Topology and K-Theory. Lectures by Daniel Quillen 9783030439958, 9783030439965

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Lecture Notes in Mathematics 2262 History of Mathematics Subseries

Robert Penner

Topology and K-Theory Lectures by Daniel Quillen

Lecture Notes in Mathematics History of Mathematics Subseries Volume 2262

Series Editor Patrick Popescu-Pampu, CNRS, UMR 8524 - Laboratoire Paul Painlevé, Université de Lille, Lille, France

More information about this subseries at http://www.springer.com/series/8909

Robert Penner

Topology and K-Theory Lectures by Daniel Quillen

With Contribution by Mikhail Kapranov, Kavli Institute for the Physics and Mathematics of the Universe, Kashiwa, Chiba, Japan

123

Robert Penner Institut des Hautes Études Scientifiques Bures-sur-Yvette, France

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISSN 2193-1771 ISSN 2625-7157 (electronic) History of Mathematics Subseries ISBN 978-3-030-43995-8 ISBN 978-3-030-43996-5 (eBook) https://doi.org/10.1007/978-3-030-43996-5 Mathematics Subject Classification (2010): 19-01, 55-01 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

These are notes from a graduate student course on algebraic topology and K-theory given by Daniel Quillen at the Massachusetts Institute of Technology during 1979– 1980. He had just received the Fields Medal for his work on these topics, among others. As a second-semester graduate student myself, it seemed an opportunity to see what all this meant, what a Fields medalist looked and sounded like, what was this exciting new mathematics. Among the gaggle of graduate students, there was also one senior faculty member in attendance, namely Giancarlo Rota. Dan Quillen was funny and playful with a confident humility from the start. There were points during lectures where he might get stuck and just abandon a proof midstream with a casual never mind, and there was always much joking and laughter, enormous energy. A particularly funny moment about which I have periodically chuckled over the years occurred when he was asked some question or other, thought deeply a long moment and answered with a smile that from a sufficiently enlightened point of view it is obvious. Giancarlo was also characteristically funny and playful. He early in the semester asked if we could share class notes, especially when he was absent. At that time in my studies, I was able to scrawl every word uttered in class, and given the attention from Giancarlo, I was driven to revise and legibly copy my notes like never before or since. Furiously taking notes in Sweden a few years later, Peter Jones pulled postdoc me aside and explained that gentlemen don’t take notes, and I gave up the practice altogether then and there except for an occasional reference or formula. Giancarlo and I would discuss the lectures which sometimes informed my revisions, and we quickly became friends despite our obviously different circumstances. This was only further cemented a bit less than a decade later when he was a regular Visiting Scholar at the University of Southern California, where I was an assistant professor. We used to joke that we had gone to graduate school together because of the course memorialized here. There were furthermore several occasions from days when I was apparently absent comprising Giancarlo’s own notes. The handwritten notes, mine, ours, and Giancarlo’s, are actually in complete sentences with acceptable grammar, and they surfaced recently from a drawer when I moved from my home of thirty years. These are not meant to be polished lecture v

vi

Preface

notes, rather, I have tried to present things as did Quillen, reflected in the handwritten notes, resisting any temptation to change or add notation, details, or elaborations. Indeed, I have been faithful to Quillen’s own exposition, even respecting the board-like presentation of formulae, diagrams, and proofs, omitting numbering theorems in favor of names and so on. This is meant to be Quillen on Quillen as it happened forty years ago, an informal text for a second-semester graduate student. The intellectual pace of the lectures, namely fast and lively, is Quillen himself, and part of the point here is to capture some of this intimacy. I remember especially the last lecture with its abrupt end, a kind of charmingly embarrassed sayonara, having shared so much of himself during the course. Despite the avowed goal to present these lectures in their native form, a few insignificant and obvious errors have been corrected. Quillen’s own writings are sheer perfection, and the same standard cannot be applied here in the informal context of lecture notes where much ground is covered quickly. The reader is warned, therefore, that there may be small inconsistencies remaining though I and we have done our best in this regard and hope that the overall flow and temperament of these notes might compensate whatever errors may remain. To be sure, much has happened since then from this categorical perspective started by Grothendieck, and I am grateful to my friend Misha Kapranov for contributing an Afterword to this volume in order to make it more useful to current students and also for refining and correcting my own notes. It is likewise a pleasure to thank Cécile Gourgues for her superb transcription from the handwritten notes to beautiful LATEX and the Institut des Hautes Études Scientifiques for supporting this project. Thanks also to Geoff Taylor and Tony Philp for their hospitality in Fiji, where this manuscript was ultimately completed. Let me finally dedicate this little volume to the memories of my friend Gianco and my love Lexy. Savusavu, Fiji November 2018

Robert Penner

Contents

1

Group Extensions and Cohomology . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Categories and Their Nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3

Simplicial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4

Normalization and Conical Contractibility . . . . . . . . . . . . . . . . . . .

15

5

Effaceable d-Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

6

(Co)homology of Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

7

An Application to the Schur–Zassenhaus Theorem . . . . . . . . . . . . .

33

8

The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

9

Kan Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

10 Abelian and Additive Categories . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

11 Diagram Chasing in Abelian Categories . . . . . . . . . . . . . . . . . . . . .

57

12 Fibered and Cofibered Categories . . . . . . . . . . . . . . . . . . . . . . . . . .

61

13 Examples of Fibered Categories . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

14 Projective Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

15 Analogues of Homotopy Liftings . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

16 The Mapping Cylinder and Mapping Cone . . . . . . . . . . . . . . . . . .

83

17 Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

18 The First Homotopy Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

19 Group Completions and Grothendieck Groups . . . . . . . . . . . . . . . . 101 20 Devissage and Resolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . 105

vii

viii

Contents

21 Exact Sequences of Homotopy Classes . . . . . . . . . . . . . . . . . . . . . . 109 22 Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 23 Spectral Sequences Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 24 Hyper-Homology Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . 125 25 Generalized Kan Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 26 The Hochschild–Serre Spectral Sequence . . . . . . . . . . . . . . . . . . . . 137 27 Resolution for Exact Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 28 K 0 A ffi K 0 A½T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 29 Classifying Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 30 Higher K-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 31 The Category QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 32 Homotopy Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 33 A Filtration of QðP A Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 34 Bi-simplicial Sets and Dold–Thom . . . . . . . . . . . . . . . . . . . . . . . . . 179 35 Homology of QðP A Þ and the Tits Complex . . . . . . . . . . . . . . . . . . . 185 36 Long Exact Sequences of K-Groups . . . . . . . . . . . . . . . . . . . . . . . . 189 37 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 38 The Plus Construction, K 1 and K 2 . . . . . . . . . . . . . . . . . . . . . . . . . 201 Afterword by Mikhail Karpranov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Chapter 1

Group Extensions and Cohomology

Consider an extension E of the group G by the group N p

i

∗ → N − → E −→ G −→ ∗, so i : N → E is an injection, p : E  G a surjection and the kernel Ker p of p equals the image Im i of i, i.e., a short exact sequence. The obvious notion of isomorphisms of extensions is given by a commutative diagram ∗

N

E ∼ =

id ≡

∗

N

∗

G id ≡

E

∗

G

and we let E(G, N ) denote the collection of isomorphism classes. If u : G  → G is a homomorphism, then there is a pull back u ∗ : E(G, N ) → E(G  , N ) defined by the diagram ∗

N

π2

E ×G G  π1

∗

N

E

G

∗

u p

G

∗

where E ×G G  = {(a, b) : a ∈ E, b ∈ G  and pa = ub} and π1 is induced by projection onto the first factor. © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_1

1

2

1 Group Extensions and Cohomology

The pull back of an epimorphism is again an epimorphism, the pull back preserves fibers, i.e., cosets of N , and we can check directly that the induced map N → N in the diagram is the identity map. p

i

Given an extension ∗ −→ N − → E −→ G −→ ∗ and given g ∈ G, choose some h ∈ E so that p(h) = g. Then h acts on E by conjugation and in particular on N since N  E. Thus we have Out N

G p

Inn E

E

Aut N

i

N

Inn N

where Aut denotes the automorphism group, Inn  Aut the inner automorphism group, and the quotient Out = Aut/Inn is the outer automorphism group. In particular for N abelian, we have Aut N = Out N and get a homomorphism G −→ Aut N g −→ (n → i −1 (h i(n) h −1 )) where h ∈ E projects to g = p(h). Thus for N abelian as we assume from now on, N is a G-module and  Eθ (G, N ) , E(G, N ) = θ

G −→AutN

where Eθ (G, N ) denotes the extensions compatible with the homomorphism θ : G → Aut N . Eθ (G, N ) is covariant in the G-module N for G-equivariant maps in the sense that given two G-actions θ on N and θ on N  , a G-equivariant u : N → N  induces Eθ (G, N ) → Eθ (g, N  ), i.e., ∗

