo-triple. We define the surpassing relation ◦ on A by: a ◦ b if and only if b = a + c◦ for some c ∈ A. Likewise, we define the surpassing relation ◦ on an A-module M, by: a ◦ b if and only if b = a + c◦ for some c ∈ A. Definition 2.13. We define the following subset of A: ANull = {b ∈ A : b + b + b , ∀b ∈ A}. Remark 2.14. (i) One can easily check that ANull is a submodule of A. Furthermore, since 0 ∈ M, for an A-module M, MNull has the following simpler description MNull = {b ∈ M : b + 0}. ◦

(ii) MNull = M , when  is ◦ as given in Definition 2.12. (iii) In the hypergroup setting of [56, §3.6, Definition 4.23], ANull consists of those sets containing 0, which is the version usually considered in the hypergroup literature, for instance, in [24]. (iv) ANull = A \ {0} for the Green relation [56, Remark 3.1(i)], so in this case the theory degenerates. 3 One might prefer to require A only to be a T -module† with ϕ(a) ∈ T  for all a ∈ T , but A A we want to permit projections of the free module to be module homomorphisms.

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A motivating example from classical algebra (for which  is just equality) is for A to be an associative algebra graded by a monoid; T could be the subset of homogeneous elements, in particular, a submonoid of (A, ·). A transparent example is when T is the multiplicative subgroup of a field A. We are more interested in the non-classical situation, involving semirings which are not rings. Our structure of choice is as follows: Definition 2.15. A system (resp. pseudo-system) is a quadruple (A, T , (−), ), where  is a surpassing relation on the triple (resp. pseudo-triple) (A, T , (−)), which is uniquely negated in the sense that if a + a ∈ TNull for a, a ∈ T , then a = (−)a. (Compare with Definition 2.8.) A semiring system is a system where A is a semiring. A semiring-group system is a semiring system where T is a group. The default ◦-pseudo-system of a pseudo-triple (A, T , (−)) is (A, T , (−), ◦ ). T -systems, etc., are defined analogously, where we assume that T ⊆ A. It is convenient to modify , generalizing Definition 2.12. Definition 2.16. Given a pseudo-system (A, T , (−), ), define Null by b Null b if b = b + c for some c ∈ ANull . The default -pseudo-system of a pseudo-system (A, T , (−)) is (A, T , (−), Null ). Null is a PO if it satisfies the condition, called upper bound (ub), that a + b + c = a for b, c ∈ ANull implies a + b = a. When ANull =A◦ , then Null is just ◦ , which happens in virtually all of our applications, in which case we can skip the technicality of utilizing the system in defining another surpassing relation. Systems are the main subject in [56], employed there to establish basic connections with tropical structures, hypergroups, and fuzzy rings, by means of the following examples: • (The standard supertropical T -triple) (A, T , (−)) where A = T ∪ A◦ and (−) is the identity map. (A◦ is called the set of “ghost elements” G.) We get the default ◦-system, and  is a PO. • (The hypersystem [56, §3.6.1]) Let T be a hyperfield. Then one can associate a triple (S(T ), T , (−)), where P(T ) is its power set (with componentwise operations), S(T ) is the additive sub-semigroup of P(T ) spanned by the singletons, and (−) on the power set is induced from the hypernegation. Here  is ⊆, which is a PO. • Symmetrized systems. (See §2.2 below.) • Fuzzy rings are described as default ◦-systems, in [56, Appendix B]. This wealth of examples motivates a further development of the algebraic theory of T -systems, which is the rationale for this paper. Remark 2.17. Recently [7], [8, §1.2.1] defined a tract to be a pair (G; NG ) consisting of an abelian group G (written multiplicatively), together with a subset NG (called the nullset of the tract) of the group semiring N [G] satisfying: • The zero element of N [G] belongs to NG , and the identity element 1 of G is not in NG .

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• NG is closed under the natural action of G on N [G]. • There is a unique element ε of G with 1 + ε ∈ NG . Tracts are special cases of semiring-group T -systems, where T is the given abelian group G, A = N[G], ε = (−)1, and NG can be taken to be ANull , often taken to be A◦ . Likewise, pasteurized blueprints [8] are semiring T -systems with unique negation, when the map  is a PO. 2.2. Symmetrization and the twist action. In this subsection, we utilize an idea of Gaubert [21] to provide a negation map and surpassing relation for an arbitrary T -module A, when it is lacking. Definition 2.18. Let T = T0 ∪ T1 . A T -module A is said to be a T -super module if A is a Z2 -graded semigroup A0 ⊕ A1 , satisfying Ti Aj ⊆ Ai+j , where Ti = T ∩ Ai , subscripts modulo 2. From now on, A) denotes A × A, i.e., A0 = A1 = A. Definition 2.19. The switch map on A) is given by (b0 , b1 ) → (b1 , b0 ). In order to identify the second component as the negation of the first, we employ an idea dating back to Bourbaki [11] and the Grothendieck group completion, as well as [10, 12, 21, 33, 35, 56] (done here in generality for T -modules), which comes from the familiar construction of Z from N. The idea arises from the elementary computation: (a0 − a1 )(b0 − b1 ) = (a0 b0 + a1 b1 ) − (a0 b1 + a1 b0 ). Definition 2.20. A) is called the symmetrization of A. For any T -module A, the twist action, denoted ·tw , on A) over T) is given by the super-action, namely (2.2)

(a0 , a1 ) ·tw (b0 , b1 ) = (a0 b0 + a1 b1 , a0 b1 + a1 b0 ).

When T is a monoid, we view T) as a monoid, also via the twist action as in (2.2), where ai , bi ∈ T , the unit element of T) being (1T , 0). Lemma 2.21. A) is a T) -module. When A is a T -monoid module, A) is a T) monoid module. Proof. To see that the twist action is associative over T) , we note for ai , bi ∈ T and ci ∈ M that (2.3) ((a0 , a1 ) ·tw (b0 , b1 )) ·tw (c0 , c1 ) = (a0 b0 + a1 b1 , a0 b1 + a1 b0 ) ·tw (c0 , c1 ) = (a0 b0 c0 + a1 b1 c0 + a0 b1 c1 + a1 b0 c1 , a0 b0 c1 + a1 b1 c1 + a0 b1 c0 + a1 b0 c0 ), in which we see that the subscript 0 appears an odd number of times on the left and an even number of times on the right, independently of the original placement of parentheses.  Remark 2.22. For any (a0 , a1 ) ∈ T) , {(a0 , a1 )}·tw A) ⊆ A) yields a T -submodule of A.

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Symmetrization plays an especially important role in geometry, cf. §4.3.1, and in homology theory, to be seen in [38]. But T) is too big for our purposes (since it is not a group), so we take instead TA = (T × {0}) ∪ ({0} × T ). Lemma 2.23. If T is a group, then TA also is a group under the twist action. Proof. In TA, (a, 0)−1 = (a−1 , 0) and (0, a)−1 = (0, a−1 ).



Definition 2.24. Let A be a T -module. ) T , (−)) is a T -pseudo-triple, (i) The symmetrized pseudo-triple (A, A ) where A = A × A with componentwise addition, and with multiplication T)A × A) → A) given by the twist action. Here we take (−) to be the switch map. (ii) The symmetrized pseudo-system is the default ◦-system of the symmetrized pseudo-triple. Here A)Null = A)◦ = {(b, b) : b ∈ A}. We identify A inside A) via the injection a → (a, 0). ) T , (−)) is a triple with unique negation and T) ◦ = {(a, a) : Lemma 2.25. (A, A a ∈ T \ {0}}. Proof. The first assertion is clear, and T) ◦ = {(a, a) : a ∈ T \ {0}} since for any (a, 0) (resp. (0, a)), (−)(a, 0) = (0, a) (resp. (−)(0, a) = (a, 0)) and hence (a, a) = (a, 0) + (0, a) = (a, 0)(−)(a, 0).  The triple A) is not meta-tangible. This can be rectified by redefining addition on TA to make it meta-tangible as done in [21] or [56, Example 3.9], but here we ) to make it find it convenient to use the natural (componentwise) addition on A, applicable for congruences. Here is an important application of the twist, whose role in tropical geometry is featured in [33]. Definition 2.26. The symmetrized semiring† A) := A × A of a semiring A is A) viewed as a symmetrized TA-module, and made into a semiring via the “twisted” multiplication of (2.2) for ai , bi ∈ A. 2.3. Ground systems and systemic modules. Representation theory often is described in terms of an abelian category, such as the class of modules over a given ring. Analogously, there are two main ways of utilizing pseudo-systems. 2.3.1. Ground systems. We call a triple (resp. system) (A, T , (−)) a ground triple (resp. ground system) when we study it as a small category with a single object in its own right, often a commutative semidomain. In short, our overall strategy is to fix a ground triple (A, T , (−), ), often (−)-bipotent, and then consider its “prime” homomorphic images, as well as the module systems over this ground T -system, to be defined presently.

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2.3.2. Systemic modules over ground systems. One defines a module (called semimodule in [25]) over a semiring A, in analogy to modules over rings. Definition 2.27. (i) Let A = (A, TA , (−)) be a T -semiring triple. A TA -module M := (M, TM , (−)) with negation is said to be a left A-module over a monoid triple A = (A, TA , (−)) if M is an A-module such that TA acts on TM and satisfies the following condition: ((−)a)b = a((−)b) = (−)(ab),

∀a ∈ TA , b ∈ TM .

A (systemic) module over a ground T -system A = (A, T , (−), ), is an A-module (M, T , (−)) with a surpassing relation, also denoted , satisfying (a) ((−)a)m = (−)(am) for a ∈ A, m ∈ M. (b) am  a m for a  a ∈ A, m  m ∈ M. Analogously, we define a right (systemic) A-module from the other side, and an (A, A )-(systemic) bimodule. (ii) A group module is an A-module for which TA is a group. So we study systemic modules over a fixed ground triple. We use M instead of A to denote a systemic module. A (systemic) submodule of M is required to contain MNull . 0 over A) Remark 2.28. The twist action (Definition 2.20) on the module M 0 Indeed, suppose (x0 , x1 ) ∈ M 0 and (a0 , 0) ∈ extends the TA -module action on M. 0 0 TA . Then (a0 , 0) ·tw (x0 , x1 ) = (a0 x0 , a0 x1 ) ∈ M. Proposition 2.29. If N is a sub-triple of an A-module triple M with negation map (−), then N becomes a T0 A -submodule of M under the action (a0 , a1 )x = a0 x(−)a1 x. Proof. (a(a0 , a1 ))x = aa0 x(−)aa1 x = a((a0 , a1 )x). Likewise for addition.



We note a conflict between the switch map on A) and a given negation map on 1 0 , a1 ) = ((−)a0 , (−)a1 ) unless a1 = (−)a0 . A, which do not match; (a1 , a0 ) = (−)(a Fortunately this does not affect Proposition 2.29 since (a1 , a0 )x = a1 x(−)a0 x = (−)(a0 x(−)a1 x) = (−)((a0 , a1 )x) = (((−)a0 , (−)a1 )x). 2.3.3. The characteristic sub-triple. Definition 2.30. A sub-T -triple of a T -triple (A, TA , (−)) is a triple (A , TA , (−)) where TA is a subset of TA (with the relevant structure) and A is the sub-T -module of A generated by TA . Example 2.31. Suppose (A, TA , (−)) is a triple. The characteristic subtriple A1 is the sub-triple generated by 1, which is A1 := {1, (−)1, e := 1(−)1, . . . } and TA1 := {1, (−)1}. If A is a semiring then clearly (A, TA , (−)) is a module triple over A1 . This ties in with other approaches to tropical algebra, and to some fundamental hyperfields, as follows. If e = 1 then we have the Boolean semifield, so assume that e = 1. If e + 1 = 1 then e behaves like 0, and we wind up with A1 isomorphic to Z or Z/nZ for some n > 1.

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So assume that e + 1 = 1. In height 2, e + 1 ∈ {(−)1, e}. Recall that a negation map is of the first kind if (−)a = a for any a ∈ T , and of the second kind if (−)a = a for any a ∈ T ([56, Definition 2.22])). (i) If (−) is of the first kind (as in the supertropical case), then e + 1 = e. A1 is {1, e}. This corresponds to the “Krasner hyperfield” K = {0, 1} with the usual operations of Boolean algebra, except that now 1  1 = {0, 1}, and we can identify {0, 1} with 1ν . (ii) If (−) is of the second kind, the we have two cases. (a) In the (−)-bipotent case A1 = {1, (−)1, e}, with 1 + 1 = 1, which is the symmetrized triple of the trivial idempotent triple {1}. This corresponds to the “hyperfield of signs” S := {0, 1, −1} with the usual multiplication law and hyperaddition defined by 1  1 = {1}, −1  −1 = {−1}, x  0 = 0  x = {x}, ∀x, and 1  −1 = −1  1 = {0, 1, −1} = S. (b) (−) is of the second kind but non-(−)-bipotent. Then e + 1 = (−)1, which leads to a strange structure of characteristic 4 since 2 = (e(−)1) + 1 = e + e = (−)1(−)1 = (−)2, e + 2 = 1(−)1 = e, and e + e + e = (e + 1) + e + (−)1 = e. In either case one could adjoin {0}, of course. Height > 2 is more intricate, involving layered structures. 2.3.4. The systemic Hom module. Hom(M, N ) denotes the set of morphisms from M to N and Hom(M, N )T be the subset of T -morphisms. We use the given module negation map to define a negation map on Hom. Remark 2.32. Suppose M := (M, TM , (−), ) and N := (N, TN , (−) ) are systemic modules over a ground T -system (A, T , (−), ). If M, N are T -monoid modules then so is Hom(M, N ), and, for A commutative, T acts on HomT (M, N ) via the left multiplication map "a given by "a (f ) = af. The action is elementwise: (af )(x) := af (x). We view THom(M,N ) := {f ∈ Hom(M, N ) : f (TM ) ⊆ TN } in Hom(M, N ) via these left multiplication maps. Proposition 2.33. (1) Hom(M, N ) := (Hom(M, N ), THom(M,N ) , (−)) is a right module, where (−) is defined elementwise, i.e., ((−)f )(x) = (−)(f (x)). (ii) Hom(M, N ) has unique negation if (N , TN , (−) ) has unique negation. (iii) Hom(M, N ) := (Hom(M, N ), THom(M,N ) , (−), ) has a surpassing relation, where f  g if and only if f (x)  g(x) for all x ∈ M. (iv) If A is a T -semiring system, then (Hom(M, M), THom(M,M) , (−), ) is a semiring system. Proof. (i) Check all properties elementwise (ii) Suppose f + g = h◦ . Then f (x) + g(x) = h(x)◦ for each x ∈ TM , implying g(x) = (−)f (x), and thus g = (−)f. (iii) Check the conditions of Definition 2.10 . (iv) We define the product f g by f g(x) = f (g(x)). 

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Definition 2.34. For a module M = (M, TM , (−), ) write M∗ for ∗ for {f |TM : f ∈ M∗ with f (TM ) ⊆ TA }. Hom(M, A), and TM Proposition 2.35. Let S = A(I) , where A = (A, T , (−), ) is a system. For I finite, the dual module (S ∗ , TS∗ , (−), ) also is a module, where (−) and  are defined elementwise. Proof. Let 1i denote the vector whose only component = 0 is 1 in the i-th position. Given a morphism f : S → A, we define fi ∈ THom(TS ,TA ) to be the map sending 1i to f (1i ), and Ti to 0 for all other components i = i. It is easy to see  that f is generated by the fi . (We need the hypothesis that I is finite in order for TS∗ to generate S ∗ .) As in usual linear algebra, when A is commutative as well as associative, we ∗ (I) ∗ ∗ ∗ can embed S into S . Write a for (ai ) ∈ T , and define a ∈ S by a ((bi )) = a · (bi ) = ai bi . Let ei denote the vector with 1 in the i position and 0 elsewhere.  Proposition 2.36. Suppose a = (ai ) ∈ S = A(I) . Then a∗ = i ai e∗i ∈ S ∗ is spanned over T by the e∗i . There is an injection (S, TS , (−)) → (S ∗ , TS∗ , (−)) given by a → a∗ , which is onto when I is finite. Proof. Just as in the classical case, noting that negation is not used in its proof.  2.4. Morphisms of systems. We have two kinds of morphisms. Definition 2.37. A -morphism of pseudo-systems ϕ : (A, T , (−), ) → (A , T  , (−) ,  ) is a map ϕ : A → A together with ϕ : T → T  satisfying the following properties for a ∈ T and b  b , bi in A: (i) ϕ((−)b1 )  (−)ϕ(b1 ); (ii) ϕ(b1 + b2 )  ϕ(b1 ) + ϕ(b2 ); (iii) ϕ(ab)  ϕ(a)ϕ(b). (iv) ϕ(b)  ϕ(b ). (v) ϕ(ANull ) ⊆ A Null . (vi) ϕ(0A ) = 0A . A homomorphism of pseudo-systems ϕ : (A, T , (−), ) → (A , T  , (−) ,  ) is defined in the same way, but with equality holding in (i),(ii) and (iii). These will be cast in terms of universal algebra in §9.1.2. Remark 2.38. Let ϕ : (A, T , (−), ) → (A , T  , (−) ,  ) be a -morphism of pseudo-systems. Conditions (ii) and (vi) imply (v) when 0 ∈ A. Even if (A, T , (−)) is a system, ϕ(A) ∩ TA need not generate A , so we add this stipulation for morphisms of systems. As in classical module theory, when treating -morphisms of systemic modules over a given ground system, one always assumes that ϕ is the identity on T , so (iii) becomes ϕ(ab)  aϕ(b). Lemma 2.39. The map a → a◦ is a ◦ -morphism of T -semiring systems.

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Proof. a◦ b◦ = 2(ab)◦ = (ab)◦ + c◦ where c = ab, so (ab)◦ ◦ a◦ b◦ . Also  (a + b)◦ = a◦ + b◦ . Although condition (ii) works well for hypersystems, (iii) does not fit in so well intuitively, but fortunately the following easy result will provide equality for (iii) in Proposition 2.42. We say that a T -module homomorphism φ : A → A is invertible if there is some T -module homomorphism ψ : A → A such that ψφ = 1A = φψ. Proposition 2.40. Suppose that φ : A → A is an invertible homomorphism, and  is a surpassing PO. Then f (φ(b)) = φ(f (b)), for any -morphism f : A → A satisfying f (φ(b))  φ(f (b)) ∀b ∈ A. Proof. φ(f (b)) = φ(f (ψφ(b)))  φψ(φ(f (b))) = φ(f (b)), so we get equality.  We have the following consequences at our disposal, unifying several ad hoc observations in [56]. Proposition 2.41. Any -morphism f satisfies f ((−)b) = (−)f (b). Proof. (−) is an invertible homomorphism of additive semigroups, so Proposition 2.40 is applicable.  Proposition 2.42. Any -morphism f of T -group modules satisfies f (ab) = af (b) for all a ∈ T and b ∈ A. Proof. The left multiplication map "a by a ∈ T is invertible on T , having the  inverse "a−1 , and thus is invertible on A. Lemma 2.43. Any -morphism f with respect to a surpassing PO satisfies the following convexity condition: If f (b0 ) = f (b1 ) and b0  b  b1 , then f (b0 ) = f (b). Proof. f (b1 ) = f (b0 )  f (b)  f (b1 ), so equality holds at each stage.



It follows that every -morphism “collapses” intervals, so triples, systems, etc., do not provide varieties (since they are not closed under arbitrary homomorphic images). At times we need to decide whether to take homomorphisms or -morphisms. Here is a compromise. Definition 2.44. Let M and N be A-systemic modules.  A -morphism  f : M → N is T -admissible when it satisfies the condition that if ti=1 ai = uj=1 aj   for ai , aj ∈ TM , then ti=1 f (ai ) = uj=1 f (aj ). Lemma 2.45. Every homomorphism is T -admissible. 

t u     Proof. If f (ai ) = f ( i ai ) = f = j aj i=1 ai = j=1 aj , then    f (aj ). 2.5. Function triples. Here is a wide-ranging example needed for geometry and linear algebra, unifying polynomials and Laurent polynomials, cf. [6, Example 2.19], [25], [31, §3.5]. It is convenient to assume that A is with 0.

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Definition 2.46. For a T -module (A, +) over T and a given set S, we define AS to be the set of functions from S to A 4 , also written as Fun(S, A). For c ∈ A, the constant function c˜ is given by c˜(s) = c for all s ∈ S. In ˜ is given by 0(s) ˜ particular, the zero function 0 = 0 for all s ∈ S. Definition 2.47. Given f ∈ AS we define its support supp(f ) := {s ∈ S : f (s) = 0}, and supp(AS ) for {supp(f ) : f ∈ AS }. Lemma 2.48. For any f, g ∈ AS , we have the following: (i) supp(f + g) ⊆ supp(f ) ∪ supp(g). (ii) (Under componentwise multiplication) supp(f g) ⊆ supp(f ) ∩ supp(g). Proof. For the first statement, one can see that f (s) = 0 = g(s) implies f (s) + g(s) = 0. The second statement is clear; f (s) = 0 or g(s) = 0 implies f (s)g(s) = 0.  We must cope with a delicate issue. We have required for T -triples that TA generates (A, +). Thus, AS a priori is only a pseudo-triple. AS is a triple when A is a triple of finite height. Alternatively, those maps having finite support play a special role. Definition 2.49. We introduce the following notations: • A(S) := {f ∈ AS : supp(f ) is finite}. • A monomial is an element f ∈ A(S) for which | supp(f )| = 1. • TAS = {f ∈ T (S) : | supp(f )| = 1}. Explicitly, we get polynomials when S = N, and Laurent polynomials when S = Z. Lemma 2.50. If (A, T , (−)) is a pseudo-triple, then (A(S) , TAS , (−)) also is a pseudo-triple, where TAS := {f ∈ T (S) : | supp(f )| = 1}, and (−)f (s) = (−)f (s) for f ∈ AS and s ∈ S. Proof. This is clear, noting that any element of A(S) is a finite sum of monomials.  If A is a triple, then A(S) also is a triple since the monomials span A(S) . For systems, we define  componentwise on AS by putting f  g when f (s)  g(s) for each s in S. 2.5.1. Direct sums and powers. Definition 2.51. The direct sum ⊕i∈I (Ai , TAi , (−)) of pseudo-triples is defined as (⊕Ai , T⊕Ai , (−)) where T⊕Ai = ∪TAi , viewed in ⊕TAi via νi : Ai → ⊕Ai being the canonical homomorphism.  TAi , viewed in ⊕Ai , but this Another natural possibility  for T⊕Ai would be essentially is the same, since i TAi is generated by ∪i TAi . Definition 2.51 works out better for systems. Proposition 2.52. The direct sum ⊕(Ai , Ti , (−)) of Ti -triples is a T⊕Ai -triple. Proof. T⊕Ai generates ⊕Ai , and unique negation is obtained componentwise.  4 For a T -module† (A, +) without 0, one would take AS to be the set of partial functions from S to A, and define supp(f ) to be those s on which f is defined.

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Example 2.53. When all Ai = A, this takes us back to A(S) , now viewed as a TA(S) -triple via the componentwise negation map. We often write I = S and call (I) (A(I) , TA , (−)) the free A-module triple, to stress the role of I as an index set and the analogy with the free module. This provides the free systemic module (A(I) , T (I) , (−), ), cf. [56, Definition 2.6]. Remark 2.54. When (M, TM , (−)) is an A-module triple, then (M(S) , TM(S) , (−)) is an A-module triple, under the action (af )(s) = af (s). Remark 2.55. A(I) is not meta-tangible when |I| > 1, and the theory of module triples is quite different from that of meta-tangible triples, much as module theory differs from the structure theory of rings. 2.5.2. The convolution product: Polynomials and Laurent polynomials. If (S, +) is a monoid, we can define the convolution product f ∗ g for f : S → T and g : S → A by  f (u)g(v), (f ∗ g)(s) = u+v=s

which makes sense in A since there are only finitely many u ∈ supp(f ) and v ∈ supp(g) with u + v = s. (S)

Lemma 2.56. Under convolution, supp(f ∗ g) ⊆ supp(f ) + supp(g). Proof. f (u)g(v) = 0 requires u ∈ supp f and v ∈ supp g, which is necessary for u + v ∈ supp(f ∗ g).  Lemma 2.57. If T is a monoid then TA(S) also is a monoid under the convolution product. When T is a group, TA(I) is a group. Proof. The product of monomials is a monomial.



Proposition 2.58. If {A, T , (−)} is a semiring-group triple, then {A(S) , TA(S) , (−)} is also a semiring-group triple (with the convolution product). Proof. This follows directly from the previous two lemmas.



The convolution product unifies polynomial-type constructions. Definition 2.59. The polynomial system (A(N) , TA(N) , (−), ) over a system (A, T , (−), ) is taken with A(N) endowed with the convolution product, TA(N) the set of monomials with tangible coefficients, and (−) and  defined componentwise (i.e., according to the corresponding monomials). One can iterate this construction to define A[λ1 , . . . λn ] (and then take direct limits to handle an infinite number of indeterminates). There is a subtlety here which should be addressed. When defining the module of monomials Aλ, we could view λ either as a formal indeterminate, or as a placemark for a function f : λ → A given by choosing a and defining f : λ → a. These are not the same, since different formal polynomials could agree as functions. Our point of view is the functional one. Another intriguing example of functor categories (not pursued here) is the exterior product where the functions in T (N) satisfy f (u)g(v) = (−1)uv g(v)f (u) for u, v, ∈ N.

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2.6. The role of universal algebra. As in [56, §5], universal algebra provides a guide for the definitions, especially with regard to the roles of possible multiplication on T . The notions of universal algebra are particularly appropriate here since we have a simultaneous double structure, of A and its designated subset T of tangible elements, together with a negation map (−). We recall briefly that a carrier, called a universe in [49], is a t-tuple of sets {A1 , A2 , . . . , At } for some given t. A set of operators is a set Ω := ∪m∈N Ω(m), where each Ω(m) in turn is a set of formal m-ary symbols {ωm,j = ωm,j (x1,j , . . . , xm,j ) : j ∈ JΩ(m) }, which are maps ωm,j : Aij,1 × · · · × Aij,m → Aij . The 0-ary operators are just distinguished elements called constants. (Here A1 = A, and A2 = T , and the ωm,j include the various operations needed to define T -modules. In particular, 0 is assumed to be a constant of Aj when 0 ∈ Aj .) The algebraic structure has universal relations (otherwise called identities in the literature), such as associativity, distributivity, which are expressed in terms of the operators, and this package of the carriers, operators and universal relations is called the signature of the carrier. For example, in classical algebra one might take the signature to be a ring or a module, endowed with various operations, together with identities written as universal relations. 2.6.1. The category of a variety in universal algebra. The class of carriers of a given structure is called an algebraic variety. These are well known to be characterized by being closed under sub-algebras, homomorphic images, and direct products. To view universal algebra more categorically, we work with a given variety (of a given signature). The objects of our category C are the carriers A of that signature (which clearly are sets), and its morphisms are the homomorphisms, which are maps f : A → A satisfying, for all operators ωm,j : Aij,1 × · · · × Aij,m → Aim,j . f (ωm,j (a1 , . . . , am )) = ωm,j (f (a1 ), . . . , f (am )),

∀ak ∈ Aij,k .

For a system, the signature includes A2 = T , usually a multiplicative monoid, and A1 = A, a T -module. In this philosophy, any homomorphism preserves constants and tangible elements. For example, if we fix T then the class of T -modules is an algebraic variety, and there are many other general examples, cf. [56, §5.6.1], although as explained in [56, §5.6.2] some major tropical-oriented axioms do not define varieties. We can extend the convolution product to morphisms: Definition 2.60. Given a semiring T and maps h1 : A1 → T and h2 : B1 → T , (S) (S) define the convolution product h1 ∗ h2 : A1 × B1 → T by  ((h1 ∗ h2 )(f1 , f2 ))(s) = h1 (f1 (u))h2 (f2 (v)). u+v=s

Example 2.61. (The trivial case) When S is the trivial category consisting of (S) one object 0, and writing ci = hi (0), then Ai = Ai , and the convolution product is just given by h1 ∗ h2 (c1 , c2 ) = h1 (c1 )h2 (c2 ). This viewpoint will be useful when we consider tensor products.

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2.6.2. Triples (−)-layered by a semiring L. Here is an example paralleling graded algebras, which both relates to the symmetrized structure and is needed in differential calculus of T -systems. Definition 2.62. A pseudo-triple A is (−)-layered by a monoid (L, ·) if A= A ; (−)A = A(−) ∈L

where the union of the A is disjoint, and A1 A2 ⊆ A1 2 . Example 2.63. (−)-layering also provides “layered” structures, as described in [3, §2.1]. Namely, a T -monoid triple layered by a semiring† L is viewed as a special case of Definition 2.62, where all the A are the same (taking (−) on A to be the identity map). This can be viewed as AL 1 , viewing a monoid L as a small category, with TA = A1 . Addition in the layered triple A of A by L is given in [56, Example 3.7]. This is treated in [3, §2.1], which also considers other subtleties concerning layering of triples in general. We also note that there is a  (L) T -homomorphism A1 → A given by (a ) → (", a ). 3. The structure theory of ground triples via congruences We are ready to embark on the structure theory of ground triples, in analogy to the structure theory of rings and integral domains. Our objective in this section is to modify the ideal theory (which does not work for homomorphisms over semirings) and the corresponding factor-module theory to an analog which is robust enough to support the structure theory of ground T -systems. To do this, we first ignore the issue of negatives, and then, as in [56], use “symmetry” (which is formal negation) instead of negatives. 3.1. The role of congruences. Unfortunately everything starts to unravel at once. For starters, [26, §1.6.2] is too restrictive for our purposes. Factor TA -modules (or factor semirings† ) are a serious obstacle, since cosets need not be disjoint (this fact relying on additive cancellation, which fails in the max-plus algebra, since 1 + 3 = 2 + 3 = 3). We also have the following problematic homomorphism, if we want to use the preimage of 0 in the theory. Example 3.1. Define the homomorphism f : A × A → A × A by f (a0 , a1 ) = (a0 + a1 , a0 + a1 ). Then f is not 1 : 1 over the max-plus algebra, but f −1 (0, 0) = (0, 0). This example also blocks the naive definitions of kernels and cokernels. (In what follows, the null elements will take the place of the element 0 in A.) This is solved in universal algebra via the use of congruences, which are equivalence relations respecting the given algebraic structure. Congruences can also be viewed as subalgebras of A × A which are also equivalence relations. ) we define Definition 3.2. Viewing a congruence of A as a substructure of A, the trivial congruence as follows: DiagA := {(a, a) : a ∈ A}.

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Clearly every congruence contains DiagA . Furthermore, since we are viewing (−) in the signature, we require that ((−)a1 , (−)a2 ) ∈ Φ whenever (a1 , a2 ) ∈ Φ. Since we are incorporating the switch morphism into the signature, we require for an A-module M that if (a, b) ∈ M then (b, a) = (−)(a, b) ∈ M. This is a very mild condition; for example, if 0 ∈ A then (b, a) = (0, 1)(a, b) ∈ M. Lemma 3.3. Any congruence is closed under the twist action. Proof. Let Φ be a congruence. By applying (i) again, we obtain that (0, a1 ) ·tw (x0 , x1 ) = (a1 x0 , a1 x1 ) ∈ Φ. But, since Φ is a congruence (in particular symmetric), we have that (a1 x1 , a1 x0 ) ∈  Φ and hence the sum (a0 x0 + a1 x1 , a0 x1 + a1 x0 ) ∈ Φ, yielding the assertion. 3.1.1. The twist action on congruences. In universal algebra, in particular semirings, congruences are more important to us than ideals. But TA -module congruences are difficult to work with, since they 0 = M × M over A. The next concept eases this difficulty are TA -submodules of M by bringing in the twist action as a negation map on semirings† . We write Φ1 ·tw Φ2 for the congruence generated by {(a0 , a1 ) ·tw (b0 , b1 ) : ai ∈ Φ1 , bi ∈ Φ2 }, cf. (2.2). From associativity, it makes sense to write Φn for the twist product of n copies of Φ. 3.1.2. T -congruences. We restrict our congruences somewhat, to give T its proper role. Definition 3.4. For any congruence Φ, we define Φ|T = {(a, a ) ∈ Φ : a, a ∈ T0 }. We say that a congruence Φ is a T -congruence if and only if Φ is additively generated by elements of the form (a, b) where a, b ∈ T . Clearly DiagA is a T -congruence. As with rings and modules, there are two notions of factoring out T -congruences. If Φ is a T -congruence on a triple A, we can form the factor triple A/Φ, generated by T /Φ|T := {([a1 ], [a2 ]) : (a1 , a2 ) ∈ T × T }, where the equivalence classes are taken with respect to Φ. Lemma 3.5. If Φ|T is a T -congruence on a triple (A, T , (−)), then (A/Φ|T , T /Φ|T , (−)) is a triple, where one defines (−)[a] = [(−)a], and there is a morphism A → A/Φ (as A-module triples) given by a → [a]. (

to be well-defined on A/Φ|T , which means that if  We need (−)    Proof.  a1i , a2i ) ∈ Φ|T , then ( (−)a1i , (−)a2i ) ∈ Φ|T . But this is patent.

On the other hand, if Φ2 ⊆ Φ1 are T -congruences, we can define the factor T -congruence Φ1 /Φ2 generated by {([a1 ], [a2 ]) : (a1 , a2 ) ∈ Φ1 } where the equivalence classes are taken with respect to Φ2 . Φ/DiagA is just Φ. If Φ1 is a T -congruence then so is Φ1 /Φ2 .

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3.2. Prime T -systems and prime T -congruences. Since classical algebra focuses on algebras over integral domains (i.e., prime commutative rings), we look for their systemic generalization. The idea is taken from Jo´o and Mincheva [33] as well as Berkovich [9]. In [9], Berkovich defined a notion of the prime spectrum Spec(A) when A is a commutative monoid. Berkovich’s definition of a prime ideal of a commutative monoid A is a congruence relation ∼ on A such that the monoid A/ ∼ is nontrivial and cancellative. Remark 3.6. For a semiring A, Jo´ o and Mincheva defined a prime congruence for A as a congruence P of A such that if a ·tw b ∈ P then a ∈ P or b ∈ P for ) This definition implies the definition of Berkovich in the following sense: a, b ∈ A. Let A be an idempotent commutative T -semiring triple. Then, for any prime T congruence P (defined as in [33]), the T /P -semiring triple A/P is cancellative. The case when A is an idempotent semiring is proved in [33, Proposition 2.8] although the converse is not true in general (cf. [33, Theorem 2.12]). Let us modify this, to get both directions. We drop the assumption of commutativity whenever the proofs are essentially the same. Definition 3.7. Let (A, T , (−)) be a T -semiring† triple. (i) A congruence Φ = A) of a semiring† triple A is prime if Φ ·tw Φ ⊆ Φ for congruences implies Φ ⊆ Φ or Φ ⊆ Φ. (ii) A T -congruence Φ = A) of a semiring† triple A is T -prime if Φ ·tw Φ ⊆ Φ for T -congruences implies Φ ⊆ Φ or Φ ⊆ Φ. A T -congruence Φ is T semiprime if (Φ )2 ⊆ Φ implies Φ ⊆ Φ. Semiprime T -congruences are called radical when A is commutative. (iii) The triple (A, T , (−)) is prime (resp. semiprime if the trivial T -congruence DiagA is a prime (resp. -semiprime) T -congruence. (iv) The triple (A, T , (−)) is T -prime (resp. semiprime if the trivial T congruence DiagA is a T -prime (resp. T -semiprime) T -congruence. A commutative semiprime triple is called reduced, in analogy with the classical theory. (v) The triple (A, T , (−)) is T -irreducible if the intersection of nontrivial T -congruences is nontrivial. For notational convenience, we write congruences (instead of T -congruences) if the context is clear. Prime congruences arise naturally as follows (which could be formulated much more generally): ) there Proposition 3.8. (i) Given any ·tw -multiplicative subset S of A, is a T -congruence Φ maximal with respect to Φ ∩ S = ∅, and the T congruence Φ is T -prime. ) there is a congruence Φ max(ii) Given any ·tw -multiplicative subset S of A, imal with respect to Φ ∩ S = ∅, and the congruence Φ is T -prime. Proof. (i) The union of a chain of T -congruences is a T -congruence, so the existence of a maximal such T -congruence is by Zorn’s lemma. For the second assertion, suppose that Φ ·tw Φ ⊆ Φ, but Φ ⊆ Φ and Φ ⊆ Φ. Then S ∩ Φ (resp. S ∩ Φ ) contains an element s (resp. s ). It follows that S ∩ Φ ·tw Φ contains s ·tw s ∈ S. This contradicts to our assumption that Φ is disjoint from S and hence Φ is prime. (ii) Analogous to (i). 

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Corollary 3.9. If S is a submonoid of T , then there is a congruence Φ maximal with respect to Φ ∩ ((S × {0}) ∪ ({0} × S) = ∅, and it is prime. Proof. (S × {0}) ∪ ({0} × S) is a monoid under ·tw .



We get all prime T -congruences in this way, because of the following observation: Remark 3.10. For any prime T -congruence Φ of A, S := A \ Φ is a ·tw submonoid of A × A, and obviously Φ is maximal with respect to Φ ∩ S = ∅. We say that a T -congruence Φ is maximal if there is no T -congruence Φ with Φ|T ⊂ Φ |T ⊂ TA. Lemma 3.11. The T -congruence Φ is maximal if and only if it is maximal with respect to the condition that (1, 0) ∈ / Φ. Proof. If (1, 0) ∈ Φ then (b0 , b1 ) = (b0 , b1 ) ·tw (1, 0) ∈ Φ.



Corollary 3.12. Every maximal T -congruence Φ is prime. Lemma 3.13. A triple is prime if and only if it is semiprime and T -irreducible. Proof. (⇒) Semiprime is a fortiori. But if Φ∩Φ is trivial then Φ·tw Φ ⊆ Φ∩Φ is trivial, implying Φ or Φ is trivial. (⇐) If Φ ·tw Φ is trivial, then (Φ ∩ Φ )2 is trivial, implying Φ ∩ Φ is trivial, so  Φ or Φ is trivial. The following assertions are straightforward. Lemma 3.14. (i) For a T -congruence Φ , Φ|T ⊆ Φ if and only if Φ ⊆ Φ. (ii) The intersection of prime T -congruences is semiprime. Jo´o and Mincheva [33] showed that any irreducible, cancellative commutative B-algebra A is prime, and it follows that the polynomial system A[Λ] is prime. We say that A satisfies the ACC on T -congruences if for every ascending chain {Φi : i ∈ I} there is some i such that Φi = Φi for all i > i. Proposition 3.15. If A satisfies the ACC on T -congruences, then every T congruence Φ contains a finite product of prime T -congruences, and in particular there is a finite set of prime T -congruences whose product is trivial. Proof. A standard argument on Noetherian induction: We take a maximal counterexample Φ. If Φ is not already prime, then there are two T -congruences Φ , Φ whose intersection with T is not in Φ, but whose product is in Φ. By Noetherian induction applied to Φ/Φ in A/Φ , and to Φ/Φ in A/Φ , we get finite sets of prime T -congruences whose respective products are in Φ and Φ , so the product of all of them is in Φ.  √ For A commutative with a T -congruence Φ, define Φ to be the T -congruence generated by the set {(a0 , a1 ) ∈ T) : (a0 , a1 )n ∈ Φ for some n}.

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Lemma 3.16. In a commutative T -semiring system, (i) Φ is prime if and only if it satisfies the condition that (a0 , a1 ) ·tw (b0 , b1 ) ∈ Φ implies (a0 , a1 ) ∈ Φ or (b0 , b1 ) ∈ Φ for (a0 , a1 ), (b0 , b1 ) ∈ T) . (ii) Φ is T -radical if and only if (a0 , a1 )2 ∈ Φ implies (a0 , a1 ) ∈ Φ for (a0 , a1 ) ∈ T) . √ (iii) If Φ is a T -congruence, then Φ is a radical T -congruence. Proposition 3.17. For every T -congruence Φ on a commutative T -semiring √ system, Φ is an intersection of prime T -congruences. Proof. For any given a = (a0 , a1 ) ∈ T) , let Sa = {(a0 ,√ a1 )n : n ∈ N}. If Sa ∩ Φ = ∅, Zorn’s lemma gives us a T -congruence containing Φ, maximal with respect to being disjoint from Sa , and easily seen √ to be prime, so their intersection is disjoint from all such a, which is precisely Φ.  3.3. Annihilators, maximal T -congruences, and simple T -modules. Definition 3.18. Suppose M is a systemic module over a T -semiring† system A. For any S ⊆ M, the annihilator AnnT (S) is the T -congruence generated by 00 : a0 s = a1 s, ∀si ∈ S}. {(a0 , a1 ) ∈ T Likewise, suppose Φ is a T -congruence on the T -module system M. For S ⊆ Φ, define the annihilator AnnT (S) to be the T -congruence generated by {(a0 , a1 ) ∈ T) : a0 s0 + a1 s1 = a0 s1 + a1 s0 , ∀s ∈ S}. In other words, AnnT (S)S is trivial, under the twist multiplication. Definition 3.19. An A-module M is simple if MNull is the only proper submodule. As in classical algebra, these are the building blocks in the structure theory of modules. Remark 3.20. For any s ∈ TM , the map b → bs induces an isomorphism A/AnnT (s) ∼ = As. Consequently, AnnT (s) is a maximal congruence iff the pseudosystem (As, T s, (−), ) is simple. Proposition 3.21. If M is a simple T -module, then AnnT (M) is a T -prime T -congruence of A. Proof. If ΦΦ ⊆ AnnT (M) for T -congruences Φ, Φ , then Φ(Φ M) is trivial. Thus either Φ ⊆ AnnA (M) or Φ M = M, in which case ΦM is trivial and hence  Φ ⊆ AnnA (M). Proposition 3.22. For A commutative, M is a simple systemic module if and only if AnnT (M) is a maximal T -congruence of A. Proof. As in the usual commutative theory, AnnT (M) = AnnT ({s}) for any nonzero s ∈ M, since As = M.  In the noncommutative case, one could go on to define primitive T -congruences of a ground T -system to be the annihilators of simple systemic modules, but that is outside the scope of this work.