N

E

G

∗

E

G

∗

u

∗

N

where E  = N  × N E = N  × E modulo the action of N , i.e., (n  n, e) ∼ (n  , ne). Carry on to define multiplication in E  by (c(n 1 , e1 )) · (c(n 2 , e2 )) = (c(n 1 · p(e1 ) n 2 , e1 e2 )) , d

1 Group Extensions and Cohomology

3

where c means equivalence class, the motivation being n 1 e1 n 2 e2 = n 1 e1 n 2 e1−1 e1 e2 , and we check that this is well-defined. Description by Means of Cocycles i

p

→ E −→ G −→ ∗, choose a section s : G → E of Given the extension ∗ −→ N − p, that is, a map of sets so that ps = id G . Then every element of E can be uniquely written as i(u) s(g) , for n ∈ N and g ∈ G . The multiplication is d

[i(n 1 ) s(g1 )] [i(n 2 ) s(g2 )] = i(n 1 + g1 n 2 ) s(g1 ) s(g2 ) . The notation is that N is additive, E and G multiplicative, and we denote the action of g on N multiplicatively, so the product can be written = i(n 1 + g1 n 2 ) i( f (g1 , g2 )) s(g1 g2 ) = i(n 1 + g1 n 2 + f (g1 , g2 )) s(g1 g2 ) , where f : G × G → N depending on the choice of s. Conclusion the group operation on E is determined by f where s(g1 ) s(g2 ) = i( f (g1 , g2 )) s(g1 g2 ) .

Exercise 1.1 Associativity in E implies (∗1 )

g1 f (g2 , g3 ) − f (g1 g2 , g3 ) + f (g1 , g2 g3 ) − f (g1 , g2 ) = 0 .

Hint: (s(g1 ) s(g2 )) s(g3 ) gives the second s(g1 ) (s(g2 ) s(g3 )) gives the first and third.

and

fourth

terms,

and

Such an f : G × G → N is called a 2-cocycle of G with values in the G-module N. Exercise 1.2 Suppose s˜ : G → E is another section of p. Then we get h : G → N by s˜ (g) = i(h(g)) s(g). Show (∗2 )

f˜(g1 , g2 ) − f (g1 , g2 ) = g1 h(g2 ) − h(g1 g2 ) + h(g1 ) .

Hint: s˜ (g1 ) s˜ (g2 ) = i h(g1 ) s(g2 ) i h(g2 ) s(g2 ).

4

1 Group Extensions and Cohomology

The right-hand side of (∗2 ) is called the 1-coboundary of h. Conclusion Can attach to ∗ → N → E → G → ∗, for N a G-module, a welldefined element of d

H 2 (G, N ) = (group of 2-cocycles with values in N )/(group of 1-coboundaries). An element of the denominator is a 2-cocycle of the form (g1 , g2 ) → g1 h(g2 ) − h(g1 g2 ) + h(g1 ) . Theorem In the above way, the isomorphism classes of extensions of G by the G-module N are in 1 − 1 correspondence with elements of H 2 (G, N ). 

Proof A messy but routine exercise. General Formulae for the Cochain Complex of G with Values in the G-Module N Consider the cochain complex δ

δ

δ

→ C 1 (G, N ) − → C 2 (G, N ) − → ··· , · · · −→ 0 −→ C 0 (G, N ) − with C q (G, N ) = Maps(G q , N ), where G q is the q-fold Cartesian product and Maps(X, Y ) denotes all set maps from X to Y with C 0 (G, N ) = N , and for f ∈ C q d

(δ f )(g1 , . . . , gq+1 ) = g1 f (g2 , . . . , gq+1 ) − f (g1 g2 , g3 , . . . , gq+1 ) + f (g1 , g2 g3 , g4 , . . . , gq+1 ) − · · · ± f (g1 , . . . , gq ) . A routine but messy exercise which we shall explicate later shows that δ 2 = 0, and we define H q (G, N ) = (Kernel of δ on C q )/(Image of C q−1 under δ) . To compute H 0 , if n ∈ C 0 (G, N ) ≡ N , then (δ 0 n)(g) = gn − n, and so H 0 (G, N ) = {n : gn = n for all g ∈ G} = subgroup of elements of N fixed by G = NG . And by the Theorem, H 2 (G, N ) is the collection of isomorphism classes of extensions of G by N .

Chapter 2

Categories and Their Nerves

Let us next compute H 1 (G, N ) for N a G-module. h ∈ Z 1 (G, N ) = cycles in C 1 = Ker(δ : C 1 → C 2 ) means h(g1 g2 ) = g1 h(g2 ) + h(g1 ) , and h is called either a derivation or a crossed homomorphism in this case. They can be interpreted as follows: Let θ be an automorphism of an extension E of G by N , so that ∗

N

i

p

E

G

∗

G

∗

θ

∗

N

i

E

p

Consider E → E where e → θ(e)e−1 so p(θ(e)e−1 ) = 1 while θ(e · i(n))(e · i(n))−1 = θ(e)θ(i(n)) i(n)−1 e−1 = θ(e) e−1 . So this map is constant on cosets of N , and there is some h : G → N so that θ(e) e−1 = i(h( pe)) . We claim that this h is a derivation. To see this, take g1 , g2 ∈ G, lift to e1 , e2 ∈ E and compare the two sides of θ(e1 e2 ) = θ(e1 ) θ(e2 ) = i h(g1 ) e1 i h(g2 ) e2 . The left-hand side is i(h(g1 g2 )) e1 e2 , while the right-hand side is © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_2

5

6

2 Categories and Their Nerves

i h(g1 ) i(g1 h(g2 )) e1 e2 = i(h(g1 ) + g1 h(g2 )) e1 e2 , as desired. Conversely, defining θ(e) = i h( pe) e gives an automorphism. Thus Z 1 (G, N ) = group of automorphisms of any extension of G by N . Exercise B 1 (G, N ) = coboundaries in C 1 = Im{δ : C 0 → C 1 } is the subgroup consisting of all inner automorphisms by elements of N . Note that for G acting trivially on N , we have δ0 ≡ 0, and a derivation is just a homomorphism, so H 0 (G, N ) = N , H 1 (G, N ) = group homomorphisms from G to N and H 2 (G, N ) = isomorphism classes of central extensions of G by the abelian group N . Categories A category C consists of a class Ob C of objects, and for X, Y ∈ Ob C we are given a set HomC (X, Y ) of maps or morphisms, and for X, Y, Z ∈ Ob C a composition map HomC (X, Y ) × HomC (Y, Z ) −→ HomC (X, Z ) f × g −→ g f If f ∈ H om C (X, Y ), then we sometimes write simply f : X → Y and may call f an arrow in C. Axioms (1) associativity of composition, (2) existence of identity maps, (3) the sets HomC (X, Y ) are disjoint as X, Y varies in Ob C. A small category is a category C so that Ob C is a set. Example 2.1 Given a partially ordered set (I, ≤), called a poset for short, we get a small category I˜ so that Ob I˜ = I and  Hom I˜ (X, Y ) =

{(x, y)} (a 1-element set), if x ≤ y, ∅, if (x ≤ y) does not hold.

Example 2.2 For a monoid M, so the operation is associative and there are identity  with maps, we get a category M  = {∗}, Ob M

Hom M (∗, ∗) = M,

where composition is multiplication in M. In a category C, f : X → Y is an isomorphism if there is g : Y → X so that g f = id X and f g = idY .

2 Categories and Their Nerves

7

A groupoid is a category in which every morphism is an isomorphism. Note that I˜ is  is an isomorphism groupoid if and only if all elements are incomparable. Also m ∈ M  is a groupoid if and only if M is a group. if m −1 exists, and M Let C be a small category. Let N0 (C) be the set of objects and 

N1 (C) =

HomC (X, Y ).

X,Y ∈Ob C

We have maps source

N1 (C)

N0 (C) .

target identity map

Let now Nq (C) = diagrams in C of composable arrows of length q and call an element of Nq (C) a q-simplex. We have maps Nq (C)

d j , j=0,...,q

Nq−1 (C)

where  dj

 fj f j+1 X0 ← − · · · ←−− X j ←−−− X j+1 · · · ←− X q   f j−1 f j f j+1 = X 0 ←− · · · ←−−− X j−1 ←−−−− X j+1 ←− · · · X q .

We also have maps Nq−1 (C)

s j , j=0,...,q−1

Nq (C)

where   id s j (X 0 ←− · · · ←− X j · · · ←− X q−1 ) = X 0 ←− · · · ←− X j ←−− X j ←− · · · X q .

This so-called simplicial set is the nerve of the category C. A functor F : C1 → C2 induces a map N (C1 ) −→ N (C2 ) . Consider the following posets. op → C is the same thing as a  [q]op = {0, 1, . . . , q} with the wrong order. A functor [q] op is the category of Example 1 derived from the poset [q] op .   q-simplex in C , where [q]

8

2 Categories and Their Nerves θ

An order preserving map [ p]op − → [q]op will induce a map θ∗

Nq (C) −→ N p (C), and any θ∗ has a canonical presentation si1 . . . si p d j1 . . . d jk where i 1 > . . . > i p and j1 < . . . < jk using the commutation relations.