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4. The geometry of prime systems As in classical algebra, the “prime” ground systems lend themselves to affine geometry. 4.1. Primeness of the polynomial system A[λ]. Let us consider when AS is prime. We need a slightly different definition of “trivial.” Definition 4.1. Let (S)

AΦ := {(f, g) ∈ A(S) × A(S) | (f (s), g(s)) ∈ Φ, ∀s ∈ S}. ASΦ is defined analogously, using AS instead of A(S) . An element (f, g) ∈ ASΦ is functionally trivial if (f (s), g(s)) ∈ DiagA , ∀s ∈ S. A T -congruence Φ is functionally trivial if any pair (f, g) ∈ ASΦ is functionally trivial. Proposition 4.2. Let Φ be a T -congruence on A. If Φ is radical then so are (S) the congruences AΦ on A(S) , and ASΦ on AS , for any small category S, and in particular so is A[λ1 , . . . , λn ]Φ . (S)

(S)

Proof. Let (f, g) ∈ AΦ and suppose that (f, g)2 = (f 2 , g 2 ) ∈ AΦ , i.e., one has (f 2 (s), g 2 (s)) ∈ Φ2 , ∀s ∈ S. (S)

It follows that (f (s), g(s)) ∈ Φ since Φ is radical, and hence (f, g) ∈ AΦ . This (S)  proves that AΦ is radical. The case of ASΦ is analogous. On the other hand, the other ingredient, irreducibility, is harder to attain. Given a T -congruence Φ on A(S) (in particular, Φ is a subset of A(S) × A(S) ), we define SΦ := {s ∈ S : (f (s), g(s)) ∈ DiagA , ∀(f, g) ∈ Φ}. This leads to a kind of consideration of density. Lemma 4.3. Suppose that A has T -congruences Φ, Φ with Φ ∩ Φ functionally trivial. If A is irreducible, then SA(S) ∩ SA(S) = ∅. Φ

Φ



Proof. For each s ∈ S, since Φ ∩ Φ is functionally trivial, either s ∈ SA(S) or Φ s ∈ SA(S) . This implies that SA(S) ∩ SA(S) = ∅.  Φ

Φ

Φ

It is well-known by means of a Vandermonde determinant argument that over an integral domain, any nonzero polynomial of degree n cannot have n + 1 distinct zeros. The analog for semirings also holds for triples, using ideas from [2]. Namely, we recall [56, Definition 6.20]: Definition 4.4. Suppose A has a negation map (−). For a permutation π, write  a : π even; π (−) a = (−)a : π odd. (i) The (−)-determinant |A| of a matrix A is 

 (−)π ai,π(i) . π∈Sn

i

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(ii) The n × n Vandermonde matrix V (a1 , . . . , an ) is defined to be ⎛ ⎞ 1 a1 a21 . . . an−1 1 ⎜1 a2 a22 . . . an−1 ⎟ 2 ⎜ ⎟ ⎜. .. .. . . .. ⎟ . ⎝ .. . . . . ⎠ 2 n−1 1 an an . . . an (iii) Write ai,j for the (−)-determinant of the j, i minor of a matrix A. The (−)-adjoint matrix adj(A) is (ai,j ). We have the following adjoint formula from [3, Theorem 1.57]. Lemma 4.5. n (i) |A| = j=1 (−)i+j ai,j ai,j , for any given i.  (ii) |V (a1 , . . . , an )| = i>j (aj (−)ai ). Proof. This is well-known for rings, so is an application of the transfer principle in [2]. (Put another way, one could view the ai as indeterminates, so the assertion holds formally.)  We say b is a ◦-root of f ∈ T [λ] if f (b) ∈ A◦ , i.e., f (b) +◦ 0. This definition was the underlying approach to supertropical affine varieties in [31]. Theorem 4.6. Over a commutative prime triple (A, T , (−)) with unique negation, any nonzero polynomial f ∈ T [λ] of degree n cannot have n+1 distinct ◦-roots in T . n i Proof. Write f = i=0 bi λ for bi ∈ T . Suppose on the contrary that a1 , . . . , an+1 are distinct ◦-roots. Write v for the column vector (a0 , . . . , an ). Then Av is the column vector (f (a1 ), . . . , f (an )) which is a quasi-zero, so  (aj (−)ai )v = |A|v = adj(A)Av ∈ adj(A)A◦  A◦ , implying



i>j i>j (aj (−)ai )

∈ A◦ , contrary to a1 , . . . , an+1 being distinct.



Corollary 4.7. If (A, T , (−)) is a prime commutative triple with T infinite, then so is (A[λ], TA[λ] , (−)). Proof. Follows from Lemma 4.3, since any finite set of distinct nonzero polynomials cannot have infinitely many common roots.  Example 4.8. We will need the congruence version of Definition 4.4, for which we turn to symmetrization, as treated in [3]. Namely, according to Definition 2.24, ) T) , (−)) with multiplication T) × we embed (A, T ) into the symmetrized system (A, A) → A) given by the twist action, (−) is the switch map, and  is Null . Given (f (λ), g(λ)) ∈ T) [λ], we define (f, g)(b0 , b1 ) = (f (b0 ) + g(b1 ), f (b1 ) + g(b0 )), ) The element (b0 , b1 ) is a symmetrized root of for any element (b0 , b1 ) of A. (f (λ), g(λ)) ∈ T) [λ] if (f, g)(b0 , b1 ) ∈ A)◦ , evaluated under twist multiplication. (In particular, for b1 = 0, this means f (b0 ) = g(b0 ).) Now one can define the symmetrized determinant to be the (−)-determinant in this sense, and of the j, i minor of a matrix A. The symmetrized adjoint matrix is the (−)-adjoint matrix.

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In this context, Theorem 4.6 says that over a commutative prime triple A, any pair of polynomials (f, g) ∈ T1 [λ] of degree n cannot have n + 1 distinct symmetrized roots in T) . 4.2. Localization. We refer to [56, §6.8], where we used Bourbaki’s standard technique of localization [11], to pass from commutative metatangible T -semiring systems (resp. T monoid module triples) to metatangible T -systems (resp. T - triples) over groups. We assume that T is a monoid and S is a central submonoid of A \ ANull (i.e., sa = as for all a ∈ A, s ∈ S). Often S ⊆ T . Recall that one defines the equivalence (s1 , b1 ) ≡ (s2 , b2 ) when s(s1 b2 ) = s(s2 b1 ) for some s ∈ S, and we write s−1 b or b s for the equivalence class of (s, b). We might as well assume that 1 ∈ S since b bs 1 = s . We localize a T -semiring triple (A, T , (−)) with respect to S by imposing multiplication: −1  −1 b0 b1 , (s−1 1 b0 )(s2 b0 ) = (s1 s2 ) and addition: −1  −1 (s2 b0 + s1 b0 ). (s−1 1 b0 ) + (s2 b0 ) = (s1 s2 )

The standard ring-theoretic facts are mirrored in the systemic situation. Remark 4.9. Any finite set of fractions as11 , as22 , . . . , asnn has a common denominator s = s1 · · · sn , since ai s1 · · · si−1 ai si+1 · · · sn . = si s We say that a ∈ T is -regular if ab1  ab2 implies b1  b2 . We say that a ∈ S is regular if ab1 = ab2 implies b1 = b2 , and S is regular if each of its elements is regular. Lemma 4.10. When  is a PO, then -regular implies regular. Proof. ab1 = ab2 implies ab1  ab2 and ab1 + ab2 , so b1  b2 and b1 + b2 ,  implying b1 = b2 . Proposition 4.11. Let (A, T , (−)) be a pseudo-triple with unique quasi-negatives, and S be a multiplicative submonoid of A. Then the following hold. (i) (S −1 A, S −1 T , (−)) is a pseudo-triple with unique quasi-negatives. (ii) If (A, T , (−)) is a T -triple with unique quasi-negatives, then (S −1 A, S −1 T , (−)) is also a T -triple which has unique quasi-negatives. (iii) There is a canonical homomorphism S −1 : (A, T , (−)) −→ (S −1 A, S −1 T , (−)),

b →

b , 1

whose congruence kernel is the following T -congruence Φ = {(b0 , b1 ) : sb0 = sb1 for some s ∈ S}. −1

(iv) The map S −1 induces an isomorphism S −1 A/S −1 Φ ∼ = S1 (A/Φ). (v) If S is regular then the map of (iii) is an injection. (vi) If T is regular then (A, T , (−)) injects into the triple (T −1 A, T −1 T , (−)) over the group T −1 T .

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Proof. (i): One can easily check that (S −1 A, S −1 T , (−)) is a pseudo-triple. This is standard (where (−)(s−1 a) := s−1 ((−)a), s ∈ S)). For the assertion −1 about unique quasi-negatives, suppose s−1 a. Then 1 a1 is a quasi-negative of s −1 a ∈ (S −1 A)◦ , (ss1 )−1 (sa1 + s1 a) = s−1 1 a1 + s

implying sa1 + s1 a ∈ A◦ , and thus sa1 = (−)s1 a = s1 ((−)a), and s−1 1 a1 = s−1 ((−)a). (ii): The proof is essentially the same as (i). We only have to check that S −1 T ∩ (S −1 A)◦ = ∅. Suppose that sb ∈ S −1 T ∩ (S −1 A)◦ . In particular, sb = s1 a(−)s1 a a a for some sa1 ∈ S −1 A. It follows that s s1 a(−)s s1 a ∈ T for s1 (−) s1 = s21 some s ∈ S. However, this implies that T ∩ A◦ = ∅, which contradicts to the assumption that (A, T , (−)) is a T -triple. (iii): Clearly S −1 is a homomorphism and Φ is the congruence kernel of S −1 . (iv): The congruence kernel of the map is S −1 Φ. (v): Φ is trivial, by (iii) and the definition of regular.  (vi): T −1 T is a group. Lemma 4.12. Any surpassing relation  on a monoid system (A, T , (−), ) extends to (S −1 A, S −1 T , (−)), by putting s−1 b  (s )−1 b whenever s b  sb . Proof. We verify the conditions of Definition 2.10. (i): Suppose that sb , and sc ∈ S −1 T Then s b  s b + sc◦ , implying s b c◦ b s b + sc◦ b =   = +  .  s ss ss s s (ii): Suppose that showing that

b s



b s .

Then we have s b  sb and hence (−)s b  (−)sb ,

b b (−)b (−)b = (−)  (−)  = . s s s s (iii): Suppose that follows that

b1 s1



b1 s1

and

b2 s2



b2 s2 .

Then b1 s1  s1 b1 and b2 s2  s2 b2 . It

s1 s2 (b1 s2 +s1 b2 ) = s2 (s1 b1 )s2 +s1 (s2 b2 )s1  s2 s1 b1 s2 +s1 s2 b2 s1 = (s1 s2 )(s2 b1 +s1 b2 ), which shows that b1 b2 s2 b1 + s1 b2 s b + s b b b + =  2 1   1 2 = 1 + 2 . s1 s2 s1 s2 s1 s2 s1 s2 (iv): Suppose that S⊆T,

b s



b s .

Then we have that bs  sb . In particular, since

as2 (s1 b1 )  as2 (s1 b1 ). for any a ∈ T and s2 ∈ S. It follows that for any

a s2

∈ S −1 T , we have

ab1 a b1 ab1 a b1 =  = . s2 s1 s2 s1 s2 s1 s2 s1 (v): Suppose that as11  as22 for asii ∈ S −1 T . This implies that s2 a1  s1 a2 , however s2 a1 , s1 a2 ∈ T and hence s2 a1 = s1 a2 , showing that as11 = as22 . 

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4.2.1. Localization of T -congruences. Next, we introduce localization for T -congruences. Again we take S a submonoid of T , but the flavor is different. Let (b0 , b1 ) and (b0 , b1 ) be elements of a T -congruence Φ. One defines the following equivalence (b0 , b1 ) ≡ (b0 , b1 ) if and only if s(b0 , b1 ) = s (b0 , b1 ) for some s, s ∈ S. Definition 4.13. S −1 Φ = {( bs0 , bs1 ) : (b0 , b1 ) ∈ Φ}. Remark 4.14. If Φ is a T -congruence of (A, T , (−)), then S −1 Φ is a T congruence of (S −1 A, S −1 T , (−)). Lemma 4.15. Any T -congruence of (S −1 A, S −1 T , (−)) has the form S −1 Φ, where Φ is a T -congruence of (A, T , (−)). Proof. Given the T -congruence Φ of S −1 A, define b0 b1 Φ = {(b0 , b1 ) ∈ A : ( , ) ∈ Φ }. 1 1 If (b0 , b1 ) ∈ Φ then writing bi = sbi for bi ∈ A we have s0 s1 (b0 , b1 ) = (s1 b0 , s0 b1 ) ∈ i  Φ, so Φ = S −1 Φ. From now on we assume for convenience that S ⊆ T is a submonoid of Φregular elements, in the sense that (sb0 , sb1 ) ∈ Φ implies (b0 , b1 ) ∈ Φ for any s ∈ S. Proposition 4.16. If Φ is a prime T -congruence of (A, T , (−)), then S −1 Φ is a prime T -congruence of (S −1 A, S −1 T , (−)). Proof. Suppose S −1 Φ ·tw S −1 Φ ⊆ S −1 Φ. Then clearly Φ ·tw Φ ⊆ Φ by regularity, so Φ ⊆ Φ or Φ ⊆ Φ, implying S −1 Φ ⊆ S −1 Φ or S −1 Φ ⊆ S −1 Φ.  Example 4.17. Take S = {s ∈ T : (s, 0) ∈ / Φ}. If Φ is prime then S is Φregular. Hence S −1 Φ is maximal with respect to being disjoint from S −1 T , in the sense of Corollary 3.9. 4.3. Extensions of systems. Definition 4.18. When we have a homomorphism ϕ : (A, TA , (−)) → (A , TA , (−)) of pseudo-triples whose congruence kernel is trivial, we say that (A , TA , (−)) is an extension of (A, TA , (−)). In other words, (A, TA , (−)) can be viewed as a sub-triple of (A , TA , (−)). As in the classical theory, for ai ∈ TA , i ∈ I, we write A[ai : i ∈ I] for the sub-triple of A generated by the ai (where TA[ai :i∈I] is the set of monomials in the ai with coefficients in T ). For B, B  ⊆ A , we write B  B  , and say B  -generates B, if for each b ∈ B there is b ∈ B  such that b  b . Definition 4.19. We say that A is -affine over A if T   T [ai : i ∈ I] for I finite. (In other words, taking I = {1, . . . , n}, we write A = A[a1 , . . . , an ] , where for any a ∈ T  there is f (λ1 , . . . , λn ) ∈ T [a1 , . . . , an ] such that a  f (a1 , . . . , an ).) Remark 4.20. For any extension (A , TA , (−)) of (A, TA , (−)), and ai ∈ A, There is a natural homomorphism A[λi : i ∈ I] → A given by λi → ai , whose kernel is prime if and only if A[ai : i ∈ I] is prime.

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Remark 4.21. Given a prime system (A, T , (−), ), we form A[Λ], where Λ = {λi : i ∈ I} and some subset Λ = {λi : i ∈ I  }, and we take S to be the submonoid of monomials in {λi : i ∈ I  }, and T  to be the submonoid of monomials in {λi : i ∈ I  }. Then S −1 T  is a group, and we have the prime system S −1 Λ = (S −1 A[Λ], S −1 T , (−), ). In this way, we “expand” T to S −1 T  and lower the number of indeterminates under consideration. Lemma 4.22. If A is -affine over A and A is -affine over A, then A is -affine over A. Proof. Write A = A [a1 , . . . , am ] and A = A[a1 , . . . , an ] . Then writing a  f (a1 , . . . , an ) and aj  gj (a1 , . . . , an ), we have a  f (g1 (a1 , . . . , an ), . . . ,  gm (a1 , . . . , an )). 4.3.1. Weak nullstellensatz. Having introduced affine systems, we would like to develop techniques to analyze them, and present some results related to the Nullstellensatz. To this end, we need an observation about spanning sets. Definition  4.23. Elements {vi : i ∈ I} of an A-module M -span a submodule N if N  T vi .  bi vi ∈ Elements {vi : i ∈ I} are -independent over a submodule N if NNull implies each bi ∈ ANull . A -base is a -independent -spanning set. A symmetric base is a -base for the symmetrized module triple. Even though dependence is not transitive for modules over semirings, we do have the following result. Lemma 4.24. If the extension (A , TA , (−)) of (A , TA , (−)) has symmetric base b1 , . . . , bm and the extension (A , TA , (−)) of (A, TA , (−)) has symmetric base b1 , . . . , bm , then the extension (A , TA , (−)) over (A, TA , (−)) has symmetric base {bi bj : 1 ≤ i ≤ m , 1 ≤ j ≤ m }.      aj bj , and aj  Proof. Any b ∈ A satisfies b  j i ai,j bi , implying  b  i,j ai,j bi bj , proving -spanning. For -independence, suppose   ai,j bi bj = ( ai,j bi )bj ∈ ANull . Then each (



i,j

 i ai,j bi )

∈

j

ANull ,

i

implying each ai,j ∈ ANull .



Lemma 4.25. Suppose A is a module over a system A, and has a -base B = {vi : i ∈ I} over T . If (H, TH , (−)) is a sub-semiring system of A containing ANull , over which B still -spans A over T , then A  H.  Proof. For any element w of A , we can write wv1  hi vi for suitable hi ∈ H ⊆ A . Hence, we have that (h1 (−)w)v1 + h2 v2 + · · · + hn vn ∈ ANull . It follows from linear dependence of the vi over A that each coefficient must be in  ANull ; in particular, w  w + (h1 (−)w) = h1 + w◦ , proving A  H. Theorem 4.26 (Artin-Tate lemma, -version). Suppose A = A[a1 , a2 , . . . , an ] is a -affine system over A, and K a subsystem of A , with A having a -base v1 = 1, . . . , vd of A over K. Then K is -affine over A.

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Proof. There are suitable αijk , αiv ∈ K such that (4.1)

vi vj 

d  k=1

αijk vk ,

au 

d 

αuk vk ,

1 ≤ i, j ≤ d, 1 ≤ u ≤ n.

k=1

Let H = A[αijk , αuk : 1 ≤ i, j, k ≤ d, 1 ≤ u ≤ n] ⊆ K, and A0 := {v ∈ A :  v  di=1 Hvi }. The relations (4.1) imply that A0 is closed under multiplication, and thus is a subalgebra of A containing a1 , . . . , an , implying A  A0 , which is obviously -affine.  These results can all be viewed in terms of the symmetrized triple. To continue, we need a workable definition of “algebraic.” Presumably an algebraic element b should be a symmetric root of a tangible pair (f, g) of polynomials. This is tricky since we need to identify the tangible polynomials, since examining its coefficients leads to the difficulty that (λ+1)(λ(−)1) = λ2 +1◦ λ(−)1. To exclude such examples leads us to functional considerations, when T is infinite. Suppose we are given an extension (A , TA , (−)) of (A, TA , (−)). Definition 4.27. A polynomial f ∈ A[λ1 , . . . , am ] is functionally tangible (with respect to T  ) if f (a1 , . . . , am ) ∈ T for almost all a1 , . . . , am in T  . In the symmetric situation, a pair of polynomials (f, g) ∈ A[λ1 , . . . , λm ] is symmetrically functionally tangible if for each b ∈ T there are only finitely many b ∈ T for which f (b) = g(b ). Remark 4.28. Suppose T is infinite, and (A, T , (−)) is a prime triple. Then it is enough to check when b ∈ T (so that T  is irrelevant), and the set of symmetrically functionally tangible pairs is a submonoid of A[λ1 , . . . , λm ]. Definition 4.29. (i) An element a ∈ T  := TA is symmetrically al gebraic if there is a symmetrically functionally tangible pair (f, g) ∈ A[λ 1] for which f (a) = g(a). (ii) Given an extension (A , TA , (−)) of (A, TA , (−)), and a ∈ TA , we define the a-denominator set Sa = {g(a) : g is a symmetrically functionally tangible polynomial with g(a) ∈ T }. (iii) An -affine extension (A , TA , (−)) of (A, TA , (−)) is fractionally closed over a if every element of Sa is invertible in TA . Lemma 4.30. If a is symmetrically algebraic, i.e., f (a) = g(a), and t is the largest number such that at least one of the coefficients of λn in f and g is tangible, then 1, a, . . . , at−1 is a symmetric base of A[a]. Proof. 1, a, . . . , at−1 are independent, by choice of t. But at is dependent on 1, a, . . . , an−1 , and continuing inductively, each am is dependent on 1, a, . . . , at−1 .  Remark 4.31. Sa is a monoid, so we can localize, and (Sa−1 A , Sa−1 TA , (−)) is fractionally closed over a. Lemma 4.32. Suppose T has the property that for any a ∈ T , the set {a+c : c ∈ T } is infinite, and (A , T  , (−)) is a symmetrized triple, fractionally closed over a, with S ⊇ T a subgroup of A , and a ∈ T  . If the module of fractions QS (A[a ]) of A[a ] is -affine over A, then QS (A[a ]) = A[a ].

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3 2  ) fn (a ) Proof. We may assume A = QS (A[a ]). Writing A = A fg11 (a , . . . ,   (a ) gn (a ) , for fi , gi ∈ A [λ], gi symmetrically algebraic, note by Remark4.9(ii) that we may assume that all the denominators are equal, i.e., g1 (a ) = g2 (a ) = · · · = gn (a ), which we write as g(a ). Any element of A can be written with denominator a power of g(a ), which means in particular, for any c ∈ T , 1 f (a ) = g(a ) + c g(a )m for suitable m and suitable f ∈ F [λ]. Thus, f (a )(g(a )+c) = g(a )m . Consequently, a is a symmetric root of (f (λ)(g(λ) + c), g(λ)m ). Hence f (λ)(g(λ) + c) = g(λ)m as functions, for infinitely many c ∈ T , which is impossible for g nonconstant (by  comparing factorizations). Hence g is a constant, and QS (A[a ]) = A[a ]. The following assertion sometimes is called the “weak Nullstellensatz,” since in classical mathematics it can be used to prove Hilbert’s Nullstellensatz cf. [54, Theorem 10.11]. Theorem 4.33. If (A, T , (−)) is a semiring-group system over T and (A = A[a1 , . . . , am ], T  , (−)) is a -affine semiring-group system, in which (f, g)(ai , 0) is invertible for every symmetrically functionally tangible pair (f, g) of polynomials, 0 has a symmetric base over T . then A Proof. We follow the proof given in [56, Theorem A], based on the Artin0 , T −1  , (−)), where S = {(f, g)(a1 , 0) : Tate lemma. Namely, take K to be (S −1 A S A (f, g) symmetrically functionally tangible}. By induction on n, A has a symmetric base over K, so is -affine by Theorem 4.26, and thus has a symmetric base. We conclude with Lemmas 4.24 and 4.25.  4.4. Classical Krull dimension. Definition 4.34. The height of a chain P0 ⊇ P1 ⊇ · · · ⊇ Pt of prime T congruences is t. The (Krull) dimension is the maximal length n of a chain of prime T -congruences of A. T -Homomorphisms from polynomial triples have an especially nice form. Definition 4.35. The transcendence degree of A[[λ1 , . . . , λn ]] over a semiring-group system A is n. A congruence Φ on A[[λ1 , . . . , λn ]] is projectively T -trivial if all of the elements of Φ|T have the form (a0 h, a1 h) where ai ∈ T and h is a monomial. A substitution homomorphism ϕ : A[[λ1 , . . . , λn ]] → (A , T  , (−)) is a T homomorphism determined by substitution of λ1 , . . . , λn to elements of T  . Lemma 4.36. If Φ is a non-projectively trivial prime T -congruence of the Laurent polynomial triple (A[[λ1 , . . . , λn ]], TA[[λ1 ,...,λn ]] , (−)), and A := (A[[λ1 , . . . , λn ]], TA[[λ1 ,...,λn ]] , (−))/Φ, then A is isomorphic to a Laurent polynomial triple of transcendence degree < n, and the natural T -homomorphism (A[[λ1 , . . . , λn ]], TA[[λ1 ,...,λn ]] , (−)) → A is a substitution homomorphism.

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Proof. We view the images of λ1 , . . . , λn as Laurent monomials, in the sense that (a0 h0 , a1 h1 ) is identified with (a0 a1 )−1 hh01 where ai ∈ T and hi are pure monomials in the λi . The hypothesis that Φ is non-projectively trivial means that the image of some λi can be solved in terms of the others, so the transcendence degree decreases.  Because of our restricted definition of homomorphism (sending monomials to monomials), the next theorem comes easily. Theorem 4.37. Both the polynomial A[λ1 , . . . , λn ] and Laurent polynomial systems A[[λ1 , . . . , λn ]] in n commuting indeterminates over a T -semiring-group system have dimension n. Proof. By Lemma 4.36, the chain of prime T -congruences of A[[λ1 , . . . , λn ]] correspond to a homomorphic chain A[[λ1 , . . . , λn ]] → A[[λ1 , . . . , λn−1 ]] → A[[λ1 , . . . , λn−2 ]] → . . . (reordering the indices if necessary) and this must stop after n steps.



5. Tensor products Two of the most important functors in the category theory of modules are the tensor product ⊗ and Hom . We turn to triples and systems emerging over a given ground triple (A, T , (−)), and the categories (⊗ and Hom) that arise from them. Both appertain to systems, but each with somewhat unexpected difficulties. Hom was studied in §2.3.4, so we focus on tensor products, a very well-known process in general category theory [27, 40], as well as over semirings [41, 59], which has been studied formally in the context of monoidal categories, for example in [18, Chapter 2]. 5.1. Tensor products of systems. Here we need the tensor product of systems over a ground T -system. These are described (for semirings) in terms of congruences, as given for example in [41, Definition 3] or, in our notation, [42, §3]. This material also is a special case of [26, § 1.4.5], but we present details which are specific to systems, to see just how far we can go with -morphisms and the negation map. En route we also hit a technical glitch in applying universal algebra, which historically appeared before tensor categories. The tensor product, which exists for systems, t cannot be described directly in universal algebra, since the length t of a sum i=1 ai ⊗bi of simple tensors need not be bounded. So one needs a “monoidal” universal algebra, where the signature contains the tensor products of the original structures, which is beyond the scope of this paper. Let us work with a right A-systemic module M1 and left A-systemic module M2 over a given ground T -system A. The following observations are well known. Definition 5.1. A map Φ : M1 × M2 → N is bilinear if       (5.1) Φ x1,j , x2,k = Φ x1,j , x2,k , Φ(x1 a, x1 ) = Φ(x1 , ax1 ), j

∀xi,j ∈ Mi , a ∈ A.

k

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One defines the tensor product M1 ⊗A M2 of M1 and M2 in the usual way, to be (F1 ⊕ F2 )/Φ, where Fi is the free system (respectively right or left) with base Mi (and TFi = Mi ), and Φ is the congruence generated by all     

   x1,j , x2,k , (x1 a, x2 ), (x1 , ax2 ) x1,j , x2,k , (5.2) j

k

j,k

∀xi,j , xi,k ∈ Mi , a ∈ A. ¯ : Remark 5.2. Any bilinear map Ψ : M1 × M2 → N induces a map Ψ ¯ M1 ⊗ M2 → N given by Ψ(a1 ⊗ a2 ) = Ψ(a1 , a2 ), since Ψ passes through the defining congruence Φ of (5.2). To handle negation maps, we take a slightly more technical version emphasizing TA . Definition 5.3. The TA -tensor product M1 ⊗TA M2 of a right TA -systemic module M1 and a left TA -systemic module M2 is (F1 ⊕ F2 )/Φ, where Fi is the free system with base Mi (and TFi = Mi ), and Φ is the congruence generated as in (5.2), but now with a ∈ TA . If M1 , M2 have negation maps (−), then we define a negated tensor product by further imposing the extra axiom ((−)x) ⊗ y = x ⊗ ((−)y). Note that this is done by modding out by the congruence generated by all elements ((−)x ⊗ y, x ⊗ (−)y), x, y ∈ TM , in the congruence defining the tensor product in the universal algebra framework. From now on, the notation M1 ⊗ M2 includes this negated tensor product stipulation, and A and TA are understood. We can incorporate the negation map into the tensor product, defining (−)(v ⊗ w) := ((−)v) ⊗ w. Remark 5.4. As in the classical theory, if M1 is an (A, A ) systemic bimodule, then M1 ⊗ M2 is an A-systemic module. In particular, this happens when A is commutative and the right and left actions on M1 are the same; then we take A = A.5 Since there are more relations in the defining congruence, the TA -tensor product maps down onto the negated tensor product. Definition 5.5. The tensor product of triples (A, T , (−)) and (A , T  , (−) ) is the triple (A ⊗ A , {a1 ⊗ a2 : a1 ∈ T , a2 ∈ T  }, (−) ⊗ 1A ). In order to be able to apply the theory of monoidal categories, we need to be able to show that the module tensor product is functorial; i.e., given morphisms fi : Mi → Ni for i = 1, 2, we want a well-defined morphism f1 ⊗ f2 : M1 ⊗ M2 → N1 ⊗ N2 . We would define the tensor product f1 ⊗f2 of -morphisms by (f1 ⊗f2 )(a⊗b) = f (a1 ) ⊗ f2 (a2 ), as a case of f1 ∗ f2 in Example 2.61, but we run into immediate difficulties, even over free modules. 5 There is some universal algebra lurking beneath the surface, since one must define an abelian carrier. We are indebted to D. Stanovsky and M. Bonato for pointing out the references [19] and [50, Definitions 3.5, 3.7], which are rather intricate; also see [49, Definition 4.146].

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Example 5.6. Consider the polynomial triple A[λ1 , λ2 ] and the -morphism f : A[λ1 , λ2 ] → A[λ1 , λ2 ] given by taking f to be the identity on all monomials and f (q) = 0 whenever q is a sum of at least two nonconstant monomials. Then λ1 ⊗λ1 +λ1 ⊗λ2 +λ2 ⊗λ2 = λ1 ⊗(λ1 +λ2 )+λ2 ⊗λ2 = λ1 ⊗λ1 +(λ1 +λ2 )⊗λ2 , so λ2 ⊗ λ2 = (f ⊗ f )(λ1 ⊗ λ1 + λ1 ⊗ λ2 + λ2 ⊗ λ2 ) = λ1 ⊗ λ1 , so f ⊗ f is not well-defined. The same argument shows that even f ⊗ 1 is not well-defined. As is well known, the process does work for homomorphisms, as a consequence of Remark 5.2: Proposition 5.7. Suppose that fi : Mi → Ni are homomorphisms. Then the map f1 ⊗ f2 : M1 ⊗ M2 → N1 ⊗ N2   given by (f1 ⊗ f2 )( i a1,i ⊗ a2,i ) = f1 (a1,i ) ⊗ f2 (a2,i ) is a well-defined homomorphism. Remark 5.8. In view of [41], also cf. [25, Proposition 17.15], one has the natural adjoint isomorphism HomA (M1 ⊗B M2 , M3 ) → HomB (M1 , HomA (M2 , M3 ), since the standard proof given for algebras does not involve negation. This respects tangible homomorphisms, so yields an isomorphism of triples. 5.1.1. The tensor semialgebra triple. Next, as usual, given a bimodule V over TA , one defines V ⊗(1) = V, and inductively V ⊗(k) = V ⊗ V ⊗(k−1) . From what we just described, if V has a negation map (−) then V ⊗(k) also has a natural negation map, and often is a triple when V is a triple. 5.9. [56, Remark 6.35] Define the tensor semialgebra T (V ) =  Definition ⊗(k) V (adjoining a copy of TA if we want to have a unit element), with the k usual multiplication. Given a TA -module triple (M, TM , (−)), the tensor semialgebra triple (T (M), TT (M) , (−)) of M is defined by using the negated tensor products of Definition 5.3 to define T (M), induced from the negation maps on M⊗(k) ; writing a ˜k = ak,1 ⊗ · · · ⊗ ak,k for ak,j ∈ TT (M) , we put (−)(˜ ak ) = (−)(ak,1 ⊗ · · · ⊗ ak,k ). ak :}, the set of simple multi-tensors. TT (M) is ∪{˜ We can now view the polynomial semiring† (Definition 2.59) in this context. Example 5.10. Suppose A = (A, T , (−)) is a triple. The polynomial semiring† A[λ] now is defined as a special case of the tensor semialgebra. TA[λ] again is the set of monomials with coefficients in T . 6. The structure theory of systemic modules via congruences Aiming for a representation theory, we take a category of module systems over our ground triples. Throughout, we take T -module systems over a ground system with a fixed signature.

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6.1. Submodules versus subcongruences. Definition 6.1. A systemic submodule of (M, TM , (−), ) is a submodule N of M, satisfying the following conditions, where a ∈ TM : (i) Write TN ,0 for TN ∪ {0}. (N , TN , (−), ) is a submodule of (M, TM , (−), ). (In particular, TN ,0 generates (N , +).) ◦ ⊆ N (and thus M◦ ⊆ N ). (ii) TM (iii) If a  b + v, for v ∈ N , then there is w ∈ TN ,0 for which a  b + w. Note that (ii) implies that {0} is not a systemic submodule, for we want systemic submodules to contain the null elements. In what follows, N always denotes a systemic submodule of M. Remark 6.2. The definition implicitly includes the condition that (−)TN ,0 = TN ,0 , since (−)a = ((−)1)a. Lemma 6.3. If a ∈ N and a ◦ b, then b ∈ N . Proof. Just write b = a + c◦ , noting that c◦ ∈ N .



The first stab at defining a T -module of a T -congruence Φ might be to take {a(−)b : (a, b) ∈ Φ}, which works in classical algebra. We will modify this slightly, but the real difficulty lies in the other direction. The natural candidate for the T -congruence of a T -submodule N might be {(a, b) : a(−)b ∈ N }, but it fails to satisfy transitivity! Definition 6.4. (1) Given a T -submodule N of M, define ΦN on N by   the T -congruence a ≡ b if and only if we can write a = j aj and b = j bj for aj , bj ∈ TN ,0 such that aj  bj + vj for vj ∈ TN ,0 , each j. (2) Given a T -congruence Φ, define NΦ to be the additive sub-semigroup of M generated by all c ∈ TM,0 such that c = a(−)b for some a, b ∈ TN ,0 such that (a, b) ∈ Φ. Example 6.5. When the system M is meta-tangible, then in the definition of ΦN , either bj = (−)vj in which case aj  vj◦ ∈ TN◦ , or aj = bj (yielding the diagonal) or aj = vj ∈ TN . The results from [56] carry over, with the same proofs, to systemic modules. Definition 6.6. A T -systemic module M = (M, TM , (−), ) is TM -reversible if a1  a2 + b implies a2  a1 (−)b for a1 , a2 ∈ TM and b ∈ M. Lemma 6.7 ([56, Proposition 6.12]). In a T -reversible system, a ≡ b (with respect to ΦN ) for a, b ∈ TN , if and only if either a = b or TN contains an element v such that v  a(−)b. Remark 6.8 ([56, Remark 8.29]). In a T -reversible system, Condition (iii) of Definition 6.1 yields w  a(−)b. Likewise, in Definition 6.4, aj  bj + vj implies bj  aj (−)vj . Proposition 6.9 ([56, Proposition 8.30]). In a T -reversible system, ΦN is a T -congruence for any T -submodule N . For any T -congruence Φ, NΦ is a T submodule. Furthermore, ΦNΦ ⊇ Φ and NΦN = N .

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6.2. The N-category of systemic modules. Since we lack the classical negative, the trivial subcategory of 0 morphisms and 0 objects is replaced by a more extensive subcategory of quasi-zero morphisms and quasi-zero objects. The quasi-zero morphisms have been treated formally in [26, §1.3] under the name of N-category and homological category, with the terminology “null morphisms” and “null objects,” and with the null subcategory designated as Null6 . To view systems A = (A, T , (−), ) in the context of N-categories, we must identify the null objects ANull . The intuitive choice might be A◦ , but ANull often seems to be more inclusive. We turn to systemic modules (M, TM , (−), ), to which we refer merely as M for shorthand. From now on we take =◦ in order to simplify the exposition. Definition 6.10. (i) A chain of T -morphisms of systemic modules is a sequence g

f

··· → K → M → N → ··· such that (f g)(k) ∈ NNull for all k ∈ K. (ii) (Compare with [1]) The chain is exact at M if g(K) = {b ∈ M : f (b) ∈ NNull }. One can easily see from the above definitions that if K and N are null (i.e., K = KNull and N = NNull ) and the chain is exact, then M is null as well. Definition 6.11. Let f : M → N be a morphism of T -modules. • The T -module kernel T -ker f of f is {a ∈ T : f (a) ∈ NNull }. • f is null if f (a) ∈ NNull for all a ∈ TM , i.e. T -ker f = TM . • The T -module image fTM (M) is the T -submodule spanned by {f (a) : a ∈ TM }. • f is Null-monic if f (a0 ) = f (a1 ) implies that a0 (−)a1 ∈ MNull . • f is Null-onto if fTM (M) + NNull = N . Thus, the null morphisms are closed null in the categorical sense, and take the place of the 0 morphism. Since 0 ∈ / TM , we might take f (TM ) to be any tangible constant z, i.e., z ∈ TN . 6.3. The category of congruences of systemic modules. We have the analogous results for T -congruences. The next definition takes into account that any T -congruence contains the diagonal. Definition 6.12. For any T -congruence Φ and any morphism f : M → N , define the congruence image f (M)cong to be the T -congruence  * 

     f (ai ), f (aj ) : ai , aj ∈ Φ . (6.1) DiagN + i

6 [26,

j

§1.3] uses the notation (E, N ), but we already have used N otherwise.

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The T -congruence kernel Tcong - ker(f ) is the T -congruence of M, generated by {(a0 , a1 ) ∈ TM : f (a0 ) = f (a1 )}. For f , the term in the summation in (6.1) is {(f (x0 ), f (x1 )) : xi ∈ M, (x0 , x1 ) ∈ Φ}.

% 

   &  In particular, f (M)cong is DiagN + f (a ), f (a ) : a = aj , i i j i j the T -congruence of N generated by DiagN and {(f (a1 ), f (a2 )) : ai ∈ T0,M }. The fact that f (Φ) is more complicated for -morphisms presents serious obstacles later on. Definition 6.13. A congruence morphism f : Φ → Φ is trivial if f (0) = 0 and f (Φ) ⊆ DiagΦ . Lemma 6.14. For any morphism, f : M → N , Tcong - ker(f ) = {(x, x ) ∈ M : f (x) = f (x )}.        Proof. Write x = xi and x = xj and y = y i and y = yj . If  f (xi ) + f (yi ) = f (xj ) + f (x) = f (x ) and f (y) = f (y  ), then f (x + y) =      f (yj ) = f (x + y ).

Lemma 6.15. For any TM -morphism f : M → N , the induced morphism 0→N ) is Null-onto if and only if f is epic. f) : M 0 is trivial, then gf 0 (M) = g)(M), 0→N ) is Null-onto and gf 0 Proof. If f) : M implying g) is trivial. 0→N ) is not Null-onto. Then take the canonical map Conversely, assume that f) : M  g : N → N /f (M)cong to get gf trivial, but g is not trivial, so f is not epic.

To construct cokernels, we define N → N /f (M)cong , which will turn out to be the categorical cokernel of f. Lemma 6.16. For any morphism f : M → N , there is a Null-monic f¯ : f

M/Tcong - ker(f ) → N , given by f ([a]) = f (a), where [a] is the equivalence class of a ∈ T under the congruence kernel Tcong - ker(f ). Proof. If (a, a ) ∈ Tcong - ker(f ), then f (a) = f (a ), showing that f is welldefined. Clearly f is a T -morphism since T - ker(f ) is a T -congruence. Finally, one can easily check that f is a monic as it is injective. 

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Lemma 6.17. A TM -morphism f : M → N is Null-monic if and only if Tcong - ker(f ) is diagonal. Proof. Assume -congruence Tcong - ker(f ) is not diagonal, i.e.,    first that the T b = αi = aj = b although f (ai ) = f (aj ). Define the cokernel f¯ :   ¯ f (ai ) where a = ai . f f¯ is trivial, implying Tcong - ker(f ) → M by f (a) = Tcong - ker(f ) is trivial. On the other hand, if Tcong - ker(f ) is diagonal, and f g is trivial, then g(M) is diagonal, i.e., g is trivial.  Lemma 6.18. Any morphism f : M → N is composed as M → M/Tcong - ker(f ) f

→ N , where the first map is the canonical morphism, and the second map is given in Lemma 6.16. Proof. The first map sends a ∈ M to a ¯, where a ¯ is the equivalence class of a under Tcong - ker(f ). This is clearly a T -morphism and since Tcong - ker(f ) is a congruence relation. The second map is just Lemma 6.16.  We call f¯ the monic associated to the T -congruence kernel.