Chapter 3

Simplicial Objects

Let ´ be the small category with objects Ob  = {[p]|p ≥ 0}, where [p] = {0, . . . , p} with the usual order and Hom ´ the set of order-preserving maps. If C is any category, then a simplicial object in C is a contravariant functor X : ´ → C. In particular the nerve of a small category C is such, and the nerve itself is a functor on ´. Morphisms in ´ There is a nice system of generators as follows faces ∂i : [p − 1] → [p], the unique order-preserving injective map with i omitted from the image, degeneracies σj : [p + 1] → [p], the unique order-preserving surjective map identifying j and j + 1. Now, any injective map ε : [p] → [q] has a canonical factorization ε = ∂ir . . . ∂i1 , where i1 < . . . < ir are the elements omitted in the image of ε. Likewise any surjective map η : [p] → [q] has a canonical factorization η = σj1 . . . σjs , where j1 < . . . < js are the elements so that η(j) = η(j + 1). © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_3

9

10

3 Simplicial Objects

Thus any map θ in ´ factors as θ = ∂ir · · · ∂i1 σj1 · · · σjs . Axioms for simplicial objects are (essentially) rules for composition: a≤b

implies

∂a ∂b = ∂b+1 ∂a ,

a≤b

implies

σb σa = σa σb+1 , ⎧ ⎨ id, if a ∈ {b, b − 1}, a > b, σa ∂b = ∂b σa−1 , if ⎩ ∂b−1 σa , if a < b − 1.

for all a, b,

Proposition A simplicial object X : ´ → C in a category C is a collection Xp ∈ Ob C , for p ≥ 0 , together with morphisms di : Xp −→ Xp−1 , for i = 0, . . . , p, si : Xp −→ Xp+1 , for i = 0, . . . , p, so that a≤b

implies

db da = da db+1 ,

a≤b

implies

sa sb = sb+1 sa , ⎧ ⎨ id, for b ∈ {a, a + 1}, for a > b, db sa = sa−1 db , ⎩ sa db−1 , for a < b − 1.

for all a, b,

Given a morphism θ : [p] −→ [q], there is an induced θ∗ : Xp ←− Xq , where θ∗ is a composition. . . s d . . d . θ∗ = s .  . indices go down indices go up

Chains Let C be a small category and let F : C → Ab be a covariant functor to the category Ab of abelian groups. We shall presently define a simplicial abelian group. A motivating example is described by the diagram

···

X0 ←X1 ←X2

Precisely with



F(X2 )

X0 ←X1

= ⊕ the direct sum, define

F(X1 )

X0

F(X0 ) .

3 Simplicial Objects

11



d

Cp (C, F) =

F(Xp ) .

X0 ←...←Xp ∈ Np (C)

Given a monotone θ : [p] → [q], define θ∗ : Cq (C, F) −→ Cp (C, F) as follows. Suppose α is in the summand belonging to X0 ←− · · · ←− Xq inj

and let Xj −−→



Xj be the canonical map. Then

j

α = in(X0 ←···←Xq ) ξ , for ξ ∈ F(Xq ) , and we have θ∗ (X0 ←− · · · Xθ(p) · · · ←− Xq ) = Xθ(0) ←− · · · ←− Xθ(p) . u

Then there is a map Xq −→ Xθ(p) as part of this simplex, and we define θ∗ in(X0 ←···←Xq ) α = inθ∗ (X0 ←···←Xq ) F(u)(α) . Exercise If F as above is contravariant, then modify the construction to make sense of    F(X0 ) F(X0 ) F(X0 ) . ··· X0 ←X1 ←X2

X0 ←X1

X0

A co-simplicial object in C is a covariant functor ´ → C. Co-chains Suppose F : C → Ab is covariant. Construct

F(X0 )

X0

∂0

∂1





F(X0 )

X0 ←X1

as before, namely C p (C, F) =

F(X0 ) · · ·

X0 ←X1 ←X2



F(X0 ) ,

X0 ←···←Xp ∈Np C

and given θ : [p] → [q], define θ∗ : C p (C, F) → C q (C, F), for α ∈ follows

Np C

F(X0 ), as

12

3 Simplicial Objects

pr X0 ←···←Xq θ∗ α −→ F(u) pr θ∗ (X0 ←···← Xq ) α , where pr (X0 ←······←Xq ) denotes the canonical map. u  Xθ(0)



Let now α∈

F(X0 ) ∈ C p (C, F)

X0 ←···←Xp ∈Np C

and write α(X0 ← · · · ← Xp ) for pr X0 ←···←Xp α, so given ∂j : [p] → [p + 1], we have  (∂j α)(X0 ←− · · · ←− Xp+1 ) =

j ←− · · · ←− Xp+1 ) , for j = 0, α (X0 ←− · · · X for j = 0. F(u) α(X1 ←− · · · ←− Xp+1 ) ,

Suppose now given a simplicial abelian group · · · C3 and define d=

C2

C0 ,

C1

p−1  (−1)j dj : Cp −→ Cp−1 . j=0

Claim d 2 = 0. Proof For j ≥ i, we have dj di = di dj+1 , and d2 =

p−1 p   (−1)j dj (−1)k dk j=0

=



k=0

(−1)

j+k

dj dk

0≤j≤p−1 0≤k≤p

=



0≤j X p . If K is a simplicial complex, then let I be the poset of simplexes in K . Then the simplicial complex associated to I is the barycentric sub-division of K . Note that the usual complex of chains on the simplicial complex belonging to I I , A). is the non-degenerate part of C∗ ( Reference Dold–Puppe, Ann. Inst. Fourier (1961). Normalization Theorem Let {C p } be a simplicial abelian group. Then Cp =

p−1 

Im {s j : C p−1 −→ C p }

j=0

⊕

p 

Ker {d j : C p −→ C p−1 }

j=1 d

d

is a decomposition of the complex · · · → C p+1 −→ C p −→ C p−1 → · · · . Moredeg deg over calling the first summand C p , the degenerate sub-complex, {C p } has trivial homology. © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_4

15

16

4 Normalization and Conical Contractibility

Consequence C∗ −→ C∗ /C∗deg induces a homology isomorphism and in particular I , A) C p (



I , A)/C p ( I , A)deg C p (



A

homology isomorphism

x0 ≥...≥x p

A

x0 >...>x p

Conclusion For a constant functor A, H∗ ( I , A) is the same as the homology of the simplicial complex, and likewise for cohomology. Example 4.1 [ p] as a simplicial complex is ´( p) so that  H∗ ([ p], A) =

A, ∗ = 0, 0, else.

Example 4.2 The poset described by the Hasse diagram • •

• • has the “same” simplicial complex so  H∗ =

A, ∗ = 0, 1, 0, else.

Exercise Suppose {C p } is the simplicial abelian group . . . C2

C1

C0

and suppose there is s−1 : C p → C p+1 , for all p ≥ 0, so that the usual identities hold. Then H∗ C = 0 for ∗ > 0 . For example, given a simplicial abelian group

4 Normalization and Conical Contractibility

A3

17

A2

A0 ,

A1

define C p = A p+1 with d j , s j for C given respectively by d j+1 , s j+1 for A . Then {C p } is a simplicial abelian group and s−1 exists as in the exercise, so then H∗ C = 0. Why is H1 C = 0? Well, we have d

d

C2 −→ C1 −→ C0 with d s−1 = (d0 − d1 + d2 ) s−1 = id − s−1 d0 + s−1 d1 , and s−1 d = s−1 (d0 − d1 ) = s−1 d0 − s−1 d1 , so that d s−1 + s−1 d = id which implies H1 C = 0 . Exercise Generalize this. In general, the extra degeneracy map gives a chain homotopy id  0, and C is said to be conically contractible. Application  Suppose that C is a small category and Y a fixed object of C. Take A for A an object in Ab. Then C∗ (C, F) is F(X ) = X ←Y







A

X 0 ←X 1 ←X 2 X 2 ←Y

so we have

 X 0 ←X 1 ←X 2 ←Y





A

X 0 ←X 1 X 1 ←Y

A

 X 0 ←X 1 ←Y

A = C0 (C, F) ,

X 0 ←Y

A

d0

d1



A.

X 0 ←Y

We get an extra degeneracy map, where we have s p+1 here, so acyclic as above since conically contractible. Note that the direct sum of this construction over all Y gives the construction of the previous exercise. Also note C C together with H∗ C = 0 implies H∗ C = 0.

18

4 Normalization and Conical Contractibility

Cohomology Consider C ∗ (C, F) with F covariant 



∂0

F(X 0 )

∂1

X0

and take for fixed Y , F(X ) =

F(X 0 )

···

X 0 ←X 1



A, the product over all arrows, i.e., over all

Y ←X

morphisms. For this F, C ∗ (C, F) is 

σ0



A

Y ←X 0

σ1

··· ,

A

Y ←X 0 ←X 1

and we again have the extra degeneracy id

(σ−1 f )(Y ←− X 0 ←− · · · ←− X p ) = f (Y ←− Y ←− · · · ←− X p ), so the complex is acyclic. Thus we have constructed functors for H∗ and H ∗ which kill homology in that Theorem For all ∗ > 0, we have

  H∗ C, X −→ A = 0, X ←Y

 A = 0. H ∗ C, X −→ Y ←X

 be the category associated to the group G. Consider Example 4.1 Let C = G X −→



A

Y ←X

i.e., the image is



A, so this functor is the G-module Maps(G, A), where G acts

g∈G

on the right of G to give a left action on Maps(G, A). Example 4.2 X −→

 X ←Y

A=



A = Z[G] ⊗Z A ,

g

where we multiply on the left in Z[G] for the module structure.

4 Normalization and Conical Contractibility

Corollary

19

H ∗ (G, Maps(G, A)) = 0 for all ∗ > 0, H∗ (G, Z[G] ⊗Z A) = 0 for all ∗ > 0 .

Note that this corollary plus long exact sequences allow us to compute in practice.

Chapter 5

Effaceable δ-Functors

Suppose 0 → M  → M → M  → 0 is an exact sequence of G-modules. Then we get an exact sequence of complexes of cochains on G 0 −→ C ∗ (G, M  ) −→ C ∗ (G, M) −→ C ∗ (G, M  ) −→ 0 with values in these modules and hence get the usual long exact sequence in cohomology δ

0 −→ H 0 (G, M  ) −→ H 0 (G, M) −→ H 0 (G, M  ) −→ H 1 (G, M  ) −→ · · · which is natural

0

H0 M

H0M

H 0 M 

H1 M

···

0

H0N

H0N

H 0 N 

H1N

···

with respect to maps of exact sequences in the usual sense. Last time we saw that if A ∈ Ab, then the G-module Maps(G, A), that is, set maps, with the action (g f )(x) = f (xg), had the property that H ∗ (G, Maps(G, A)) = 0, for all ∗ > 0. A functor T : C → Ab is effaceable if for any object X in C there exists an injection X → Y in C so that T (Y ) = 0. Say a cohomology theory itself is effaceable if it is so in positive dimensions. Proposition H ∗ (G, ·) : Mod G → Ab is effaceable, for all ∗ > 0, where Mod G denotes the category of G-modules.

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_5

21

5 Effaceable δ-Functors

22

Proof Consider M −→ Mapsset (G, M) m −→ (x −→ xm) and check this is a G-module map; it is clearly an embedding.



Example of Free Groups Let G be the free group on {gi : i ∈ I }. Then a G-module is the same thing as an abelian group M with a family of automorphisms indexed by I . Recall that H 0 (G, M) = M G and H 2 (G, M) = isomorphism classes of extension of G by M. gi ∈ E. Since G is free, we get a section s to p Lift each gi ∈ G back to 

∗

M

E

p

G

∗,

s

and s is a group homomorphism. This says the extension splits, and E is the semidirect product of G acting on M. Note that in general split extensions of G by M correspond to the zero element in H 2 (G, M). The reason is that s(g1 ) s(g2 ) = i( f (g1 , g2 )) s(g1 g2 ) where f is the two-cocycle belonging to the extension, and f = 0 since s is a group map. Thus, H 2 (G, M) = 0, for all M if G is free. Claim H ∗ (G, M) = 0, for all M and all ∗ ≥ 2 if G is free. Proof Given M, embed M in N where H ∗ (G, N ) = 0 for all ∗ > 0 giving 0 −→ M −→ N −→ M1 −→ 0 . Take the long exact sequence. The proof follows from dimension shift and induction.  Recall that H 1 (G, M) = Z 1 (G, M)/B 1 (G, M) where Z 1 (G, M) = Der (G, M) = derivations G → M . Claim Der (G, M) = Maps (I, M). Proof A derivation is determined by the Dgi , and these can be assigned arbitrarily. To get D(g1−1 ), note that D(e) = 0 so 0 = Dg1−1 + g1−1 Dg1 so that Dg1−1 = −g1−1 D(g1 ). 

5 Effaceable δ-Functors

23

The exact sequence for cohomology of free groups gives m −→ (i −→ gi m − m) 0 −→ H (G, M) −→ M −→ Maps (I, M ) −→ H 1 (G, M) −→ 0 . 0

 MG

 Der (G,M)

For instance if M = Z with trivial G-action, then H 0 (G, Z) = Z and H 1 (G, Z) ∼ = Zn where n is the number of generators of G. This concludes our discussion for the example of free groups. Definition A δ-functor on the category of G-modules to abelian groups is a collection of so-called additive functors T q , for q ≥ 0, together with, for any sequence 0 → M  → M → M  → 0 of G-modules, an assignment of connecting homomorphisms δ : T q M  → T q+1 M  so that δ

(i) T 0 M  → T 0 M → T 0 M  − → T 1 (M  ) → · · · is a complex, (ii) naturality of connecting homomorphisms with respect to maps of exact sequences. As an exercise, generalize this definition so that the domain of the functor is any abelian category. In this definition, an additive functor means Hom G-mod (M, N ) −→ HomAb (T q M, T q N ) u −→ T q (u) is a homomorphism of abelian groups. Example 5.1 N −→ Hom G -mod (M, N ) is additive. Example 5.2 N −→ N ⊗Z[G] M is additive. Example 5.3 N −→ M ⊗G N is not additive. Theorem Suppose {T q } and {U q } are δ-functors from G-modules to Ab. Assume T q is effaceable for all q > 0 and {T q } is exact, i.e., (i) in the definition above is in fact exact. Then any natural transformation θ : T 0 → U 0 extends uniquely to a natural transformation of δ-functors. We shall prove this theorem in the next Chapter. Corollary Any two exact effaceable functors that are equal in dimension 0 agree everywhere.

5 Effaceable δ-Functors

24

Proof of Corollary Given M, find 0 → M → N → M1 → 0 so that T ∗ N = 0 for all ∗ > 0. Then T0M θ

U0M

T0N θ

U0N

T 0 M

δ

θ

U 0 M

T 1M

T 1 /N = 0

exists uniquely

δ

U1M

U1N

Now induct. It remains as an exercise to check functoriality and independence of choice of N . 

Chapter 6

(Co)homology of Cyclic Groups

Homology of Cyclic Groups Suppose G = Z/n Z, so Z [G] = Z [T ]/(T n − 1), T some indeterminate. This holds since Homring (Z[G], A) = Homgroups (G, A• ) where A• = groups of units of A and for G = Z/n Z A• = {a ∈ A : a n = 1} = Homrings (Z [T ]/(T n − 1), A). Now, T n − 1 = (T − 1)(T n−1 + . . . + 1), so we get two exact sequences of groups and the following identifications: (∗1 ) 0

(T − 1)/(T n − 1)

mult by T −1

Z [T ]/(T n − 1)

Z [T ]/(T − 1)

Z[T ]/(T n−1 + . . . + 1)

0

Z

where the isomorphism Z[T ]/(T n−1 + . . . + 1) → (T − 1)/(T n − 1) is given by 1 → T − 1; (∗2 ) 0

(T n−1 + . . . + 1)/(T n − 1)

mult by (T n−1 +...+1)

Z [T ]/(T n − 1)

Z [T ]/(T n−1 + . . . + 1)

0

Z[T ]/(T − 1) Z

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_6

25

26

6 (Co)homology of Cyclic Groups

where the isomorphism Z[T ]/(T − 1) → (T n−1 + . . . + 1)/(T n − 1) is given by 1 → (T n−1 + . . . + 1). Note that both these sequences split over Z since the terms are free. Note also that (∗2 ) is a “resolution” of the leftmost term in (∗1 ). Define I (G) = Z[T ]/(T n−1 + . . . + 1) = augmentation ideal for G = Z/n Z, Z(G) = Z[T ]/(T n − 1) = Z[G] for G = Z/n Z, as above. The sequences (∗1 ) and (∗2 ) become (∗∗1 )

0 −→ I (G) −→ Z(G) −→ Z −→ 0 ,

(∗∗2 )

0 −→ Z −→ Z(G) −→ I (G) −→ 0 .