7. Functors among semiring ground triples and systems In this section we recapitulate the previous connections among the notions of systems, viewed categorically. We focus on ground triples, but at the end indicate some of the important functors for systemic modules, in prelude to [38]. Notation. Let us introduce the following notations: • • • • •

• • • • •

Rings= the category of commutative rings. Doms= the category of integral domains. SRings= the category of commutative semirings. SDoms= the category of commutative semidomains. SDM= the category whose objects are pairs (cS, T ) consisting of a semiring S and a multiplicative submonoid T of S which (additively) generates S and does not contain 0S . A homomorphism from (S1 , T1 ) to (S2 , T2 ) is a semiring homomorphism f : S1 → S2 such that f (T1 ) ⊆ T2 and f (T1 ) generates S2 . SRTT = the category of T -triples, with ◦ -morphisms, whose objects are semirings. HDoms= the category of hyperrings without multiplicative zero-divisors, but with -morphisms. HFields= the category of hyperfields. FRingsw = the category of fuzzy rings with -morphisms (cf. [24]). FRingsstr = the category of coherent fuzzy rings (cf. [24] and [56] for the notion of coherence).

The following diagram illustrates how various categories are related (note that this is not a commutative diagram, for instance, e ◦ t ◦ i = c ◦ j).

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(7.1) Doms

i

SDoms

j

HFields

k

t

a

HDoms

c

g

e

SDM

SRTT

d

FRingsw

(If we are willing to bypass SDM in this diagram, then we could generalize the first two terms of the top row to Rings and SRings and accordingly, in the second row, HDoms can be generalized to the category of hyperrings.) In the following propositions, we explain the functors in the above diagram. All of these functors are stipulated to preserve the negation map. First, one can easily see that the functors i, j, k are simply embeddings. To be precise: Proposition 7.1. (The functors i, j, and k) (1) The functor i : Doms → SDoms, sending an object A to A and a homomorphism f ∈ HomRings (A, B) to f ∈ HomSRings (i(A), i(B)), is fully faithful. (2) The functor j : Doms → HDoms, sending an object A to A and a homomorphism f ∈ HomRings (A, B) to f ∈ HomHRings (j(A), j(B)), is fully faithful. (3) The functor k : Hfields → HDoms, sending an object A to A and a homomorphism f ∈ HomHfields (A, B) to f ∈ HomHRings (k(A), k(B)), is faithful. Proof. This is straightforward.



Remark 7.2. The functor k : Hfields → HDoms cannot be full since morphisms exist which are not homomorphisms, cf. [34]. Now, for a commutative semidomain A, we let t(A) = (A, A − {0A }). It is crucial that A is a semidomain for A − {0A } to be a monoid. For a semiring homomorphism f : A1 → A2 , we define t(f ) : (A1 , A1 − {0A1 }) → (A2 , A2 − {0A2 }) which is induced by f . Then, clearly t is a functor from SDoms to SDM. In fact, we have the following: Proposition 7.3. The functor t : SDoms → SDM is fully faithful. Proof. This is clear from the definition of t.



Remark 7.4. One may notice that our construction of t is not canonical since any semiring may have different sets of monoid generators. For instance, the coordinate ring of an affine tropical scheme may have different sets of monoid generators depending on torus embeddings, cf. [23].

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The functors a and g are already constructed in [24]. For the sake of completeness, we recall the construction. Let H be a hyperring. Then one can define the following set: 4  n hi | hi ∈ H, n ∈ N ⊆ P(H), (7.2) S(H) := i=1

where P(H) is the power set of H. By [56, Theorem 2.5], [24], S(H) is a semiring with multiplication and addition as follows: "! m " ! n    hi hj = hi hj ∈ S(H), (7.3) i=1

j=1

i,j

Bigg(

n  i=1

" hi

! +

m  i=1j

" hj

=



(hi + hi ) ∈ S(H).

i,j

Now, the functor a : Hfields → SDM sends any hyperfield H to (S(H), H × ), i.e., a(H) = (S(H), H × ). Also, if f : H1 → H2 is a homomorphism of hyperfields, then f canonically induces a homomorphism " ! n n   × × a(f ) : (S(H1 ), H1 , −) → (S(H1 ), H1 , (−)), a(f ) hi = f (hi ). i=1

i=1

We emphasize that since the subcategory Hfields of HDoms only has homomorphisms, a becomes a functor. The construction of the functor g is similar to a; we use the powerset P(H) of H instead of S(H) in this case. For details, we refer the readers to [24]. It is proved in [24] that when one restricts the functors to hyperfields, the functors a and g are faithful, but not full. Remark 7.5. Since a fuzzy ring assumes weaker axioms than semirings, the functor g can be defined for all hyperrings, whereas the functor a can be only defined for hyperfields with homomorphisms. Next, we construct the functor e : SDM → SYST . To this end, we need to fix a negation map of interest, so a priori the functor e is not canonical. For an object (S, M ) of SDM, we let e(S, M ) be the T -system (A, T , (−), ), where A = S, T = M , and (−) is the identity map and =◦ . Since, we choose (−) to be equality, any homomorphism f : (S1 , M1 ) → (S2 , M2 ) induces a homomorphism e(f ). The functor c is defined as follows: For a hyperring R without zero-divisors, we associate S(R) as in (7.2) and also impose addition and multiplication as in (7.3). Now, the T -system c(R) = (A, T , (−), ) consists of A = S(R), T = R, (−) : S(R) → S(R) sending A to −A := {−a | a ∈ A}, where − is the negation in R, and  is set inclusion ⊆. One checks easily that c(R) is indeed a T -system and any homomorphism f : R1 → R2 of hyperrings induces a morphism c(f ) of T -systems. Finally, we review the functor d : FRingsw → SYST , defining (−)a to be εa. [56, Lemma 14.5] shows how this can be retracted at times. One can easily see that the definition of coherent fuzzy rings is similar to T -systems; we only have

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to specify a negation map (−) and a surpassing relation . To be precise, let F be a fuzzy ring. The system d(F ) = (A, T , (−), ) consists of A = F , T = A× , (−) : A → A sending a to ε · a, and  is defined to be the equality. One can easily check that any -morphism f : F1 → F2 of fuzzy rings induces a homomorphism d(f ) of the corresponding T -systems. Remark 7.6. In the commutative diagram, one can think of forgetful functors in opposite directions. For instance, the functor t has a forgetful functor (forgetting T ) as an adjoint functor. Remark 7.7. Although we do not pursue them in this paper, we point out two possible links to partial fields, first introduced by C. Semple and G. Whittle [58] (see, also [52] to study representability of matroids). Recall that a commutative partial field P = (R, G) is a commutative ring R together with a subgroup G ≤ R× of the group of multiplicative units of R such that −1 ∈ G. (1) If one considers the subring R of R which is generated by G, then the pair (R , G, −, =) becomes a system. (2) Any commutative partial field P = (R, G) gives rise to a quotient hyperring R/G. This defines a functor from the category of commutative partial fields to the category of hyperrings. Remark 7.8. Since any semiring† containing T is a T -module, we have forgetful functors from T -semiring systems to T -module systems. We also have the tensor functor and Hom functors. 7.1. Valuations of semirings via systems. We briefly mention one potential application of systems. In [36], the first author introduced the notion of valuations for semirings by implementing the idea of hyperrings, and this was put in the context of systems in [56, Definition 8.8]. Definition 7.9. Let T := R ∪ {−∞}, where R is the set of real numbers. The multiplication of T is the usual addition of real numbers such that a (−∞) = (−∞) for all a ∈ T. Hyperaddition is defined as follows:  max{a, b} if a = b ab= [−∞, a] if a = b, For a commutative ring A, a homomorphism from A to T is what sometimes is called a “semivaluation,” (but in a different context from which we have used the prefix “semi”) i.e., which could have a non-trivial kernel. Inspired by this observation, in [36], the first author proposed the following definition to study tropical curves by means of valuations; this is analogous to the classical construction of abstract nonsingular curve via discrete valuations. Definition 7.10. Let S be an idempotent semiring and T be the tropical hyperfield. A valuation on S is a function ν : S → T such that ν(a · b) = ν(a) ν(b),

ν(0S ) = −∞,

ν(a + b) ∈ ν(a)  ν(b),

ν(S) = {−∞}.

The implementation of systems, through the aforementioned functors, allows one to reinterpret Definition 7.10 as a morphism in the category of systems. To be precise, let S be a semiring which is additively generated by a multiplicative submonoid M . This gives rise to the T -system e(S, M ) = (S, M, id, =) via the

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functor e. Also, for the hyperfield T, via the functor e◦a, we obtain a T -system, say AT . Then a semiring valuation on S is simply a morphism ν : (S, M, id, =) → AT of T -systems. 7.2. Functors among module triples and systems. We conclude with some important functors needed to study module triples and systems. Given a T -system (A, TA , (−), ), define A-Mod to be the category of (A, TA , (−), )-systemic modules. ) T , (−), Remark 7.11. The symmetrizing functor injecting A-Mod into (A, A ) )-Mod is given by f → (f, 0), with the reverse direction (A, TA, (−), )-Mod to A-Mod given by (f0 , f1 ) → f0 (−)f1 . This functor respects the universal algebra approach since TM → TM . Other important functors in this vein are the tensor functor and the Hom functor. 8. Appendix A: Interface between systems and tropical mathematics We relate our approach to tropical mathematics in view of systems to other approaches taken in tropical mathematics. 8.1. Tropical versus supertropical. 8.1.1. The “standard” tropical approach. Let Rmax be the tropical semifield with the maximum convention. One often works in the polynomial semiring Rmax [Λ], although here we replace Rmax by any ordered semigroup (Γ, ·), with Γ0 := Γ ∪ {0} where 0a = 0 for all a ∈ Γ. For i1 i in i = (i1 , . . . , in ) ∈ N(n) , we  writei Λ for λ1 · · · λn . A tropical hypersurface of a tropical polynomial f = i αi Λ ∈ Γ[Λ] is defined as the set of points in which two monomials take on the same dominant value, which is the same thing as the supertropical value of f being a ghost.  Definition 8.1 ([23, Definition 5.1.1]). Given a polynomial f = αi Λi ∈ Γ[Λ], define supp(f ) to be all the tuples i = (i1 · · · in ) for which the monomial in f has nonzero coefficient αi , and for any such monomial h, write fhˆ for the polynomial obtained from deleting h from the summation. The bend relation of f is a congruence relation on Γ[Λ] which is generated by the following set where fh indicates that the monomial h is deleted: {f ≡bend fh=α i : i ∈ supp(f )}. ˆ iΛ The point of this definition is that the tropical variety V defined by a tropical polynomial is defined by two monomials (not necessarily the same throughout) taking equal dominant values at each point of V , and then the bend relation reflects the equality of these values on V , thence the relation. 8.1.2. Tropicalization and tropical ideals. In many cases, one relates tropical algebra to Puiseux series via the following tropicalization map. Definition 8.2. For any additive group F , one can define the additive group F {{t}} of Puiseux series (actually due on the variable t, which is the  to Newton) k/N c t where N ∈ N, " ∈ Z, and ck ∈ K. set of formal series of the form p = ∞ k= k

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Remark 8.3. If F is an algebra, then F {{t}} is also an algebra under the usual convolution product. For Γ = (Q, +), one has a canonical semigroup homomorphism (the Puiseux valuation) as follows: (8.1)

val : F {{t}} \ {0} → Γ,

p=

∞ 

ck tk/N → min {k/N }. ck =0

k=

This induces the semigroup homomorphism (8.2)

trop : F {{t}}[Λ] → Γ[Λ],

p(λ1 · · · λn ) :=



pi λi →



val(pi )λi .

The map trop is called tropicalization. Definition 8.4. [23] With the same notations as above, suppose I is an ideal of F {{t}}[Λ]. The bend congruence on {trop (f ) : f ∈ I} is called the tropicalization congruence trop (I).  Suppose F is a field. We can normalize a Puiseux series p = pi λi11 · · · λinn at any given index i ∈ supp(p) by dividing through by pi ; then the normalized   coefficient is 1. Given two Puiseux series p = pi λi11 · · · λinn , q = qi λi11 · · · λinn i1 i in having a common monomial Λ = λ1 · · · λn in their support, one can normalize both and assume that pi = qi = 1, and remove this monomial from their difference p − q, i.e., the coefficient of Λi in val(p − q) is 0 (= −∞). Accordingly,  a tropical ideal of Γ[Λ] is an ideal I such that for any two  gi λi ∈ I having a common monomial Λi there is polynomials f = fi λi , g =  h= hi λi11 · · · λinn ∈ I whose coefficient of Λi is 0, for which hi ≥ min{af fi , bg gi } for suitable af , bg ∈ F. For any tropical ideal I, the sets of minimal indices of supports constitutes the set of circuits of a matroid. This can be formulated in terms of valuated matroids, defined in [16, Definition 1.1] as follows: A valuated matroid of rank m is a pair (E, v) where E is a set and v : E m → Γ is a map satisfying the following properties: (i) There exist e1 , . . . , em ∈ E with v(e1 , . . . , em ) = 0. (ii) v(e1 , . . . , em ) = v(eπ(1) , . . . , eπ(m) ) for each e1 , . . . , em ∈ E and every permutation π. Furthermore, v(e1 , . . . , em ) = 0 in case some ei = ej . (iii) For (e0 , . . . , em , e2 , . . . , em ∈ E there exists some i with 1 ≤ i ≤ m and v(e1 , . . . , em )v(e0 , e2 , . . . , em ) ≤ v(e0 , . . . , ei−1 , ei+1 , . . . , em )v(ei , e2 , . . . , em ). This information is encapsulated in the following result. Theorem 8.5. [46, Theorem 1.1] Let K be a field with a valuation val : K → Γ0 , and let Y be a closed subvariety defined of (K × )n defined by an ideal I  ±1 K[λ±1 1 , . . . , λn ]. Then any of the following three objects determines the others: ±1 (i) The bend congruence T(I) on the semiring S := Γ[λ±1 1 , . . . , λn ] of tropical Laurent polynomials; (ii) The ideal trop (I) in S; (iii) The set of valuated matroids of the vector spaces Ihd , where Ihd is the degree d part of the homogenization of the tropical ideal I.

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8.1.3. The supertropical approach. In supertropical mathematics, the definitions run somewhat more smoothly. A tropical variety V was defined in terms of T ◦ , so for f, g ∈ T [Λ] we define the ◦-equivalence f ≡◦ g on T [Λ] if and only if f ◦ = g ◦ , i.e. f (a)◦ = g(a)◦ for each a∈T. Proposition 8.6. The bend relation implies ◦-equivalence, in the sense that if f ≡bend g then f ≡◦ g, for any polynomials in f, g ∈ T [Λ]. Proof. The bend relation is obtained from a sequence of steps, each removing or adding on a monomial which takes on the same value of some polynomial f . Thus the defining relations of the bend  relation areall ◦ relations. Conversely, fi and g = gj for given a ◦ relation f ◦ = g ◦ , where f = f ≡bend g1 +f ≡bend g1 +g2 +f ≡bend · · · ≡bend g+f ≡bend g+fhˆ ≡bend · · · ≡bend g, implying f ≡bend g.



 In supertropical algebra, given a polynomial f = αi Λi , define ν-supp(f ) to be all the tuples i = (i1 · · · in ) for which the monomial in f has coefficient αi ∈ T . The supertropical version of tropical ideal is that if f, g ∈ I and i ∈ ν-supp(f )∩ / ν-supp(af f + bg g). ν-supp(g), then, by normalizing, there are af , bg such that i ∈ This is somewhat stronger than the claim of the previous paragraph, since it specifies the desired element. Supertropical “d-bases” over a super-semifield are treated in [29], where vectors are defined to be independent iff no tangible linear combination is a ghost. If a tropical variety V is defined as the set {v ∈ F (n) : fj (v) ∈ ν(F ), ∀j ∈ J} for a set {fj : j ∈ J} of homogeneous polynomials of degree m, then taking I = {f : f (V ) ∈ ν(F )} and Im to be its polynomials of degree m, one sees that the d-bases of Im of cardinality m comprise a matroid (whose circuits are those polynomials of minimal support), by [29, Lemma 4.10]. On the other hand, submodules of free modules can fail to satisfy Steinitz’ exchange property ([29, Examples 4.18,4.9]), so there is room for considerable further investigation. “Supertropicalization” then is the same tropicalization map as trop , now taken to the standard supertropical semifield strop : Γ ∪ G0 (where T = Γ). In view of Proposition 7.5, the analogous proof of [46, Theorem 1.1] yields the corresponding result: Theorem 8.7. Let K be a field with a valuation val : K → Γ0 , and let Y be ±1 a closed subvariety defined of (K × )n defined by an ideal I  K[λ±1 1 , . . . , λn ]. Then any of the following objects determines the others: (i) The congruence given by ◦-equivalence on I = strop (I) in the supertrop±1 ical semiring† S := Γ[λ±1 1 , . . . , λn ] of tropical Laurent polynomials; (ii) The ideal trop (I) in S; (iii) The set of valuated matroids of the vector spaces Ihd , where Ihd is the degree d part of the homogenization of the tropical ideal I. 8.1.4. The systemic approach. The supertropical approach can be generalized directly to the systemic approach, which also includes hyperfields and fuzzy rings. We assume (A, T , (−), ) is a system.

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Definition 8.8. The ◦-equivalence on Fun(S, A) is defined by, f ≡◦ g if and only if f ◦ = g ◦ , i.e. f (s)(b)◦ = g(s)(b)◦ for each s ∈ S and b ∈ A. This matches the bend congruence. Definition 8.9. Given f ∈ Fun(S, A) define ◦-supp(f ) = {s ∈ S : f (s) ∈ T }. The systemic version of tropical ideal is that if f, g ∈ I and s ∈ ◦-supp(f ) ∩ ◦/ ◦-supp(af f (−)bg g). supp(g), then there are af , bg ∈ T such that s ∈ Now one can view tropicalization as a functor as in [56, §10], and define the appropriate valuated matroid. Then one can address the recent work on matroids and valuated matroids, and formulate them over systems. Presumably, as in [3], in the presence of various assumptions, one might be able to carry out the proofs of these assertions, but we have not yet had the opportunity to carry out this program. 9. Appendix B: The categorical underpinning In this appendix we present the ideas more categorically, with an emphasis on universal algebra. 9.1. Categorical aspects of systemic theory. One can apply categorical concepts to better understand triples and T -systems and their categories. 9.1.1. Categories with a negation functor and categorical triples. We introduce another categorical notion, to compensate for lack of negatives. Definition 9.1. Let C be a category. A negation functor is an endofunctor (−) : C → C satisfying (−)A = A for each object A, and, for all morphisms f, g: (i) (−)((−)f ) = f . (ii) (−)(f g) = ((−)f )g = f ((−)g), i.e., any composite of morphisms (if defined) commutes with (−). We write f (−)g for f + ((−)g), and f ◦ for f (−)f. Remark 9.2. In each situation, the negation map gives rise to a negation functor in the category arising from universal algebra, defining (−)f for any morphism f to be given by ((−)f )(a) = (−)(f (a)). The identity functor obviously is a negation functor, since these conditions become tautological. But categories in general may fail to have a natural non-identity negation functor. For example, a nontrivial negation functor for the usual category Ring might be expected to contain “negated homomorphisms” −f where (−f )(a) := f (−a). Note then that (−f )(a1 a2 ) = −(−f )(a1 )(−f )(a2 ), so −f is not a homomorphism unless f = −f. In such a situation, we must expand the set of morphisms to contain “negated homomorphisms.” Also (−f ) · (−g) = f · g, where · denotes pointwise multiplication. 9.1.2. -morphisms in the context of universal algebra. One of our main ideas is to bring the surpassing relation  into the picture (although it is not an identity), and in the universal algebra context we require  to be defined on each Aj .

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˜ = (A1 , . . . Am ) be carriers (of S) with surpassing Let A˜ = (A1 , . . . Am ) and B ˜ is a set of maps fj : Aj → POs  and  , respectively. A -morphism f : A˜ → B Aj , 1 ≤ j ≤ m, satisfying the properties for every operator ω and all bi , ci ∈ Aji ,: (i) f (ω(b1 , . . . , bm ))  ω(fj1 (b1 ), . . . , fjm (bm )), (ii) If bi  ci , then ω(fj1 (b1 ), . . . , fjm (bm ))  ω(fj1 (c1 ), . . . , fjm (cm )). (iii) fj (0) = 0 whenever 0 ∈ Aj . We call f an ω-homomorphism when equality holds in (i). A homomorphism is an ω- homomorphism for each operator ω. Any homomorphism is a -morphism when we take  to be the identity map. This is a delicate issue, since although -morphisms play an important structural role, as indicated in [37], homomorphisms fit in better with general monoidal category theory, as we shall see. Thus, our default terminology is according to the standard universal algebra version. Definition 9.3. An abstract surpassing relation on a system A = (A, T , (−), ) is a relation  satisfying the following properties: for a, b ∈ A, viewed as a carrier in universal algebra: • 0  a. • If ai  bi for 1 ≤ i ≤ m then ωm,j (a1,j , . . . , am,j )  ωm,j (b1,j , . . . , bm,j ). An abstract surpassing PO is an abstract surpassing relation that is a PO (partial order). In particular, if a  0 for an abstract surpassing PO, then a = 0. Surpassing relations can play a role in modifying universal algebra. This can be formulated categorically, but we state it for systems. Definition 9.4. The surpassing relation on morphisms  for categories of systems is defined by putting f  g if f (b)  g(b) for each b ∈ A. Example 9.5. f ◦ g when g = f + h◦ for some morphism h, but there could be other instances where we cannot obtain h from the values of f and g. 9.2. Functor categories. Definition 9.6. Let S be a small category, i.e., Obj(S) and Hom(S) are sets. Let C be a category. We define C S to be the category whose objects are the sets of functors from S to C and morphisms are natural transformations. Let S be a small category. For any ground T -system A = (A, T , (−), ), considered as a carrier, we view the carrier in the functor category AS (A viewed as a small category) as the carrier {AS1 , AS2 , . . . , ASt }, where ASi denotes the morphisms from S to Ai and, given an operator ωm,j : Aij,1 × · · · × Aij,m → Aij , we define the operator ω ˜ : ASij,1 × · · · × ASij,m → ASim,j “componentwise,” by ω ˜ (f1 , . . . , ft )(s) = ω(f1 (s), . . . ft (s)),

∀s ∈ S.

Likewise for A . Universal relations clearly pass from A to AS and A(S) , verified componentwise. For convenience, we assume that the signature includes the operation + together with the distinguished zero element 0. (S)

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9.2.1. N-categories. Since we lack negatives in semirings and their modules, Hom(A, B) is not a group under addition, but rather a semigroup, and the zero morphism loses its special role, to be supplanted by a more general notion. Grandis in [26] generalizes the usual categorical definitions to “N -categories.” Definition 9.7. Let C be a category. (i) A left absorbing set of morphisms of C is a collection of sets of morphisms I such that if f belongs to I, then any composite gf (if defined) belongs to I. (ii) A right absorbing set of morphisms of C is a collection of sets of morphisms I such that if f belongs to I, then any composite f g (if defined) belongs to I. (iii) An absorbing set of morphisms is a left and right absorbing set of morphisms. ([26, § 1.3.1] calls this an “ideal” but we prefer to reserve this terminology for semirings.) For any given absorbing set N of morphisms of a category C, one can associate the set O(N ) of null objects as follows: O(N ) := {A ∈ Obj(C) | 1A ∈ N }. Conversely, we can fix a class O of null objects in C, and can associate the absorbing morphisms N (O) as follows: N (O) := {f ∈ Hom(C) | f factors through some object in O}. Then one clearly has the following: O ⊆ ON (O),

N O(N ) ⊆ N.

For details, we refer the reader to [26, §1.3]. Definition 9.8. Let C be a category. An absorbing set N of C is closed if N = N (O) for some set of objects O. The null morphisms Null (depending on a fixed class of null objects) are an example of an absorbing set of morphisms; the null morphisms in Hom(A, B) are designated as NullA,B , which will take the role of {0}. Such a category with a designated class of null objects and of null morphisms is called a closed N-category; we delete the word “closed” for brevity. Definition 9.9. [26, § 1.3.1] Let C be a closed N-category with a fixed class of null objects O and the corresponding null morphisms N . (i) The kernel with respect to N of a morphism f : A → B, denoted by ker f , is a monic which satisfies the universal property of a categorical kernel with respect to N , i.e., (a) f (ker f ) is null. (b) If f g is null then g uniquely factors through ker f in the sense that g = (ker f )h for some h. (ii) The cokernel with respect to N of a morphism f : A → B, denoted by coker f , is defined dually, i.e., coker f is an epic satisfying (a) (coker f )f is null. (b) If gf is null then g uniquely factors through coker f in the sense that g = h(coker f ) for some h.

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(iii) Products and coproducts (direct sums) of morphisms also are defined in the usual way, cf. [18, Definition 1.2.1(A2)]. 9.2.2. T -linear categories with a negation functor. Since most of this paper involves T -modules over semirings, let us pause to see in what direction we want to proceed. Definition 9.10. Let C be a category. (i) C is T -linear over a monoid T if it satisfies the following two properties: (a) Composition is bi-additive, i.e., (g + h)f = gf + hf, k(g + h) = kg + kh ˜ ˜ ˜ → C, ˜ and k : C˜ → D. ˜ In for all morphisms f : A → B, g, h : B † ˜ ˜ particular, Mor(A, A) is a semiring , where multiplication is given by composition. (b) T acts naturally on Mor(A, B) in the sense that the action commutes with morphisms. (In practice, the objects will be T -modules, and the action will be by left multiplication.) (ii) A T -linear N-category C with categorical sums is called semi-additive. 9.2.3. Systemic categories. Definition 9.11. (i) A T -linear category with negation is a T -linear category with a negation functor (−). (ii) A semi-additive category with negation is a semi-additive category with a negation functor. Example 9.12. Any additive category (in the classical sense) is Z-linear and semi-additive with negation. Proposition 9.13. For any T -linear category with negation, the “quasi-zero” morphisms (of the form f ◦ := f + ((−)f )) comprise an absorbing set. Proof. One can easily observe: (f (−)f )g = f g(−)f g and g(f (−)f ) = gf (−)gf and this shows the desired result.  Example 9.14. Let C be a category with a negation functor. We can impose an N-category structure on C by defining the absorbing set N of null morphisms to be the morphisms of the form f ◦ ; this may or may not be a closed N -category. On the other hand, later in §6.2, we will introduce a closed N-category structure by means of elements of the form a◦ . Expressing unique negation categorically enables us both to relate to the work [34, §2] of the first author on hyperrings, and also appeal to the categorical literature. Definition 9.15. A semi-abelian category (resp. with negation) is a semi-additive category (resp. with negation) C satisfying the following extra property: Let N be the set of null morphisms of C. Every morphism of C can be written as the composite of a cokernel with respect to N and a kernel with respect N . (This property is called “semiexact” in [26, § 1.3.3].)

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One can define the monoidal property in terms of the adjoint isomorphism, cf. [47]. In short, our discussion also fits into the well-known theory of “rigid” monoidal categories as described in [18], and makes the system category amenable to [60]. Proposition 9.16. The category of systemic modules over A is a monoidal semi-abelian category, with respect to negated T -tensor products. Proof. Immediate from Lemma 6.18 and Remark 5.8.



Remark 9.17. The following properties pass from the category C to C S , seen componentwise: T -linear, semi-additive, semi-additive with negation, semi-abelian. There is a very well-developed theory of abelian categories, which one would like to utilize by generalizing to semi-abelian categories. This has already been done for a large part in [26], so one main task should be to arrange for the category of systems to fit into Grandis’ hypotheses. Acknowledgment The authors would like to thank Oliver Lorscheid for helpful comments concerning the first draft of the paper. References [1] J. Abuhlail, Exact sequences of semimodules over semirings, arXiv preprint arXiv:1210.4566, 2012. [2] Marianne Akian, St´ ephane Gaubert, and Alexander Guterman, Linear independence over tropical semirings and beyond, Tropical and idempotent mathematics, Contemp. Math., vol. 495, Amer. Math. Soc., Providence, RI, 2009, pp. 1–38, DOI 10.1090/conm/495/09689. MR2581511 [3] M. Akian, S. Gaubert, and L. Rowen, Linear algebra over systems, preprint (2018). [4] S. A. Amitsur, A general theory of radicals. I. Radicals in complete lattices, Amer. J. Math. 74 (1952), 774–786, DOI 10.2307/2372225. MR50563 [5] S. A. Amitsur, A general theory of radicals. II. Radicals in rings and bicategories, Amer. J. Math. 76 (1954), 100–125, DOI 10.2307/2372403. MR59256 [6] M. Baker and N. Bowler, Matroids over hyperfields, arXiv:1601.01204v4, 2016. [7] Matthew Baker and Nathan Bowler, Matroids over partial hyperstructures, Adv. Math. 343 (2019), 821–863, DOI 10.1016/j.aim.2018.12.004. MR3891757 [8] M. Baker and O. Lorscheid, The moduli space of matroids, arXiv:1809.03542v1 [math.AG], 2018. [9] V. Berkovich, Algebraic and analytic geometry over the field with one element, preprint, 2014. [10] Aaron Bertram and Robert Easton, The tropical Nullstellensatz for congruences, Adv. Math. 308 (2017), 36–82, DOI 10.1016/j.aim.2016.12.004. MR3600055 [11] N. Bourbaki, Commutative Algebra, Paris and Reading, 1972. [12] Alain Connes and Caterina Consani, Homological algebra in characteristic one, High. Struct. 3 (2019), no. 1, 155–247. MR3939048 [13] Guillermo Corti˜ nas, Christian Haesemeyer, Mark E. Walker, and Charles Weibel, Toric varieties, monoid schemes and cdh descent, J. Reine Angew. Math. 698 (2015), 1–54, DOI 10.1515/crelle-2012-0123. MR3294649 [14] A. Almeida Costa, Sur la th´ eorie g´ en´ erale des demi-anneaux (French), Publ. Math. Debrecen 10 (1963), 14–29. MR168608 [15] Andreas W. M. Dress, Duality theory for finite and infinite matroids with coefficients, Adv. in Math. 59 (1986), no. 2, 97–123, DOI 10.1016/0001-8708(86)90047-2. MR834224 [16] Andreas W. M. Dress and Walter Wenzel, Valuated matroids, Adv. Math. 93 (1992), no. 2, 214–250, DOI 10.1016/0001-8708(92)90028-J. MR1164708

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Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15117

Quasi-Baer module hulls and their examples Jae Keol Park and S. Tariq Rizvi Dedicated to Professor S. K. Jain on his 80th birthday Abstract. The notion of various types of module hulls has been of importance since the discovery of injective hull of a given module M by Shoda in 1952 and independently by Eckmann and Schopf in 1953. While these hulls are very useful as they have better properties than the module M , it is most often not easy to construct or find explicit examples of a hull belonging to a particular class. In this expository paper, we present a number of explicit examples and counterexamples related to quasi-Baer hulls of modules. The notion of quasi-Baer rings was introduced by Clark from a characterization of finite dimensional algebras over an algebraically closed field in terms of twisted semigroup algebras. We exhibit the connections of a quasi-Baer rings to a unital boundedly centrally closed C ∗ -algebra. The design of the notion of a quasi-Baer module and why it makes sense is discussed. Relevant properties of quasi-Baer modules are presented. In the main Section 3, a number of results for quasi-Baer modules are included. Various examples and counterexamples of quasi-Baer hulls of modules are constructed or presented to illustrate the notion and delimit its properties. Quasi-Baer hulls are compared with Baer and Rickart hulls. We expect the applications of the results and examples to motivate further research on this topic. The included examples and results are intended to make this paper as self-contained as possible.

1. Introduction A quasi-Baer ring (i.e., a ring for which the left annihilator of every ideal is generated by an idempotent) was defined by Clark [13] while characterizing a finite dimensional algebra over an algebraically closed field. Clark [13] also showed that any finite distributive lattice is isomorphic to a sublattice of the lattice of all ideals of an artinian quasi-Baer ring. It is shown in [8] that there is a strong connection between quasi-Baer rings and unital boundedly centrally closed C ∗ algebras. Indeed, it is proven that a unital C ∗ -algebra A is boundedly centrally closed if and only if A is a quasi-Baer ring (see also [9]). More recently, the notion of a Baer ring (i.e., a ring for which the left annihilator of every nonempty set is generated by an idempotent) was extended to an analogous module theoretic notion via the endomorphism ring of a module by Rizvi and Roman [40] in view of the Morita context. 2010 Mathematics Subject Classification. 16D25, 16D40, 16D80, 16E60, 16R20, 16U10. Key words and phrases. quasi-Baer module, quasi-Baer hull, Baer module, Baer hull, Rickart module, Rickart hull, FI-extending module, extending module, quasi-retractable. c 2020 American Mathematical Society

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In Section 2, designing notions of Baer and Rickart modules are discussed briefly (see [38] for details on notions of Baer and Rickart modules). Then the notion of quasi-Baer modules is introduced related to that of Baer modules. A module MR is called quasi-Baer module if, for any N M , there exists e2 = e ∈ S := End(MR ) with "S (N ) = Se, where "S (N ) = {f ∈ S | f (N ) = 0}. We note that a ring R is a (quasi-)Baer ring if and only if RR is a (quasi-)Baer module. Also a ring R is right Rickart if and only if RR is a Rickart module. A module M is called extending if every submodule of M is essential in a direct summand of M . A ring R is said to be right (resp., left) extending if RR (resp., R R) is extending. It is known that there are close connections between an extending ring and a Baer ring (see [12]). In [40] it was shown that every K-nonsingular extending module is Baer and that every Baer module under a dual condition becomes extending. A number of interesting properties of (quasi-)Baer and Rickart modules are shown in [28], [29], [30], [40], [41], and [42]. Assume that MR is a module. We fix an injective hull E(MR ) of MR . Let M be a class of modules. We call, when it exists, a module HR the M hull of MR if HR is the smallest extension of MR in E(MR ) that belongs to M (see [9, Definition 8.4.1]). For a given module M , the smallest quasi-Baer (resp., Baer, Rickart) overmodule of M in E(M ) is called the quasi-Baer (resp., Baer, Rickart) hull of M . One of the difficulties in dealing with the (quasi-)Baer and the Rickart hulls of MR is the interplay of the scalar multiplication of M with R on one side of M and the endomorphism ring S = End(MR ) on the other side of M . Such an overmodule of M not only has to satisfy the condition to be a quasi-Baer (resp., Baer, Rickart) module but also to be the smallest such overmodule of M to become a hull. The quasi-Baer (resp., Baer, Rickart) property of the hull thus necessitates a consideration of endomorphism rings of all overmodules of M in E(M ) before we can locate the smallest quasi-Baer (resp., Baer, Rickart) overmodule of M . In [27], [34], [35], [37], and [38], quasi-Baer, Rickart, and Baer hulls of a certain class of modules were studied. Further, these hulls were described explicitly and various examples were provided. In this expository paper, we will mainly focus on quasi-Baer module hulls and their explicit examples. We include relevant results (without proofs) but provide explicit constructions and details of quasi-Baer module hulls examples. These examples (in not such details) and results can be found in [27], [34], [35], [36], [37], and [38]. We have made an attempt for this paper to be self-contained as possible. The examples will explicitly illustrate the results and it is hoped that the examples will motivate further research. All rings are assumed to have identity and all modules are assumed to be unitary. For R-modules MR and NR , we use Hom(MR , NR ), HomR (M, N ), or Hom(M, N ) to denote the set of all R-module homomorphisms from MR to NR . Likewise, End(MR ), EndR (M ), or End(M ) denotes the endomorphism ring of an R-module M . For an R-module homomorphism f ∈ HomR (M, N ), Ker(f ) denotes the kernel of f . We use E(MR ) or E(M ) to denote an injective hull of a module MR . For a module M , K ≤ M , L M , N ≤ess M , and U ≤⊕ M denote that K is a submodule of M , L is a fully invariant submodule of M , N is an essential submodule of M ,

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and U is a direct summand of M , respectively. If M is an R-module, AnnR (M ) stands for the annihilator of M in R. For a module M and a nonempty set Λ, let M (Λ) denote the direct sum of |Λ| copies of M . When Λ is finite with |Λ| = n and n > 1, M (n) stands for M (Λ) . We use CFMΛ (R) to denote the Λ × Λ column finite matrix ring over a ring R. Let Matn (R) denote the n × n matrix ring over a ring R. Let M be a right R-module. We set rR (U ) = {a ∈ R | U a = 0} for ∅ = U ⊆ M. Similarly, for a left S-module M, we denote "S (M ) = {s ∈ S | sM = 0}, and for any ∅ = V ⊆ M, we let "S (V ) = {s ∈ S | sV = 0}. Let R be a ring and ∅ = X ⊆ R, we let "R (X) = {a ∈ R | aX = 0} and rR (X) = {a ∈ R | Xa = 0} which are called the left annihilator of X in R and the right annihilator of X in R, respectively. For a ring R, we let Q(R) stand for the maximal right ring of quotients of R. The symbols C, R, Q, Z, and Zn (n > 1) denote the field of complex numbers, the field of real numbers, the field of rational numbers, the ring of integers, and the ring of integers modulo n, respectively. Ideals of a ring without the adjective “left” or “right” mean two-sided ideals. For a ring R, the notation I R denotes that I is an ideal of R. 2. Quasi-Baer Rings and Modules In this section, first we introduce historical background of the notion of quasiBaer rings defined by Clark [13] in characterizing a finite dimensional algebra over an algebraically closed field as a twisted semigroup algebra of some matrix unit semigroup. Some properties of quasi-Baer rings are provided. Then we discuss a strong connection from quasi-Baer rings to unital boundedly centrally closed C ∗ algebras, that is, a unital C ∗ -algebra A is boundedly centrally closed if and only if A is a quasi-Baer ring. Some relevant properties of Baer rings and Rickart rings are included. Designing the notion of Baer modules in module theoretic setting in view of the Morita context from the notion of Baer rings is discussed briefly. Then designing the notion of quasi-Baer modules is explained. Basic properties and various useful results on quasi-Baer, Baer, and Ricakrt modules for Section 3 are included. Various interesting examples of quasi-Baer, Baer, and Rickart modules are provided. All results and examples of this section are adopted from [7], [8], [9], [13], [23], [24], [28], [29], [30], [40], [41], and [42]. Let S be a semigroup with zero, F a field, and let ϕ : S × S → F satisfy the following: (i) ϕ(s, t) = 0 if and only if st = 0 for s, t ∈ S. (ii) ϕ(r, st)ϕ(s, t) = ϕ(rs, t)ϕ(r, s) whenever rst = 0, for r, s, t ∈ S. Further, let Fϕ [S] denote the vector space of all formal finite linear combina tions αi si , where αi ∈ F , si ∈ S, and si = 0 for each i. Define a multiplication by s · t = ϕ(s, t)st for s and t nonzero elements of S, and extend this multiplication linearly to all of Fϕ [S]. Then we see that (ii) above is exactly what is required to ensure the associativity of Fϕ [S]. In this case, Fϕ [S] is called a twisted semigroup algebra of S over F . If S has no zero, Fϕ [S] is defined similarly with the obvious modification.

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Say n is a positive integer and let M U (n) denote the full semigroup of matrix units {eij | 1 ≤ i, j ≤ n}∪{0}, where ehi ejk = δij ehk and δij is the Kronecker delta. By a matrix units semigroup is meant a subsemigroup of M U (n) which contains e11 , . . . , enn . In the next result, a finite dimensional algebra over an algebraically closed field which is a twisted semigroup algebra is characterized. This motivates the definition of a quasi-Baer ring. Theorem 2.1 ([13, Clark]). Let R be a finite dimensional algebra over an algebraically closed field F . Then the following are equivalent. (i) R ∼ = Fϕ [S] for some matrix units semigroup S. (ii) The left annihilator of every ideal of R is generated by an idempotent and R has a finite ideal lattice. Furthermore, every finite distributive lattice is isomorphic to a certain sublattice of the lattice of all ideals of an artinian ring with condition (i) of Theorem 2.2 as follows. Theorem 2.2 ([13, Clark]). Let L be a finite distributive lattice. Then there exists an artinian ring R such that: (i) the left annihilator of any ideal of R is generated by an idempotent. (ii) the lattice L is isomorphic to the sublattice {"R (I) | R I ≤ R R} of the lattice of all ideals of R. Condition (ii) of Theorem 2.1 and Condition (i) of Theorem 2.2 motivate the following definition. Definition 2.3 ([13, Clark]). A ring R is called quasi-Baer if the left annihilator of every ideal of R is generated by an idempotent of R. For another motivation of the notion of a quasi-Baer ring, the next remark is useful. Remark 2.4. (i) We recall the definition of a crossed product of a group G over a ring R: Given a group homomorphism α : G → Aut(R) (where Aut(R) is the group of ring automorphisms of R), and a map γ : G × G → U (R), U (R) the units of R, such that γ(x, y)γ(xy, z) = γ(y, z)α(x) γ(x, yz) and γ(x, y)r α(xy) = r α(y)α(x) γ(x, y) for all x, y, z ∈ G and r ∈ R, where γ(y, z)α(x) , etc., is the image of γ(y, z) under α of all formal finite α(x), etc., we define the crossed product Rγ [G] to be the set sums of the forms rx x with rx ∈ R and x ∈ G (i.e., Rγα [G] is a free module with a basis {x | x ∈ G} over R). Hence the addition is componentwise. The multiplication is given by the rule (rx x) (ry y) = rx yyα(x) γ(x, y)xy. This makes Rγα [G] an associative ring with identity γ(1, 1)−1 1. When α(x) is the identity map of R for every x ∈ G, the crossed product is called the twisted group ring of G over R, and γ is called a factor set on G in R. On the other hand, if γ(x, y) = 1 for all x, y ∈ G the crossed product is called the skew group ring of G over R.

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If α(x) is the identity map of R for every x ∈ G and γ(x, y) = 1 for all x, y ∈ G, the crossed product becomes the group ring of G over R. Assume that R[G] be the group ring of a group G over a ring R. Then for any normal subgroup H of G, R[G] is a crossed product of the group G/H over R[H] (see [4]). (ii) Let K be a finite dimensional normal extension field of a field F with Galois group G and let γ be a factor set of G in K. Then the crossed product Kγα [G], where α is the embedding from G into Aut(K), is a simple ring with the center F , that is, a central simple algebra over F (see [20, Theorem 4.4.1]). (iii) Theorem 2.1 and Theorem 2.2 illustrate that the notion of quasi-Baer rings is not just a simple generalization of that of Baer rings. In fact, Theorem 2.1 shows that a quasi-Baer ring originates from a characterization of finite dimensional algebras over an algebraically closed field. Furthermore, the notion of quasi-Baer rings is closely related to a twisted semigroup algebras. Historically, it is interesting to note that the Hamilton quaternion division algebra H is a twisted group algebra of the Klein four group over R. The next result shows the left-right symmetry of quasi-Baer rings. Theorem 2.5. Let R be a ring. Then the following are equivalent. (i) R is a quasi-Baer ring. (ii) For each I R, there exists e2 = e ∈ R such that rR (I) = eR. Some examples of quasi-Baer rings include the following. Example 2.6. (i) Every Baer ring is a quasi-Baer ring (see Definition 2.11). (ii) Every prime ring is a quasi-Baer ring. (iii) The endomorphism ring of a projective (hence a free) module over a quasiBaer ring is a quasi-Baer ring. (iv) If R is a quasi-Baer ring, then Matn (R) and Tn (R) are quasi-Baer rings for every positive integer n, where Tn (R) is the n × n upper triangular matrix ring over R (see [9, Theorem 5.6.7]). (v) Mat2 (Z[x]) is a quasi-Baer ring (see [9, Theorem 6.2.4]), but it is neither a Baer ring nor a Rickart ring (see [9, Example 3.1.28]). (vi) If a ring R is quasi-Baer, then R[x] is quasi-Baer (see [9, Theorem 6.2.4]). (vii) Any group algebra F [G] of a polycyclic-by-finite group G over a field F with characteristic zero is quasi-Baer (see [9, Corollary 6.3.4]). There are natural connections of Baer and quasi-Baer rings to Functional Analysis. We show some of these briefly in the following. Assume that R is a ring. Recall that a map ∗ : R → R is called an involution if (i) (x + y)∗ = x∗ + y ∗ ; (ii) (xy)∗ = y ∗ x∗ ; and (iii) (x∗ )∗ = x for all x, y ∈ R. A ring R with an involution ∗ is called a ∗-ring. An element e ∈ R is called a projection if e2 = e and e∗ = e (i.e., e is a self-adjoint idempotent). A Banach algebra is a complex normed algebra A which is complete (as a topological space) such that ||ab|| ≤ ||a|| ||b|| for all a, b ∈ A. A Banach ∗-algebra is a complex Banach algebra A with an involution ∗ satisfying that (λa)∗ = λ a∗ for λ ∈ C and a ∈ A, where λ is the conjugate of λ ∈ C. A C ∗ -algebra A is a Banach ∗-algebra with the additional norm condition that ∗ ||a a|| = ||a||2 for all a ∈ A. A C ∗ -algebra is called unital if it has identity.

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By Kaplansky [23], the notion of AW ∗ -algebras was defined for the study to abstract W ∗ -algebras (also called von Neumann algebras). Definition 2.7 ([23, Kaplansky]). An AW ∗ -algebra A is a C ∗ -algebra which satisfies the following. (i) In the partially ordered set of projections of A, any set of orthogonal projections has a least upper bound. (ii) Any maximal commutative self-adjoint subalgebra B (i.e., B ∗ = B) is generated by its projections (that is, B is equal to the smallest closed subalgebra containing its projections). It is shown in [23] that an AW ∗ -algebra is a unital C ∗ -algebra. There are several characterizations of an AW ∗ -algebra. One of the most useful characterizations is the following (see [23] and [10, p.243]). Theorem 2.8 ([23, Kaplansky]). A C ∗ -algebra A is an AW ∗ -algebra if and only if the left annihilator of any nonempty subset of A is Ae for some projection e of A if and only if the right annihilator of any nonempty subset of A is f A for some projection f of A. The next theorem, essentially due to Baer [5], provides the basis for defining the notion of a Baer ring by Kaplansky [24]. Theorem 2.9. Assume that MR is a semisimple R-module and S = End(MR ). Then the left annihilator of any nonempty subset of S is Se for some e2 = e ∈ S. The right annihilator of any nonempty subset of S is also f S for some f 2 = f ∈ S. Theorem 2.10 ([24, Theorem 3, p.2]). In a ring R, not necessarily with identity, any two of the following statements imply the third. (i) For each ∅ = X ⊆ R, there exists e2 = e ∈ R such that "R (X) = Re. (ii) For each ∅ = Y ⊆ R, there exists f 2 = f ∈ R such that rR (Y ) = f R. (iii) R has an identity. From Theorems 2.9 and 2.10, Kaplansky [24] defined the notion of Baer rings as follows. Definition 2.11. A ring R is called Baer if R satisfies any two (hence, all three) of the statements of Theorem 2.4. From Theorem 2.10, any Baer ring always has an identity, and that a Baer ring is left-right symmetric. Let R be a ∗-ring. Then R is called a Baer ∗-ring if and only if rR (X) = eR with e ∈ R a projection for every ∅ = X ⊆ R. Thus by Theorem 2.8, a C ∗ -algebra A is an AW ∗ -algebra if and only if A is a Baer ∗-ring. Example 2.12. (i) Every AW ∗ -algebra is a Baer ring. (ii) The endomorphism ring of a semisimple module is a Baer ring. (iii) The ring of all sequences of complex numbers which are eventually real numbers is a Baer ring. (iv) Any right self-injective von Neumann regular ring is a Baer ring. (v) Let R be a domain and n be an integer such that n > 1. Then Tn (R) is a Baer ring if and only if R is a division ring ([24, Exercises 2, p.16]), where Tn (R) is the n × n upper triangular matrix ring over R. In [31], Maeda defined Rickart rings as in Definition 2.13 below. Hattori [19] introduced the notion of a right PP ring (i.e., a ring with the property that every

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principal right ideal is projective). It was later shown that right PP rings are precisely right Rickart rings (see Example 2.14(v)). Definition 2.13. A ring R is called right Rickart if the right annihilator of any element is generated by an idempotent. A left Rickart ring is defined similarly. The Rickart notion is not left-right symmetric and a right and left Rickart ring is called a Rickart ring. Example 2.14. (i) Every von Neumann regular ring is a right (and left) Rickart ring. (ii) Every Baer ring is a right (and left) Rickart ring. (iii) The center of a right Rickart ring is a Rickart ring. (iv) Let An = Z for n = 1, 2, . . . , and put *  ∞  An | an is constant eventually , R = (an )∞ n=1 ∈ ∞

n=1

a subring of n=1 An . Then R is a Rickart ring. Indeed, say α = (an )∞ n=1 ∈ R. Let e = (en )∞ n=1 , where en = 1 if an = 0 and en = 0 if an = 0. Since an is constant eventually, so is en . Also e2 = e ∈ R and rR (α) = eR. But R is neither Baer nor von Neumann regular. (v) A ring R is right Rickart if and only if R is right PP. While Baer rings are derived from AW ∗ -algebras in Functional Analysis, quasiBaer rings are connected to boundedly centrally closed C ∗ -algebras in Functional Analysis (see Theorem 2.17). We now discuss this connection from quasi-Baer rings to Functional Analysis and show that a unital C ∗ -algebra A is boundedly centrally closed if and only if A is a quasi-Baer ring. For this, we start with the following. Assume that A is a C ∗ -algebra. Then A is a semiprime ring. Indeed, assume aAa = 0 with a ∈ A. Then (a∗ a)∗ A(a∗ a) = a∗ aAa∗ a = a∗ a(Aa∗ )a ⊆ a∗ aAa = 0. Therefore (a∗ a)∗ A(a∗ a) = 0. In particular, (a∗ a)∗ (a∗ a) = 0 and hence a∗ a = 0, so a = 0 because ∗ is positive-definite. Thus A is semiprime. Let F be the set of all ideals of A such that AnnA (I) = 0. We take Qs (A) = {q ∈ Q(A) | qI + Iq ⊆ A for some I ∈ F}. The ring Qs (A) is known as the symmetric ring of quotients of A. We let I be a norm closed (i.e., I = I) essential ideal of the C ∗ -algebra A, where I is the topological closure of I. We put M (I) = {q ∈ Qs (A) | qI + Iq ⊆ I}, which is called the idealizer of I in Qs (A). Let Ice be the set of all norm closed essential ideals of A. If I ∈ Ice , then I is a C ∗ -algebra. Also in this case M (I) is a C ∗ -algebra (see [2, Proposition 2.1.3]). There exists the algebraic direct limit of {M (I) | I ∈ Ice }. Let alg.limI∈Ice M (I) denote the algebraic direct limit of {M (I) | I ∈ Ice }. Then there exists a unique isometric ∗-isomorphism from alg.limI∈Ice M (I) onto Qb (A) (see [2, Proposition 2.2.2]). Hence alg.limI∈Ice M (I) is identified with Qb (A) and Qb (A) is called the bounded symmetric algebra of quotients of the C ∗ -algebra A.

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The topological completion Mloc (A) of Qb (A) is called the local multiplier algebra of A. By Mloc (A) we denote the norm closure of Qb (A). See [2] and [9] for more details on local multiplier algebras of C ∗ -algebras. Definition 2.15. Let A be a C ∗ -algebra and Cen(Qb (A)) be the center of Qb (A). Then the C ∗ -algebra ACen(Qb (A)) in Mloc (A) is called the bounded central closure of A. We say that the C ∗ -algebra A is boundedly centrally closed if A = ACen(Qb (A)). In the following, we exhibit some examples of boundedly centrally closed C ∗ algebras (see also [2]). Example 2.16. (i) A commutative unital C ∗ -algebra is boundedly centrally closed if and only if it is an algebra of continuous functions on a Stonean space. (ii) The bounded central closure of a C ∗ -algebra is boundedly centrally closed. (iii) Every AW ∗ -algebra (in particular, every von Neumann algebra) is boundedly centrally closed. (iv) Every unital prime C ∗ -algebra is boundedly centrally closed. (v) The local multiplier algebra Mloc (A) of a C ∗ -algebra A is boundedly centrally closed. A ∗-ring R is called a quasi-Baer ∗-ring if rR (I) = eR with e ∈ R a projection for every I R. The following theorem shows the connection of quasi-Baer rings to boundedly centrally closed C ∗ -algebras in Functional Analysis. Indeed, a unital boundedly centrally closed C ∗ -algebra is precisely a C ∗ -algebra which is a quasiBaer ring (see [8] and [9]). Theorem 2.17. Let A be a unital C ∗ -algebra. Then the following are equivalent. (i) A is boundedly centrally closed. (ii) A is a quasi-Baer ∗-ring. (iii) A is a quasi-Baer ring. The notion of Baer modules was established using the fact that the endomorphism ring of a semisimple module is a Baer ring (see [5]). Hence in view of the Morita context the notion of a Baer module is expected to be designed such that any semisimple module is a Baer module. Furthermore, the endomorphism ring of a Baer module should be a Baer ring so that the decomposition by type theory of the endomorphism ring can be applied to the decomposition of a Baer module. Motivated by these considerations, the notion of a Baer module is defined as follows. For more details on design of the notion of Baer modules, see [38]. Definition 2.18 ([40, Rizvi and Roman]). Let M be a right R-module with S = End(MR ). Then M is called a Baer module if for any ∅ = X ⊆ M , "S (X) = Se for some e2 = e ∈ S. Let M be a right R-module and ∅ = X ⊆ M . Put N = XR ≤ M . Then "S (X) = "S (XR) = "S (N ). Hence Definition 2.18 is equivalent to the following definition. Definition 2.19. Let M be a right R-module with S = End(MR ). Then M is called a Baer module if for any N ≤ M , "S (N ) = Se for some e2 = e ∈ S. The Baer property of rings is left-right symmetric. The following result shows a module theoretic analogue of this fact.

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Theorem 2.20. The following are equivalent for a right R-module M. (i) M is a Baer module. (ii) For any left ideal I of S, rM (I) = f M with f 2 = f ∈ S. (iii) For any ∅ = Y ⊆ S, rM (Y ) = gM with g 2 = g ∈ S. Let R be a ring. Then R is a Baer ring if and only if RR is a Baer module from Theorem 2.20. Further, all semisimple modules are Baer modules (see [38, Theorem 2.20]). Moreover, the endomorphism ring of a Baer module is a Baer ring (Theorem 3.9). Several examples of Baer modules will follow from our results later (see also Section 3). Definition 2.21. Let MR be a right R-module and S = End(MR ). Then M is called K-nonsingular if, for φ ∈ S, rM (φ) = Ker(φ) ≤ess M implies φ = 0. Proposition 2.22. Every nonsingular module is K-nonsingular. The following example shows that the converse of Proposition 2.22 does not hold true. But when M = R, the K-nonsingularity and the nonsingularity of M coincide. Example 2.23. The Z-module Zp (p is a prime integer) is K-nonsingular. However, Zp is not nonsingular. Definition 2.24. A right R-module M is called K-cononsingular if, for N ≤ M, "S (N ) = 0 implies N ≤ess M. A module M is called extending if every submodule of M is essential in a direct summand of M . A ring R is said to be right (resp., left) extending if RR (resp., R R) is extending. It is easy to see that every extending module is K-cononsingular. By Chatters and Khuri [12], a right extending right nonsingular ring is precisely a Baer and right cononsingular ring. For a module theoretic analogue of this result, we have the following due to Rizvi and Roman [40], which shows strong connections between extending modules and Baer modules. Theorem 2.25. Let MR be a right R-module. Then the following are equivalent. (i) MR is extending and K-nonsingular. (ii) MR is Baer and K-cononsingular. To design a suitable notion of a Rickart module, the following should be kept in mind: (i) Any Baer module must be a Rickart module; and (ii) For a ring R, RR is a Rickart module if and only if R is a right Rickart ring. By this consideration, a Rickart module is defined as follows, as suggested from Theorem 2.20. For more details on designing the notion of a Rickart module, see [38]. Definition 2.26. A module MR with S = End(MR ) is called a Rickart module if, for any φ ∈ S, rM (φ) = eM for some e2 = e ∈ S. Note in Definition 2.26 that rM (φ) = Ker(φ). For a ring R, by Definition 2.26, RR is a Rickart module if and only if R is a right Rickart ring. From Definition 2.26, we also see that any Baer module is a Rickart module. Suggested from Definition 2.19, the following definition is given. Definition 2.27. Let MR be a right R-module and S = End(MR ). Then MR is called a quasi-Baer module if, for any N M , "S (N ) = Se for some e2 = e ∈ S.

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From Definition 2.27, a ring R is a quasi-Baer ring if and only if RR is a quasiBaer module. Furthermore, every Baer module is a quasi-Baer module. Similar to the case of Baer modules, we get the following. Theorem 2.28. The following are equivalent for a module M. (i) M is a quasi-Baer module. (ii) For any I S, rM (I) = f M with f 2 = f ∈ S, where S = End(M ). The next results show that the endomorphism ring of a quasi-Baer module inherits the property just as do the direct summands. Theorem 2.29. (i) The endomorphism ring of a quasi-Baer module is a quasiBaer ring. (ii) Let MR be a quasi-retractable. Then MR is quasi-Baer if and only if End(MR ) is a quasi-Baer ring. Theorem 2.30. Any direct summand of a quasi-Baer module is a quasi-Baer module. As a consequence, we obtain an alternate way to show that the corner ring inherits the quasi-Baer property as well. Corollary 2.31. Let R be a quasi-Baer ring and 0 = e2 = e ∈ R. Then: (i) eRR is a quasi-Baer module. (ii) End(eRR ) = eRe is a quasi-Baer ring. The next result, due to Rizvi and Roman [40], is necessary for the study of projective modules over a quasi-Baer ring. Theorem 2.32. Let {Mi }i∈Λ be a set of quasi-Baer modules. If Mi is subisomorphic to (i.e., isomorphic to a submodule of ) Mj for all i, j ∈ Λ with i = j. Then M = ⊕i∈Λ Mi is quasi-Baer. Corollary 2.33. Let R be a ring. Then the following are equivalent. (i) R is a quasi-Baer ring. (ii) Any free right R-module is a quasi-Baer module. (iii) Any projective right R-module is a quasi-Baer module. (iv) CFMΛ (R) is a quasi-Baer ring for any nonempty set Λ. Similar to the case of Baer modules, quasi-Baer modules also satisfy a weak nonsingularity condition (see Definition 2.36). This condition will allow us to exhibit close links between quasi-Baer modules and generalized extending modules, in Theorem 2.41. For this we consider the following definition. Definition 2.34. A module M is called FI-extending (i.e., fully invariant extending) if every fully invariant submodule of M is essential in a direct summand of M. A ring R is called right FI-extending if RR is FI-extending. Similarly, a left FI-extending ring is defined. The notion of FI-extending modules generalizes that of extending modules by requiring that only every fully invariant submodule of M is essential in a direct summand rather than every submodule of M . See [7] (also [9]) for more details on FI-extending modules. In the following result, any direct sum of FI-extending modules is FI-extending without any additional requirements. Thus, while a direct sum of extending modules may not be extending (for example, Z2 ⊕ Z8 ), it does satisfy the extending

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property for all its fully invariant submodules (which include many well-known submodules of any given module M ) such as J(M ), Z(M ), and Soc(M ), etc. Theorem 2.35. Any direct sum of FI-extending modules is FI-extending. A direct consequence of Theorem 2.35 is that any direct sum of uniform (or extending) modules is FI-extending. Definition 2.36. A right R-module M is called FI-K-nonsingular if for any I S, rM (I) ≤ess eM with e2 = e ∈ S implies Ie = 0. Definition 2.37. A right R-module M is called FI-K-cononsingular if for any N M, rM ("S (N )) ≤⊕ M implies N ≤ess rM ("S (N )). The next two propositions provide motivations for Definitions 2.36 and 2.37. Proposition 2.38. The following are equivalent for a right R-module M. (i) M is K-nonsingular. (ii) For each left ideal I of S, rM (I) ≤ess eM with e2 = e ∈ S implies I ∩Se = 0. (iii) For every left ideal J of S, rM (J) ≤ess M implies J = 0. Proposition 2.39. A right R-module M is K-cononsingular if and only if, for N ≤ M, rM ("S (N )) ≤⊕ M implies N ≤ess rM ("S (N )). From Proposition 2.38 and Proposition 2.39, every K-nonsingular (resp., Kcononsingular) module is FI-K-nonsingular (resp., FI-K-cononsingular). Example 2.40. (i) If a ring R is semiprime, then RR is FI-K-nonsingular. To see this, say I R such that rR (I) ≤ess eRR with e2 = e ∈ R. Assume on the contrary that Ie = 0. Since R is semiprime and I R, eI = 0 and so there is x ∈ I with 0 = ex ∈ eI ⊆ eR. Hence there exists r ∈ R with 0 = exr ∈ rR (I) because rR (I) ≤ess eRR . Thus 0 = exr ∈ rR (I) ∩ I = 0, a contradiction. Hence Ie = 0. Therefore RR is FI-K-nonsingular. (ii) There exists a ring R such that RR is FI-K-nonsingular, but RR is not K-nonsingular. In fact, by [11, 26, 33] there is a prime ring R with Z(RR ) = 0, where Z(RR ) is the singular submodule of RR (see [9, Example 3.2.7] adopted from [33]). Then RR is not K-nonsingular by comments after Proposition 2.22. But note that, RR is FI-K-nonsingular from part (i). (iii) There is a ring R such that RR is FI-K-cononsingular, but RR is not K-cononsingular. For example, consider the ring,

C C R= . 0 R Then R is a Baer ring since T2 (C) is a Baer ring and R contains all idempotents of T2 (C). Furthermore, R is a right FI-extending ring by a direct computation. Let

0 i α= , 0 0 where i is the imaginary unit. There is no idempotent e ∈ R with αRR ≤ess eRR . Thus R is not right extending. Then RR is not K-cononsingular by Theorem 2.25. But RR is FI-K-cononsingular. In general, any module which is Baer and FIextending, but not extending has the property that it is FI-K-cononsingular but not K-cononsingular.

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The following result shows strong connections between quasi-Baer modules and FI-extending modules. Theorem 2.41. Let MR be a right R-module. Then the following are equivalent. (i) M is FI-extending and FI-K-nonsingular. (ii) M is quasi-Baer and FI-K-cononsingular. 3. Quasi-Baer Hulls, Examples, and Counterexamples In this section, we present various examples and counterexamples related to quasi-Baer, Baer, and Rickart hulls of modules. These hulls are compared then with extending and FI-extending hulls via explicit examples. We include relevant results on quasi-Baer, Baer, and Rickart hulls. A number of motivating examples for the study of quasi-Baer hulls are also exhibited. As an application, we also show that if LR is a direct summand of a module NR , then the quasi-Baer hull of L may not be a direct summand of the quasi-Baer hull of N , even if the ring R is a Dedekind domain. Further, we provide modules U and V such that the quasi-Baer hull of U ⊕ V is not isomorphic to the direct sum of the quasi-Baer hull of U and that of V even if all the quasi-Baer hull of U ⊕ V , the quasi-Baer hull of U , and the quasi-Baer hull of V exist. Most results, examples, and counterexamples are adopted from [27], [34], [35], [36], and [37]. Definition 3.1 ([9, Definition 8.4.1]). Let MR be a module. We fix an injective hull E(MR ) of MR . Let M be a class of modules. We call, when it exists, a module HR the M hull of MR if HR is the smallest extension of MR in E(MR ) that belongs to M. Assume that M is a module. By Definition 3.1, the quasi-Baer (resp., Baer etc.) hull of M is the smallest quasi-Baer (resp., Baer, etc.) essential extension of M in E(M ). We use qB(−), B(−), Ric(−), Ex(−), and FI(−) to denote the quasi-Baer, the Baer, the Rickart, the extending, and the FI-extending hull of a module, respectively when they exist. We mention that in most cases it is challenging to find various module hulls. However with the help of some new methods and results we have been able to locate and describe quasi-Baer and some other types of hulls explicitly. Examples of these hulls are the objectives of this paper. For a ring R, Cen(R) denotes the center of R. Recall from [3] that a ring R is said to be ideal intrinsic over Cen(R) if I ∩ Cen(R) = 0 for any 0 = I R. For a semiprime ring which is ideal intrinsic over Cen(R), it is known that R is left and right nonsingular by [3, Proposition 1.2], and Cen(Q(R)) = Q(Cen(R)) by [1, Lemma 1.1] or [3, Theorem 3.4]. Semiprime rings which are ideal intrinsic over their centers include, for example, biregular rings; reduced rings with essential socles; von Neumann regular right selfinjective directly finite rings [18, Theorem 9.25, p.105]; and R[x] where R is a simple ring. For an algebra R over a commutative ring C, we can form the enveloping algebra Re = R ⊗C Ro , where Ro denotes the algebra opposite to R. In this case, R has a left Re -module induced by (x ⊗ y)r = xry for x, r ∈ R and y ∈ Ro . The

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algebra R is called separable if R is a projective left Re -module. If R is a separable algebra, then R is ideal intrinsic over Cen(R) (see [9, pp.77–78]). Recall that a ring R is called a PI-ring if R satisfies a polynomial identity. We note that any commutative ring is obviously a PI-ring. It is shown in [44] that any semiprime PI-ring R is ideal intrinsic over Cen(R). According to [3]: (i) a ring R is called an almost PI-ring if every prime factor ring of R is a PI-ring; and (ii) a ring R is called an intrinsically PI-ring if every nonzero ideal contains a nonzero PI-ideal of R. For a semiprime ring, the following implications hold which are not reversible (see [3] for more detail): PI ⇒ almost PI ⇒ intrinsically PI. By [3, Theorem 1.17] , if a ring R is semiprime intrinsically PI, then R is ideal intrinsic over Cen(R). For a semiprime R which is not intrinsically PI, but R is ideal intrinsic over its center, take R = W1 [F ] the first Weyl algebra over a field F of characteristic 0. Then R is a simple domain, so R is ideal intrinsic over Cen(R) = F . Note that Q(R) is a division ring with center F and Q(R) is infinite dimensional over F . By Posner’s Theorem (see [9, Theorem 3.2.19]), R is not a PI-ring. So R is not intrinsically PI because R is simple. For a ring R, let B(Q(R)) be the set of all central idempotents of Q(R) and RB(Q(R)) the subring of Q(R) generated by R and B(Q(R)). The ring RB(Q(R)) is called the idempotent closure of R (see [6] and [9]). Let R be a semiprime ring. Then the ring RB(Q(R)) is the smallest quasiBaer intermediate ring between R and Q(R) (see [9, Corollary 8.3.19]). When R is a semiprime ring, RB(Q(R))R is a quasi-Baer module by Definition 2.27 since EndR (RB(Q(R))) = RB(Q(R)) is a quasi-Baer ring. Therefore for every positive (n) (n) integer n and e2 = e ∈ End(RR ), eRB(Q(R))R is a quasi-Baer module by Theorem 2.30 and Theorem 2.32. Theorem 3.2. Let a ring R be semiprime and ideal intrinsic over Cen(R), n (n) (n) (n) be a positive integer, and e2 = e ∈ End(RR ). Then qB(eRR ) = eRB(Q(R))R . Therefore, any finitely generated projective module over R has a quasi-Baer hull. Theorem 3.3 ([9, Theorem 8.4.15 and Corollary 8.3.19]). Assume that a ring R (n) (n) is semiprime, n is a positive integer, and e2 = e ∈ End(RR ). Then FI(eRR ) = (n) eRB(Q(R))R . Hence any finitely generated projective module over a semiprime ring has an FI-extending hull. From Theorem 3.2 and Theorem 3.3, the next corollary follows immediately. Corollary 3.4. Let a ring R be semiprime and ideal intrinsic over Cen(R), and let PR be a finitely generated projective module over R. Then qB(PR ) = FI(PR ). Recall that a commutative domain R is called Pr¨ ufer if R is semihereditary (i.e., every finitely generated ideal is projective). Theorem 3.2 does not hold for the existence of the Baer hull or the Rickart hull of a finitely generated projective module over a ring R even when R is a commutative domain (see Theorem 3.17). For this, some preparations are necessary.

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Theorem 3.5. (see [9, Theorem 6.1.4]) Let R be a commutative domain. Then the following are equivalent. (i) R is a Pr¨ ufer domain. (ii) Matn (R) is a Baer (Rickart) ring for every positive integer n. (iii) Matk (R) is a Baer (Rickart) ring for some integer k > 1. (iv) Mat2 (R) is a Baer (Rickart) ring. Furthermore, in (ii), (iii), and (iv), “Baer (Rickart) ring” can be can be replaced by “(right) extending ring”. Recall that a module M is called retractable if Hom(M, N ) = 0, for any 0 = N ≤ M . Examples of retractable modules include free modules and semisimple modules. Obviously, retractable modules are quasi-retractable. The following notion is introduced which is defined by Rizvi and Roman [42]. Definition 3.6. Let M be a right R-module and S = End(M ). Then M is called quasi-retractable if Hom(M, rM (I)) = 0 for every left ideal I of S with rM (I) = 0, where rM (I) = {m ∈ M | Im = 0}. We remark that a module MR is quasi-retractable if and only if rS (I) = 0 for every left ideal I of S with rM (I) = 0. In the following, we provide some other classes of quasi-retractable modules. Theorem 3.7. (i) If R is a ring and A is an intermediate ring between R and (n) Q(R), then AR is a quasi-retractable R-module for any positive integer n. (ii) If R is a prime PI-ring (e.g., R is a commutative domain) and A is an inter(Λ) mediate ring between R and Q(R), then AR is quasi-retractable for any nonempty (Λ) (Λ) set Λ. Furthermore, End(AR ) = End(AA ) = CFMΛ (A). (iii) If R is a right nonsingular ring and M is an intermediate (R, R)-bimodule (n) between R and Q(R), then MR is quasi-retractable for any positive integer n. Remark 3.8. Let R be a commutative domain with the field of fractions F . (Λ) For an intermediate domain A between R and F , AR is quasi-retractable for any (Λ) nonempty set Λ by Theorem 3.7(ii). But AR may not be retractable. For example, QZ is quasi-retractable. But QZ is not retractable because Hom(QZ , ZZ ) = 0. Let p be a prime integer and Zp∞ be the Pr¨ ufer p-group. We put M = Zp∞ as a Z-module. Then End(M ) is the ring of p-adic integer, which is a commutative domain. Hence End(M ) is a Baer ring. However, M is not Baer as a Z-module. The following, due to Rizvi and Roman [42], is a full characterization of a Baer module via its endomorphism ring and the quasi-retractability. Theorem 3.9. A module M is a Baer module if and only if S = End(M ) is a Baer ring and M is quasi-retractable. The next theorem is useful for the study of eRR when R is a Baer ring and e2 = e ∈ R. Theorem 3.10. Every direct summand of a Baer module is a Baer module. Theorem 3.9 and Theorem 3.10 yield the following corollary. Corollary 3.11. Let R be a Baer ring and 0 = e2 = e ∈ R. Then: (i) eRR is a Baer module. (ii) End(eRR ) = eRe is a Baer ring.

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The proof of the next result is shown in [30] (see also [9, Theorem 4.2.18]). The result is useful for the study of module hulls. Theorem 3.12. Let V be a nonsingular injective module and M be a nonsingular extending module. Then V ⊕ M is a Baer module. The following results are for the direct summands and the endomorphism ring of a Rickart module. Theorem 3.13. (i) Every direct summand of a Rickart module is a Rickart module. (ii) The endomorphism ring of a Rickart right R-module is a right Rickart ring. Theorem 3.13 yields the following corollary immediately. Corollary 3.14. Let R be a right Rickart ring and e2 = e ∈ R. Then: (i) eRR is a Rickart module. (ii) End(eRR ) = eRe is a right Rickart ring. Theorem 3.15. Assume that MR is a quasi-retractable R-module. Then MR is a Rickart module if and only if End(MR ) is a right Rickart ring. Every free module is retractable and so it is quasi-retractable. Using this fact, the following corollary is obtained. Corollary 3.16. Let R be a ring and Λ be a nonempty set. Then the following are equivalent. (Λ) (i) RR is a Rickart module. (ii) CFMΛ (R) is a right Rickart ring. Theorem 3.17. Let R be a commutative domain and let n be an integer such that n > 1. Then the following are equivalent. (n) (i) RR has a Baer hull. (n) (ii) RR has a Rickart hull. (iii) R is a Pr¨ ufer domain. (n)

Proof. (i)⇔(iii) Assume that RR has a Baer hull, say VR . Let Λ = {1, 2, . . . , n} and F be the field of fractions of R. Then by Theorem 3.12, (R ⊕ F (Λ\{i}) )R is a Baer module for each i, 1 ≤ i ≤ n. Thus V ⊆ ∩ni=1 (R ⊕ F (Λ\{i}) ) = (n) (n) R(n) , so V = R(n) . Hence RR is a Baer module. By Theorem 3.9, EndR (RR ) = ufer domain by Theorem 3.5 since n > 1. Matn (R) is a Baer ring. So R is a Pr¨ Conversely, assume R is a Pr¨ ufer domain. Then EndR (R(n) ) = Matn (R) is a (n) (n) Baer ring by Theorem 3.5. Note RR is quasi-retractable (because RR is a free (n) (n) R-module). Hence RR is a Baer module from Theorem 3.9, so RR itself is the (n) Baer hull of RR . (n) ufer domain (ii)⇔(iii) Similarly, RR has a Rickart hull if and only if R is a Pr¨ by arguments used in the proof of (i)⇔(iii).  Now the next example shows that Theorem 3.2 does not hold for the case of Baer hulls and the case of Rickart hulls. Remark 3.18. By Theorem 3.17, the finitely generated free module Z[x]module Z[x] ⊕ Z[x] has no Baer hull and has no Rickart hull because the commutative domain Z[x] is not Pr¨ ufer. But, whenever R is a commutative domain, (n) (n) RR itself is the quasi-Baer hull of RR from Theorem 2.32 or Theorem 3.2.

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A module M is called quasi-injective if M E(M ). An extending module M is called quasi-continuous if for any direct summands M1 and M2 of M , M1 ∩ M2 = 0 implies M1 ⊕ M2 is also a direct summand of M . An extending module M is called continuous if any submodule of M which is isomorphic to a direct summand of M is a direct summand of M . Let M be a module. By Johnson and Wong [22], ΛM is the quasi-injective hull of M , where Λ = End(E(M )). On the other hand, let Ω be the subring of End(E(M )) generated by all the idempotents of End(E(M )). By Goel and Jain [17], ΩM is the quasi-continuous hull of M . However, by M¨ uller and Rizvi [32] there exists a module which has no continuous hull (see also [9, Example 8.4.4]). In contrast to Theorem 3.17, the following result is obtained from Theorem 3.2 for the existence of Baer hulls of finitely generated projective modules over the ring Matk (A), where A is a Boolean ring and k is a positive integer. Corollary 3.19. Assume that A is a Boolean ring and R = Matk (A), where k is a positive integer. Let PR be a finitely generated projective module over R. Then we have the following. (i) PR has a Baer hull. (ii) PR has an extending hull. (iii) The quasi-Baer hull, the Baer hull, the injective hull, the quasi-injective hull, the continuous hull, the quasi-continuous hull, the extending hull, and the FI-extending hull of PR all exist and coincide. Let A be a Boolean ring, R = Matk (A), and k a positive integer. Assume that PR is a finitely generated projective module over R. In view of Corollary 3.19, one may expect that qB(RR ) = Ric(RR ) (recall that Ric(−) denotes the Rickart hull of a module). But the following example shows that this is impossible.  Example 3.20. Let A = {(an ) ∈ ∞ n=1 Z2 | an is eventually constant}. Then A is a Boolean ring. Put R = Matk (A), where k is any positive ∞ integer. We note that Q(Matk (A)) = Matk (Q(A)) (see [47]) and Q(A) = n=1 Z2 . Therefore by Corollary 3.19, !∞ "  Z2 . qB(RR ) = B(RR ) = Ex(RR ) = FI(RR ) = E(RR ) = Matk n=1

Since A is a Boolean ring, R is von Neumann regular, so RR is Rickart. Hence Ric(RR ) = RR = E(RR ). Therefore qB(RR ) = Ric(RR ). For a module M over a commutative domain R, we use t(M ) to denote the torsion submodule of M , that is t(M ) = {m ∈ M | ma = 0 for some nonzero a ∈ R}. When t(M ) = 0, M is called torsion-free. When M = t(M ), we say that M is a torsion module. As another motivation for the study of quasi-Baer hulls of modules, we consider the following result. Proposition 3.21. Let N be a finitely generated Z-module. Then the following are equivalent. (i) N is a quasi-Baer module. (ii) N is a Rickart module.

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(iii) N is a Baer module. (iv) N is semisimple or torsion-free. Let N = Zp ⊕ Z (p a prime integer) as a Z-module. Then N is not quasi-Baer, not Baer, and is not Rickart by Proposition 3.21. From [27] and [34] (also see [38]), B(N ) = Ric(N ) = Zp ⊕ Z[1/p]. Hence it is interesting to discuss the existence of qB(N ) and the description of qB(N ) if it exists. For our discussion, let R be a commutative noetherian domain and F its field of fractions. Assume N = MR ⊕ (⊕i∈Λ Ki ), where {Ki }i∈Λ is a set of nonzero submodules of FR . In the following result, we characterize all intermediate quasiBaer, Baer, and Rickart modules between N and E(N ). Motivated by the example of Zp ⊕ Z, in the following, we assume that M is semisimple with only a finite number of homogeneous components. Theorem 3.22. Let R be a commutative noetherian domain which is not a field. Assume that M is a semisimple R-module with only a finite number of homogeneous components, and {Ki | i ∈ Λ} is a set of nonzero submodules of FR , where F is the field of fractions of R. Let VR be an essential extension of MR ⊕ (⊕i∈Λ Ki )R . Then the following are equivalent. (i) V is a quasi-Baer (resp., Rickart, Baer) module. (ii) (1) V = M ⊕ W for some quasi-Baer (resp., Rickart, Baer) essential extension W of (⊕i∈Λ Ki )R . (2) HomR (W, M ) = 0. We note that QZ is quasi-retractable by Theorem 3.7 and EndZ (Q) = Q is a Baer ring. Hence QZ is a Baer module from Theorem 3.9. For N = Zp ⊕ Z (p a prime integer), we note that HomZ (Q, Zp ) = 0. Hence by Theorem 3.22, Zp ⊕ Q is the largest quasi-Baer essential extension of N because E(ZZ ) = Q. Therefore it is interesting to find the smallest quasi-Baer essential extension (i.e., the quasi-Baer hull) of N if it exists. Next let V = Zpn ⊕ Z (p is a prime integer and n is an integer such that n > 1). Note that V is not quasi-Baer by Proposition 3.21. So it is also interesting to study the possibility for the existence of a quasi-Baer hull of V . Let R be a commutative domain with the field of fractions F . For an ideal B of R, we put B −1 = {q ∈ F | qB ⊆ R}. It is well-known that for a nonzero ideal I of a commutative domain R, IR is projective if and only if II −1 = R. In this case, IR is finitely generated. Recall that a commutative domain R is called Dedekind if every ideal of R is projective as an R-module. Therefore a Dedekind domain is precisely a commutative hereditary domain. Hence for each nonzero ideal I of a Dedekind domain R, it follows that II −1 = R because IR is projective. Thus a Dedekind domain is noetherian because every ideal is projective (hence every ideal is finitely generated). See [25], [39], and [45] for more details on Dedekind domains. Proposition 3.23. Let R be a Dedekind domain which is not a field. Assume that M is an R-module such that AnnR (M ) = 0, and {Ki | i ∈ Λ} is a set of nonzero submodules of FR , where F is the field of fractions of R. If MR ⊕ (⊕i∈Λ Ki )R has a quasi-Baer essential extension, then MR is semisimple. Remark 3.24. By Proposition 3.23, V = Zpn ⊕ Z (p is a prime integer and n is an integer such that n > 1) has no quasi-Baer hull.