Now suppose M is a G-module, G = Z/n Z, and tensor (∗∗1 ) and (∗∗2 ) with M to get two exact sequences of G-modules (†1 ) 0

I (G) ⊗Z M

(T −1)⊗m →1⊗m

Z[G] ⊗Z M

1⊗m →m

M

0

I (G) ⊗Z M

0

(†2 ) m →

0

M

n−1  i=0

Ti⊗m

Z[G] ⊗Z M

1⊗m →(T −1)⊗m

where the G-action on M is as usual and the G-module structure of A ⊗Z B is g(a ⊗ b) = ga ⊗ gb; check these are G-module sequences. Moreover, if M is a G-module, then Z[G] ⊗Z M with action x(g ⊗ m) = xg ⊗ m is isomorphic to Z[G] ⊗Z M with action x(g ⊗ m) = xg ⊗ xm, where g ⊗ m → g ⊗ gm and g ⊗ g −1 m  ↤ g ⊗ m  . Recall we showed H∗ (G, Z[G] ⊗Z A) = 0, for all ∗ > 0. We use this and long exact sequences to compute the homology of cyclic groups. Long exact sequences of (†1 ) and (†2 ) respectively give ∂

Hq (G, Z[G] ⊗ M) −→ Hq (G, M) −→ Hq−1 (G, I [G] ⊗ M) −→ Hq−1 (G, Z[G] ⊗ M)

so the first ∂ is an isomorphism for q ≥ 2, and ∂

Hq (G, Z[G] ⊗ M) −→ Hq (G, I (G) ⊗ M) −→ Hq−1 (G, M) −→ Hq−1 (G, Z[G] ⊗ M)

so the second ∂ is an isomorphism for q ≥ 2.

6 (Co)homology of Cyclic Groups

27

We thus get canonical isomorphisms Hq (G, M) −→ Hq−2 (G, M) ,

for all q > 2

and we finally compute Hq (G, M), for q = 0, 1, 2. Exercise H0 (G, M) = Z



M = M/{(g − 1)m}

Z[G]

i.e., the largest quotient on which the group acts trivially. Indeed, C1 (G, M)



C0 (G, M)

d0

M

M

d1

g

∗← −∗

and d0 : g ⊗ m −→ m , d1 : g ⊗ m −→ gm . Thus, H0 (G, Z[G] ⊗ M) = Z



(Z[G] ⊗Z M) = M .

Z[G]

Now, we have 0

H1 (G, I ⊗ M)

∂

H0 (G, M)

H0 (G, Z[G] ⊗ M)

H0 (G, I ⊗ M)

M/{(g − 1)m}

M/(T − 1)M mult by

n−1 

M Ti

i=0

The map given by multiplication by

n−1  i=0

T i is called the norm: M → M.

0

28

6 (Co)homology of Cyclic Groups

Thus H0 (G, I ⊗ M) = M/Im{norm : M → M} and H1 (G, I ⊗ M) =

Ker{norm : M → M} . Im{(T − 1) : M → M}

Since H2 (G, M) = H1 (G, I ⊗ M), we get H2 (G, M) =

Ker{norm : M → M} Im{(T − 1) : M → M}

H1 (G, M) =

Ker{(T − 1) : M → M} Im{norm : M → M}

Exercise

Cohomology of Cyclic Groups Recall H ∗ (G, Maps (G, A)) = 0, for all ∗ > 0, and for G finite abelian Maps (G, M) = HomZ (Z[G], M) = HomZ (Z[G], Z) ⊗Z M . Thus, from (∗∗1 ) and (∗∗2 ), which we recall are split, we get two exact sequences M

0

HomZ (Z, M)

HomZ (Z[G], M)

HomZ (I, M)

0

0

HomZ (I, M)

HomZ (Z[G], M)

HomZ (Z, M)

0

M and as above HomZ (Z[G], M) has trivial higher cohomology. Exercise (if interested) ∼ =

H q (G, M) −→ H q+2 (G, M) ,

for q ≥ 1,

H 2 (G, M) =

Ker {(T − 1) : M → M} , Im {norm : M → M}

H 1 (G, M) =

Ker {norm : M → M} , Im {(T − 1) : M → M}

6 (Co)homology of Cyclic Groups

29

and we know H 0 (G, M) = M G , as before, completing this discussion of cyclic groups. Recall the Theorem Suppose {T q , q ≥ 0} is an exact δ-functor and T q is effaceable for all q > 0 and {U q , q ≥ 0} is any δ-functor. Then any natural transformation θ0 : T 0 → U 0 extends uniquely to a natural transformation of δ-functors. Proof It suffices to extend to q = 1 and check properties by induction. 1: Uniqueness: Suppose given M and find exact 0 −→ M −→ M  −→ M  −→ 0 so that T 1 M  = 0 . Then we have T 0 M

T 0 M 

θ0

U 0 M

δ

θ0

U 0 M 

T 1M

T 1 M = 0

θ1 δ

U1M

with the top row exact. A diagram chase gives uniqueness of θ1 . Defining θ1 as before in the obvious way, we must check: Claim 6.2 Well-defined. Claim 6.3 Natural transformation. Claim 6.4 Compatible with δ. Check 6.2 We have the Claim Given M → M  and M → M1 with T 1 M  = T 1 M1 = 0, then we can embed these maps in a diagram M M

M1

M2

with T 1 M2 = 0. 2 = M  ⊕ M1 /(Im M ⊕ −Im M) and embed Proof Take the push out, i.e., take M   2 → M2 by effaceability.  it M

30

6 (Co)homology of Cyclic Groups

Thus to show θ1 : T 1 M → U 1 M is the same for M → M  and M → M1 , we may assume M M

M1

M commutes. Thus we have 0

M

M

M 

0

0

M

M1

M1

0

which gives the following cube of arrows δ

TM0 

T 1M

id

id δ

T 0 (M1 )

θ0

θ0

U 0 M 

δ

θ1

T 1 (M)

U1M

θ11 id

id

U 0 (M  )

U 1 (M)

δ

and this all commutes by definition except perhaps the rightmost face, which thus also commutes, as desired. Check 6.3 Suppose we have 0

M

M

M 

0

M1

M1

0

u

0

M1

We get the same cube as above with identity maps replaced by u ∗ and all appearances of M in parentheses having subscript “1”. Check 6.4 We must check that an exact 0 −→ M −→ N −→ Q −→ 0

6 (Co)homology of Cyclic Groups

31

gives a commutative T0Q

δ

T 1M

θ0

U0Q

θ1 δ

U1M

To this end we resolve M

N

M so that T 1 M  = 0. Exercise Finish the argument.



Chapter 7

An Application to the Schur–Zassenhaus Theorem

We have the Theorem Given {T q } an exact δ-functor with T q effaceable, for q > 0, and {U q } any δ-functor. Then any natural transformation θ0 : T 0 → U 0 extends uniquely to a natural transformation of δ-functors. Example 7.1 Suppose f : H → G is a group homomorphism. Then any G-module M is also an H -module, denoted f ∗ M. We define two δ-functors T q (M) = H q (G, M) U q (M) = H q (H, M) with M regarded as f ∗ M. from the category of G-modules to the category of abelian groups. Automatically a short exact sequence of G-modules is a short exact sequence of H -modules. Then MG 

MH 

H 0 (G, M) ⊂ H 0 (H, M) is a natural transformation of δ-functors. By the theorem, there exists a unique morphism of δ-functors Res f :H →G : H q (G, M) −→ H q (H, M) extending from G to H . u

f

Formulae Given K −→ H −→ G, we have Res K →M Res H →G = Res K →G by the uniqueness assertion, since it is true in degree 0. We could define the restriction on the cochain level, and this must agree, again by uniqueness. Example 7.2 Transfer Map Let H < G be a subgroup of finite index and let M be a G-module. We have the transfer H 0 (H, M) −→ H 0 (G, M) ,  MH

 MG

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_7

33

34

7 An Application to the Schur–Zassenhaus Theorem

defined as follows. If m ∈ M H and g ∈ G, then gm depends only on the coset g H of g, where (gh)m = g(hm) = gm. Thus if g1 , . . . , gn are coset representatives, so n  G = gi H , then the element i=1

n 

gi m

i=1

is independent of the choice of coset representatives. Write it as

 g H ∈G/H

(g H ) m.

This element is invariant under all g ∈ G. In this way we get a homomorphism M H −→ M G called the transfer, denoted variously by Ver H →G , Ind H →G . Check the hypotheses of the theorem so as to extend to higher dimensions. Exactness is clear, and the only point that remains to be checked is that H q (H, ·) from the category of G-modules is effaceable. As before, we have M −→ Maps (G, M) , and it is enough to show that for any abelian group A that as an H -module 

Maps (G, A) =

Maps (g H, A)

g H ∈G/H



 Maps (H, A)

g H ∈G/H

and Maps (H, A) has trivial H + (H, Maps(H, A)), as proved at the end of Chapter 4. We thus obtain Ver H →G : H ∗ (H, M) → H ∗ (G, M). Formulae Exercise 7.1 Transitivity If K < H < G with [H : K ] < ∞

[H : G] < ∞ ,

then Ver H →G Ver K →H = Ver K →G .