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Assume that R is a commutative domain with the field of fractions F . Let B be a nonzero ideal of R. We put B 0 = R. For each non-negative integer ", let We take T (B) =

#

[R : B  ] = (B  )−1 = {q ∈ F | qB  ⊂ R}. ≥0

[R : B  ]. Then   T (B) = [R : B  ] = (B  )−1 ≥0

≥0

since R = [R : B ] ⊆ [R : B] ⊆ [R : B ] ⊆ . . . . We remark that T (B) is an intermediate domain between R and F . The commutative domain T (B) is called the Nagata transform (or the ideal transform ) of B (see [14, p.490] and [16, p.325]). For an invertible ideal I of a commutative domain R, let 0

2

I −2 = I −1 I −1 , I −3 = I −1 I −1 I −1 , and so on. Assume that R is a commutative domain with the field of fractions F . It is well-known that if R is Pr¨ ufer, then any intermediate domain between R and F is also Pr¨ ufer. Proposition 3.25. Let R be a commutative domain. Then we have the following. (i) For invertible ideals I1 , . . . , In of R, (I1 · · · In )−1 = In−1 · · · I1−1 . (ii) If I is an invertible ideal of R, then (I  )−1 = I − for any non-negative integer ".  (iii) If I is an invertible ideal of R, then T (I) = ≥0 I − and n

T (I) = R[q1 , q2 , . . . , qn ],

where 1 = i=1 ri qi for some ri ∈ I and qi ∈ I −1 with 1 ≤ i ≤ n. Additionally, if R is a Dedekind (resp., Pr¨ ufer) domain, then T (I) is a Dedekind (resp., Pr¨ ufer) domain. In the following theorem, we give necessary and sufficient conditions for the existence of quasi-Baer hulls of certain modules over a Dedekind domain. Furthermore, the following theorem gives explicit description of such quasi-Baer hulls. Theorem 3.26. Let R be a Dedekind domain. Assume that M is an R-module such that I := AnnR (M ) = 0, and {Ki | i ∈ Λ} is a set of nonzero submodules of FR , where F is the field of fractions of R. Then the following are equivalent. (i) MR ⊕ (⊕i∈Λ Ki )R has a quasi-Baer hull. (ii) MR ⊕ (⊕i∈Λ Ki )R has a quasi-Baer essential extension. (iii) MR is semisimple. Ki )R ) = MR ⊕ (⊕i∈Λ Ki T (I))R . Furthermore, In this case, qB(MR ⊕ (⊕i∈Λ  T (I) = R[q1 , q2 , . . . , qn ], where 1 = nk=1 ak qk with ak ∈ I and qk ∈ I −1 , 1 ≤ k ≤ n for some positive integer n. Lemma 3.27. Let R be a right hereditary ring. Then every submodule of a free right R-module is isomorphic to a direct sum of right ideals of R (as R-modules). See [43, Theorem 4.17, p.121] for the proof of Lemma 3.27. Let N be a module over a Dedekind domain R such that N/t(N ) is projective. Then N ∼ = = t(N ) ⊕ N/t(N ) as N/t(N ) is projective. By Lemma 3.27, N/t(N ) ∼

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⊕i∈Λ Ki , where {Ki | i ∈ Λ} is a set of right ideals of R. Since R is Dedekind, R is noetherian and hence each Ki is finitely generated as an R-module. Hence Theorem 3.26 yields the following. Corollary 3.28. Let R be a Dedekind domain. Assume that N is an R-module with N/t(N ) projective and AnnR (t(N )) = 0. Then the following are equivalent. (i) N has a quasi-Baer hull. (ii) N has a quasi-Baer essential extension. (iii) t(N ) is semisimple. In this case, qB(NR ) ∼ = t(N ) ⊕ (N/t(N )T (AnnR (t(N ))). Theorem 3.29. Let R be a Dedekind domain, M an R-module with I := AnnR (M ) = 0, and let {Ki | i ∈ Λ} be a set of nonzero finitely generated submodules of FR , where F is the field of fractions of R. Then the following are equivalent. (i) MR ⊕ (⊕i∈Λ Ki )R has a Rickart hull. (ii) MR ⊕ (⊕i∈Λ Ki )R has a Rickart essential extension. (iii) MR is semisimple. Ki )R ) = MR ⊕ (⊕i∈Λ Ki T (I))R . Furthermore, In this case, Ric(MR ⊕ (⊕i∈Λ T (I) = R[q1 , q2 , . . . , qn ], where 1 = nk=1 ak qk with ak ∈ I and qk ∈ I −1 , 1 ≤ k ≤ n for some positive integer n. A somewhat surprising consequence of Theorem 3.26 and Theorem 3.29 that follows immediately, is that the quasi-Baer hull and the Rickart hull for certain modules are same. Corollary 3.30. Let R be a Dedekind domain, M an R-module such that I := AnnR (M ) = 0, and let {Ki | i ∈ Λ} be a set of nonzero finitely generated submodules of FR , where F is the field of fractions of R. Put N = MR ⊕(⊕i∈Λ Ki )R . Then the following are equivalent. (i) N has a quasi-Baer hull. (ii) N has a Rickart hull. (iii) M is semisimple. In this case, qB(N ) = Ric(N  ) = MR ⊕ (⊕i∈Λ Ki T (I))R . Furthermore, T (I) = R[q1 , q2 , . . . , qn ], where 1 = nk=1 ak qk with ak ∈ I and qk ∈ I −1 , 1 ≤ k ≤ n for some positive integer n. We note that the next result is a restatement of Theorem 3.29. Theorem 3.31. Let R be a Dedekind domain. Assume that N is an R-module with N/t(N ) projective and AnnR (t(N )) = 0. Then the following are equivalent. (i) N has a Rickart hull. (ii) N has a Rickart essential extension. (iii) t(N ) is semisimple. ∼ t(N ) ⊕ (N/t(N )T (AnnR (t(N ))). In this case, Ric(NR ) = The following result is for the case of Baer hulls of a certain class of modules over a Dedekind domain. Theorem 3.32. Let R be a Dedekind domain with F the field of fractions. Assume that M is an R-module with I := AnnR (M ) = 0, and {K1 , . . . , Km } is a

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finite set of finitely generated nonzero R-submodules of FR . Then the following are equivalent. (i) MR ⊕ (⊕m j=1 Kj )R has a Baer hull. (ii) MR ⊕ (⊕m j=1 Kj )R has an essential Baer extension. (iii) MR is semisimple. m In this case, B(MR ⊕ (⊕m j )R ) = MR ⊕ (⊕j=1 Kj T (I))R . Furthermore, j=1 K n T (I) = R[q1 , q2 , . . . , qn ], where 1 = k=1 rk qk with rk ∈ I and qk ∈ I −1 , 1 ≤ k ≤ n for some positive integer n.

yright 2020. AMS. rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

Theorem 3.26, Theorem 3.29 and Theorem 3.32 now yield the following corollary immediately. Corollary 3.33. Let R be a Dedekind domain with F the field of fractions. Assume that M is an R-module with I := AnnR (M ) = 0, and {K1 , . . . , Km } is a finite set of finitely generated nonzero R-submodules of FR . We put N = MR ⊕ (⊕m j=1 Kj )R . Then the following are equivalent. (i) N has a quasi-Baer hull. (ii) N has a Rickart hull. (iii) N has a Baer hull. (iv) M is semisimple. In this case, qB(N ) = Ric(N ) = B(N ) = MR ⊕(⊕m j=1 Kj T (I))R . Furthermore, n T (I) = R[q1 , q2 , . . . , qn ], where 1 = k=1 rk qk with rk ∈ I and qk ∈ I −1 , 1 ≤ k ≤ n for some positive integer n. Let R be a commutative domain with the field of fractions F . An R-submodule K of FR is called a fractional ideal of R if there exists 0 = r ∈ R such that rK ⊆ R. Every finitely generated R-submodule of FR is a fractional ideal of R. Further, if R is noetherian, then a submodule K of FR is fractional ideal of R if and only if K is finitely generated as an R-module. The next theorem details the structure of finitely generated modules over a Dedekind domain. Theorem 3.34. (see [45, Theorem 6.16, p.177]) Let R be a Dedekind domain and N a finitely generated R-module. Then there exist positive integers n1 , . . . , nk (k is a non-negative integer), nonzero maximal ideals P1 , . . . , Pk , and nonzero fractional ideals K1 , . . . , Km (m is a non-negative integer) of R such that N∼ = (⊕ki=1 R/Pini ) ⊕ (⊕m j=1 Kj ) as R-modules. We remark that if N is a finitely generated module over a Dedekind domain R, then AnnR (t(N )) = 0 and N/t(N ) is a finitely generated projective module over R by Theorem 3.34. Corollary 3.33 and Theorem 3.34 yield the coincidence of the quasi-Baer hull, the Rickart hull, and the Baer hull of a finitely generated module over a Dedekind domain. Corollary 3.35. Let R be a Dedekind domain and N be a finitely generated R-module. Then the following are equivalent. (i) N has a quasi-Baer hull. (ii) N has a Rickart hull.

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(iii) N has a Baer hull. (iv) t(N ) is semisimple. In this case, qB(N ) = Ric(N ) = B(N ) ∼ = t(N ) ⊕ (N/t(N ))T (AnnR (t(N ))). Example 3.36. Let p be a prime integer and let N = Zp ⊕ Z as a Z-module. As was discussed below Theorem 3.22, Z ⊕ Q is the largest quasi-Baer essential extension of N . Since I := AnnZ (pZ) = pZ, it follows that T (I) = Z[1/p]. Therefore from Theorem 3.26, qB(N ) = Zp ⊕ ZT (I) = Zp ⊕ Z[1/p]. Furthermore, qB(N ) = Ric(N ) = B(N ) by Corollary 3.33. The following example illustrates some applications of Theorem 3.26, Corollary 3.30, and Corollary 3.33 in describing the quasi-Baer hulls explicitly for certain modules. Example 3.37. Assume that Γi , i = 1, 2, 3, are nonempty sets, and assume (Γ ) (Γ ) (Γ ) that M = Z2 1 ⊕ Z3 2 ⊕ Z5 3 . (i) For any nonempty set Ω, let NΩ = M ⊕ Z(Ω) . Then qB(NΩ ) = Ric(NΩ ) = M ⊕ Z[1/30](Ω) by Corollary 3.30 as AnnZ (M ) = 30Z. (ii) For any positive integer m, let Vm = M ⊕ Z(m) . Then qB(Vm ) = Ric(Vm ) = B(N ) = M ⊕ Z[1/30](m) from Corollary 3.33 since AnnZ (M ) = 30Z. In view of Theorem 3.26, one may expect that the assumption “{Ki | i ∈ Λ} is a set of nonzero finitely generated submodules of FR ” in Theorem 3.29 can be extended to “{Ki | i ∈ Λ} is a set of nonzero submodules of FR ”. However part (iii) of the following example shows that this is impossible. Also we may expect that the assumption “{K1 , · · · , Km } is a finite set of finitely generated nonzero R-submodules of FR ” in Theorem 3.32 can be extended to “{K1 , . . . , Km } is a finite set of of nonzero R-submodules (not necessarily finitely generated) of FR ”. However, the following example shows that these two expectations are impossible. Example 3.38. (see [27, Example 3.5]) Let N = Z2 ⊕ Z[1/3] ⊕ Z[1/2] as a Z-module. Then we have the following. (i) N is neither quasi-Baer nor Rickart. For this, define h : Z[1/3] ⊕ Z[1/2] → Z2 by h[(m/3n , k/2 )] = m for (m/3n , k/2 ) ∈ Z[1/3] ⊕ Z[1/2], where m, k are integers, n, " are non-negative integers, and m is the image of m in Z2 . Then we have that 0 = h ∈ Hom(Z[1/3] ⊕ Z[1/2], Z2 ). Hence Hom(Z[1/3] ⊕ Z[1/2], Z2 ) = 0. From Theorem 3.22, N is neither quasi-Baer nor Rickart. (ii) From Theorem 3.26, it follows that qB(N ) = Z2 ⊕ Z[1/3]T (I) ⊕ Z[1/2]T (I), where I = AnnZ (Z2 ) = 2Z. Therefore T (I) = Z[1/2], the Nagata transform of I. So qB(N ) = Z2 ⊕ Z[1/6] ⊕ Z[1/2].

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(iii) We show that N has no Rickart hull. For this, first W := Z[1/6] ⊕ Z[1/2] is not a Rickart module (see [38, Example 3.23]). Therefore, Z2 ⊕ Z[1/6] ⊕ Z[1/2] is not a Rickart module by Theorem 3.13(i). On the contrary assume that N has a Rickart hull, say V . By Theorem 3.22, V = Z2 ⊕ W with W ≤ V such that Z[1/3] ⊕ Z[1/2] ≤ WZ ≤ Q ⊕ Q and WZ is Rickart. Also from Theorem 3.22, HomZ (W, Z2 ) = 0. Now we claim that W = 2W . Indeed, from the exact sequence 0 → 2W → W → W/2W → 0, we obtain the following exact sequence 0 → HomZ (W/2W, Z/2Z) → HomZ (W, Z/2Z) → HomZ (2W, Z/2Z) because HomZ (−, Z/2Z) is a left exact contravariant functor. As HomZ (W, Z/2Z) = HomZ (W, Z2 ) = 0, from the preceding exact sequence we have that HomZ (W/2W, Z/2Z) = 0. We notice that W/2W is a Z/2Z-module, which is induced from the Z-module structure of W/2W . Further, HomZ/2Z (W/2W, Z/2Z) = HomZ (W/2W, Z/2Z) = 0. Now W/2W is a vector space over the field Z/2Z. Therefore W/2W = 0, and thus W = 2W .  We let I = 2Z, and so W I = W . Put T (I) = ≥0 I − , the Nagata transform of I. Then T (I) = Z[1/2]. We claim that Z[1/3]T (I) ⊕ Z[1/2]T (I) ⊆ W by similar arguments used in [38, Example 3.15] . For this, we let J = I  , where " is a non-negative integer. Because W I = W , W I  = W and so W J = W . Take β ∈ Z[1/3]. Then (β, 0) ∈ Z[1/3] ⊕ Z[1/2] ⊆ W = W J. n Hence there exists a positive integer n such that (β, 0) = i=1 wi ai , where wi ∈ W and ai ∈ J for i, 1 ≤ i ≤ n. Put wi = (xi , yi ) ∈ Q ⊕ Q for i, 1 ≤ i ≤ n. Then ! n " n n    wi ai = xi a i , yi ai . (β, 0) = So β =

n i=1

i=1

xi ai and 0 =

n

i=1

i=1

i=1

yi ai .

n n Now we take any q ∈ J −1 . Then βq = i=1 xi qai and 0 = i=1 yi qai . Since each ai ∈ J and qJ ⊆ Z, each qai ∈ Z. So " ! n n n n     xi qai , yi qai = (xi , yi )qai = wi qai . (βq, 0) = i=1

i=1

i=1

i=1

 Hence (βq, 0) ∈ ni=1 wi Z ⊆ W . Therefore (ZJ −1 , 0) ⊆ W . By Proposition 3.25, J −1 = I − . Thus (Z[1/3]I − , 0) ⊆ W for all non-negative integers ". Hence (Z[1/3]T (I), 0) ⊆ W. Similarly, (0, Z[1/2]T (I)) ⊆ W . Therefore Z[1/3]T (I) ⊕ Z[1/2]T (I) ⊆ W . Because T (I) = Z[1/2], we have that Z[1/3]T (I) ⊕ Z[1/2]T (I) = Z[1/3]Z[1/2] ⊕ Z[1/2]Z[1/2] = Z[1/6] ⊕ Z[1/2].

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Consequently, Z[1/6] ⊕ Z[1/2] ⊆ W. Hence Z2 ⊕ Z[1/6] ⊕ Z[1/2] ≤ Z2 ⊕ W = V. So V is the smallest Rickart module such that Z2 ⊕ Z[1/6] ⊕ Z[1/2] ≤ V ≤ E(N ). By Theorem 3.12, Q ⊕ Z[1/2] and Z[1/6] ⊕ Q are Baer as Z-modules. Since Hom(Q, Z2 ) = 0 and Hom(Z[1/2], Z2 ) = 0, Hom(Q ⊕ Z[1/2], Z2 ) = 0. Similarly, Hom(Z[1/6] ⊕ Q, Z2 ) = 0. Therefore from Theorem 3.22, Z2 ⊕ Q ⊕ Z[1/2] and Z2 ⊕ Z[1/6] ⊕ Q are Rickart modules. Note that Z2 ⊕ Z[1/6] ⊕ Z[1/2] ≤ Z2 ⊕ Q ⊕ Z[1/2] and Z2 ⊕ Z[1/6] ⊕ Z[1/2] ≤ Z2 ⊕ Z[1/6] ⊕ Q. Hence V ≤ (Z2 ⊕ Q ⊕ Z[1/2]) ∩ (Z2 ⊕ Z[1/6] ⊕ Q) = Z2 ⊕ Z[1/6] ⊕ Z[1/2]. Therefore V = Z2 ⊕ Z[1/6] ⊕ Z[1/2], and thus Z2 ⊕ Z[1/6] ⊕ Z[1/2] is a Rickart module, a contradiction by the preceding argument (see [38, Example 3.23]). Thus N has no Rickart hull. (iv) Similarly, N has no Baer hull. Motivated by Example 3.38(iii) and (iv), the next result for Rickart and Baer hulls of M ⊕ (⊕i∈Λ Ki ) when M is a module with AnnR (M ) = 0 and {Ki | i ∈ Λ} is a set of nonzero submodules of FR , where F is the field of fractions of a Dedekind domain R, is obtained. It is of interest to compare this with Theorem 3.26. Theorem 3.39. Let R be a Dedekind domain with F the field of fractions, M an R-module with I := AnnR (M ) = 0, and let {Ki | i ∈ Λ} be a set of nonzero submodules of FR . Put N = M ⊕ (⊕i∈Λ Ki ). Then the following are equivalent. (i) N has a Rickart (resp., Baer) hull. (ii) M is semisimple and ⊕i∈Λ Ki T (I) is a Rickart (resp., Baer) module. In this case, Ric(N ) = M ⊕(⊕i∈Λ Ki T (I)) (resp., B(N ) = M ⊕(⊕i∈Λ Ki T (I))). Recall that a ring R is said to be semiprimary if R/J(R) is artinian and J(R) is nilpotent, where J(R) is the Jacobson radical of R. Theorem 3.40 ([46, Theorem 2] and [42, Theorem 3.3]). Let R be a ring. Then the following are equivalent. (i) R is semiprimary right (left) hereditary. (ii) CFMΛ (R) is a Baer ring for any nonempty set Λ. In view of Theorem 3.29, one may expect that the assumption “{K1 , · · · , Km } is a finite set of finitely generated nonzero R-submodules of FR ” in Theorem 3.32 can be extended to “{Ki | i ∈ Λ} is a (not necessarily finite) set of finitely generated nonzero R-submodules of FR ”. However, the following example illustrates that this is impossible. Example 3.41. Let p be a prime integer. Since Z[1/p] is not a field, Z[1/p] is not semiprimary. By Theorem 3.40, there exists a nonempty set Λ such that CFMΛ (Z[1/p]) is not a Baer ring. We let N = Zp ⊕ Z(Λ) as a Z-module. Then we have the following. (i) qB(N ) = Ric(N ) = Zp ⊕ Z[1/p](Λ) by Corollary 3.30.

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(ii) N/t(N ) is not finitely generated. (iii) N has no Baer hull. For (ii), note that by Theorem 3.7(ii), Z[1/p](Λ) is quasi-retractable as a Zmodule, and EndZ (Z[1/p](Λ) ) = EndZ[1/p] (Z[1/p](Λ) ) = CFMΛ (Z[1/p]). Since EndZ (Z[1/p](Λ) ) = CFMΛ (Z[1/p]) is not a Baer ring, Z[1/p](Λ) is not a Baer Z-module by Theorem 3.9. Now we prove that Λ is an infinite set. Assume on the contrary that Λ is a finite set, say |Λ| = m, where m is a positive integer. Then we see that m > 1 as ufer EndZ (Z[1/p](Λ) ) = CFMΛ (Z[1/p]) is not a Baer ring. Note that Z[1/p] is a Pr¨ domain. So EndZ (Z[1/p](Λ) ) = Matm (Z[1/p]) is a Baer ring by Theorem 3.5. Thus we get a contradiction since m > 1. Therefore Λ is infinite, so N/t(N ) ∼ = Z(Λ) is not finitely generated as a Z-module. (iii) From the proof of (ii), Z[1/p](Λ) is not a Baer Z-module. By Theorem 3.39, N has no Baer hull. As an application, we study connections between a quasi-Baer hull of a module N and a quasi-Baer hull of a direct summand of N . Let R be a Dedekind domain. Assume that L and N are R-modules such that L ≤⊕ N . Assume that qB(N ) exists. Then we may raise the following question: Is qB(L) ≤⊕ qB(N )? The following examples answer in the negative this question. Moreover, surprisingly qB(L) does not exist even while qB(N ) exists. Example 3.42. Assume that N = Z3 ⊕ Z2 ⊕ Z[1/3] ⊕ Z[1/2] and assume that L = Z2 ⊕ Z[1/3] ⊕ Z[1/2] as Z-modules. Then: (i) L ≤⊕ N as Z-modules. (ii) qB(N ) = Ric(N ) = B(N ) = Z3 ⊕ Z2 ⊕ Z[1/6](2) by Theorem 3.26 and Theorem 3.39. (iii) L has no Rickart hull and has no Baer hull from Example 3.38. (iv) qB(L) = Z2 ⊕ Z[1/6] ⊕ Z[1/2] from Theorem 3.26 (see also Example 3.38). (v) qB(L) ≤⊕ qB(N ) as Z-modules. On the contrary, assume that qB(L) ≤⊕ qB(N ) as Z-modules. Then there exists W ≤ qB(N ) such that qB(N ) = qB(L) ⊕ W . Therefore from (ii) and (iv), Z2 ⊕ W ⊕ Z[1/6] ⊕ Z[1/2] = Z3 ⊕ Z2 ⊕ Z[1/6] ⊕ Z[1/6]. Hence udim(W ) = 1, where udim(−) denotes the uniform dimension. Furthermore, Z2 ⊕ t(W ) = Z3 ⊕ Z2 . Therefore 2(Z2 ⊕ t(W )) = 2(Z3 ⊕ Z2 ), and thus 2t(W ) = 2Z3 = Z3 . So udim(2t(W )) = udim(Z3 ) = 1. Because udim(W ) = 1, 2t(W ) ≤ess W as Z-module. Hence Z3 ≤ess W , and thus W ≤ E(Z3 ) = Z3∞ , the Pr¨ ufer 3-group. So W is a torsion group. Hence W = t(W ). From Z3 ⊕ Z2 ⊕ Z[1/6] ⊕ Z[1/6] = Z2 ⊕ W ⊕ Z[1/6] ⊕ Z[1/2], we have that Z[1/6] ⊕ Z[1/6] = Z[1/6] ⊕ Z[1/2].

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Now take (1/6, 1/6) ∈ Z[1/6] ⊕ Z[1/6]. Then there exists (α, β) ∈ Z[1/6] ⊕ Z[1/2] with α ∈ Z[1/6] and β ∈ Z[1/2] such that (1/6, 1/6) = (α, β). Therefore it follows that 1/6 = β ∈ Z[1/2]. Put β = m/2n , where m, n are integers, n is non-negative, and m is an odd integer if n ≥ 1. Then 1/6 = m/2n . If n = 0, then 1/6 = m, which is impossible. If n = 1, then 1/6 = m/2 and so 2 = 6m. Hence 1 = 3m, which is impossible. Finally, assume that n ≥ 2. Then from 1/6 = m/2n , 2n = 6m = 2 · 3m, so 2n−1 = 3m, this is also impossible because m is an odd integer. Consequently, qB(N ) ≤⊕ qB(L). Next, suppose that U and V are R-modules such that qB(U ), qB(V ), and qB(U ⊕V ) exist. Then one may expect the following: qB(U ⊕V ) = qB(U )⊕qB(V ). However, the following example eliminate our expectation. Example 3.43. Let U = Z2 and V = Z[1/3] ⊕ Z[1/2] as Z-modules. By Theorem 3.26, qB(U ⊕ V ) = Z2 ⊕ Z[1/6] ⊕ Z[1/2] (see also Example 3.38). From Theorem 3.26, qB(V ) = V . Note qB(U ) = U . Hence qB(U ) ⊕ qB(V ) = U ⊕ V = Z2 ⊕ Z[1/3] ⊕ Z[1/2], which is not quasi-Baer (see Example 3.38). Therefore qB(U ⊕ V ) ∼ qB(U ) ⊕ qB(V ). = Here additionally note that Z[1/6]⊕Z[1/2] is not Rickart from Example 3.38. Hence by Theorem 3.39, B(U ⊕ V ) and Ric(U ⊕ V ) do not exist. We consider here the “isomorphism problem” for quasi-Baer hulls, given as follows: Let N1 and N2 be two modules. Is it true that N1 ∼ = N2 if and only if qB(N1 ) ∼ = qB(N2 )? Isomorphism problem for Baer and Rickart hulls is discussed in [27] and [34] (see also [38]). Example 3.44. If N1 ∼ = N2 , then clearly qB(N1 ) ∼ = qB(N2 ). However, there N2 . exist modules N1 and N2 such that qB(N1 ) = qB(N2 ), but N1 ∼ = For example, let N1 = Z2 ⊕ Z3 ⊕ Z and N2 = Z2 ⊕ Z3 ⊕ Z[1/3]. From Theorem 3.26, qB(N1 ) = Z2 ⊕ Z3 ⊕ Z[1/6]. On the other hand, by Theorem 3.26, qB(N2 ) = Z2 ⊕ Z3 ⊕ Z[1/6]. Therefore N2 since Z ∼ Z[1/3] as Z-modules. Thus the answer qB(N1 ) = qB(N2 ). But, N1 ∼ = = to the isomorphism problem for the case of quasi-Baer module hulls is negative. While there exist disparities between quasi-Baer, Baer, and Rickart modules, the following theorem shows that the quasi-Baer, Baer, and Rickart notions coincide for the case when a module is finitely generated over a Dedekind domain. This result extends Proposition 3.21. Theorem 3.45. Let R be a Dedekind domain and N be a finitely generated R-module. Then the following are equivalent. (i) N is quasi-Baer. (ii) N is Rickart. (iii) N is Baer. (iv) N is semisimple or torsion-free. In the following theorem, quasi-Baer modules and Rickart modules, which are direct sums of finitely generated modules over a Dedekind domain are characterized. In contrast to the Z-module Z2 ⊕Z[1/6]⊕Z[1/2] in Example 3.38, these two notions coincide in this case.

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Theorem 3.46. Let R be a Dedekind domain and N be a direct sum of finitely generated R-modules. Then the following are equivalent. (i) N is quasi-Baer. (ii) N is Rickart. (iii) N is semisimple or torsion-free. Lemma 3.47. (see [45, Theorem 6.11, p.171 and Theorem 6.14, p.174]) Assume that R is a Dedekind domain and M is an R-module with AnnR (M ) = 0. Then there exists a unique family {Pj , nj }i∈Γ such that: (i) The Pj are maximal ideals of R and there are only finitely many distinct ones. (ii) {nj | j ∈ Γ} is a bounded family of positive integers. n (iii) M ∼ = ⊕j∈Γ (R/Pj j ) as R-modules. Theorem 3.46 and Lemma 3.47 yield the following immediately Corollary 3.48. Let R be a Dedekind domain. Assume that N is an R-module with N/t(N ) projective and AnnR (t(N )) = 0. Then the following are equivalent. (i) N is quasi-Baer. (ii) N is Rickart. (iii) N is semisimple or torsion-free. The following example illustrates Theorem 3.45 and Theorem 3.46. Example 3.49. Let R be a Dedekind domain. When N is an R-module which is not finitely generated, Theorem 3.45 does not hold true. In fact, let Λ be an uncountable set and take N = Z(Λ) as a Z-module. Then we have the following. (i) qB(N ) = Ric(N ) = N by Corollary 3.30. Hence N is quasi-Baer and Rickart. Further, note that N is torsion-free. (ii) N is not a Baer module by [48, Remark, p.32]. Additionally, N has no Baer hull. Assume on the contrary that N has a Baer hull, say V . By Theorem 3.12, Z ⊕ Q(Λ\{α}) is Baer for each α ∈ Λ. So V ⊆ ∩α∈Λ (Z ⊕ Q(Λ\{α}) ) = Z(Λ) . Hence V = Z(Λ) = N , and thus N is a Baer module, a contradiction. As another application of our results, we obtain the following consequence for the existence and description of the quasi-Baer, Rickart and Baer hull of a finitely generated module N over a Pr¨ ufer domain R when t(N ) is semisimple and AnnR (t(N )) is a finitely generated ideal of R. Proposition 3.50. Let N be a finitely generated module over a Pr¨ ufer domain R. If t(N ) is semisimple and AnnR (t(N )) is finitely generated, then N has a quasiBaer, a Rickart, and a Baer hulls. In this case, qB(N ) = Ric(N ) = B(N ) ∼ = t(N ) ⊕ (N/t(N ))T (AnnR (t(N ))). In Example 3.51, we exhibit that a finitely generated module V over a Pr¨ ufer domain R such that V is a quasi-Baer, Rickart, and Baer module. But V is neither semisimple nor torsion-free. Thereby, Theorem 3.45 does not hold true for finitely generated modules over a Pr¨ ufer domain. Furthermore, Example 3.51 shows that Corollary 3.28, Theorem 3.31, and Corollary 3.33 do not extend to the case when R is a Pr¨ ufer domain. Example 3.51 (see [27, Example 4.18]). Consider R = Z+xQ[[x]], the subring of the formal power series ring Q[[x]] over Q. Then R is a Pr¨ ufer domain (see [15, Example 3.6, p.102]).

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(i) R is not Dedekind since xQ[[x]] is a nonzero prime ideal of R, but xQ[[x]] is not maximal (note that every nonzero prime ideal of a Dedekind domain is maximal). (ii) All the maximal ideals of R are principal, and the only non-maximal nonzero prime ideal of R is xQ[[x]] (see [15, Example 3.6, p.102]). (iii) We prove that any finitely generated module N over R with t(N ) semisimple has a Baer, a quasi-Baer, and a Rickart hull. In fact, first we observe that N∼ = t(N )⊕N/t(N ) by Kaplansky (see [15, Corollary 2.9, p.155]). Say t(N ) = M1 ⊕ · · · ⊕ Mk , where each Mi is simple and k is a positive integer. Put Pi = AnnR (Mi ) for each i. Then each Pi is a maximal ideal. Also AnnR (t(N )) = P1 ∩ · · · ∩ Pk . Now we claim that AnnR (t(N )) = P1 · · · Pk by induction on k. If k = 1, then we are done. Suppose k > 1. If (P1 · · · Pk−1 ) + Pk is a proper ideal of R, then there exists a maximal ideal P of R with (P1 · · · Pk−1 ) + Pk ⊆ P , thus P1 · · · Pk−1 ⊆ P and Pk ⊆ P . So Pi ⊆ P for some i with 1 ≤ i ≤ k − 1 and Pk = P . Also Pi = P and hence R = Pi + Pk = P , a contradiction. Hence (P1 · · · Pk−1 ) + Pk = R. So there exists a ∈ P1 · · · Pk−1 and b ∈ Pk such that a + b = 1. Let r ∈ P1 ∩ · · · ∩ Pk−1 ∩ Pk . Then ra ∈ (P1 ∩ · · · ∩ Pk−1 ∩ Pk )(P1 · · · Pk−1 ) ⊆ Pk P1 · · · Pk−1 = P1 · · · Pk−1 Pk . By induction P1 ∩ · · · ∩ Pk−1 = P1 · · · Pk−1 . Thus rb ∈ (P1 ∩ · · · ∩ Pk−1 ∩ Pk )Pk = ((P1 · · · Pk−1 ) ∩ Pk )Pk ⊆ (P1 · · · Pk−1 )Pk . Hence r = ra + rb ∈ P1 · · · Pk . So P1 ∩ · · · ∩ Pk ⊆ P1 · · · Pk . Obviously, we have that P1 · · · Pk ⊆ P1 ∩ · · · ∩ Pk . Hence AnnR (t(N )) = P1 · · · Pk = P1 ∩ · · · ∩ Pk . Since each Pi is principal by part (ii), P1 · · · Pk is also principal, so AnnR (t(N )) is a principal ideal. By Proposition 3.50, N has quasi-Baer, Rickart, and Baer hulls. Further, qB(N ) = Ric(N ) = B(N ). (iv) Put V = R/xQ[[x]] which is an R-module. Then S := EndR (V ) ∼ = Z (as rings). Thus, for 0 = UR ≤ VR , we have "S (U ) = 0. So V is a (cyclic) Baer R-module. However, V is neither semisimple nor torsion-free (cf. Theorem 3.45). Now V /t(V ) is projective as V /t(V ) = 0. We note that AnnR (t(V )) = AnnR (V ) = xQ[[x]], and so we have that AnnR (t(V )) = 0. Also t(V ) is not semisimple. However, qB(V ) = Ric(V ) = B(V ) = V (cf. Corollary 3.35). (v) We show that AnnR (t(V )) = xQ[[x]] is not finitely generated as an Rmodule. For this, we assume on the contrary that AnnR (t(V )) = xQ[[x]] is finitely generated. Then there exists n1 + xf1 (x), . . . , nk + xfk (x) ∈ Z + xQ[[x]], where ni ∈ Z and fi (x) ∈ Q[[x]] for 1 ≤ i ≤ k, such that xQ[[x]] = (n1 + xf1 (x)(Z + xQ[[x]]) + · · · + (nk + xfk (x))(Z + xQ[[x]]). Therefore we have that n1 Z + · · · + nk Z = 0, so n1 = · · · = nk = 0. Hence xQ[[x]] = xf1 (x)(Z + xQ[[x]]) + · · · + xfk (x)(Z + xQ[[x]]). Thus Q[[x]] = f1 (x)(Z + xQ[[x]]) + · · · + fk (x)(Z + xQ[[x]]).

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So Q = f1 (0)Z + · · · + fk (0)Z with each fi (0) ∈ Q, so Q is finitely generated as a Z-module, a contradiction. Hence AnnR (t(V )) is not finitely generated. So the converse of Proposition 3.50 does not hold true. The Nagata transform T (AnnR (t(N ))) of the ideal AnnR (t(N )), has been a critical tool in our study. In particular, we used it in Theorem 3.26, Theorem 3.29, and Theorem 3.32, etc. for the existence and description of the quasi-Baer hull, the Rickart hull, and the Baer hull of a module N over a Dedekind domain R for the case when t(N ) is semisimple, AnnR (t(N )) = 0, and N/t(N ) projective. In contrast, for the case when R is a Pr¨ ufer domain, the next example shows that the Nagata transform T (AnnR (t(N ))) provides no useful information for the existence or description of the quasi-Baer, Rickart or Baer hulls of a finitely generated module N when AnnR (t(N )) is not finitely generated, even though t(N ) may be simple. Example 3.52 (see [27, Example 4.19]). Let R be a Pr¨ ufer domain with a maximal ideal P , which is not finitely generated (see [16, Example 42.6, p.516]). Therefore P −1 = [R : P ] = R by [21, Corollary 3.4]. Let q ∈ [R : P 2 ]. Then qP 2 ⊆ R, so qP ⊆ [R : P ] = R. Thus q ∈ [R : P ] = R, and hence [R : P 2 ] = R. Similarly, [R : P  ] = R for all non-negative integer ". Therefore  [R : P  ] = R. T (P ) = ≥0

Let M = R/P , a simple R-module, and take N = M ⊕ R. Now AnnR (t(N )) = AnnR (M ) = P is not finitely generated. So P is not invertible. In contrast to Proposition 3.50, M ⊕ RT (P ) is neither the quasi-Baer hull, nor the Rickart hull, and nor the Baer hull of N . Let S = EndR (N ). Since HomR (M, R) = 0,



EndR (M ) HomR (R, M ) ∼ EndR (M ) M . S= = 0 EndR (R) 0 R So (M ⊕ P )R (M ⊕ R)R , but there is no e2 = e ∈ S such that "S (M ⊕ P ) = Se. Thus M ⊕ R = M ⊕ RT (P ) is not quasi-Baer (hence not Baer). Next let φ ∈ S be defined by φ(m, r) = (r + P, 0) for (m, r) ∈ M ⊕ R. Then we see that Ker(φ) = M ⊕ P , which is not a direct summand of M ⊕ R, therefore M ⊕ R = M ⊕ RT (P ) is not Rickart. The following theorem, which is [36, Theorem 2.38], describes the extending hulls of certain modules over a commutative domain. Theorem 3.53. Assume that R is a commutative domain with the field of fractions F . Let A be an intermediate domain between R and F . Then the following are equivalent. (i) A is a Pr¨ ufer domain. (n) (n) (ii) E(MR ) ⊕ AR is the extending hull of MR ⊕ AR for any R-module M with AnnR (M ) = 0 and for any positive integer n. (2) (iii) AR is an extending module. (2) (iv) AR is a Baer module.

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In spite of Theorem 2.25 and Theorem 2.41, we have the following example, which shows different types of hulls explicitly. These are consequences of the various results presented as listed below. Example 3.54 (cf. [36, Example 2.42]). Let p be a prime integer. Then we have the following. (i) Zp∞ ⊕ Z is the extending hull of Zp ⊕ Z by Theorem 3.53. (ii) Zp ⊕ Z[1/p] is the Baer, the quasi-Baer, and the Rickart hull of Zp ⊕ Z from Corollary 3.33. (iii) Zp2 ⊕ Z has an extending hull, which is Zp∞ ⊕ Z by Theorem 3.53. (iv) From Corollary 3.33, the Z-module Zp2 ⊕ Z has no quasi-Baer, no Rickart, and no Baer hull because Zp2 is not semisimple. (v) Zp ⊕Z itself is the FI-extending hull of Zp ⊕Z because Zp ⊕Z is FI-extending as a Z-module by Theorem 2.35. (vi) Zp2 ⊕ Z itself is the FI-extending hull of Zp2 ⊕ Z because Zp2 ⊕ Z is FIextending as a Z-module by Theorem 2.35. Acknowledgments. The authors are grateful for the partial grant support received from OSU-Lima, the Office of International Affairs, Mathematics Research Institute and the ASC College, The Ohio State University. The authors also thank each others’ institutions for the hospitality received during the research work on this paper. References Pere Ara, Centers of maximal quotient rings, Arch. Math. (Basel) 50 (1988), no. 4, 342–347, DOI 10.1007/BF01190229. MR937337 [2] Pere Ara and Martin Mathieu, Local multipliers of C ∗ -algebras, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2003. MR1940428 [3] Efraim P. Armendariz, Gary F. Birkenmeier, and Jae Keol Park, Corrigendum to “Ideal intrinsic extensions with connections to PI-rings” [J. Pure Appl. Algebra 213 (2009) 1756–1776] [MR2518175], J. Pure Appl. Algebra 215 (2011), no. 1, 99–100, DOI 10.1016/j.jpaa.2010.04.009. MR2678703 [4] Efraim P. Armendariz, Hyeng Keun Koo, and Jae Keol Park, Compressible group algebras, Comm. Algebra 13 (1985), no. 8, 1763–1777, DOI 10.1080/00927878508823251. MR792561 [5] Reinhold Baer, Linear algebra and projective geometry, Academic Press Inc., New York, N. Y., 1952. MR0052795 [6] Konstantin Beidar and Robert Wisbauer, Strongly and properly semiprime modules and rings, Ring theory (Granville, OH, 1992), World Sci. Publ., River Edge, NJ, 1993, pp. 58–94. MR1344223 [7] Gary F. Birkenmeier, Bruno J. M¨ uller, and S. Tariq Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Comm. Algebra 30 (2002), no. 3, 1395–1415, DOI 10.1081/AGB-120004878. MR1892606 [8] Gary F. Birkenmeier, Jae Keol Park, and S. Tariq Rizvi, Hulls of semiprime rings with applications to C ∗ -algebras, J. Algebra 322 (2009), no. 2, 327–352, DOI 10.1016/j.jalgebra.2009.03.036. MR2529092 [9] Gary F. Birkenmeier, Jae Keol Park, and S. Tariq Rizvi, Extensions of rings and modules, Birkh¨ auser/Springer, New York, 2013. MR3099829 [10] B. Blackadar, Operator algebras: Theory of C ∗ -algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Operator Algebras and Non-commutative Geometry, III. MR2188261 [11] Kenneth A. Brown, The singular ideals of group rings, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 109, 41–60, DOI 10.1093/qmath/28.1.41. MR447316 [12] A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc. (2) 21 (1980), no. 3, 434–444, DOI 10.1112/jlms/s2-21.3.434. MR577719 [1]

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Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15085

Strict Mittag-Leffler modules and purely generated classes Philipp Rothmaler To SK Jain Abstract. We study versions of strict Mittag-Leffler modules relativized to a class K (of modules), that is, strict versions (in the technical sense of Raynaud and Gruson) of K-Mittag-Leffler modules, as investigated in the preceding paper, Mittag-Leffler modules and definable subcategories, in this very series.

Goodearl [Goo] noticed (in different terms) that a module is RR -Mittag-Leffler if and only if the inclusion of any of its finitely generated submodules factors through a finitely presented module. (This can be found, among other more or less relative Mittag-Leffler properties, in the preceding paper [Rot4].) Raynaud and Gruson [RG] characterized strict Mittag-Leffler modules as those which can be mapped down to any finitely generated submodule in such a way that this map pointwise fixes the given finitely generated submodule and factors through a finitely presented module. There is a major difference here: in the strict version one must be able to map the entire module each time one examines a finitely generated submodule. What is in common though is a certain local behavior concerning finitely generated submodules or, one could say, finite subsets (the generator sets of them). One way of dealing with local behavior is to consider separation properties. Baer [Bae] introduced separable abelian groups as those in which every finite subset can be ‘separated’ by (is contained in) a direct summand which is completely decomposable, see [F]. Whatever these completely decomposable groups may be, inspired by Baer’s original investigations, a theory of separation has emerged that has taken other classes of groups as the pool from which to choose the separating direct summands, for instance free groups, see [EM]. For our purposes the most natural choice of separation module is the class R-mod of finitely presented modules, as in [GIRT] and here in Definition 4.1(3) and (5) (and thereafter). As was shown in [GIRT, Thm. 1], all strict Mittag-Leffler modules are so separable if and only if all pure-projective modules are direct sums of finitely presented modules—while, in general, a module is pure-projective if and only if it is a direct summand of a direct sum of finitely presented modules. This leads us to proving that, in general, a module is strict Mittag-Leffler if and only if it is a direct summand of a separable module, Corollary 6.19. And while this has 2010 Mathematics Subject Classification. Primary 16D40, 16B70, 16D80, 16S90. Supported in part by PSC CUNY Awards 64595-00 42 and 68835-00 46, BCCF Faculty Scholarship Support Grant (Spring Cycle 2018). c 2020 American Mathematical Society

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been known in this form since [HT], we prove it for arbitrary relativizations in a rather general context, Corollary 6.8, as a consequence of a more general result, our first separation theorem 6.7. The other part of this theorem characterizes the (non-strict) relativized Mittag-Leffler modules in terms of another separability, that by pure submodules in place of direct summands. The unrelativized case of it says that a module is Mittag-Leffler if and only if it is a direct summand of a pure separable module, i.e., of a module all of whose finite subsets can be contained in a pure submodule that is finitely presented, Corollary 6.19. The second separation theorem 6.13 describes when the clause ‘direct summand’ is superfluous in Theorem 6.7. The most transparent form this assumes is in the classical, unrelativized case: all strict Mittag-Leffler modules are separable if and only if all Mittag-Leffler modules are pure separable—and this is the case precisely when also in the description of pure-projective modules ‘direct summand’ is superfluous, see Corollary 6.20. What are all these relativizations and what the rather more general context? As for the context, the most satisfying results, in §6, have been achieved in purely generated classes, in particular, in purely resolving classes (cf. Definition 3.1), as developed in [HR]. Relativizations to classes of modules occurred very naturally in the study of Mittag-Leffler modules in the preceding paper [Rot4]. So it is only natural to ask what the corresponding relativizations should be in the strict case. Taking into account the description of (the unrestricted) strict Mittag-Leffler modules, given in [GIRT, Prop. 1], as those in which every tuple freely realizes a certain pp formula, the correct relativization, it seems, should be obtained by relativizing the scope of ‘freeness’ in the free realizations part of the definition, see Definitions 2.1(2) and 2.2. Another approach to achieve a similar goal can be found in [AH] in the shape of ‘stationary’ modules. In the (non-strict) Mittag-Leffler case a fundamental feature was that the relativizations to a certain class depended only on the definable subcategory this class generates, which greatly simplifies many considerations and formulations. It also allowed for free transfer between formulations for classes K of right modules and classes L of left modules by simply applying elementary duality to the defining pairs of pp formulas, as was first done in [Her]. So, a module is K-Mittag-Leffler if and only if it is L-atomic, whenever K and L generate elementarily dual definable subcategories, cf. [Rot4, Thm.3.1]. For the strict versions the situation is different, and I found no particular choice of module on the other side naturally lending itself to a similar transfer as in the non-strict case, which is why I mostly stay on the same side as the original module. This results in strict L-atomic modules having preference over strict K-Mittag-Leffler modules. Even more, I define the latter concept only for definable subcategories K—namely, as the strict L-atomic modules for L, the definable subcategory elementarily dual to K, cf. Definition 2.2. An actual example showing that the state of affairs is indeed different in the strict case was provided only by an anonymous referee, who pointed out the role of pure-injective modules for this study and how this could be used to obtain two classes of modules that generate the same definable subcategory, yet yield different notions of strict atomicity, see Remark 2.7 at the beginning of §2.3. More on this issue can be found in §5 and the final §7.