7 An Application to the Schur–Zassenhaus Theorem

35

Exercise 7.2 If H < G is finite index, then Res H →G

Ver H →G

H ∗ (G, M) −−−−−→ H ∗ (H, M) −−−−−→ H ∗ (G, M) is multiplication by [G : H ]. Exercise 7.3 Mackey Formula Suppose −→ h ghg −1 K ∩ g H g −1 −→ H ∩ ∩ finite index K −→ G inclusion

Then Res K →G Ver H →G =



. K ∩ g H g −1 → H −1 x → g xg

Ver K ∩g H g−1 →K Res

K g H ∈K \G/H

This is not fantastically useful. Interesting special case: H = K  G a normal subgroup of finite index, then Res H →G Ver H →G =



. H→H h → g −1 hg

Res

g H ∈G/H

Proof of 2 Check in degree zero and use the Theorem: Res

Ver

M G −−→ M H −−→ m −→ m −→

G M (g H ) m = [G : H ] m for m ∈ M G .

g H ∈G/H

 Corollary of 2 Suppose that G is a finite group and M is a G-module so that multiplication by |G| is an isomorphism of M. Then H + (G, M) = 0, where H + means H ∗ for ∗ > 0. Proof Take H = {e}. Then we have the commutative H q (e, M) Res

H q (G, M)

Ver mult by |G|

H q (G, M)

Multiplication by |G| is an isomorphism and H ∗ (e, M) = 0, for q > 0, as the trivial group e is hilariously free. 

36

7 An Application to the Schur–Zassenhaus Theorem p

Theorem (Schur–Zassenhaus) Any extension ∗ −→ H −→ G −→ Q −→ ∗ of finite groups, where |H | and |Q| are relatively prime, splits, i.e., there exists a homomorphism s : Q → G so that ps = id. In particular, if H  G, for G finite and |H | relatively prime to [G : H ], then there exists a subgroup complementary to H . Proof (in the spirit of finite group-theory) Let G be a minimal counter-example. Let P be a Sylow p-subgroup of G for a prime p dividing |H |. Then since |H | and |Q| are relatively prime, we must have P ⊂ H . Claim NG (P) maps onto Q, for, given q ∈ Q, lift it to g; compare g Pg −1 and P, both Sylow p-subgroups of H , so they are conjugate in H and there exists h ∈ H with g Pg −1 = h Ph −1 which implies h −1 g P(h −1 g)−1 = P, so h −1 g ∈ NG H , the p normalizer of H in G, and h −1 g −→ q. Thus we have ∗

NH P

NG P

Q

∗

∗

H

G

Q

∗

where the kernel of the top map is N H P. By minimality, if NG (P) < G, then NG (P) would contain a subgroup mapping onto Q, a contradiction since the top sequence would then split, whence the bottom as well. We conclude that NG P = G . Transitivity argument Suppose we can find a subgroup K so that 1 < K < H with K  G. Then ∗ −→ H/K −→ G/K −→ Q −→ ∗ . By minimality, this splits. Let A ⊂ G/K map isomorphically onto Q. Then A = B/K where B  G with B  Q. It is therefore an extension of Q by H ∩ B, which is of order prime to Q. By minimality again, this extension ∗ −→ B ∩ H −→ B −−− Q −→ ∗ splits, giving a section Q → B and hence a section Q → G, contradiction. Thus K as above does not exist. We conclude that for a minimal counter-example G, H has no proper subgroups K which are normal in G. In particular, P, which is the unique Sylow p-subgroup of G since it is normal in G must be all of H , so P = H . Since P has no characteristic subgroups, i.e., invariant under all automorphisms of the group, it must be an elementary abelian p-group.

7 An Application to the Schur–Zassenhaus Theorem

37

Now, use that H 2 (Q, H ) classifies the extensions for H abelian. H is a p-group and p is relatively prime to |Q|, so the Corollary of 2 above implies that H + (Q, H ) = 0. A minimal counter-example therefore cannot exist. 

Chapter 8

The Yoneda Lemma

Let C be some category and fix some X ∈ Ob C. Define C op = opposed category, which has the same objects but with HomC op (X, Y ) = HomC (Y, X ). Define h X : C op → Sets, i.e., just a contravariant functor C → Sets, given by h X : Y −→ HomC (Y, X ) . Define h X : C −→ Sets Y −→ HomC (X, Y ) . Yoneda Lemma If F : C op → Sets is a functor, then the set NatTrans(h X , F) of natural transformations from h X to F is in one-to-one correspondence with F(X ) given by (θ : h X → F) −→ θ(X )(id X ) θ so that θ(Y )( f ) = F( f )ξ ←− ξ. In a diagram, Hom(Y, X ) = h X (Y )

θ(Y )

F( f )

f∗

Hom(X, X ) = h X (X )

F(Y )

θ(X )

F(X )

where θ(Y )( f ) = F( f ) θ(X ) (id X ) .

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8 The Yoneda Lemma

Corollary NatTrans(h X , h Y ) is in one-to-one correspondence with Hom(X, Y ). Modulo set-theory, we have a functor as follows C −→ Funct(C op , Sets) X −→ h X , where Funct as a category has its morphisms given by natural transformations, and the corollary says HomFunct (C op ,Sets) (h X , h Y ) = HomC (X, Y ) . Thus X → h X gives an embedding of C in Funct(C op , Sets). Definition A functor F : C op → Sets is representable if it is isomorphic to a functor of the form h X . According to Yoneda’s Lemma, an isomorphism of h X  F is given by some ξ ∈ F(X ). Thus F is representable when there exist X ∈ Ob C and ξ ∈ F(X ) so that for all Y ∈ Ob C, 

HomC (Y, X ) −→ F(Y ) f −→ F( f ) ξ . This says (X, ξ) have the following universal property: for any Y ∈ Ob C and η ∈ F(Y ) there is a unique map f : Y → X so that η is induced from ξ. General exercise Re-write universal constructions in terms of representable functors. Example 8.1 A final (or terminal) object of C, call it e, is an object so that for all Y ∈ C there exists a unique Y → e, and e represents the functor Y → ∗. Example 8.2 The product  I X i ∈ Ob C of X i ∈ Ob C, for i ∈ I , equipped with maps pri :  I X i → X i , i ∈ I , has the universal property that for all Y and f i : Y → f

X i , for all i ∈ I , thereexists a unique map Y −→ X i so that pri f = f i . The functor represented is h X i . i∈I

Example 8.3 Let I be a small category and i → X i a functor from I to C. Then lim X i denotes an object in C equipped with maps ← − pi : lim X i −→ X i ← − compatible with the morphisms in I in the sense that for all α : i → i  in I , the diagram

8 The Yoneda Lemma

41

pi

lim X i ← −

Xi

pi

X i

α∗

commutes and enjoying the obvious universal property. The “new” terminology for this is simply lim = lim and lim = colim. The functor being so represented assigns ← − − → to Y all  Hom(Y, X i ) ( fi ) ∈ i ∈ Ob I

so that for all α : i → i  we have α∗ f i = f i  . We can write this as ⎧ ⎨  Ker Hom(Y, X i ) ⎩ i∈Ob C



Hom(Y, X i  )

α

i  ←− i

⎫ ⎬ ⎭



  α i  ←− i −→ f i   

f (i) α −→ i  ←− i −→ α∗ f i −→

where Ker here denotes equalizer, i.e., elements having the same image under each map. If this kernel is written lim Hom(Y, X i ) then ← − I

 HomC Y, lim X i = lim HomC (Y, X i ) . ← − ← − I

Note that if F : C → Sets, then lim F = Ker ← −





F(X 0 )

X 0 ←X 1

X0

C

F(X 0 )

,

and this coincides with H 0 (C, F) for F an abelian group functor, i.e., in general H 0 (C, F) = lim F . ← − C

For example, a G-set is a functor  −→ Sets G ∗ −→ S

42

8 The Yoneda Lemma

and then lim S = S G . ← −  G

In general by Yondea, we have HomFunct (C,Sets) (h X , F) = F(X ) and get an embedding C op −→ Funct(C, Sets) X −→ h X . An initial object φ is so that Hom(φ, Y ) = ∗ for all Y . For instance:  in i  (1) Direct sum I X i : for all i ∈ I , we have X i −−→ I X i with the appropriate universal property. in i

(2) Direct limit (or colimit): lim X i together with Yi −−→ lim X i compatible with the − → − → I I arrows of I and the appropriate universal property.    (1) represents Hom  I X i , Y =  I Hom(X i , Y ), and (2) represents HomC lim X i , Y − → I

= lim HomC (X i , Y ). ← − I

 a functor S : G  → Sets is the same as a G-set S, and a Example With C = G,  functor M : G → Ab is a G-module M, where lim S = S G and lim M = M G . ← − ← −  G