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Some results seem new even in the classical situation of L being the entire module category. For instance, the pp definability of finitely generated endosubmodules of any strict Mittag-Leffler module, Corollary 2.13, which had been previously known for pure-projective modules, cf. [P2, Cor. 2.1.27]. Another such result is the passage ‘from elements to tuples:’ in order for a module to be strict Mittag-Leffler it suffices that every element be a free realization of some (unary) pp formula, Corollary 3.7. This is done in the larger context of modules purely generated by a strict relatively atomic module, see Proposition 3.6. In [HR] axiomatizations of purely generated classes by (usually infinitary) implications of pp formulas were found that provide a better handle on the problems arising with relativizations. They show that purely generated classes are F-classes in the sense of [PRZ]. In §3 we pick up this theory and specify it to fit purely resolving classes, cf. Lemma 3.11. The resulting more special axiomatization entails some downward L¨ owenheim-Skolem-like behavior, called self-separation, see §§5.2–5.4. A typical application of this is Theorem 5.12 showing that relative atomic modules in a purely resolving class are already (fully) Mittag-Leffler.

I would like to thank Ivo Herzog and Martin Ziegler for inspiring discussions. Further thanks are due to an anonymous referee for her or his contribution to §2.3 and other helpful comments improving the paper.

Preliminaries. The reader is assumed to be somewhat familiar with [Rot4]— its main techniques and definitions—in particular, with pp formulas and types, their finite generation and free realization, and definable subcategories and the like. Throughout, K denotes a nonempty class of right R-modules and L a nonempty class of left R-modules. ‘Module’ means left R-module unless otherwise specified. R-Inj stands for the class of injective (left) R-modules, # for the class of absolutely pure modules (with subscript R to indicate the side when necessary), and $ for that of flat modules (again, possibly, with a left or right subscript R). Map means (homo) morphism, at least when it is between modules. (M, N ) is a shorthand for Hom(M, N ), so (M, M ) means End M . We often identify a tuple with the (finite) set of its entries, which will be completely clear from the context. All tuples and formulas are assumed to be matching. Herzog [Her, §4] discovered that the map that sends a pp pair ϕ/ψ to the pair Dψ/Dϕ can be used to extend elementary duality from pp formulas to closed sets of the Ziegler spectrum (cf. [P2, Thm.5.4.1]) and hence to definable subcategories, and even to arbitrary theories of modules. Ever since we have the notion of elementarily dual definable subcategories that the reader is assumed to be acquainted with (cf. [P2, §3.4.2]). Recall from [Rot4, §2.5], that DK denotes the class of all character duals (K, Q/Z) of K ∈ K. The definable subcategory DK turns out to be elementarily dual to K, see [Z-HZ], [PRZ] or simply [P2, Cor.3.4.17]. In [Rot4, Convention 2.7] it was stipulated that the classes K and L were definably dual in the sense that the definable subcategories they generate, K and L, are elementarily dual. While no such convention will be made now, as definable subcategories play a different role here, we will freely use definably dual classes in the sense just specified.

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1. Purity 1.1. Relativized purity. The first part of the following definition is discussed in great detail in [Rot4, §5.1]. The second is its natural dual, but seems not to have been discussed much. (A stronger version of it occurs in [Rot4, §5.4].) Definition 1.1. (1) The map f : A −→ B is an L-pure monomorphism if for every tuple a in A and every pp formula ϕ its image satisfies in B, there is a pp formula ψ ≤L ϕ it satisfies in A. A submodule A of a module B is locally L-pure if the identical inclusion is. (2) The map g : B −→ C is an L-pure epimorphism if for every tuple c in C and every pp formula ϕ it satisfies, there is a g-preimage b in B and a pp formula ψ ≤L ϕ it satisfies. As the partial order does not change when passing from L to the definable subcategory it generates, L-pure and L-pure are the same. Beware, L-pure monomorphisms may not be monomorphisms. E.g., in the category of abelian groups the zero map on the group of two elements is an L-pure monomorphism for L, the class of torsionfree abelian groups. Remark 1.2. If the source of any of these maps is in L, the two formulas are equivalent in that source, which yields the usual purity (where ψ can be taken to be ϕ). In particular, (a) L-pure submodules that are in L are pure submodules, (b) L-pure epimorphic images of modules in L are pure epimorphic images. 1.2. Local splitting. Recall Azumaya’s notions, [Azu1], [Azu2]: a monomorphism f : N !→ M is called locally split if for every tuple n in N there is a local retract, i.e., a homomorphism gn : M −→ N with gn f (n) = n. As usual, one calls a submodule N of a module M locally split if the identical embedding is locally split. (Notice, this makes f a monomorphism anyway.) For completeness, let me mention that, dually, an epimorphism g : N  M is called locally split if for every tuple m in M there is a local section, i.e., a homomorphism hm : M −→ N with ghm (m) = m. (Again, this makes g an epimorphism anyway.) Remark 1.3. Clearly, in both definitions, the tuples n and f (n), respectively m and hm (m), have the same pp type. Hence locally split monomorphisms (resp., locally split epimorphisms) are pure. It is not surprising that in order for a monomorphism to be locally split it suffices to check the definition on a generating subset: Remark 1.4. Suppose N is generated by the set C ⊆ N and f : N −→ M is a map such that every tuple c in C has a local retract, i.e., a homomorphism gc : M −→ N with gc h(c) = c. Then f is a locally split monomorphism. 1.3. Relativized local splitting. Just as we were naturally led to weakenings of purity in [Rot3] and [Rot4], we are here led to weakenings of local splitting by the investigations of how much one can enlarge a class L without disturbing the property of being strict L-atomic (or strict K-Mittag-Leffler), see §2.3 below, the only section where this is used.

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Definition 1.5. Suppose P is a class of (left R-) modules. A homomorphism f : A −→ B is locally P-split if for every tuple a from A and every tuple p from some P ∈ P, every map g : (P, p) −→ (B, f (a)) factors through f so that p goes to a. More precisely, for every such g : (P, p) −→ (B, f (a)) there is h : (P, p) −→ (A, a) such that g(p) = f h(p). A locally P-split source of B is a module A for which there is such a locally P-split homomorphism f : A −→ B. A submodule A of a module B is locally P-split if the identical inclusion is. Remark 1.6. Just as R-Mod-purity was the usual purity, R-Mod-locally split submodules are the usual locally split submodules. Further, the (R-mod)-locally split submodules are precisely the pure submodules. Thus, so long as R-mod ⊆ P, all P-locally split submodules are pure submodules. 2. Strict Mittag-Leffler and atomic modules 2.1. Free realizations of formulas. Definition 2.1. (1) [Rot3, Def.2.2(1)] A free realization for L of a pp formula ϕ, or an L-free realization of ϕ, is a pointed module (M, m) such that m is a (matching) tuple satisfying ϕ in M and, whenever a tuple c satisfies ϕ in a module L ∈ L, then there is a map M −→ L sending m to c.1 (2) M is said to be a strict L-atomic if every tuple in M freely realizes some pp formula for L. (3) We omit the attribute L if it is all of R-Mod. Note that every strict L-atomic module is L-atomic in the sense of [R] (or [Rot4])—as it should be. For, if (M, m) freely realizes ϕ for L and ψ ∈ ppM (m), then, in any L ∈ L, if (L, a) satisfies ϕ, there is a map (M, m) −→ (L, a), hence (L, a) satisfies also ψ. Consequently, ϕ L-generates ppM (m). This property for every m in M is what is called L-atomicity. Now, the main theorem of [R] (or [Rot4]) says that in case the classes K and L are definably dual in the sense specified in the preliminaries above (which, recall, means that the definable subcategories they generate are elementarily dual in the sense of [Her]), the two properties, Latomicity and that of being K-Mittag-Leffler, are the same—and so we can take it as a definition of K-Mittag-Leffler. Similarly, [GIRT, Prop. 2.1] established that the two properties of strict atomicity (i.e., strict R-Mod-atomicity in the sense just introduced) and of being strict Mittag-Leffler (in the sense of [RG]) are the same. So it seems only legitimate to extend this to arbitrary definable subcategories and make it a definition, redundant as it may seem. Definition 2.2. Given a definable subcategory K of right R-modules, a module is said to be a strict K-Mittag-Leffler if it is strict L-atomic for L, the definable subcategory of left modules elementarily dual to K in the sense of Herzog (see Preliminaries). We will still prefer the terms (strict) L-atomic over their Mittag-Leffler version for three reasons: first it allows us to stay on the same side of the ring, second, and 1 This differs from the definition made in [HR, §1.4] in that M is not required to be a member of L. Both differ from Prest’s original definition in that the requirement on the underlying module of being finitely presented is also abandoned.

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more importantly, it makes available our main techniques involving pp formulas, third, and most importantly, it depends here on how we apply elementary duality to a class of modules, as the concepts may no longer be invariant under definable subcategories. At the same time though, we want to keep in mind their connotations ‘on the other side:’ as strict K-Mittag-Leffler modules—provided K and L are definable subcategories that are elementarily dual. Remark 2.3. The strict Mittag-Leffler modules of [RG] are the strict KMittag-Leffler modules for K = Mod-R (i.e., the strictly L-atomic modules for L = R-Mod), [GIRT, Prop. 2.1], or, equivalently, [Azu2], the locally pure-projective modules, i.e., the modules such that every pure epimorphism onto them locally splits in the sense of §1.2. It is not hard to see that within a definable subcategory D, for a D-atomic module to have the strict property it suffices to check it for single modules in D one by one: # Lemma 2.4. Let M be an L-atomic module and L = i∈I Li ⊆ L. If M is strict Li -atomic for every i ∈ I, then it is strict L -atomic. Proof. By hypothesis, every tuple a in M satisfies a pp formula ϕ that Lgenerates—hence also L-generates—its type ppM (a), and there are ϕi in that type that a freely realizes for Li , i ∈ I. As mentioned before, this implies that ϕi ≤Li ϕ. But Li ⊆ L implies also ϕ ≤Li ϕi , which shows that we may take ϕ  for all the ϕi . Consequently, (M, a) freely realizes ϕ for L , as required. 2.2. Pure submodules of atomic modules. It is clear from the definition that the class of strict L-atomic modules is closed under arbitrary direct sum and pure submodules. Here is a refinement of the latter. Lemma 2.5. If f : N −→ M is an L-pure monomorphism with M strict Latomic, then N is strict L-atomic. In particular, L-pure submodules of strict L-atomic modules are strict L-atomic. (Without ‘strict’ this is [Rot4, Cor. 6.3].) Proof. For notational simplicity, assume N is an L-pure submodule of the strict L-atomic module M and f is the identical inclusion. Let n be a tuple in N . Then (M, f (n)) freely realizes some formula ϕ for L which is L-equivalent to some ψ in ppN (n). To see that (N, n) freely realizes ψ for L, consider L ∈ L and a tuple c therein that satisfies ψ. By L-equivalence, it also satisfies ϕ, whence we have a  map (M, f (n)) −→ (L, c). This yields the desired map (N, n) −→ (L, c). Lemma 2.6. Suppose f : N −→ M is an L-pure monomorphism with M strict L-atomic. If N is itself in L, then f is a locally split (and hence pure2 ) monomorphism. If N is, in addition, finitely generated, f splits. In particular, L-pure submodules of strict L-atomic modules that are themselves in L are locally split (and hence pure) submodules. If they are, in addition, finitely generated, they are even direct summands. Proof. Let n be any tuple in N and let (M, f (n)) L-freely realize the pp formula ϕ. As N ∈ L, by L-purity, ϕ is contained in the type of n in N , and so we  have (M, n) −→ (N, n), which is the desired local retract for n. 2 It’s

pure anyway, see Remark 1.2.

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2.3. Closure of the target class. First a crucial observation about pureinjectives provided by the referee, which culminates in an example showing that, in contrast to L-atomicity (and K-Mittag-Leffler modules), which depends only on the definable subcategory L (or K) generates, strict L-atomicity is, in general, different from L-atomicity. Remark 2.7. (a) If L consists of pure-injective modules only, L-atomic implies strict Latomic. For, if the pp type ppM (a) in M is L-generated by ϕ which b in L ∈ L satisfies, then ppM (a) ⊆ ppL (b), hence pure injectivity of L guarantees a map (M, a) −→ (L, b), as desired (see e.g. [P2, Thm.4.3.9]). (b) If L ⊆ L ⊆ L are such that L \ L consists of pure-injectives only, then every strict L-atomic module is strict L -atomic. This follows from the implications strict L-atomic =⇒ L-atomic =⇒ L-atomic =⇒ L \ L-atomic =⇒ strict L \ L-atomic (by (1)), which, together with strict L-atomicity, yields strict L -atomicity by Lemma 2.4. (c) Let D be a definable subcategory and L be the class of all of pure-injectives in D. If strict L-atomic implies strict L-atomic, then D-atomic implies strict D-atomic. This follows from the implications D-atomic =⇒ L-atomic =⇒ strict L-atomic =⇒ strict L-atomic ⇐⇒ strict D-atomic, the second of which is (1); for the last notice L = D, for any module is in the same definable subcategories as is its pure-injective hull. (d) Consequently, every definable subcategory D with a D-atomic member which is not strictly D-atomic gives rise to a subclass L (namely, the class of all pure-injective members of D) such that strict L-atomicity is stronger than strict L-atomicity. (e) As a concrete example, take L to be the class of all pure-injective abelian groups. The definable subcategory it generates is the category of all abelian groups. Since there are Mittag-Leffler abelian groups that are not strict Mittag-Leffler, e.g., Example 6.24 below, not every strict L-atomic abelian group is strict L-atomic (= strict Mittag-Leffler). The obstacle in extending strict L-atomicity to strict L-atomicity is the closure of L under pure submodules (note, every module is pure in its pure-injective envelope). If we strengthen purity to a certain weak local splitting, however, we can prove this. Again, the additional clause about pure-injectives is due to the referee. Proposition 2.8. Suppose L is any class of modules and P is the class of strict L-atomic modules. Let L be the union of L and the class of all pure-injectives in L, and let L be the closure of L under direct product, direct limit and local P-split sources (in particular, under local P-split submodules). Then every strict L-atomic module is strict L-atomic. Proof. Suppose M is strict L-atomic, hence, by (2) of the previous remark, also strict L -atomic. That this condition is preserved in direct products of the target modules follows easily from the fact that pp formulas are p-functors, i.e., commute with direct product. Namely, let m ∈ ϕ(M ) freely realize a pp formula ϕ for L . If N is the product of some Li ∈ L and a tuple n satisfies ϕ in N , then each coordinate of n in Li , call it li , satisfies ϕ. So there are maps fi : (M, m) −→ (Li , li ), whose product sends (M, m) to (N, n), as desired.

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To show that this condition is preserved in direct limits, suppose the Li form a directed system with direct limit N and the tuple n satisfies ϕ in N . This is possible only if ϕ was true along some tail of n in the direct system. In particular, there is a preimage li of n, under a canonical map, which satisfies ϕ. By hypothesis, there is a map (M, m) −→ (Li , li ), which, composed with that canonical map, yields a map (M, m) −→ (N, n), as desired. Finally, suppose f : N −→ L is locally P-split, L ∈ L , and (N, n) satisfies ϕ. Being existential, ϕ is also satisfied in (L, f (n)). By hypothesis on M , there is a map (M, m) −→ (L, f (n)), which, by hypothesis on N , factors through f , and thus  yields the desired map to (N, n). A simple argument using R-mod ⊆ P shows that L is closed under local P-split sources. The other operations being part of the definition of definable subcategory, one infers that L is contained in the definable subcategory L. It might be short of being the whole thing by leaving out some pure submodules that are not locally P-split. In fact, we have the following. Lemma 2.9. Suppose A is a pure submodule of some L ∈ L such that every strict L-atomic module is also strict A-atomic. Then A is locally P-split in L (with P, the class of strict L-atomic modules). Proof. Suppose A ⊆ L and g is a map from P ∈ P to L with g(p) = a for given tuples a in A and p in P . As P ∈ P, the tuple p freely realizes some pp formula ϕ for L in P . As g preserves pp formulas, a satisfies ϕ in L, and, by purity, also in A. Since P is strict L-atomic, by hypothesis, it is also strict A-atomic, so (P, p) must freely realize some pp formula ψ for A. Then ϕ ≤L ψ, and as A is in L, the  tuple a satisfies ψ in A, which yields the desired map (P, p) −→ (A, a). We will return to this issue in §§5.2 and 7. 2.4. Countably generated atomic modules are strict. [Rot3, Prop. 2.4] says that every tuple in a countably generated L-atomic module freely realizes some pp formula for L. This is so basic a fact for this topic that we restate it as follows and make use of it without much mention. Lemma 2.10 (The ‘Strict’ Lemma). [Rot3, Prop. 2.4] Countably generated L-atomic modules are strict L-atomic. Consequently, countably generated strict L-atomic modules are strict L-atomic. Proof. The first part of the statement is [Rot3, Prop. 2.4]. Invoking [Rot4, Cor. 3.6(a)] saying that L-atomic is the same as L-atomic, the second follows at once.  2.5. Traces. The role of traces in this context has been made clear in [Z-H] and [Gar], see also [P2, Cor. 1.2.17]. Given a tuple m in a module M , by the trace of m in a module N we mean the set (M, N ) m := {h(m) | h ∈ (M, N )}. By an L-trace we mean a trace in a module from L. Lemma 2.11. (M, m) is a free realization for L of the pp formula ϕ if and only if m satisfies ϕ in M and ϕ defines all L-traces of m, i.e., (M, L) m = ϕ(L) for all L ∈ L.

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Proof. As morphisms preserve pp formulas, m ∈ ϕ(M ) implies (M, F ) m ⊆ ϕ(F ), for any module F . The inverse inclusion holds if and only if m can be mapped to every element of ϕ(F ).  Note, if M ∈ L then m is in its own L-trace and therefore automatically satisfies any pp formula that defines all its L-traces. Proposition 2.12. The following are equivalent for any (left R-) module M . (i) M is strict L-atomic. (ii) Every tuple m in M satisfies a pp formula that defines all traces of m in modules from L. (iii) Every tuple m in M satisfies a pp formula ϕ that is freely realized in some module N with ϕ(N ) = (M, N ) m. Proof. (i) and (ii) are equivalent by the lemma. Since every pp formula has a (total) free realization somewhere (in some module N , even a finitely presented one), (iii) is equivalent as well.  This allows us to extend a result known for pure-projective modules to strict Mittag-Leffler modules, cf. [P2, Cor. 2.1.27]. Corollary 2.13. Every finitely generated endosubmodule of a strict MittagLeffler module is pp-definable. Proof. Let M be strict Mittag-Leffler, that is, strict (R-Mod-) atomic. An endosubmodule generated by elements mi ∈ M (i ∈ I) is the sum of all the traces (End M ) mi = (M, M ) mi . These being pp-definable, their sum is too (so long as I is finite).  Clearly, the same is true for finite sums of traces of k-tuples (M, M ) mi , so that also finitely generated submodules of M k —regarded as a module under the diagonal action of End M again—are pp-definable (in M ). Remark 2.14. Every finitely generated endosubmodule of a module M is ppdefinable if and only if M is strict M -atomic, for the finitely generated endosubmodules of M are precisely the traces under End M = (M, M ) of the generator tuples. 2.6. Strict R #-atomic modules. Over left coherent rings, where $R and R # form definable subcategories, [P2, Thm.3.4.24], which are elementarily dual, these are, by definition, the strict $R -Mittag-Leffler modules. However, we prefer to work with R #-atomic modules, which make sense over any ring. (So does the duality of $R and R #, namely, Herzog showed that these classes are what we call definably dual, i.e., they generate elementarily dual subcategories [Her, §12], cf. [P2, Prop.3.4.26].) It has been known since [Goo] that all modules are RR -Mittag-Leffler (equivalently, $R -Mittag-Leffler, or R #-atomic) if and only if R is left noetherian, cf. also [Rot4, Cor. 4.3]. In our terminology, R is left noetherian if and only if every module is R #-atomic. We are going to strengthen this by showing that then all modules are even strict R #-atomic. First a general observation, whose original proof became obsolete with the referee’s Remark 2.7(1) above. Remark 2.15. RR -Mittag-Leffler (=R #-atomic) modules are strict R-Injatomic.

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Proposition 2.16. A ring R is left noetherian if and only if every module is strict R #-atomic. In particular, every abelian group has this property. Proof. In the noetherian case, R # = R-Inj, so the implication from left to right follows from the remark (and the fact that then every module is R #-atomic). As mentioned, for the converse it suffices to have all modules (plain) R #-atomic (=  RR -Mittag-Leffler). Example 2.17. By [AF, Prop. 7], an abelian group M is Mittag-Leffler iff it has trivial first Ulm group (i.e., the intersection of all nM is 0) and M/tM is Mittag-Leffler. Hence no divisible group is Mittag-Leffler. But all of them are strict R #-atomic. 2.7. Strict R $-atomic modules. Dually to what was said at the beginning of the previous section, now it is the right coherent rings over which these are the strict #R -Mittag-Leffler modules. But again, we wish to work with arbitrary rings and stick to strict R $-atomic modules instead. Example 2.18. (Of strict R $-atomic abelian groups that are not Mittag-Leffler.) By [Rot3, Prop. 5.1], every torsion abelian group is R $-atomic—in fact, by [Rot3, Thm. 6.8 or Cor. 6.12], every group M with M/t(M ) Mittag-Leffler is. Hence every countable such group is even strict R $-atomic, cf. Lemma 2.10. Now one gets a host of such groups which are not Mittag-Leffler from [AF, Prop. 7], see the previous Example. Namely, take any countable torsion group with non-trivial first Ulm group, like the Pr¨ ufer groups. In contrast, flat strict R $-atomic modules turn out to be always strict atomic (i.e., strict Mittag-Leffler), as will be shown in Corollary 3.5 below. More will be said in the introduction to §6.6 as a consequence of some separation results for purely generated classes. 3. Pure generation Definition 3.1. [HR, §2] (1) A module is purely generated by a class B if it is a pure epimorphic image of a direct sum of modules from B. A class is purely generated by B if every module in it is. The class of all modules purely generated by B is denoted PGen B. (2) Following [HR, §2], we let C = Add B and C˜ be the class of all pure epimorphic images of modules from C (equivalently, from ⊕B). So C˜ is a shorthand for PGen B. (3) [HR, Rem. 2.4(a)]. A class is purely resolving if it is purely generated by pure-projective modules. (4) For the ease of exposition, we always assume B to be closed under finite direct sum. Note, a class is pure resolving whenever it is purely generated by locally pureprojective modules, [HR, Cor. 2.5], which means we may allow B to consist of strict Mittag-Leffler modules and still have C˜ = PGen B purely resolving.

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3.1. Basic properties. Proposition 3.2. Suppose M is a pure-epimorphic image of a strict G-atomic module L ∈ L. If a tuple m in M freely realizes some pp formula for L, then it freely realizes also some (possibly different) pp formula for G. Consequently, if M is strict L-atomic, it is also strict G-atomic. Proof. Consider a pure epimorphism h : L −→ M . Let m (in M ) freely realizes a pp formula ϕ for L and choose a preimage l ∈ ϕ(L) of m by purity. By hypothesis on ϕ, there is a map f : (M, m) −→ (L, l). Being strict G-atomic, (L, l) freely realizes a pp formula ψ for G. Application of h shows that (M, m) also satisfies ψ. To see that it does so freely for G, let G ∈ G and g be a tuple that satisfies ψ in G. Then there is a map (L, l) −→ (G, g). Composed with f , this  yields the desired map (M, m) −→ (G, g). Corollary 3.3. Let L be a class of strict G-atomic modules and C˜ a class purely generated by L, [HR, §2]. Then every strict L-atomic member of C˜ is strict G-atomic. Special cases are: Corollary 3.4. If L is a purely resolving class, then every strict L-atomic member of L is strict atomic, i.e., strict Mittag-Leffler.  Corollary 3.5. Every flat strict strict Mittag-Leffler.

R $-atomic

module is strict atomic, hence

Proof. The class R $ of flat modules is purely resolving, in fact, purely generated by the finitely generated projectives, cf. [HR, before Thm.2.1]. Now the previous corollary applies.  Compare this to [Rot3, Thm. 3.10], which says the same without ‘strict.’ 3.2. From elements to tuples. It is always desirable to reduce a condition on tuples to the same condition on just elements. Proposition 3.6. Suppose M is a pure-epimorphic image of a strict L-atomic module in L. Then the following are equivalent. (i) M is strict L-atomic. (ii) Every element in M freely realizes a certain pp formula for L. Proof. We prove the nontrivial direction by induction on the length of the tuple. Suppose m0 is a tuple in M L-freely realizing a pp formula ϕ0 and m ∈ M . We are going to find a pp formula ϕ that the concatenated tuple (m0 , m) freely realizes for L (in M ). By hypothesis, there is a pure epimorphism g from a strict L-atomic module L ∈ L onto M . Using purity of g, choose a preimage b0 of m0 in L satisfying ϕ0 . By choice of ϕ0 , there’s a map f0 : (M, m0 ) −→ (L, b0 ). Note, gf0 (m0 ) = m0 . By (ii), there is a pp formula ϕ that the element m := m − gf0 (m) freely realizes for L in M , and, by purity of g again, a preimage b in L realizing it. Then there is a map f  : (M, m ) −→ (L, b ). Again, gf  (m ) = m .

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Let f be the map f  + f0 − f  gf0 . An easy calculation shows that f (m0 ) = b0 and f (m) = f  (m ) + f0 (m), and another one that therefore gf fixes both m0 and m. Thus (m0 , m) and f (m0 , m) have the same pp type (in M and L, resp.). It remains to note that, by choice of L, the latter type is generated by a single pp formula, say ϕ = ϕ(x0 , x), that (L, b0 , f (m)) freely realizes for L. Composing with the map f , we finally conclude that (M, m0 , m) freely realizes that same ϕ for L, as desired.  Warfield showed every module is a pure-epimorphic image of a pure-projective module (one can, in fact, take a direct sum of finitely presented modules), [W, 33.5]. Hence the lemma holds for L = R-Mod. Corollary 3.7. A module is strict Mittag-Leffler if and only if every element freely realizes some (unary) pp formula.  3.3. Axiomatizability. Consider F-classes as introduced in [PRZ], which are classes of modules 5 that can be axiomatized by F-sentences, i.e., implications of the form ϕ −→ ϕi , where ϕ and the ϕi are pp formulas. Here the conclusio is allowed to be an infinitary disjunction, but such that the ϕi are closed under  finite sum so that this implication is equivalent to the implication ϕ −→ ϕi , cf. [HR, §2]. The ‘F’ comes from flat, for the class of flat modules is an F-class, see next section. [HR, Thm. 2.1] exhibits an axiomatization of purely resolving classes by Fsentences. To describe the specific axioms, consider such a class, C˜ = PGen B. Let ppf B stand for the set of pp formulas that are freely realized (for all of R-Mod) by a tuple in some module from B. Given a pp formula ϕ, let ppf B ϕ be the set of all formulas (of same arity) in ppf B that are below ϕ, i.e., ppf B ϕ = {θ ∈ ppf B | θ ≤ ϕ}. 5 The axioms for C˜ are now all F-sentences of the form ϕ −→ ppf B ϕ, where ϕ runs over all pp formulas (it suffices to consider 1-place pp formulas). The overall assumption that B be closed under finite direct sum guarantees that ppf B (hence also ppf B ϕ) is closed under (finite) sums of pp formulas, so that the disjunction in the conclusio can, again,  be replaced by the sum. In other words, the above axiom is equivalent to ϕ −→ ppf B ϕ. Remark 3.8. One can generalize this axiomatization result to classes that are purely generated by relativized strict atomic modules, however those will no longer have the special features singled out in the next section. 3.4. Special F-classes. In general, one may take ϕi ∧ ϕ in the conclusio of an F-sentence, to ensure true: then 5 5 that the implication’s converse is automatically the implication ϕ −→ ϕi entails the equivalence of ϕ and ϕi . (Beware, this may lead outside the realm of a specific shape of formula. For instance, the conjunction of a formula from ppf B with an arbitrary pp formula ϕ may no longer be in ppf B . This is the reason, why, in §3.3, we worked with ppf B ϕ instead, cf. the proof of Lemma 3.11 below.) We will see shortly that R $ is so axiomatized. But the axioms for the class of flat 5 modules have yet another special feature. Given another such axiom, ψ −→ ψi with 5 ψi ≤ ψ, if every disjunct ϕi of ϕ is equivalent to one of the ψi , then ϕ −→ ϕi implies ϕ −→ ψ. Plainly speaking, one disjunction (trivially) implies another disjunction provided all disjuncts of the former are disjuncts of the latter. The special feature we are after now is when this is the only way that the former disjunction can imply the latter. Here is the precise definition.

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Definition 3.9. 5 i. (1) An F-sentence ϕ −→ ϕi is in standard form 5 if ϕi ≤ ϕ for every 5 (2) Given F-sentences in standard form, ϕ −→ ϕi and ψ −→ ψj , we say ϕ trivially implies ψ (in F) if every disjunct ϕi (of ϕ) (logically) implies some disjunct ψj (of ψ), i.e., for all i there is a j with ϕi5≤ ψj . (3) An F-class F is special if it has a set of F-axioms ϕ −→ ϕi in standard form, one for each pp formula ϕ (this is important!), and such that for 5 5 every two axioms ϕ −→ ϕi and ψ −→ ψj from that set, we have ϕ ≤F ψ (if and) only if ϕ trivially implies ψ. 5 The standard form of ϕ −→ ϕi guarantees that ϕ is F-equivalent to the disjunction of the ϕi , and similarly for ψ. Hence, if every disjunct of ϕ implies some disjunct ψj in the above ϕ and ψ, then ϕ does imply ψ in F (trivially), thus justifying the terminology. Remark 3.10. R $ is a special F-class. This follows from [Zim1], see [P2, Thm. 2.3.9] or the explicit axioms in [PRZ] or [Rot3, Fact 1.3], which are in standard form—each disjunct of the conclusio implies the premise. Is the class of torsion-free modules a special F-class? It is an F-class, cf. [Rot3]. However, one has F-axioms only for annihilation formulas ϕ, so not every pp formula may end up being equivalent to such a disjunction. I have no counterexample at hand, but certainly the proof below breaks down when not every pp formula occurs as a premise. Lemma 3.11. Purely resolving classes are special F-classes. Proof. As all formulas in ppf B ϕ are below ϕ, the axioms given in the previous sections are in standard form. To verify that they 5 are special in the5sense of the above definition, consider two axioms ϕ −→ ppf B ϕ and ψ −→ ppf B ψ with ϕ ≤C˜ ψ. All we need is ppf B ϕ ⊆ ppf B ψ. So let θ ∈ ppf B be below ϕ. To show that it is also below ψ, let (M, m) realize θ. Pick a free realization (B, b) of θ with B ∈ B, send it to (M, m) and observe that it remains to see that ψ is already satisfied in (B, b). But this follows from B ∈ B ⊆ C˜ and the assumption ϕ ≤C˜ ψ.  This will be used in §6.4. 4. Separation Reinhold Baer introduced separability in abelian groups—of finite subsets by direct summands that are completely decomposable, see [F]. 4.1. Separabilities. We distinguish several types of separation—according to three different coordinates, cardinality of sets to be separated, the form of separation, and the kind of submodules to perform the separation. In Baer’s original separation, the second coordinate, the form of separation, was by direct summand, which is what one usually sees in the literature in the various types, see [EM]. This can be relaxed to Azumaya’s locally split,3 or to pure submodules, called local separation and pure separation, respectively. On top of that, we will have an entire family of relativized pure separabilities. 3 also

to his finitely split submodules,

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Generalizing e.g. [GIRT, §4], we make the following, essentially three, definitions. Definition 4.1. Let κ be an infinite regular cardinal and S a class of modules. (1) A module M is κ-separable by (modules from) S, or (κ, S)-separable for short, if every subset of M of cardinality < κ is contained in a submodule from S that is a direct summand of M . (2) We say locally (resp., L-pure) κ-separable by (modules from) S, or locally (resp., L-pure) (κ, S)-separable, if ‘direct summand’ is replaced by ‘locally split submodule’ (resp., by ‘L-pure submodule’). (3) If κ = ℵ0 , we simply omit it or say finite separability (for whichever version of separability at hand). (4) If κ = ℵ1 , we speak of countable separability. (5) We omit the reference to the class S when it is R-mod. By these conventions, local separability by S, or local S-separability, is local (ℵ0 , S)-separability, while L-pure separability by S, or L-pure S-separability, is Lpure (ℵ0 , S)-separability. And, for instance, a separable module is one all of whose finite subsets are contained in a finitely presented direct summand—beware, this is often handled differently in the literature, cf. [EM] and [F]. Remark 4.2. (a) Bear’s original separability is (finite) separability—by completely decomposable submodules, cf. [F]. (b) Clearly, any of the κ-separabilities entail the corresponding λ-separability for every λ ≤ κ. (c) Clearly, direct summand =⇒ locally split submodule =⇒ pure submodule =⇒ L-pure submodule, hence separable =⇒ locally separable =⇒ pure separable =⇒ L-pure separable. Our interest in separation lies in the following facts, which we single out for reference. Lemma 4.3. (1) If M is L-pure separable by L-atomic modules, M is L-atomic (hence L-atomic). (2) If M is locally separable by strict L-atomic modules, M is strict L-atomic. (3) If M is locally separable by countably generated strict L-atomic modules, M is strict L-atomic. Proof. (1) By definition N ⊆ M is L-pure iff the pp types of all tuples in N are L-equivalent, whether taken in N or in M , which is, by the Main Theorem of [R] or [Rot4] all we need. (2) First separate a given tuple in M by a locally split submodule N that is strict L-atomic. Inside N , that tuple L-freely realizes a certain pp formula ϕ. The same formula generates the tuple’s pp type in M , for locally split submodules are pure. To see that this tuple freely realizes ϕ also in M , simply combine the tuple’s local retract from M to N by any of the ‘free’ maps from N to a realization of ϕ in any other module from L. (3) Apply the (stronger half of the) ‘Strict’ Lemma and (2) above. 

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The question of a converse arises at once. This will be addressed in §6, especially in Theorem 6.7. We conclude this section with some general observations. Lemma 4.4. Suppose B ⊆ L and M a strict L-atomic module. (1) The following are equivalent for any infinite cardinal κ. (i) M is L-pure κ-separable by B. (ii) M is pure κ-separable by B. (iii) M is locally κ-separable by B. (2) If all members of B are finitely generated, M is κ-separable by B in that case. Proof. As B ⊆ L, for submodules that are in B, L-purity is the same as purity, Rem.1.2. So the first two conditions are equivalent (and this does not require atomicity of M ). To see they imply the third, and for (2), refer to Lemma 2.6.  4.2. Separation and definable subcategories. Next we show that in the situation of the previous lemma, M belongs to the definable subcategory generated by L. Lemma 4.5. (1) If M is L-pure separable by L (hence pure separable by L), then M ∈ L. (2) Suppose L is axiomatizable by implications of the form Φ −→ Ψ, where Φ is a possibly infinite conjunction of possibly infinite disjunctions of pp formulas and Ψ a possibly infinite disjunction of possibly infinite conjunctions of existential formulas, all in the same finitely many variables. If M is L-pure separable by L (hence pure separable by L), then M satisfies all those implications that are true in L. Proof. (1) is the special case of (2) where both Φ and Ψ are single pp formulas. To prove (2) consider one such implication Φ −→ Ψ and assume the tuple m satisfies the antecedent Φ. We may L-pure separate it by some L ∈ L, which is actually pure in M , Remark 1.2. Now m satisfies all conjuncts of Φ and hence one disjunct for each of these in M . Being pp, these latter are, by purity, satisfied by m in L as well. It is easy to see that m satisfies all of Φ in L. But being in L, L satisfies the implication, hence m satisfies Ψ in L. Being existential, Ψ it is also true of that very tuple in the original M , as desired.  5. Countable self-separation 5.1. A general fact. As shown in [Rot3, Prop. 3.5] or [Rot4, Cor. 6.5], a module is K-Mittag-Leffler if and only if each of its countable subsets is contained in a countably generated L-pure submodule that is K-Mittag-Leffler (where K and L are assumed to be mutually dual). Because of the stronger half of the ‘Strict’ Lemma 2.10, we can say that K-Mittag-Leffler modules are countably L-pure separable by strict K-Mittag-Leffler modules. (The easy direction also follows from Lemma 4.3.) This fundamental separation result characterizing K-Mittag-Leffler modules we now formulate as follows. Fact 5.1. A module is L-atomic if and only if it is countably L-pure separable by countably generated strict L-atomic modules (even strict L-atomic modules).

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5.2. Countable self-separation. A principal deficiency in the above general fact is that one has no reason to expect the strict L-atomic separating submodules to be themselves in L, even if the original module was. The L¨owenheim-Skolem Theorem of first-order model theory springs to mind as a remedy, and we are going to pursue this in §5.3. In order to avoid repeating a mouthful let us make a definition. Definition 5.2. We call a class L (countably) self-separating if every L-atomic module in L is countably L-pure separable by countably generated (strict) L-atomic modules in L. Lemma 5.3. Suppose L is self-separating. (1) Every strict L-atomic module in L is countably locally separable by countably generated (strict) L-atomic modules in L, hence by strict L-atomic submodules in L. (2) Every strict L-atomic module in L is strict L-atomic. Proof. For (1) use the ‘Strict’ Lemma 2.10 and Lemma 4.4(1). For (2) use (1) and Lemma 4.3(3).  5.3. Elementary classes. Recall that a class axiomatized by finitary firstorder axioms is called elementary. Lemma 5.4. Suppose L is an elementary class and κ = max(|R|, ℵ0 ). Let B be the class of strict L-atomic modules in L of power ≤ κ. (1) Every L-atomic module in L is pure κ+ -separable by L-atomic modules in L of power κ. (2) Every strict L-atomic module in L is locally κ+ -separable by B. Proof. By the L¨ owenheim-Skolem Theorem, every subset of a module M of power at most κ is contained in an elementary substructure N of M of power κ. Then N is in L and a pure submodule of M , hence L-atomic. This proves (1). For (2) apply Lemma 2.5 to see that N is strict L-atomic and thus in B. In that case, Lemma 2.6 yields that N is locally split in M .  Proposition 5.5. Suppose L is an elementary class and R is countable. A module in L is strict L-atomic if and only if it is locally countably separable by countable strict L-atomic modules in L. In particular, elementary classes over countable rings are self-separating. Proof. One direction follows from the lemma (for κ = ℵ0 ) and the ‘Strict’ Lemma 2.10. The other is immediate from (2) of Lemma 4.3.  This can be slightly strengthened if the axiomatization of L is of a specific kind. Corollary 5.6. Suppose R is countable and L is axiomatizable by implications as in Lemma 4.5 (2), but finitary (so that L¨ owenheim-Skolem applies). A module is strict L-atomic and in L if and only if it is locally countably separable by countable strict L-atomic modules in L. Proof. All that’s missing in the proposition is that if M is so separable, it has to be in L. But this follows from Lemma 4.5 (2).  For definable subcategories this can be reformulated as follows.

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Corollary 5.7. Suppose R is countable, L is a definable subcategory (i.e., L = L) and K is its elementary dual. A module M is strict L-atomic (= strict K-Mittag-Leffler) and a member of L if and only if it is countably locally separable by countably generated strict L-atomic modules that are members of L.  Question 5.8. Is this true over uncountable rings? 5.4. Special F-classes. For L, the class R $ of quasi-flat modules, the previous proposition implies that, over a countable ring, a quasi-flat module is strict R $-atomic if and only if it is countably separable by countably generated strict R $-atomic modules. However, we can drop the countability assumption on R in this case, as follows from [Rot3, Prop.3.7] and get a better separation result at the same time—namely by flats rather than quasi-flats. Analyzing that proof, we have arrived at the following abstract version that tells the gist of it. We will work in special F-class as introduced in §3.4 with terminology and notation from Definition 3.1. This is the abstract version of [Rot3, Prop. 3.7]: Proposition 5.9. Every special F-class is (countably) self-separating. Proof. Referring to the proof of [Rot3, Prop. 3.7], we start also here by modifying the F-generator ϕ of the type ppM (a) from the proof of [Rot3, Prop. 3.5] (= Fact 5.1) by a disjunct ϕia ∈ ppM (a) of the conclusio, which too is an F-generator of ppM (a). Taking witnesses for ϕia in each of the countable steps and then closing off by the countably generated submodule N of M , we see, as in [Rot3, Prop. 3.7], that N is F-atomic and F-pure in M . It remains to verify that N ∈ F, for which we will verify the 5 axioms. Let ψ −→ ψj be one of them. Suppose a ∈ N satisfies ψ. We have to show, a also satisfies one of the ψj in N . As ψ ∈ ppM (a), we have ϕ ≤F ψ (even ϕia ≤F ψ). Since F is special, this can happen only if every disjunct ϕi of ϕ implies some disjunct of ψ. In particular, ϕia does, which shows that a satisfies some disjunct of ψ, as desired.  Corollary 5.10. Every purely resolving class is (countably) self-separating. Proof. Use Lemma 3.11.



Corollary 5.11. [Rot3, Prop. 3.7] Every flat R $-atomic module is countably separable by countably generated flat (strict) R $-atomic modules.

R $-pure

Proof. By Corollary 3.5, R $ is purely resolving.



5.5. Relative Mittag-Leffler modules in purely resolving classes. Recall from Corollary 3.4: strict L-atomic modules are strict atomic—i.e., strict Mittag-Leffler—provided they are members of L and L is purely resolving. We are going to show the same for the non-stict version. Theorem 5.12. Suppose L is purely resolving. L-atomic modules in L are Mittag-Leffler. Proof. Let M be L-atomic. By Corollary 5.10, M is countably L-pure separable by strict L-atomic modules in L. As these are strict atomic and L-purity is just purity by Remark 1.2, the module M is pure separable by atomic modules, hence atomic itself, which means Mittag-Leffler. 

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Corollary 5.13. [Rot3, Thm. 3.10] Flat $-atomic modules are Mittag-Leffler.  6. Finite separation in purely generated classes 6.1. The first separation theorem. Definition 6.1. (1) A separation pair is an ordered pair (L, B) of classes of modules, B ⊆ L, where L is closed under direct sum and L-pure epimorphic images and B is a small 4 class of strict L-atomic modules closed under finite direct sum. (2) A B-free module is simply a direct sum of modules from B. The class of all of them is denoted ⊕B. (3) A large B-free module is a direct sum of direct powers of infinitely many copies of each isomorphism type of modules from B, i.e., a module of the form ⊕B∈B∗ B (κB) with all κB infinite, where B ∗ is any transversal in B ∼ =. (4) We denote by P ∗ the large B-free module P ∗ = ⊕B∈B∗ B (ω) . In other words, a large B-free module is a module of the form ⊕B∈B∗ B (κB ) with all κB infinite, where B ∗ is any transversal in B/∼ =, and P ∗ = ⊕B∈B∗ B (ω) . Notation 6.2. Throughout this subsection, (L, B) is a separation pair. Following [HR, §2], we let C = Add B and C˜ be the class of all pure epimorphic images of modules from C (equivalently, from ⊕B), i.e., the class purely generated by B, which is therefore also denoted PGen B. Remark 6.3. (a) As B is closed under finite direct sum, B-free modules are (finitely) Bseparable. (b) Relaxing pure generation to L-pure generation would not gain one anything, for C ⊆ L, and L-pure epimorphisms emanating from modules in L are pure epimorphisms. Lemma 6.4. [Rot3, Prop.2.5]. Every countably generated (strict) L-atomic module in PGen B is contained in Add B, i.e., a direct summand of a B-free module (which can be taken to be large) and thus a direct summand of a module (finitely) separable by B. Proof. Being in PGen B, the module N is a pure epimorphic image of a Bfree module P . But pure epimorphisms emanating from L—this is where B ⊆ L is needed—onto countably generated L-atomic modules split by [Rot3, Prop.2.5].  Lemma 6.5 (Eilenberg’s Trick). For every L-atomic module N ∈ PGen B there is a large B-free module P such that N ⊕ P ∼ = P. If N is, in addition, countably generated, then N ⊕ P ∗ ∼ = P ∗. Proof. By the previous lemma, we have P = N ⊕ M for some M . Now comes Eilenberg’s trick: as P (ω) ∼ = P we may rewrite P as (N ⊕M )(ω) ∼ = N ⊕(M ⊕N )(ω) ∼ = ∗  N ⊕ P . In case N is countably generated, P = P works. 4 I.e.,

containing only a set of distinct isomorphism types, in other words, B/∼ = is a set.

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Lemma 6.6 (The Separation Lemma). If M is an L-atomic module in PGen B, then M ⊕ P ∗ is L-pure separable by B. Proof. Consider a finite subset (a tuple) of M ⊕P ∗ and denote its projections onto M and P ∗ by m and p, respectively. As P ∗ is separable by modules from B, the tuple p is contained in a direct summand B ∈ B of P ∗ . By the basic properties of P ∗ , we can split it into P0∗ ⊕ B, where P0∗ ∼ = P ∗ , and then write M ⊕ P ∗ = M ⊕ P0∗ ⊕ B. It remains to L-pure-separate m in M ⊕ P0∗ ∼ = M ⊕ P ∗ by a module from B. By hypothesis, we can L-pure separate m in M by a countably generated Latomic module N ∈ PGen B, an L-pure submodule of M . Then N ⊕ P ∗ is an L-pure submodule of M ⊕ P ∗ containing m, and it therefore suffices to L-pure-separate m by a module from B in N ⊕ P ∗ . Now Eilenberg’s trick  shows us that N ⊕ P ∗ is isomorphic to P ∗ and thus (even) separable by B. Theorem 6.7. Suppose (L, B) is a separation pair. (1) Every L-atomic module in PGen B is a direct summand of a module that is (finitely) L-pure separable by B (and conversely). (2) Every strict L-atomic module in PGen B is a direct summand of a module that is locally separable by B (and conversely). Proof. The direction from left to right in (1) follows from the Separation Lemma. Namely, if M is L-atomic, M ⊕ P ∗ is L-pure separable by B. For (2), notice that then M ⊕ P ∗ is even strict L-atomic, hence even locally B-separable by Lemma 4.4. The converses follow from Lemma 4.3 (together with Lemma 2.5).  See Corollary 6.19 for the case of the classical separation pair (R-Mod, R-mod). Corollary 6.8. Suppose (L, B) is a separation pair with all modules in B finitely generated. Then every strict L-atomic module in PGen B is a direct summand of a (finitely) B-separable module (and conversely). Proof. Employ Lemma 4.4(2).



Remark 6.9. (a) One may be tempted to think that the following is true even without B ⊆ L: L-atomic modules in PGen B are direct summands of modules that are (finitely) L-pure separable by B. However, the entire series of lemmas leading to the theorem is based on the first one, Lemma 6.4, which does need it. (b) The converses of the theorem are true only within PGen B, for the separabilities in question only imply inclusion in L, not necessarily in PGen B. (c) Remember also, that PGen B ⊆ L, so the theorems are only about Latomic modules belonging to L (even PGen B, which may be less). 6.2. Finite separation and direct sum decomposition. Baer made exactly the following argument—in his special case.

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Lemma 6.10. [Baer’s Lemma] Let G be a class of modules closed under direct summand and such that every module in G is (finitely) B-separable. (1) Every module in G is countably locally separable by (countable) direct sums of modules from B. (2) Countably generated modules from G are direct sums of modules from B. Proof. Let C be a countable subset in G ∈ G. (1) List the set C as ci (i < ω). Set G0 = G and b0 = c0 . As G0 is B-separable, b0 is contained in a direct summand B0 ∈ B of G0 . Write G0 = B0 ⊕ G1 and c1 = a1 + b1 accordingly. Since G1 is again in G, hence B-separable, b1 is contained in a direct summand B1 ∈ B of G1 . Successively we decompose G into ⊕i k. Hence R = A ⊕ B, where A, B are ideals such that AR = e1 R ⊕ e2 R ⊕ ... ⊕ ek R. If e1 R is simple, A is a simple artinian ring. Suppose e1 R is not simple. If k = 1, then A = e1 R = Re1 , a local, right uniserial ring. Suppose k > 1 and e1 R is not simple. Since e1 J = 0, there exists some j such that e1 Jej = 0. We obtain a non-zero homomorphism σ : ej R → e1 J, which is not an isomorphism from ej R onto e1 R. This contradicts Lemma 15. Hence k = 1. and A is a right uniserial ring. As any semi-prime right Goldie ring is of finite right uniform dimension, the second part follows from the first.  4. Structure theorem for cai-rings In this section, we prove structure theorems for right noetherian, right cai-ring. Theorem 17. Let R be a local, right noetherian cai-ring. Then R is serial. Conversely, any noetherian, local serial ring is a cai-ring. Proof. Let R be a local, right noetherian, right cai-ring. By Proposition 13(i), R is right uniserial. Set J = J(R).Now R/J 2 is a finite length almost right self-injective module. Therefore R/J 2 is right self-injective, by the remark following Theorem 3. Thus R/J(R)2 is a QF -ring. Hence R/J 2 is also left self-injective. This proves that for any x ∈ J\J 2 , J/J 2 = Rx = xR. Hence J = xR + J 2 = Rx + J 2 , J = xR = Rx for any x ∈ J\J 2 . It follows that for a fixed x ∈ J\J 2 , given a c ∈ R, there exists a c ∈ R such that cx = xc and vice versa. In addition, if c is a unit and J = 0, then c is also a unit. We also get J n = xn R = Rxn for any n ≥ 1. Let B = ∩xn R. For any positive integer r, let Dr = {a ∈ R : xr a ∈ JB}. Then Dr contains B and Dr ⊆ Dr+1 . Since R is a right noetherian ring, there exists a positive integer m such that Dm = Dm+1 . Let z ∈ xm R ∩ B. Then z = xm u for some u ∈ R. We have xm+1 u ∈ JB, u ∈ Dm+1 = Dm . That gives z ∈ JB. Hence xm R ∩ B = JB, i.e. B = JB. Observe that if some y ∈ R\B, there exists

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a non-negative integer s such that y ∈ xs R\xs+1 R, so that y = xs v for some unit v ∈ R. As remarked above, we also have xs v = wxs , w a unit. Thus Ry = Rxs = xs R = yR. Thus R/B is both sided noetherian. For any positive integer r, define Lr = {a ∈ R : axr ∈ BJ}. Once again B ⊆ Lr ⊆ Lr+1 . By repeating the above arguments, we also get BJ = B. But B = bR. Then bJ = bR. Thus B = bR = 0. This proves that R is a noetherian serial ring. Conversely, let R be a noetherian serial ring and J = J(R). Now ∩n J n = 0. If for some m, J m = 0, then R is a local, artinian serial ring and therefore it is self-injective. Suppose J m = 0 for any positive integer m. Now J = xR = Rx, where x ∈ J\J 2 . Any non-zero element of R is of the form xm u for some unit u ∈ R and so R is a domain. It can be easily checked that R is a right cai-ring.  Lemma 18. Let e, f be two orthogonal indecomposable idempotents in a right noetherian right cai ring R such that eR is not of finite length. Then f Re = 0, that is, there is no nonzero homomorphism from eR into f R. Proof. As R is right serial, there exists an epimorphism σ : gR → eJ, where g is an indecomposable idempotent. Suppose gR eR. Then we can choose e and g to be orthogonal. By Proposition 12(ii), σ(gR) is simple, i.e. eJ is simple, which ∼ eJ is a contradiction. Hence gR ∼ = eR, eR eJ = eJ 2 . ∼ ∼ eJ Suppose f Re = 0 , and f R = eR. Then ff R J = eJ 2 . This gives an epimorphism η : f R → eJ and l(eR) = 2 as above. Hence f R eR. By Proposition 12(ii), ∼ eJ f ReR = soc(f R) ∼ = eR eJ = eJ 2 . We get an epimorphism μ : eJ → soc(eR). If f R is ∼ simple, we also get eR = f R, and eR is simple. This gives a contradiction. Thus f R is not simple. Suppose f R is not eR-injective. Then for some L < eR, there exists an isomorphism η : L → f R. This gives an embedding λ : f R → eR which is not onto. By Proposition 12(ii), f R is simple, which is a contradiction. This proves that f R is eR-injective. Thus, we get a homomorphism λ : eR → f R which extends μ. Then λ(eR)  soc(f R), which contradicts Proposition 12(ii). Hence f Re = 0.  Proposition 19. Let R be a right serial ring such that J 2 = 0. Suppose for any indecomposable idempotent e ∈ R with l(eR) = 2, eR is injective. Then R is left serial. Proof. Let e ∈ R be an indecomposble idempotent. We show that Re is uniserial. If Re is simple, there is nothing to prove. Suppose Re is not simple. As J 2 = 0, Je is completely reducible. Suppose Je is not simple. We get f1 xe, f2 ye ∈ Je such that Rf1 xe, Rf2 ye are minimal left ideals with zero intersection, where f1 , f2 are some indecomposable idempotents in R. Now f1 xeR ∼ = f2 yeR ∼ = eR/eJ. This gives that f1 R and f2 R are non-simple and have isomorphic socles. As f1 R, f2 R have length 2 and are injective, we get f1 R ∼ = f2 R. Therefore, Rf1 ∼ = Rf2 ye. Therefore, for f = Rf2 , which gives Rf1 xe ∼ = f1 , we get u, v ∈ J such that Rf ue, Rf ve are two disjoint minimal submodules of Re. We have isomorphism σ : f ueR → f veR, σ(f ue) = f ve. It extends to an isomorphism η : f R → f R. Now η(f ) = f af for some a ∈ R. Then f ve = f af f ue. As J 2 = 0, f af is a unit in f Rf . Therefore Rf ue = Rf ve, which gives a contradiction. Hence Re is uniserial. This proves that R is also left serial. 

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Theorem 20. A right artinian ring R is a right cai-ring if and only if it is finite direct sum of local serial rings and serial rings with radical square zero. Proof. Without any loss of generality, we can take R to be such that R has no non-trivial central idempotent. First suppose that R is a right cai-ring. By Proposition 13(i), R is right serial. Let e ∈ R be an indecomposable idempotent. If l(eR) > 2, by Lemma 14 e is a central idempotent and hence R = eR a local serial ring. So, we take R to be such that l(eR) ≤ 2. As any finite length, uniform almost self-injective module is quasi-injective, eR is quasi-injective, By Proposition 13(ii), eR is injective, whenever l(eR) = 2. Hence, by Theorem 17, R is a serial ring. The converse is immediate from Theorem 10.  Theorem 21. Let R be a right noetherian ring. Then all cyclic right R-modules are almost self-injective if and only if R is a finite direct sum of rings of the following types: (i) Local serial rings, 2 = 0,

(ii) Serial rings S with J(S) D M (iii) A Morita Context , where D is a local, noetherian, serial do0 S main, S is an indecomposable serial ring with J(S)2 = 0, and D MS is a bimodule such that MS is simple, D M is a torsion-free divisible module and End(MS ) is a classical quotient ring of D, and if e1 , e2 , .... , eu is a maximal orthogonal set of non-isomorphic indecomposable idempotents in S, then these can be so arranged that for J = J(S), the following conditions hold. (a) l(ei S) = 2 for i < u, (b) l(eu S) = 1, (c) MS ∼ = ee11 SJ , ∼ ei+1 S for i < u. (d) ei J = ei+1 J Proof. Let R be a right cai-ring. We may assume that R has no non-trivial central idempotent. Write 1 = f1 + f2 + ... + fn where (fi ), i = 1, ...n is a maximal set of orthogonal indecomposable idempotents. If n = 1, then R is local, and by Theorem 17, R is a local, serial ring. So, assume n ≥ 2. If for some i, fi R has finite length and l(fi R) > 2, then by Lemma 14, fi is a central idempotent, which is a contradiction. Therefore, for all i, either l(fi R) ≤ 2 or fi R is not of finite length. In case every fi R is of finite length, then by (3.4), R is a serial ring with J(R)2 = 0. Thus we now consider the case when there exists some fi R which is not of finite length. By re-indexing, we take f1 R to be of infinite length. By Lemma 18 fi Rf1 = 0 for all i > 1. This implies (1 − f1 )Rf1 = 0.If also f1 R(1 − f1 ) = 0, we obtain a non-trivial central idempotent, a contradiction to our assumption. So there exists an i > 1,say, i = 2 such that f1 Rf2 = 0. Therefore f2 R has finite length by Lemma 18, and by Proposition 14, l(f2 R) ≤ 2. By Proposition 12 (ii). we have soc(f1 R) ∼ = ff22R J . By re-indexing, we can find a positive integer w ≥ 2 such that f1 , f2 , ...., fw are pairwise non-isomorphic and any fi is isomorphic to one the fs , 1 ≤ s ≤ w. By re-indexing if need be, we find a maximal v such that fi Rfi+1 = 0, whenever i < v. Then v ≥ 2 and l(fi R) ≤ 2 for 2 ≤ i ≤ v. Then fi Rfi+1 = fi Jfi+1 = 0, gives l(fi R) = 2 and soc(fi R) = fi Rfi+1 R for i < v. As

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length of f1 R is infinite, by using (1.3), it follows that if an fj R has finite length, then f1 R is fj R-injective. We prove that l(fv R) = 1. Suppose l(fv R) = 2. Then there exists some fj , 1 ≤ j ≤ w,such that fv Jfj R = soc(fv R). By the choice of v, 1 ≤ j ≤ v, otherwise it will contradict the maximality of v. If j = v, then fv Jfv R = soc(fv R), soc(fv−1 R) ∼ = soc(fv R), fv−1 R ∼ = fv R, which is a contradiction. Hence j < v. By (3.2), fv Rf1 = 0. Therefore 1 < j < v. We get soc(fj−1 R) ∼ = soc(fv R), so fv R embeds in fj−1 R. Suppose j = 2. As f1 R is fv R-injective, fv R embeds in f1 R, which contradicts (2.4)(i). Therefore j > 2, and fj−1 , fv are isomorphic. This also a contradiction. Hence l(fv R) = 1. It also follows that for any 1 ≤ j < k ≤ v, fk Rfj = 0. Set M = soc(f1 R). As soc(f1 R) ∼ = f2 R/f2 J, we have M fi = 0 for 2 < i ≤ w, M = f1 Rf2 R = f1 R(1−f1 )R. We now prove that v = w. Consider any j > v. Suppose for some k ≤ v, fj Rfk = 0. Thus l(fj R) > 1, k > 1 and soc(fj R) ∼ = soc(fk−1 R). As l(fk−1 R) = 2, we get an embedding σ : fk−1 R → fj R, which contedicts (2.4)(i). Hence fj Rfk = 0 for k ≤ v. Suppose fk Rfj = 0 for some k ≤ v. Then k > 1. If k < v, f R f R ∼ fj R then soc(fk R) ∼ = fjj R , fk+1 = fj R , which is a contradiction, as k + 1 < j Thus k+1 R k = v. But fv R is simple, therefore fv Rfj = 0, gives fv R ∼ = fj R, which is also a contradiction. Hence we also have fj Rfk = 0. Let e be the sum of those fi each of which is isomorphic to an fk for some k ≤ v, and f be the sum of all other fj , it follows that eRf = 0 = f Re and 1 = e + f . Thus e is a central idempotent. Therefore e = 1 and f = 0. This proves that v = w. It also follows that there is no other fj R which is not of finite length. Set S = (1 − f1 )R(1 − f1 ). Since (1 − f1 )Rf1 = 0, S = (1 − f1 )R and M = f1 R(1 − f1 )R = f1 R(1 − f1 ). Now A = R(1 − f1 ) is an ideal of R and as abelian group, R = f1 Rf1 ⊕ R(1 − f1 ). Set D = f1 Rf1 , then D ∼ =R A and it is a local, right cai-ring, Hence, by Theorem 17, D is a local serial ring. Now B = f1 R is an ideal R .Therefore, any right S-module is an R-module. For 2 ≤ i < w, fi S = and S ∼ = B fi R, as an S-module, is uniserial and has length 2. Furthermore, fw S = fw R is simple. For any i, j, fi S ∼ = fj S if and only if fi R ∼ = fj R. Hence S is a right serial 2 ring with J(S) = 0. Since S is also a right cai-ring, it follows that each fi S is quasi-injective. By Theorem 17, S is serial. If we set ei = fi+1 for 1 ≤ i < w − 1, u = w − 1, it follows that e1 , e2 , .... , eu satisfy conditions (a), (b), (c) and (d). We now prove that D is a domain and D M is a torsion-free divisible D-module. Suppose for some 0 = r ∈ f1 Rf1 , 0 = x ∈ M , rx = 0. As MS is simple, we get rM = 0. Now rR ∩ M = 0, as f1 R is uniserial. Therefore M ⊆ rR. Let L = {f1 y ∈ f1 R : rf1 y ∈ M }. Then M = rL and M < L. Now L being local, is a homomorphic image of some fj R, but f1 R(1 − f1 ) ⊆ soc(f1 R), therefore L ∼ ∼ f1 R ∼ is a homomorphic image of f1 R. We get ff22R J = M = f1 J , f1 = f2 , which is a contradiction. Hence rx = 0 for any 0 = r ∈ f1 Rf1 , 0 = x ∈ M . This proves that D = f1 Rf1 is a domain and D M is torsion-free and D ⊆ End(MS ). Let 0 = σ ∈ End(MS ). Suppose σ extends to an endomorphism of f1 R. If σ(f1 ) = r, then σ ∈ D. Otherwise, by the comment after (1.3), σ is monic, it has a maximal extension η : L → e1 R, L < f1 R and η(L) = f1 R. We get an r ∈ D such that L = rR, η(r) = f1 . We see that σ = r −1 . It follows that for any σ ∈ End(MS ), either σ ∈ D or σ −1 ∈ D, which proves that End(MS ) is the classical quotient ring of D and D is a local serial ring. In addition, D M is a divisible module.

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Conversely, let the given conditions hold. In view of Theorem 17, and Theorem 20, we only need to consider a right noetherian ring R satisfying (iii). Step 1. By hypothesis, there exist two orthogonal idempotents e, f such that 1 = e + f , D = eRe is a local noetherian serial domain, S = f Rf = (1 − e)R(1 − e) = f R is a serial ring with J(S)2 = 0, f Re = 0, M = eRf is a (D, S)-module such that MS is simple, D M is a torsion-free divisible module. For any non-zero ere, ereM = M . Let z ∈ eR\M . Then z = ere + esf for some r, s ∈ R with ere = 0. As M 2 = 0, we get zM = ereM = M ; which gives that any non-zero right ideal contained in eR contains M . Because eRe is uniserial and eRe ∼ = eR M , we also get that eR is uniserial. Observe that any right ideal of S is a right ideal of R. Any non-zero right ideal contained in eR,other than M , is isomorphic to eR. Therefore, given any indecomposable idempotent g ∈ S for which l(gS) = 2, gS cannot be embeded in eR. By the hypothesis, there exists a maximal set of orthogonal, nonisomorphic, indecomposable idempotents e1 , e2 , .... , eu that can be so arranged ei+1 S 1S that MS ∼ , soc(ei S) ∼ = e1eJ(S) = ei+1 J(S) , for 1 ≤ i < u, and eu S is simple. The hypothesis gives that if an indecomposable idempotent g ∈ S is not isomorphic to e1 , then no composition factor of gS is isomorphic to MS . This implies that eR is trivially gS-injective. In case u > 1,we have soc(e1 S) MS , so eR is e1 Rinjective. In the case u = 1, e1 S is simple by the condition (b) . So once again eR is e1 R-injective. This proves that for any indecomposable idempotent g ∈ S, eR is gR-injective. Let A be any uniserial S-module. Then A is a homomorphic image of some uniserial summand of SS , therefore eR is A-injective . Let H be any finitely generated right S-module, naturally, it is an R-module. As H is a finite direct sum of uniserial S-modules, eR is H-injective. Observing, HomR (eR, f R) = 0, and any right ideal A, M < A < eR is isomorphic to eR, there does not exist a non-zero homomorphism from A into a finitely generated S-module. Further, if u > 1. Then MS cannot be embedded in SS . This proves that f R is eR-injective, whenever u > 1. If u = 1, then S is simple artinian. In this case, MS embeds in S and f R is semisimple, which gives that f R is almost eR -injective. Thus in all cases f R is almost eR-injective. Let A be any uniserial S-module. If l(A) = 2, A is a summand of f R.Therefore A is almost eR-injective. Suppose l(A) = 1 and A be not isomorphic to a summand of SS , In this case Hom(A, eR) = 0 or A ∼ = MS , using this we get A is almost eR-injective. Consider any finitely generated S-module H. Then H is a finite direct sum of uniserial modules say Aj , i ≤ j ≤ m. If we consider the family {eR, Aj }, we see that condition in Theorem 6 is satisfied. Hence eR ⊕ H is almost self-injective. For any proper homomorphic image W of eR, one sees that Hom(W, H) = 0 = Hom(H, W ). Using this, it follows from Theorem 6 that W ⊕ H is almost self-injective. Step 2. Let A be a right ideal of R containing M . Now Theorems 17 and 20 give D ⊕ S is a cai ring. As R A is a cyclic D ⊕ S-module, it is almost self-injective. Step 3. Let A be a right ideal of R such that eR ∩ A = 0. If A ⊆ S, then R A being of the form eR ⊕ H, as in Step1, is almost self-injective. Suppose A  S. We f R+A eR+A eR+A ∼ ∼ have R = eR. Therefore, R A = A + A . Now A A = X + Y , where X = eR, f R+A and Y = A is a finitely generated S-module. As in Step 1, X is Y -injective. If X ∩ Y = 0, again as in Step 1, R A is almost self-injective. Suppose X ∩ Y = 0. Then X ∩ Y is simple. Any uniserial submodule B of Y of length 2 does not embed in X, so it cannot contain X ∩ Y . Since Y is a direct sum of uniserial modules,

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we obtain Y = X ∩ Y ⊕ Y1 for some Y1 ⊆ Y . Then almost self-injective. This completes the proof.

R A

= X ⊕ Y1 . By Step1,

339 R A

is 

Acknowledgment The author is extremely thankful to his teacher Prof. S. K. Jain for suggesting the problem and for his help and contributions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR0417223 Adel Alahmadi and S. K. Jain, A note on almost injective modules, Math. J. Okayama Univ. 51 (2009), 101–109. MR2482408 Yoshitomo Baba, Note on almost M -injectives, Osaka J. Math. 26 (1989), no. 3, 687–698. MR1021440 Carl Faith, When are proper cyclics injective?, Pacific J. Math. 45 (1973), 97–112. MR320069 Carl Faith, Algebra. II, Ring theory, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 191. MR0427349 Manabu Harada, Direct sums of almost relative injective modules, Osaka J. Math. 28 (1991), no. 3, 751–758. MR1144484 Manabu Harada, Almost projective modules, J. Algebra 159 (1993), no. 1, 150–157, DOI 10.1006/jabr.1993.1151. MR1231208 Manabu Harada and Anri Tozaki, Almost M -projectives and Nakayama rings, J. Algebra 122 (1989), no. 2, 447–474, DOI 10.1016/0021-8693(89)90229-9. MR999086 S. K. Jain, Ashish K. Srivastava, and Askar A. Tuganbaev, Cyclic modules and the structure of rings, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2012. MR3087146 A. Koehler, Rings with quasi-injective cyclic modules, Quart. J. Math. Oxford Ser. (2) 25 (1974), 51–55, DOI 10.1093/qmath/25.1.51. MR354778 Lawrence S. Levy, Commutative rings whose homomorphic images are self-injective, Pacific J. Math. 18 (1966), 149–153. MR194453 B. L. Osofsky, Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964), 645–650. MR161886 B. L. Osofsky, Noninjective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), 1383–1384, DOI 10.2307/2036217. MR231857 Surjeet Singh, Almost relative injective modules, Osaka J. Math. 53 (2016), no. 2, 425–438. MR3492807 R. B. Warfield Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), no. 2, 187–222, DOI 10.1016/0021-8693(75)90074-5. MR401836 House No. 424, Sector 35A, Chandigarh-160022, India Email address: [email protected]

Contemporary Mathematics Volume 751, 2020 https://doi.org/10.1090/conm/751/15116

On isoclasses of maximal subalgebras determined by automorphisms Alex Sistko

Abstract. Let k be an algebraically-closed field, and let B = kQ/I be a basic, finite-dimensional associative k-algebra with n := dimk B < ∞. Previous work shows that the collection of maximal subalgebras of B carries the structure of a projective variety, denoted by msa(Q), which only depends on the underlying quiver Q of B. The automorphism group Autk (B) acts regularly on msa(Q). Since msa(Q) does not depend on the admissible ideal I, it is not necessarily easy to tell when two points of msa(Q) actually correspond to isomorphic subalgebras of B. One way to gain insight into this problem is to study Autk (B)-orbits of msa(Q), and attempt to understand how isoclasses of maximal subalgebras decompose as unions of Autk (B)-orbits. This paper investigates the problem for B = kQ, where Q is a type A Dynkin quiver. We show that for such B, two maximal subalgebras with connected Ext quivers are isomorphic if and only if they lie in the same Autk (B)-orbit of msa(Q).

1. Introduction Let B be a finite-dimensional, unital, associative algebra over an algebraicallyclosed field k. Then the celebrated Wedderburn-Malcev Theorem states that there exists a k-subalgebra B0 ⊂ B such that B0 ∼ = B/J(B) and B = B0 ⊕ J(B), where J(B) denotes the Jacobson radical of B. Furthermore, for any subalgebra B0 ⊂ B isomorphic to B0 , there exists a x ∈ J(B) such that (1 + x)B0 (1 + x)−1 = B0 . For more details, see for instance [2] or Theorem 11.6 of [7]. Of course, the collection of all k-algebra automorphisms of B, which we denote by Autk (B), acts on the set of subalgebras of B. For any x ∈ J(B), the map y → (1 + x)y(1 + x)−1 is an automorphism of B. So another way to state the second half of the WedderburnMalcev Theorem is to say that the isoclass of B0 in B, i.e. the set of all subalgebras of B isomorphic to B0 , is a single Autk (B)-orbit. Unsurprisingly, this statement is false for general subalgebras A of B. Nevertheless, recent investigations into maximal subalgebras of finite-dimensional algebras suggest that examples of such A are not necessarily rare [6]. It is therefore natural to ask what conditions we can impose on A to ensure that its isoclass in B is an orbit of Autk (B). More generally, one can ask whether there is any way to 2010 Mathematics Subject Classification. Primary 16S99; Secondary 16W20, 16Z99, 05E15, 05C60. Key words and phrases. Inite-dimensional algebra, maximal subalgebra, subalgebra variety, automorphism group, presentation, isoclass, type A, Dynkin quiver. c 2020 American Mathematical Society

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classify the Autk (B)-orbits of subalgebras of B, and relate them to isoclasses of subalgebras. This is one source of inspiration for the current paper. Another source of inspiration comes from the study of varieties of subalgebras, as the author has done recently in [9]. For any 1 ≤ m ≤ dimk B, the collection of all m-dimensional subalgebras of B carries the structure of a projective k-variety, which we call AlgGrm (B). The linear algebraic group Autk (B) acts regularly on this variety. Neither m or B are enough to specify AlgGrm (B) up to equivalence of varieties: in fact, if B = kQ/I is a basic algebra and m = dimk B − 1, then AlgGrm (B) only depends on Q. So it will be difficult in general to choose an admissible ideal I of kQ, and determine whether two points of AlgGrdimk kQ/I−1 (kQ/I) actually represent isomorphic subalgebras of kQ/I. Thankfully, the automorphism group Autk (kQ/I) is sensitive to the data contained in I. So, provided that one can impose reasonable conditions on the relationship between orbits and isoclasses, one can expect that orbits under this group action will yield significant information on isoclasses of subalgebras. In [9] we discuss one possible version of “reasonable conditions,” where the variety is a finite union of orbits. The purpose of this paper is to carry out this program as far as possible for a suitable “test class” of algebras. For us, these will be path algebras of type A Dynkin quivers and their maximal subalgebras. As it turns out, many maximal subalgebras of such algebras will have isoclasses that are single Autk (B)-orbits. However, we will show that even for such a nicely-behaved class, isoclasses differ from orbits in at least some circumstances. This paper is organized as follows. In Section 2 we review the basic notions associated to path algebras and their automorphisms. We also discuss the major results from [6], [9] which will be used to prove our main result. In Section 3, we discuss the problem of presenting maximal subalgebras of basic algebras. In particular, Propositions 3.1 and 3.3 provide explicit presentations for maximal subalgebras of hereditary algebras. The results of this section will be used in Section 4, where we prove the main result of this article: Theorem 1.1. Let k be an algebraically closed field, Q a type A Dynkin quiver, and B = kQ. Suppose that A, A ∈ msa(Q) have connected Ext quivers. Then A and A lie in the same Autk (B)-orbit if and only if A ∼ = A as k-algebras. This is essentially done by showing that the Ext quivers of A and A are nonisomorphic whenever they lie in different Autk (B)-orbits. We note that this theorem can be rephrased as follows: if the underlying graph of Q is an oriented tree with maximum degree 2 and B = kQ, then the isoclass of a connected maximal subalgebra of B coincides with its Autk (B)-orbit. The author does not currently know whether similar statements hold for all trees with maximum degree 3 or higher. We end on an example which shows that if the Ext quiver of A is not connected, then its isoclass can differ from its Autk (B)-orbit. 2. Background Unless otherwise stated, k will denote an algebraically-closed field. All algebras are unital, associative, finite-dimensional k-algebras, and our terminology essentially comes from [1]. Let Q be a finite quiver with vertex set Q0 , arrow set Q1 , and source (resp. target) function s (resp. t) : Q1 → Q0 . The underlying graph of Q is obtained by forgetting the orientations on the arrows. Let kQ denote

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the path algebra of Q, and let J(Q) denote the two-sided ideal in kQ generated by Q1 . For n ≥ 2, we let Tn Q := kQ/J(Q)n denote the nth truncated path algebra associated to Q. By a slight abuse of notation, for any u, v ∈ Q0 we let uQ1 v denote the set of arrows in Q with source u and target v, and we let ukQ1 v denote their k-span inside kQ. Note that if uQ1 v = ∅, then ukQ1 v = {0} and GL(ukQ1 v) is the trivial group. Similar to [4], we define V 2 (Q) = {(u, v) ∈ Q0 × Q0 | uQ1 v = ∅}. A basic algebra is an algebra of the form B = kQ/I, where I is an admissible ideal of kQ, i.e. an ideal satisfying J(Q)2 ⊃ I ⊃ J(Q) for some " ≥ 2. Note that B = kQ0 ⊕ J(B) = kQ0 ⊕ J(Q)/I, and that kQ0 ∼ = B/J(B) ∼ = k|Q0 | . We let Autk (B) denote the group of all k-algebra automorphisms of B. It is a Zariski-closed subgroup of GL(B), and hence a linear affine algebraic group. Our notation for subgroups of Autk (B) is borrowed from the notation in [8], [3], [4]. If G is a subgroup of Autk (B), we say that two subalgebras A and A are G-conjugate (in B) if there exists a φ ∈ G such that A = φ(A). For a unit u ∈ B × , we let ιu denote the corresponding inner automorphism, i.e. the map ιu (x) = uxu−1 for all x ∈ B. We let Inn(B) denote the group of all inner automorphisms, and Inn∗ (B) = {ι1+x | x ∈ J(B)} denote the group of unipotent inner automorphisms. If B = kQ/I is ˆ B = {φ ∈ Autk (B) | φ(Q0 ) = Q0 } and HB = {φ ∈ Autk (B) | φ |Q = basic, we let H 0 idQ0 }. By Theorem 10.3.6 of [5], Inn(B) acts transitively on complete collections of primitive orthogonal idempotents. Since inner automorphisms induced by units  of the form v∈Q0 λv v (where λv ∈ k× for each v) fix vertices, Inn∗ (B) is also ˆB = transitive on this set and we have a decomposition Autk (B) = Inn∗ (B) · H ∗ ˆ B ·Inn (B). If the underlying graph of Q is a tree, then Aut(Q) can be considered a H subgroup of Autk (Tn Q) for any n, and it is easy to see that we have a decomposition ˆ T Q = Aut(Q) · HT Q = HT Q · Aut(Q). Since J(Q)n = 0 for large powers n, this H n n n statement includes the fact that Aut(Q) is a subgroup of Autk (kQ). In Theorem 4.1 of [6], the author and M. C. Iovanov proved the following classification Theorem for maximal subalgebras of basic algebras: Theorem 2.1. Let B = kQ/I be a basic algebra over an algebraically-closed field k. Let A ⊂ B be a maximal subalgebra. Consider the following two classes of maximal subalgebras of B: For a two-element subset {u, v} ⊂ Q0 , we define ⎛ ⎞ ' A(u + v) := k(u + v) ⊕ ⎝ kw ⎠ ⊕ J(B). w∈Q0 \{u,v}

For an element (u, v) ∈ V (Q) and a codimension-1 subspace U ≤ ukQ1 v, we define ⎛ ⎞ ' A(u, v, U ) := kQ0 ⊕ U ⊕ ⎝ wkQ1 y ⎠ ⊕ J(B)2 . 2

(w,y)∈Q20 \{(u,v)}

Then there exists a unipotent inner automorphism ι1+x ∈ Inn∗ (B) such that either ι1+x (A) = A(u + v) or ι1+x (A) = A(u, v, U ), for some appropriate choice of u, v, and possibly U . As in [6], if A is Inn∗ (B)-conjugate to a subalgebra of the form A(u + v), then we say that A is of separable type. If A is Inn∗ (B)-conjugate to an algebra of the form A(u, v, U ), then we say that A is of split type. As an immediate consequence of Theorem 2.1, all maximal subalgebras of a basic algebra are basic, have codimension

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1, and contain the radical square. In fact, we have the following easy corollary, which first appeared in [9]: Corollary 2.2. Let B be a basic k-algebra of dimension n, and let A ⊂ B be a subalgebra. Then A satisfies the following: (1) A is also a basic algebra. (2) If A is a maximal subalgebra, then dimk A = n − 1. (3) If A is a maximal subalgebra, then J(A) is a B-subbimodule of J(B), J(A) = A ∩ J(B), and J(B)2 ⊂ J(A). (4) More generally, if m = dimk A, then J(B)2(n−m) ⊂ A. If B is any k-algebra and m is a positive integer 1 ≤ m ≤ dimk B, then the collection AlgGrm (B) of all m-dimensional subalgebras of B is a Zariski-closed subset of the usual Grassmannian Grm (B). In particular, it is a projective variety over k. For any A ∈ AlgGrm (B), we let Iso(A, B) denote the set of all A ∈ AlgGrm (B) such that A ∼ = A as k-algebras. Clearly Iso(A, B) is Autk (B)-invariant, and hence a union of Autk (B)-orbits. Suppose that B = kQ/I is a basic algebra of dimension n. By the remarks above, it follows that AlgGrn−1 (B) is the variety of maximal subalgebras of B, and that there is a (biregular) bijection between maximal subalgebras of B and maximal subalgebras of B/J(B)2 ∼ = T2 Q. In other words, AlgGrn−1 (B) only depends on the underlying quiver Q, and so we define msa(Q) := AlgGrn−1 (B). We can think of msa(Q) as the variety of maximal subalgebras of any basic algebra with Ext quiver Q. See [9] for more details. Suppose that the underlying graph of Q is a tree and B = kQ. Theorem 2.1 classifies Inn∗ (B)-orbits of msa(Q), and it is easy to see that for any such B, φ(A) = A for all φ ∈ HB . So classification of Autk (B)-orbits boils down to determining which Inn∗ (B)-orbits of msa(Q) are related by elements of Aut(Q). More specifically, it is equivalent to classifying Aut(Q)-orbits on the finite sets V 2 (Q) (for split type) and {{u, v} ⊂ Q0 | u = v} (for separable type). Although this may represent an intractable problem for general Q, it at least implies that every B-isoclass of msa(Q) is a finite union of Autk (B)-orbits. In Section 4 we will show that if Q is a type A Dynkin quiver, then each B-isoclass consisting of connected algebras is a single Autk (B)-orbit. The first step will be to find presentations for each maximal subalgebra as a bound quiver algebra, which we do below. 3. Presentations of Maximal Subalgebras Corollary 2.2 (1) states that if B is basic, then all of its maximal subalgebras are also basic. In particular, they can be presented as bound quiver algebras. Ideally, one hopes for explicit presentations of maximal subalgebras in terms of a given presentation for B. More specifically, if B is given as B = kQ/I and A is a maximal subalgebra of B, one would like a combinatorial procedure to obtain the Ext quiver of A, call it Γ, from Q, and another procedure to find generators for the kernel of the projection map kΓ → A. As it currently stands, if I ⊂ kQ is an arbitrary admissible ideal, and A ⊂ kQ/I is a maximal subalgebra of split type, then it is not clear to the author how one can explicitly reconstruct Γ from Q. Nevertheless, Theorem 2.1 provides us with some insight into the presentation problem. In fact, it is good enough to give us a full description for separable type

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subalgebras, as well as explicit presentations for all maximal subalgebras in the hereditary case, i.e. I = {0}. We start by describing presentations for maximal subalgebras of separable type. Take a 2-element subset {u, v} ⊂ Q0 , and let Γ be the quiver obtained from Q by gluing u and v together. More explicitly, Γ has vertex set Q0 \ {u, v} ∪ {u + v}, and for all w, y ∈ Q0 \ {u, v} we have (u + v)Γ1 y = uQ1 y ∪ vQ1 y, wΓ1 (u + v) = wQ1 u ∪ wQ1 v, (u + v)Γ1 (u + v) = uQ1 u ∪ uQ1 v ∪ vQ1 u ∪ vQ1 v. In other words, Γ1 is just a re-partitioning of Q1 into arrows with possibly new endpoints. This induces a bijective map φ : Q1 → Γ1 . Hence, if p = α1 · · · αd is a path in Q, then φ(p) := φ(α1 ) · · · φ(αd ) is a well-defined path in Γ. We can extend this to an algebra map φ : kQ → kΓ by defining φ(w) = w for all w ∈ Q0 \ {u, v}, φ(u) = φ(v) = u + v, and extending to k-linear combinations of arbitrary paths. Proposition 3.1. Let B = kQ/I, and A a maximal subalgebra of separable type. Suppose that A is Inn∗ (B)-conjugate to A(u+v), for some two-element subset {u, v} ⊂ Q0 . Then A ∼ = kΓ/I  , where (1) Γ is obtained from Q by gluing vertices u and v. (2) I  is generated by relations in φ(I), along with elements of the form φ(α)φ(β), where either α ∈ Q1 u and β ∈ vQ1 , or α ∈ Q1 v and β ∈ uQ1 . Proof. There is a map kΓ → A(u + v) which acts as the identity on Q0 \ {u, v}, sends u + v ∈ Γ0 to u + v ∈ A(u + v), and which acts on Γ1 via the bijection  Γ1 ↔ Q1 . The kernel of this map is precisely the admissible ideal I  . Proposition 3.2. Let B = kQ/I, and A a maximal subalgebra of split type. Suppose that A is Inn∗ (B)-conjugate to A(u, v, U ), for some (u, v) ∈ V 2 (Q) and codimension-1 subspace U ≤ ukQ1 v. Write A ∼ = kΓ/I  for a certain quiver Γ and  admissible ideal I . Then Γ0 = Q0 , and for all w, x ∈ Γ0 ,       dimk w J(A)/J(A)2 x = dimk w J(A)/J(B)2 x + dimk w J(B)2 /J(A)2 x. In particular:

  (1) For all w = u and x = v, there are w J(B)/J(B)2 x arrows from w to x in Γ,   2 v − 1 arrows from u to v, and (2) There are dimk u J(B)/J(B)    (3) There are at least dimk u J(B)/J(B)2 x (resp. dimk w J(B)/J(B)2 v) arrows from u to x (resp. from w to v).