 G

But what about lim S = ? and lim M = ? − → − →  G

 G

Now, lim S is a set together with arrows S → lim S compatible with the arrows in − → − →  G

 that is, G, S in g

lim S − → in

S Thus lim S is the set of orbits G\S, and − →  G

 G

8 The Yoneda Lemma

43

lim S = M/{gm − m : g ∈ G, m ∈ M} − →  G

= largest quotient module of M on which G acts trivially. Thus, we must specify the category in which we are computing lim, as they do not − → in general agree. Adjoint Functors Suppose C and C  are categories. A pair of adjoint functors is C

F G

C  together

with a natural bijection ∼

(∗)

HomC  (F(X ), Y ) −→ HomC (X, G(Y )).

for each X ∈ Ob C, Y ∈ Ob C  . F is the left adjoint and G right adjoint of the pair. By Yoneda, (∗) is given either by natural transformations α : F G −→ idC  or β : idC −→ G F . Easy to check that F ·β

α· F

β·G

G ·α

F = F idC −−→ F G F −−→ idC  F = F and

G = idC G −−−→ G F G −−−→ G idC  = G are both the identity. Thus we could alternatively define adjoint functors to be (F, G) together with α and β so that these composites are the identity. To see this given α, β, we have β∗

G

F

α∗

Hom(F X, Y ) −−→ Hom(G F X, GY ) −−→ Hom(X, GY ) − → Hom (F X, F GY ) −−→ Hom(F X, Y )

By Yoneda, take Y = F(X ). Then id X −→ id G F X

and

β −→ F · β −→ α F · β ,

and equivalence of the two definitions follows. Kan Formulae for Computing Adjoint Functors Let C and C  be small categories and f : C → C  . Consider f∗

Funct (C, Sets) ←−− Funct (C  , Sets)

44

i.e.,

8 The Yoneda Lemma

( f ∗ F)(X ) = F( f (X )) .

This f ∗ has two adjoints f ! and f ∗ . Theorem f ∗ has a left adjoint f ! (pronounced f shriek) and right adjoint f ∗ . We shall discuss this next time.

Chapter 9

Kan Formulae

Suppose C

F G

C  is a pair of adjoint functors where we are given natural trans-

formations HomC  (F X, Y )  HomC (X, GY ) . Example 9.1 (Sets)

F= free group G= forgetful

(Groups)

and Homgroups (F S, G) = HomSets (S, G forgotten ) . u

Example 9.2 Suppose A −→ B is a homomorphism of rings with unit, and Mod A is the category of (left) A-modules. We have u∗

Mod A ←−− Mod B where u ∗ is restriction of scalars with respect to u, i.e., M → M and we have am = u(a)m. Then u ∗ has a left adjoint given by base extension Mod A −→ Mod B , i.e., M −→ B ⊗ A M B is a right A module via u, and Hom B (B ⊗ A M, N ) = Hom A (M, u ∗ N ). And u ∗ has also a right adjoint

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_9

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46

9 Kan Formulae

Mod A −→ Mod B M −→ Hom A (B, M) = { f : B → M | f (u(a)b) = u f (b) and f is additive} where Hom A (B, M) is a left B-module by b f (b1 ) = f (b1 b), i.e., use right action on B, and

Hom B (N , Hom A (B, M)) = Hom A (u ∗ N , M) .

Exercise Check this. The idea is that the left-hand Hom A (B ⊗ B N , M) and B ⊗ B N = N ; see Cartan and Eilenberg.

side

is

Summary of Example 9.5: M −→ B ⊗ A M Mod A

u∗

Mod B

M −→ Hom A (B, M) with canonical maps; recall Yoneda’s lemma that canonical maps are given by F G → id and G F ← id u ∗ (B ⊗ A M) ←− M 1⊗m ←− m and

B ⊗ A u ∗ N −→ N b ⊗ n −→ n

u ∗ Hom A (B, N ) ←− N Hom A (B, u ∗ N ) −→ N f −→ f (1) (b → bn) ←− n.

Example 9.3 Consider the category of topological spaces and continuous maps. Let I be compact. Then Hom(X × I, Y ) =

Hom(X, Y I )

with Y I given the compact-open topology. Changing gears now, consider the following situation: C and C  are small catef∗  = Funct(C, Sets), so we have C  ←− gories and f : C → C  a functor. Set C − C , G ◦ f ← G.

9 Kan Formulae

47

Kan’s Formula1 f ∗ has left adjoint f ! and right adjoint f ∗ given by ( f ! F)(Y ) =

F(X )

lim − →

(X,u : f X →Y )

( f ∗ F)(Y )

lim ← −

F(X ) .

(X,Y → f X )

For Y ∈ Ob C  , consider the category whose objects are pairs (X, u), X ∈ Ob C and u : f X → Y a morphism in C  and fX u   ⏐ Y commutes Hom((X, u), (X  , u  )) = v : X → X  |  f(v) f X

u 

Call this the left-fiber of f : C → C  over Y . As an exercise, we can check that this is a category. Similarly, we have the right-fiber of f : C → C  over Y consisting of all pairs (X, u : Y → f X ), again a category . Each of these is different from the fiber of f : C → C  over Y , which consists of all X ∈ Ob C with f (X ) = Y , and morphisms sit over the identity of Y . The fiber is denoted simply f −1 Y . Note that f −1 Y is a subcategory of both the left fiber and the right fiber. Example 9.4 Suppose C and C  are discrete categories, i.e., objects are sets and all arrows are identity maps. Then f : C → Sets is a family of sets F(X ) indexed by X ∈ Ob C. So we can think of F as just a set over Ob C, i.e., F , so in this case Cis ↓ Ob C

the category of sets over Ob C, and given F

Ob C

G f

Ob C 

we have ( f ∗ G)(X ) = G( f X ), so f ∗ G is just the usual pull back of G via f . Furthermore f ! F = F is viewed as a set over Ob C  via f since f ! (F)(Y ) =  F(X ), and this is just Kan’s formula X ∈ f −1 Y

1 For

f ! this formula also appears in SGA4 Éxp. 1 by Grothendieck and Verdier [6].

48

9 Kan Formulae

( f ! F)(Y ) =

lim − →

F(X ) but a discrete category so OK.

(X, f X →Y )

As to f ∗ , we have

HomC (G, f ∗ F) = HomC ( f ∗ G, F)

but

with f ∗ G = f −1 Y ⊂ Ob C

Hom(G, f ∗ F) = ( f ∗ F)(Y ) and

Homsets over Ob C ( f ∗ G, F) = sections of F over f −1 Y  F(X ) . = X ∈ f −1 Y



In this case ( f ! F)(Y ) =

F(X ) ,

X ∈ f −1 Y



( f ∗ F)(Y ) =

F(X ) .

X ∈ f −1 Y

 with f : G → G  a group homomor and C  = G Example 9.5 Suppose C = G phism. Then C is the category of G-sets, C the category of G  -sets, and f!

−→ f∗

(G-sets) ←− (G  -sets) −→ f∗

where essentially f ∗ S = S ◦ f and f ∗ S = S viewed as a G-set via f . Furthermore, we have d f ! S = G  ×G S = (G × S)/(g  , gs) ∼ (g  f (g), s) and in fact

Hom G  −set (G  ×G S, T ) = Hom G-set (S, f ∗ T ).

We also have that f ∗ S = Hom G-set (G  , S) = {h : G  → S | h( f (g)g  ) = gh(g  )} is a G  -set with G  acting by (g  h)(g1 ) = h(g1 g  ) , in analogy with the case of rings.

9 Kan Formulae

49

⎧ ⎪ ⎨

The Kan formulae are

( f ! F)(Y ) = where

⎫ ⎪ ⎬

lim F(X ) = Coker F(X 0 ) ⇔ F(X 1 ) − →u ⎪ ⎪ ⎩ u ⎭ fv u (X, f X − →Y ) Y← − f X0 Y← − f X 0 ←− f X 1   d0 − B = A/(d0 b ∼ d1 b) . Coker A ⇔ − d1

For the case of groups

f ! S(∗ ) = Coker

⎧ ⎪ ⎨

S(∗) ⇔



⎪ ⎩ u∈G  u  ∈G  v∈G ∗ ←−−∗ ∗← −−−∗ ←−−∗    = Coker G × S ⇔ G  × G × S

(u f (v), s) ← (u, v, s) (u, vs) ← = G  ×G S . As an exercise, check Kan also for f ∗ , i.e., confirm that lim = Hom G  (G, S) . ← − u X,Y − →f X

⎫ ⎪ ⎬ S(∗)