Furthermore, J(B)4 ⊂ J(A)2 , so that any arrows in Γ that do not appear as arrows in Q arise from elements of J(B)2 or J(B)3 .

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Proof. A(u, v, U )/J(A(u, v, U )) = kQ0 implies that Γ0 = Q0 . The  dimension formula follows from the kQ0 -bimodule isomorphism J(A)/J(A)2 / J(B)2 /J(A)2 ∼ = J(A)/J(B)2 , and claims (1)-(3) follow from the formula ⎛ J(A) = J(B) ∩ A = U ⊕ ⎝

⎞

'

wkQ1 x⎠ ⊕ J(B)2 .

(w,x)=(u,v)



The final claim follows from Corollary 2.2 (3).

Although this corollary does not give us an explicit form for I  , we can use it to present maximal subalgebras of split type in the hereditary case, i.e. when I = {0}. Proposition 3.3. Let B = kQ for an acyclic quiver Q, and A ⊂ B a maximal subalgebra conjugate to A(u, v, U ), for some (u, v) ∈ V 2 (Q). Write A ∼ = kΓ/I  ,  for a finite quiver Γ and admissible ideal I ⊂ kΓ. Then Γ0 = Q0 , and Γ1 can be obtained from Q as follows: (1) Replace the |uQ1 v| arrows from u to v in Q with |uQ1 v|−1 arrows, indexed by a fixed basis {α1 , . . . , αd } of U ; (2) For each arrow γ with target u, add an arrow γ : s(γ) → v; (3) For each arrow γ with source v, add an arrow γ : u → t(γ). Furthermore, I  can be taken to be the ideal generated by the relations βγ − βγ, for all arrows β and γ in Q with t(β) = u and s(γ) = v. Proof. We may assume without loss of generality that A = A(u, v, U ). Find αd+1 ∈ ukQ1 v such that U ⊕ kαd+1 = ukQ1 v. Then for each γ ∈ Q1 with t(γ) = u, γαd+1 ∈ J(A) \ J(A)2 . If w = s(γ), then clearly the paths of the form γαd+1 , along with the arrows in Q1 from w to v, form a basis for w(J(A)/J(A)2 )v. Define γ = γαd+1 . A similar argument exhibits a basis for u(J(A)/J(A)2)x, and allows us to define γ = αd+1 γ if γ is an arrow in Q from v to x. The form for Γ then follows from Proposition 3.2. For all arrows β, γ ∈ Q1 with t(β) = u and s(γ) = v, βγ = βαd+1 γ = βγ. Hence, βγ − βγ is in the kernel of the projection map kΓ → A. If I  is the ideal generated by these commutation relations, then it is straightforward to check that the induced k-algebra projection kΓ/I  → A has an inverse, so that the desired isomorphism holds.  Example 3.4. Let B = kQ, where

Q=

β

α v1

v2

γ v3

v4

is an equioriented Dykin quiver of type A4 . Then any maximal subalgebra of separable type must be Inn∗ (B)-conjugate to one of the six bound quiver algebras displayed below.

ISOCLASSES DDTERMINED BY AUTOMORPHISMS

kΓ/I

Γ

347

I

α β

A(v1 + v2 )

γ α

γ

A(v1 + v3 )

(α2 ) (βα)

β β

α A(v1 + v4 )

(γα)

γ β γ

α

A(v2 + v3 )

(β 2 )

β α

A(v2 + v4 )

(γβ) γ γ β

α

A(v3 + v4 )

(γ 2 )

Any maximal subalgebra of split type must be Inn∗ (B)-conjugate to one of the three bound quiver algebras displayed below. kΓ/I

Γ

I

v2 β v3

A(v1 , v2 , {0})

γ

v4

{0}

β v1 v3 γ

α A(v2 , v3 , {0})

v1

v4 α

(αγ − αγ)

γ v2 β

A(v3 , v4 .{0})

v1

α

v3

v2

{0} β

v4

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Note: If I = {0}, then I ⊂ J(Q)2 ⊂ A allows us to consider relations in I as “generalized relations” in Γ. We say “generalized” because these elements may not actually lie in J(A)2 . In other words, we can always realize A as a generally non-admissible quotient of kΓ/I. This level of detail is sufficient for the purposes of this paper. Example 3.5. Consider B = kQ/I, where Q=

β1

α v1

v2

γ v3

β2

v4

and I = (αβ1 − αβ2 ). Then the maximal subalgebra of kQ corresponding to the triple (v2 , v3 , kβ1 ) can be presented as A = kΓ/I  , where v3 γ

α Γ = v1

v4

β2 α

γ v2



and I = (αγ − αγ). Therefore, A(v2 , v3 , kβ1 ) = A/I = A/(α − αβ2 ). It follows that α is in the radical square of A(v2 , v3 , kβ1 ). Hence, to present A(v2 , v3 , kβ1 ), we must actually bound the quiver v3 γ v1

v4

β2 γ

α v2 by the relation αβ2 γ − αγ.

4. Type A Path Algebras In this section we prove theorem 1.1. Unless otherwise stated, let Q be a type-A Dynkin quiver on n vertices, in other words a quiver whose underlying graph is of the form: ··· Figure 1. A type A Dynkin diagram. Let B = kQ. If n = 2m is even, label the vertices of Figure 1 from left to right as v−m , v−m+1 ,. . ., v−1 , v1 ,. . ., vm−1 , vm . If n = 2m + 1 is odd, label the vertices v−m ,. . ., v−1 , v0 , v1 ,. . ., vm in an analogous manner. Whenever it is understood that we are talking about vertices, we may abbreviate “vi ” as “i.” We call v−i the predecessor of v−i+1 , and v−i+1 the successor of v−i . Note that if n is even, v−1 is

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the predecessor of v1 . If we need to talk about the predecessor (resp. successor) of a vertex v, we simply denote it by pred(v) (resp. succ(v)). We recursively define predi (v) and succi (v) as follow: pred1 (v) = pred(v) and succ1 (v) = succ1 (v). If predi (v) and succ(v) have already been defined, then predi+1 (v) is the predecessor of predi (v), and succi+1 (v) is the successor to succi (v), whenever these vertices are defined. Let w denote the binary word of length n − 1 such that w(i) = +1 if the edge from vi to its successor starts at vi , and w(i) = −1 if the edge starts at the successor to vi . We treat w as a function {−m, . . . , m − 1} → {−1, +1}, or as an ordered (m − 1)-tuple w = w−m w−m+1 · · · wm−1 , where each wi ∈ {±1}. Let αi denote the edge between vi and its successor. To ease notation slightly, we use the following shorthand for maximal subalgebras of kQ: Ai := A(vi , succ(vi ), {0}) for all i < m, Ai,j := A(vi + vj ), for all vi , vj ∈ Q0 with i = j. Since any automorphism of Q induces an automorphism of its underlying Dynkin diagram, Aut(Q) ≤ C2 , the cyclic group of order 2. Define w ∗ to be the binary word of the quiver obtained from Q by applying the unique non-identity automorphism of the underlying Dynkin graph to Q. In other words, w∗ (i) = −w(pred(−i)) for all −m ≤ i ≤ m − 1. Clearly, w∗∗ = w and Aut(Q) = C2 if and only if w∗ = w. The following lemma will be used extensively throughout our proofs: Lemma 4.1. Let i and j be integers with −m ≤ i, j ≤ m−1. Suppose that Ai has a connected Ext quiver. If Ai ∼ = Aj , then there exists an isomorphism ψ : Ai → Aj such that ψ(Q0 ) = Q0 . If ψ(") = −" for all " ∈ Q0 , then Aut(Q) = C2 . Proof. Let ψ  : Ai → Aj be any k-algebra isomorphism. Since Q0 is a complete set of primitive orthogonal idempotents for both Ai and Aj , ψ  (Q0 ) is a complete set of primitive orthogonal idempotents for Aj . Since Inn(Aj ) acts transitively on such sets, there exists a ιu ∈ Inn(Aj ) such that ιu ψ  (Q0 ) = ψ(Q0 ). Setting ψ = ιu ◦ ψ  demonstrates the first claim. For the second, suppose that ψ(") = −" for all " ∈ Q0 . Then for all " = i, Proposition 3.3 implies dimk "J(B)/J(B)2 succ(") = dimk "J(Ai )/J(Ai )2 succ(") = dimk (−")J(Aj )/J(Aj )2 pred(−"). But w(") = +1 if and only if dimk "J(B)/J(B)2 succ(") = 1. But Q is a tree: since −" is adjacent to pred(−") in Q and J(Aj ) ⊂ J(B), the equality dimk (−")J(Aj )/J(Aj )2 pred(−") = 1 implies pred(−")J(B)/J(B)2 (−") = 0 and w(pred(−")) = −1. In other words, for all " = i we have w(") = −w(pred(−")). Suppose by way of contradiction that w(i) = w(pred(−i)). Without loss of generality we may assume w(i) = +1. Then Proposition 3.3 and the assumption that Ai has a connected Ext quiver together imply that at least one of the two conditions must hold: (1) pred(i) exists and w(pred(i)) = +1, or (2) succ2 (i) exists and w(succ2 (i)) = +1. Suppose that the first condition holds. Then by Corollary 2.2 (3), αpred(i) αi ∈ J(Ai ) and the vector space pred(i)J(Ai ) succ(i) is non-zero. Therefore, the vector space succ(−i)J(Aj ) pred(−i) is non-zero as well. But note that since Q is a type A Dynkin quiver, succ(−i)J(Aj ) pred(−i) = succ(i)J(B)(−i)J(B) pred(−i). But then in particular (−i)J(B) pred(−i) = 0, which holds if and only if we have pred(−i)J(B)(−i) = 0. In turn, this holds if and only if w(pred(−i)) = −1, a contradiction. So we must have w(i) = −w(pred(−i)). If the second condition

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holds, a similar argument allows us to conclude again that w(i) = −w(pred(−i)).  But then we must have w ∗ = w and hence Aut(Q) = C2 . To prove the main theorem, we prove it separately for maximal subalgebras of split type and separable type. For split type, we use Proposition 3.3 to distinguish three essential cases, depending on the form of the Ext quiver of Ai . If wpred(i) wi wsucc(i) = (+1)(+1)(+1) or (−1)(−1)(−1), then the Ext quiver will contain a commutative square:

,

where the two middle vertices are i and succ(i). In all other cases, Ai will be hereditary. Again, Proposition 3.3 implies that the underlying graph of the Ext quiver of Ai has the form:

−m

succ(i)

··· pred(i)

i

succ2 (i)

···

m

,

Figure 2 where for i = −m we take this graph to only include the edges to the right of i, for i = m − 1 we delete the section containing succ2 (i), and where dotted lines indicate that an edge may or may not be present. By the connectivity hypothesis, both dotted edges cannot be absent. Note that A−m+1 and Am−2 may themselves be path algebras over type A Dynkin quivers, say if w−m w−m+1 w−m+2 = (+1)(+1)(−1). This will be our second case. Otherwise, this graph will necessarily contain a trivalent vertex, and a unique leaf adjacent to this trivalent vertex. This will be our third and final case. We now prove our first case, where Ai is a non-hereditary algebra: Lemma 4.2. Let Q and B be as before. Let A be a non-hereditary maximal subalgebra of split type whose Ext quiver is connected. Then Iso(A, B) = Autk (B) · A. Proof. Suppose first −m ≤ i ≤ −1 and that Ai is not hereditary. Then i > −m and wpred(i) wi wsucc(i) = (+1)(+1)(+1) or (−1)(−1)(−1). By inspecting the full sub-quiver from −m to pred(i) and the full sub-quiver from succ2 (i) to m, we conclude that Ai ∼ = Aj implies either j = i or j = −i − 1. If Ai ∼ = A−i−1 , use Lemma 4.1 to find an isomorphism ψ : Ai → A−i−1 such that ψ(Q0 ) = Q0 and dimk (uJ(Ai )/J(Ai )2 v) = dimk (ψ(u)J(A−i−1 )/J(A−i−1 )2 ψ(v)) for all u, v ∈ Q0 . In other words, we may assume without loss of generality that ψ induces an automorphism of the underlying quivers of Ai and A−i−1 . But then we must have ψ(m) = −m, ψ(pred(i)) = succ(−i), and ψ(succ2 (i)) = pred(−i − 1). This forces ψ(") = −" for all " ∈ Q0 , so that again by Lemma 4.1 we have Aut(Q) = C2 . In this case, the non-identity automorphism of Q sends Ai to A−i−1 . Therefore, Autk (B) · Ai = Autk (B) · A−i−1 and so Iso(Ai , B) = Autk (B) · Ai . Otherwise Ai ∼ = A−i−1 , and again Iso(Ai , B) = Autk (B) · Ai . The i > −1 case follows from  replacing w with w ∗ and repeating the argument above.

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We now prove our second case, where Ai is hereditary but does not contain a trivalent vertex. We note again that this forces i ∈ {−m + 1, m − 2}. Lemma 4.3. Let Q and B be as before, and let A = A−m+1 or Am−2 . Suppose that A is hereditary, that the Ext quiver of A is connected, and that it does not contain a trivalent vertex. Then Iso(A, B) = Autk (B) · A. Proof. It suffices to prove the claim for A−m+1 . Without loss of generality, we may assume that w−m+1 = +1. For 1 ≤ m ≤ 3 the claim can be verified through straightforward, but tedious computations. So we will assume m > 3 throughout. By hypothesis, we must have w−m w−m+1 w−m+2 = (+1)(+1)(−1). From Proposition 3.3, the Ext quiver of A−m+1 is a type A Dynkin quiver, whose valence-1 vertices are −m+1 and m. The binary word associated to this quiver, choosing −m+1 to be the left-most vertex, is then simply u = (−1)(+1)(−1)w−m+3 · · · wm−1 . The only other i for which Ai can have a type A Dynkin quiver as its Ext quiver is i = m − 2. Therefore, we have an inclusion Iso(A−m+1 , B) ⊂ Inn(B) · A−m+1 ∪ Inn(B) · Am−2 . We claim that this inclusion is an equality if and only if Aut(Q) = C2 . If it is not an equality then A−m+1 ∼ = Am−2 , and so in particular Aut(Q) = 1. But then Iso(A, B) = Inn(B) · A = Autk (B) · A, and the claim of the lemma is true in this case. So, suppose instead that Am−2 ∼ = A. Then −m and m − 1 are the leaves in the quiver of Am−2 . Comparing the full subquivers of Am−2 and A on {−m, −m + 1, −m + 2, −m + 3}, we find that no isomorphism A → Am−2 can carry −m + 1 to −m. But then this implies that we can find an isomorphism ψ : A → Am−2 which permutes Q0 and carries −m + 1 to m − 1. This forces ψ(j) = −j for all j ∈ Q0 . By Lemma 4.1 we have Aut(Q) = C2 . In this case the non-identity element of Aut(Q), carries A−m+1 to Am−2 , and so Iso(A, B) = Autk (B) · A in this case as well.  Now it only remains to prove the hereditary trivalent case. We break the proof into the cases where |Q0 | is even or odd. For the next lemma, let i be chosen such that −m ≤ i ≤ m−1. Then according to Figure 2, exactly one of the dotted arrows must represent an arrow in the quiver of Ai . We refer to Ai as Li if the arrow from pred(i) to succ(i) is present, and Ri if the arrow from i to succ2 (i) is present. For brevity, we write Ai = Li if the former holds, and Ai = Ri if the latter holds. In either case, there is a unique trivalent vertex in the quiver of Ai , and a unique univalent vertex adjacent to it. We refer to this univalent vertex as the root of the quiver. We call the smallest connected full subquiver containing this univalent vertex and −m as the left path. Similarly, the smallest connected full subquiver containing the univalent vertex and m is called the right path. Note that the left path and right path are just type A Dynkin quivers. The length of the left/right path is just the number of arrows in it. Lemma 4.4. Let G and B be as before. Let A be a hereditary maximal subalgebra of split type, whose Ext quiver is connected and contains a trivalent vertex. Then Iso(A, B) = Autk (B) · A. Proof. Case 1: Suppose that n is even. Then in particular, succ(−1) = 1. We start by showing that for any i and j with −m ≤ i, j ≤ −1 and i = j, Ai ∼ = Aj . To see this, first note that for such i, the length of the left path in Li is i + m, and the length of the right path is m − i − 1. Similarly, length of the left path in Ri is m + i + 2, and the length of the right path is m + i − 2. In particular, the

352

ALEX SISTKO

difference between the lengths of the left and right paths in Li is odd, whereas the difference between the left and right paths in Ri is even. It follows that if either Ai = Li and Aj = Rj , or Ai = Ri and Aj = Rj , then Ai ∼ = Aj . Furthermore, Li ∼ = Lj , since for i = j, the corresponding sets of left and right path lengths are disjoint. It follows that if Ai ∼ = Aj for i, j ≤ −1, then necessarily Ai = Ri and Aj = Rj . In fact, by inspection of path lengths, the only possibility is m ≥ 3 and {i, j} = {−1, −3}. Without loss of generality, we may assume that w−1 = (+1). Then necessarily w−2 w−1 w1 = (−1)(+1)(+1), so that 1 is the root of A−1 and it is a source in the quiver of A−1 . Since A−3 = R−3 and w−2 = −1, we must also have w−3 = −1. But then −2 is the root of A−3 , and it is a sink in its quiver. This is a contradiction, and so A−1 ∼ = A−3 . The above argument implies that if −m ≤ i ≤ −1 and j is chosen such that Ai ∼ = Aj , then j ≥ 1. Note that if j ≥ 1, then the lengths of the left and right paths of Lj have an even difference, whereas they have an odd difference in Rj . Comparing path lengths, we find the following: (1) If i ∈ {−1, −3} and Ai = Ri , then Ai ∼ = Aj implies Aj = Lj and j ∈ {−1, 2}. (2) If i ∈ {−1, −3} and Ai = Ri , then Ai ∼ = Aj implies Aj = L−i−1 . (3) If Ai = Li , then Ai ∼ = Aj implies Aj = Rpred(−i) . A−3 . Similar computations show that if the We have already shown that A−1 ∼ = root of R−1 (resp. R−2 ) is a source, then the root of L2 (resp. L1 ) is a sink, and vice L2 and R−2 ∼ L1 . The remaining cases from statements versa. We conclude R−1 ∼ = = (1)-(3) can be rephrased as follows: for all i, either Iso(Ai , B) = Inn(B) · Ai or Iso(Ai , B) = Inn(B) · Ai ∪ Inn(B) · Apred(−i) . If Iso(Ai , B) = Inn(B) · Ai , there is nothing to show. So suppose i = pred(−i) and Ai ∼ = Apred(−i) , for some negative integer i. In particular i = −1, and if i = −2 we may assume A−2 = L−2 . Note that this implies that the left path of Ai must have a shorter length than the right path, whereas the left path of A−i−1 must have a longer length that its right path. Since any isomorphism Ai → A−i−1 permuting Q0 must send the trivalent vertex (resp. root) of Ai to the trivalent vertex (resp. root) of A−i−1 , it follows that such an isomorphism satisfies j → −j for all j ∈ Q0 . By Lemma 4.1 we have Aut(Q) = C2 , a contradiction (since n is even). Hence Iso(Ai , B) = Inn(B) · Ai , as we wished to show. Case 2: Suppose that n is odd. Then Q0 is just the interval [−m, m] of Z. Let i be chosen such that −m ≤ i ≤ −1. The the left path of Li has length i + m, and the right path has length m−i+1. The left path of Ri has length m+i+2, and the right path has length m − i − 1. As before, we want to start by showing that if j is any number between −m and −1 with i = j, then Ai ∼ = Aj . Suppose that Ai = Li . Then by comparing path sizes, we see that if Ai ∼ = Aj , then Aj = Ri−2 . Without loss of generality, we may assume wi = +1. Then wi−1 wi wi+1 = (+1)(+1)(−1). But since Ai−2 = Ri−2 , we must have wi−2 = +1 as well. But then the root of Ai is i, which is a sink, whereas the root of Ai−2 is i − 1, which is a source. This is a contradiction, and so Ai ∼ = Aj if Ai = Li . Otherwise Ai = Ri . If Ai ∼ = Aj and Aj = Lj we are in the previous case, so assume Aj = Rj as well. By comparing path lengths, we find that the only possibility is {i, j} = {−1, −2}. Suppose without loss of generality that w−1 = +1. Then since A−1 = R−1 and is connected hereditary, we must have w−2 w−1 w0 = (−1)(+1)(+1). But then A−2 = R−2 , since A−2 = R−2

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∼ R−2 , and it follows requires w−2 and w−1 to have the same sign. Hence R−1 = that for all −m ≤ i, j ≤ −1, Ai ∼ = Aj . Now we show that if Ai ∼ = Aj for j ≥ 0, then j = −i − 1 and Aut(Q) = C2 . Suppose first that Ai = Li . Then the other algebras Aj for j ≥ 0 which have the same set of path lengths as Ai are R−i−1 and L−i+1 . We first show that Li ∼ = L−i+1 . Without loss of generality, we let wi = +1. The root of Li is i, and the right path has a larger length than the left path. Treating the right path as a type A Dynkin quiver starting at i, its associated binary word is (−1)(+1)wi+1 · · · wm−1 . Now, the larger path in L−i+1 is the left path, and its root is vertex −i+1. Since the root of Li is a sink, −i + 1 must be a sink too. This forces w−i w−i+1 w−i+2 = (+1)(+1)(−1). Treating the left path of L−i+1 as a type A Dynkin quiver starting at −i+1, its associated binary word is (−1)(−w−i−1 ) · · · (−w−m ). If Li ∼ = L−i+1 , then we must have (−1)(+1)wi+1 · · · wm−1 = (−1)(−w−i−1 ) · · · (−w−m ). In particular, w−i−1 = −1 and for all i + 1 ≤ j ≤ m − 1, we have wj = −w−(j+1) . But then setting j = −i we find +1 = w−i = −w−(−i+1) = −wi−1 = −(+1) = −1, a contradiction. So Li ∼ = L−i+1 , as we wished to show. Hence, we assume Li ∼ = R−i−1 . Since the root of Li is a sink, it must be a sink in R−i−1 as well. This implies that if we start at the root, the longer path in R−i−1 has binary word (−1)(+1)(−1)(−w−i−2 ) · · · w−m . Therefore, we have (−1)(+1)(−w−i−2 ) · · · (−w−m ) = (−1)(+1)(wi+1 ) · · · wm−1 . This implies that wj = −w−(j+1) for i+1 ≤ j ≤ m−1. Similarly, the binary word for the short path in R−i−1 is w−i w−i+1 · · · wm−1 , whereas it is (−wi−1 )(−wi−2 ) · · · (−w−m ) for Li . Hence, wj = −w−(j+1) for −m ≤ j ≤ i−1. But wi = +1 by hypothesis, and w−i−1 = −1 since R−i−1 is a subalgebra of B isomorphic to Li . In other words, w∗ = w, and Aut(Q) = C2 . This shows that if Ai ∼ = Aj for j ≥ 0, then j = −i − 1 and Aut(Q) = C2 for the Ai = Li case. Note that by replacing w by w ∗ if necessary, it only remains to consider the case when Ai = Ri and Aj = Rj . By comparison of path lengths, Aj = R−i−3 . Note that we may assume i ≤ −3, for otherwise this reduces to a case that has been previously discussed. Comparing the binary words for the longer paths in Ri and R−i−3 , we see wi+1 · · · wm−1 = (+1)(−1)(−w−i−4 ) · · · (−w−m ). Therefore, wj = −w−(j+1) for i + 3 ≤ j ≤ m − 1. Since i ≤ −3, i + 3 ≤ 0 and so this suffices to show that w ∗ = w. But since wi = +1 and Ai = Ri , wi+1 = +1. Since we also assume A−i−3 = R−i−3 ∼ = Ri , comparison of roots yields w−i−2 = +1. But then wi+1 = −w−(i+1+1) , and so w∗ = w. This contradiction shows that Ri ∼ R−i−3 . = Putting this all together, we see that if Ai ∼ = Aj , then j = −i − 1 and Aut(Q) = C2 . But in this case Ai and A−i−1 lie in the same Autk (B)-orbit, and so the lemma is proved.  Lemmas 4.3, 4.2, and 4.4 combine to prove the following proposition: Proposition 4.5. Let Q and B be as before. If A ⊂ B is a maximal subalgebra of split type, and the Ext quiver of A is connected, then Iso(A, B) = Autk (B) · A. To finish the proof of Theorem 1.1, we need to show that Iso(Ai,j , B) = Autk (B) · Ai,j for all i and j. If i < j, then Proposition 3.1 asserts that the underlying graph of the Ext quiver of Ai,j looks as follows: where if i = −m (resp. j = m) we delete the subgraph including pred(i) and all vertices to its left (resp. succ(j) and all vertices to its right). Examining the paths from −m to i + j and i + j to m, we see Iso(Ai,j , B) ⊂ Autk (B) · Ai,j ∪ Autk (B) ·

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···

−m

pred(i)

···

i+j

m

succ(j)

,

Figure 3 A−i,−j . In particular, Iso(Ai,j , B) = Autk (B) · Ai,j if i = −j, and so without loss of generality we may replace the i < j assumption by the assumption that |i| < |j| throughout (allowing now for the possibility that i ≥ j). If Aut(Q) = C2 , then the unique non-identity automorphism of Q extends to an automorphism of B which sends vi → v−i , vj → v−j . Therefore Iso(Ai,j , B) = Autk (B) · Ai,j in this case as well, and so we may also assume Aut(Q) = 1. A−i,−j for the Under these assumptions, we will be done if we can show Ai,j ∼ = remaining cases. In other words, we must prove the following lemma: Lemma 4.6. Under the assumptions above, Ai,j ∼ = A−i,−j . Again we break this up into two arguments, depending on whether n is even or odd. First suppose that Q has n = 2m vertices. Note that if i, j ≤ −1 or i, j ≥ 1, then Ai,j ∼ A−i,−j by examination of the edge between v−1 and v1 in the quivers = of Ai,j and A−i,−j . Therefore, Iso(Ai,j , B) = Inn(B) · Ai,j in this case. Hence, we may assume without loss of generality that −m ≤ i ≤ −1 < 1 ≤ j ≤ m, so that combined with our reductions above, we have 1 ≤ −i < j ≤ m. Write w = w 1 · w2 · w3 · w4 · w 5 , where the wi are defined as follows (see Figure 4): (1) w1 is the binary word from v−m to v−j , (2) w2 is the binary word from v−j to vi , (3) w3 is the binary word from vi to v−i , (4) w4 is the binary word from v−i to vj , and (5) w5 is the binary word from vj to vm . −i w3 −m

··· w1

−j w2 i + j

w4 ··· w5

m

,

Figure 4 Proof (Even Case). Suppose there was an isomorphism Ai,j ∼ = A−i,−j by way of contradiction. Then the quivers of Ai,j and A−i,−j must be isomorphic. Since |i| < |j|, this can only be true if the following three conditions hold: (1) w1 = (w5 )∗ , (2) w2 = (w4 )∗ , and (3) either w3 · w 4 = w∗3 · w∗2 or w3 · w 4 = (w∗3 · w∗2 )∗ = w2 · w 3 .

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If w 3 · w4 = w ∗3 · w∗2 , then by applying condition (2) and cancelling w 4 from both sides, w3 = w ∗3 . Then w∗ = (w1 · w2 · w 3 · w4 · w 5 )∗ = w∗5 · w ∗4 · w∗3 · w∗2 · w∗1 = w1 · w 2 · w3 · w4 · w5 = w, a contradiction. Hence, w3 · w∗2 = w2 · w 3 . Case 1: Suppose that the length of w 2 is less than or equal to the length of w 3 . Then since w3 ·w∗2 = w2 ·w3 , we may write w2 = 1 · · · d , w 3 = 1 · · · d d+1 · · · d+f , where each a ∈ {±1}. Then (1)

(1 · · · d d+1 · · · d+f )(−d ) · · · (−1 ) = (1 · · · d )(1 · · · d d+1 · · · d+f ).

Suppose that d > f . Then by comparing terms, we have d+a = a for all 1 ≤ a ≤ f and f +b = −d−b+1 for all 1 ≤ b ≤ d. Since d + f is odd, (d + f + 1)/2 is a natural number, and f < d implies that (d + f + 1)/2 ≤ d. Now, d − b + 1 = (d + f + 1)/2 precisely when b = (d − f + 1)/2. Then f +b = −d−b+1 yields (d+f +1)/2 = −(d+f +1)/2 for this value of b, a contradiction. Note that d = f since d + f is odd, and so the only remaining possibility is that d < f . After cancelling the 1 · · · d term from both sides, we find that (2)

d+1 · · · d+f (−d ) · · · (−1 ) = (1 · · · f )(f +1 · · · d+f ).

For 1 ≤ a ≤ f , define q = q(a) to be the largest non-negative integer such that dq + a ≤ f . Now, comparing terms on the left-hand-side and the right-hand-side of the above equation, we see a = . . . = dq+a = d(q+1)+a . Since d(q + 1) + a > f , it follows that d(q+1)+a = −d+f +1−(d(q+1)+a) = −f −qd−a+1 = −f −(dq+a−1) . Notice that q = , f −a d -. If we can choose a such that a = f − (dq + a − 1), we will have the desired contradiction. Sub-Case 1(a): Say that f is odd. Then d is even, and for all s, (f − ds + 1)/2 is an integer. Define r to be the largest positive integer such that dr + 1 ≤ f . Then f < d(r + 1) + 1 < d(r + 2) + 1. Let a = (f − dr + 1)/2. It is clear that a ≤ f . The f − f −rd+1

2 < inequality dr + 1 ≤ f < d(r + 2) + 1 can be re-arranged to say that r ≤ d f −a f −a r + 1. In other words, , d - = r. But a = f − dr − a + 1 = f − d, d - − a + 1 by definition, and we have found a choice of a which works. Sub-Case 1(b): Say that f is even. Then d is odd, and for all odd s, (f −sd+1)/2 is an integer. Let r be the same as in Sub-Case 1(a). If r is odd, then again a = (f − rd+1)/2 works. Otherwise r is even, so that r−1 is odd and a = (f −d(r−1)+1)/2 is an integer. Notice that d(r−1)+1 ≤ f < d((r−1)+2)+1, so that , f −a d - = r. Now, for this choice of a, we have a = f − d(r − 1) − a + 1 = (f − d, f −a − a + 1) + d. d But since r ≥ 2, d(r − 1) + a − 1 ≥ 0 and f − d(r − 1) − a + 1 ≤ f . Hence, −f −(dr+a−1) = −f −(dr+a−1)+d = −f −(d(r−1)+a−1) , and we obtain our desired contradiction. The proof of Case 1 is now complete. Case 2: Suppose that the length of w 2 is greater than or equal to the length of w3 . Note that they cannot be equal, for otherwise w 3 · w ∗2 = w2 · w3 would imply w3 = w2 , and hence w∗3 = w3 , a contradiction. So then, we may write w 3 = 1 · · · d and w 2 = 1 · · · d d+1 · · · d+f , for some d, f > 0. As before, d = f . Hence, our equation reads

(3)

1 · · · d (−d+f ) · · · (−d+1 )(−d ) · · · (−1 ) = 1 · · · d d+1 · · · d+f 1 · · · d .

Comparing the last d-terms on both sides of this equation, we conclude that a = −d−a+1 , for 1 ≤ a ≤ d. Now, d is odd, so (d + 1)/2 is an integer ≤ d. Therefore, for a = (d + 1)/2 we have (d+1)/2 = −d−(d+1)/2+1 = −(2d−d−1+2)/2 = −(d+1)/2 , a contradiction. 

356

ALEX SISTKO

The remaining case left to consider is when Q has n = 2m + 1 vertices, m ≥ 1. Again, we assume Aut(Q) = 1 and |i| < |j|. If i, j ≤ 0 or i, j ≥ 0 and Ai,j ∼ = A−i,−j , then inspection of the long path in Figure 3 implies that w(") = −w(pred(−")) for either −m ≤ " ≤ 0 or 0 ≤ " ≤ m − 1. In either case we conclude w∗ = w, contradicting the triviality of Aut(Q). So we can assume i < 0 and 0 ≤ j in addition to |i| < |j|. Then we can decompose w into w = w1 · w2 · w 3 · w 4 · w5 as in Figure 4. Under the assumptions that Ai,j ∼ = A−i,−j and n is odd, the length of w3 must be even, and w 3 · w∗2 = w 2 · w3 . With this modified setup, we can now finish the proof of Lemma 4.6: Proof (Odd Case). Case 1: Say that the length of w3 is greater than the length of w2 . Write w 2 = 1 · · · d and w3 = 1 · · · d d+1 · · · d+f . Again, we find that equation (2) holds, and so d+a = a for 1 ≤ a ≤ f and f +b = −d−b+1 for 1 ≤ b ≤ d. Sub-Case 1(a): Suppose that f ≤ d. If d and f are both odd, then a = (f +1)/2 is an integer ≤ f . Since d + a ≥ f + a > f , we have a = d+a = f +(d+a−f ) = −d−(d+a−f )+1 = −f −a+1 . But for this choice of a, a = f − a + 1, and we arrive at a contradiction. So we may assume that d and f are both even. Applying the same logic, we find that for all a ≤ f , we have a sequence of equalities: a = d+a = −f −a+1 = −d+f −a+1 = a . Since f and d are both even, {a, d + a} ∩ {f − a + 1, d + f − a + 1} = ∅. The equalities a = −d+f −a+1 and d+a = −f −a+1 are equivalent to the statement that w∗3 = w3 , which contradicts our hypotheses. Hence, the f ≤ d case is complete. Sub-Case 1(b): Suppose that f > d. For any a ≤ f , define q = q(a) to be the largest non-negative integer such that dq + a ≤ f . Then a = . . . = dq+a = d(q+1)+a = −f −(dq+a−1) . If f and d are odd, then define r to be the largest positive integer such that dr + 1 ≤ f . If r is even, then a = (f − dr + 1)/2 is an integer ≤ f which satisfies , f −a d - = r and i = f − dr + a − 1. This implies a = −a as before. Otherwise r is odd. Then a = (f − d(r − 1) + 1)/2 satisfies , f −a d - = r, f − d(r − 1) − a + 1 ≤ f , and a = (f − dr − a + 1) + d, which implies a = −f −d(r−1)−a+1 = −a as before. It remains to consider the case that f and d are even. Then we have a = . . . = dq+a = d(q+1)+a = −f −(dq+a−1) = . . . = −f −a+1 = −d+f −a+1 for all 1 ≤ a ≤ f , so a = −d+f −(a−1) . Finally, for each 1 ≤ a ≤ d, there is a unique f < s ≤ f + d with s ≡ a (mod d). But then, s = d(q + 1) + a, and the equality d(q+1)+a = −f −(dq+a−1) tells us that b = −d+f −(b−1) for all 1 ≤ b ≤ f + d. This implies w 3 = w∗3 , contrary to our assumption that Aut(Q) = 1. Hence, the f > d case is complete. Case 2: Say that the length of w3 is less than or equal to the length of w 2 . As before, "(w3 ) = "(w2 ), so we may assume "(w3 ) < "(w2 ). Write w3 = 1 · · · d and w2 = 1 · · · d d+1 · · · f . As before, equation (3) holds. If d is odd, repeat the same argument given for the even-vertices case. Otherwise d is even, and so the equation  a = −d−a+1 for 1 ≤ a ≤ d tells us w∗3 = w3 , contrary to our hypotheses. Lemma 4.6 immediately implies the following: Proposition 4.7. Let A ⊂ B be a connected maximal subalgebra of separable type, where B = kQ and Q is a type A Dynkin quiver. Then Iso(A, B) = Autk (B) · A. Finally, Propositions 4.5 and 4.7 combine to yield Theorem 1.1.

ISOCLASSES DDTERMINED BY AUTOMORPHISMS

357

We end with an example to show that the connectedness hypothesis is necessary: Example 4.8. We note that the conclusion of Theorem 1.1 is false if A has a disconnected Ext quiver. For instance, suppose that Q is the type A4 Dynkin quiver v−2

v−1

v1

v2 .

Then Autk (B) = Inn(B) and Iso(A−2 , B) = Inn(B) · A−2 ∪ Inn(B) · A1 . Of course, A−2 and A1 lie in different Inn(B)-orbits, since their Jacobson radicals are distinct. References [1] Ibrahim Assem, Daniel Simson, and Andrzej Skowro´ nski, Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. MR2197389 [2] Farnsteiner, R. The Theorem of Wedderburn-Malcev: H 2 (A, N ) and Extensions, lecture notes, (2005). Accessed online at https://www.math.uni-bielefeld.de/ sek/select/RF6.pdf [3] Francisco Guil-Asensio and Manuel Saor´ın, The group of outer automorphisms and the Picard group of an algebra, Algebr. Represent. Theory 2 (1999), no. 4, 313–330, DOI 10.1023/A:1009973319703. MR1733381 [4] Francisco Guil-Asensio and Manuel Saor´ın, The automorphism group and the Picard group of a monomial algebra, Comm. Algebra 27 (1999), no. 2, 857–887, DOI 10.1080/00927879908826466. MR1672003 [5] Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko, Algebras, rings and modules. Vol. 2, Mathematics and Its Applications (Springer), vol. 586, Springer, Dordrecht, 2007. MR2356157 [6] Miodrag Cristian Iovanov and Alexander Harris Sistko, Maximal subalgebras of finitedimensional algebras, Forum Math. 31 (2019), no. 5, 1283–1304, DOI 10.1515/forum-20190033. MR4000588 [7] Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, SpringerVerlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR674652 [8] R. David Pollack, Algebras and their automorphism groups, Comm. Algebra 17 (1989), no. 8, 1843–1866, DOI 10.1080/00927878908823824. MR1013471 [9] Sistko, A. Automorphism Groups of Finite-Dimensional Algebras Acting on Subalgebra Varieties, arXiv:1809.09760 (2018). Department of Mathematics, University of Iowa, Iowa City, IA 52242 Email address: [email protected]

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CONM

751

ISBN 978-1-4704-4368-9

9 781470 443689 CONM/751

Categorical, Homological and Combinatorial Methods in Algebra • Srivastava et al., Editors

This book contains the proceedings of the AMS Special Session, in honor of S. K. Jain’s 80th birthday, on Categorical, Homological and Combinatorial Methods in Algebra held from March 16–18, 2018, at Ohio State University, Columbus, Ohio. The articles contained in this volume aim to showcase the current state of art in categorical, homological and combinatorial aspects of algebra.