⎪ ⎭

Chapter 10

Abelian and Additive Categories

Abelian Categories For example, Mod A = (left) A-modules or functors Funct(C, Mod A ) with C a small category. In Mod A , we have that Hom A (M, N ) is an abelian group in a natural way and have a notion of exact sequences. We have the same structures in functors Funct(C, Mod A ), namely ⎧ ⎫ ⎪ ⎪ ⎨  ⎬  HomFunct (F, G) = Ker Hom A (F(X ), G(X )) ⇒ Hom A (F(X ), G(Y )) , ⎪ ⎪ ⎩ X ∈Ob C ⎭ u X −→Y in C

where the sum of θ : F → G and η : F → G is (θ + η)(X ) = θ(X ) + η(X ). Moreover, a sequence of functors F → F  → F  from C to Mod A is exact when F(X ) → F  (X ) → F  (X ) is exact for all X ∈ Ob C. Question Can we get at the abelian group structure on Hom A (M, N ) more intrinsically in terms of the category Mod A ? If X is an object of C, then a group (respectively ring, lattice, etc.) structure on X is a group (etc.) structure on the functor h X = HomC (·, X ), i.e., for all Y ∈ Ob C we specify a group (etc.) structure on HomC (Y, X ) so that for all u : Y → Y  in C, we have a group (etc.) homomorphism HomC (Y  , X ) → HomC (Y, X ). Put still differently, a group structure on X is a lifting Groups forgetful

C op

hX

Sets

of h X to a functor C op → Groups. Suppose now C has a terminal object e and that the products X × X and X × X × X exist. Note that the group law gives h X × h X → h X , i.e., © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_10

51

52

10 Abelian and Additive Categories group mult

Hom(Y, X )× Hom(Y, X )

Hom(Y, X )



Hom(Y, X × X ) where the vertical map is given by f → (pr1 f, pr2 f ), and so by Yoneda’s lemma, this natural transformation h X ×X → h X is given by a morphism m : X × X → X . The unit element gives a natural transformation Hom(Y, e) = point → 1

Hom(Y, X ), and by Yoneda’s lemma, this comes from e −→ X . μ Finally and similarly, the group inverse is given by a map X −→ X . Then the group axioms correspond to the following diagrams commuting: id×m

X×X×X

X×X m

m×id m

X×X

X

⎫ (id,1) m X −−−→ X × X −→ X ⎪ ⎪ ⎪ ⎪ a −→ (a, 1) −→ a1⎪ ⎬ are id X , where 1 : X → X is short unique 1 ⎪ for X −−−−→ e − →X (1,id) ⎪ m ⎪ X −−−→ (X × X ) −→ X ⎪ ⎪ ⎭ a −→ (1, a) −→ 1a X

(id,μ)

X×X

m

X

1

X

(μ,id)

X×X

X

1

In Mod A , every object N is an abelian group object in a natural way because Hom A (·, N ) has a natural abelian group structure. A cogroup structure on X in C is the same as a group structure on X in C op , i.e., a group structure on Hom(X, ·) : C → Sets. Such a structure is given by X −→ X



X, comultiplication,

X −→ φ, φ = initial object, X −→ X, an inverse satisfying the appropriate diagrams.

10 Abelian and Additive Categories

53

Thus, in Mod A , every object is naturally a co-abelian group object since Hom A (M, ·) is naturally an abelian group. Example Any free group is a co-group in the category of groups since HomGroups (F(S), G) =



G,

S

where F(•) denotes free group on •. Exercise Interpret the arrows above in this setting. Theorem Suppose we have a monoid object X and a co-monoid object Y in a category C. Then the two possible monoid structures on HomC (Y, X ) coincide and this unique operation is abelian. Standard example C = category of spaces with basepoint and homotopy classes of basepoint-preserving maps. Take G to be an H -space (e.g., G = X , the loop space) and Y to be S 1 , which is a co-group because homotopy classes of maps [S 1 , X ] = π1 X . Then the theorem says the two operations on [S 1 , G] = π1 G coincide and are abelian. Proof of Theorem Denote γ

Y − →Y



Y, comultiplication

ε

Y − → φ, co-unit ε

unique

 ε :Y − → φ −−−−→ Y Thus we have

γ

Y

Y



( ε, id)

Y

(id, ε)

X

id

Apply the monoid-valued functor Hom(·, X ) to these arrows Hom(Y, X ) × Hom(Y, X ) ∼ =

Hom(Y θ



Hom(Y, X )

γ∗

Y, X )

We thus get a map M × M −→ M, where M is the monoid Hom(Y, X ) with operation coming from X × X → X . θ is a monoid homomorphism so that θ(m, 1) = m = θ(1, m) since ε behaves as a co-unit for γ. Now,

54

10 Abelian and Additive Categories

θ(m 1 , m 2 ) = θ((m 1 , 1)(1, m 2 )) = θ(m 1 , 1) θ(1, m 2 ) = m 1 m 2 , so θ has same effect as the product on M. Remains to check abelian. To see this, m

flip

m

replace X × X −→ X by X × X −−→ X × X −→ X .



In Mod A , we have φ = e since the 0 module is both initial and final. Also, the α canonical map M N −→ M × N is an isomorphism, where α is given by pr 1 α in1 = id M , pr 1 α in2 = 0 , pr 2 α in1 = 0 , pr 2 α in2 = id N . Moreover, the co-monoid (in fact, co-abelian group) structure on M is given by

α Δ M −→ M × M D GGGG M M, ∼ = and the abelian group structure is given by α−1

M × M −−→ M



fold

M −−→ M .

We are led to define an additive category as a category A so that (1) (2) (3) (4)

A has an object 0 which is both initial and final, for any M, N ∈ Ob A, M N and M × N exist, α the canonical map M N −→ M × N is an isomorphism, HomA (M, N ) is an abelian group for all M, N ∈ Ob A, with HomA (·, ·) given the natural structure as below. Notice that (2) and (3) imply that HomA (M, N ) is an abelian monoid with f + g defined as

( f,g) Δ α−1 M −−−→ N M −→ M × M −−→ M or equivalently as (fold) ◦ α−1

( f,g)

M −−−→ N × N −−−−−−→ N . We denote it as M ⊕ N = M N ≈ M × N . For A an additive category and f : M → N in A, we define a cokernel for f as a pair C, p, where p : N → C in A so that for all objects X p∗

f∗

0 −→ Hom(C, X ) −−→ Hom(N , X ) −−→ Hom(M, X ) is exact, i.e., u ◦ f zero implies u factors through C

10 Abelian and Additive Categories

55 f

M

p

N

C

u

X Dually, a kernel for f : M → N is a map i : K → M so that for all X f∗

i∗

0 −→ Hom(X, K ) −→ Hom(X, M) −−→ Hom(X, N ) is exact, i.e., f ◦ u zero implies u factors through K X u i

K

M

f

N

Now given f , take the kernel and cokernel to get Ker f

i

M

f

N

p

Cok f

unique

Cok i

unique

Ker p

Ker p is called the image Im f and Cok i the coimage Coim f of f . We define an abelian category as an additive category A so that (5) for any map f , Ker f and Cok f exist, (6) for any map f , the canonical map Coim f → Im f is an isomorphism.

Chapter 11

Diagram Chasing in Abelian Categories

Example (standard example of additive but non-abelian category) Let A be the category of topological abelian groups. Then HomA (A, B) is itself an abelian group. Checking Condition (5), suppose f : A → B, then Ker f is the usual set-theoretic kernel with the topology induced by A. Cok f is the group B/ f (A) with the quotient topology, so the topology is Hausdorff for f (A) closed. Condition (6) breaks down however: the map id

Rdiscrete −→ Rusual from R with the discrete topology to R with the usual topology has Ker = Cok = 0, while Im = Rusual , Coim = Rdiscrete , i.e., Im = f (A) has the subspace topology and Coim f = f (A) ∼ = A/Ker f has the quotient top. Example If C is a small category, then A = Funct (C, Ab) is an abelian category. We can think of functors as diagrams of abelian groups indexed by C. Suppose u : F → G is a natural transformation, then Ker u is computed as (Ker u)(X ) = Ker (u(X ) : F(X ) −→ G(X )) , and similarly,

  u(X ) (Cok u)(X ) = Cok F(X ) −−−→ G(X ) .

As for checking Condition (6), we have Ker u(X )

F(X )

Coim u(X )

u(X )

G(X )

Cok u(X ) .

Im u(X )

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11 Diagram Chasing in Abelian Categories

Note that sheaves of abelian groups on topological spaces is also an abelian category, but this requires some work that we shall skip. Let A be an abelian category. Then f : X → Y is an epimorphism if Cok f = 0 (i.e., if for all T , Hom(Y, T ) → Hom(X, T )), and is a monomorphism if Ker f = 0 (i.e., for all T , Hom(T, X ) → Hom(T, Y )). The adjectival form of epimorphism is epic and of monomorphism is monic. If f is epic, then we write A  B, and if f is monic, then we write A → B. If f is epic, then we have f

X

Ker f

Coim f

Y

0

Im f

so that Y = Cok(Ker f −→ X ). f

This has the consequence that if we are given X −− Y , in order to construct Y → T , it suffices to construct X → T that carries Ker f to zero. This is the basis of the General Principle Any of the usual diagram-chasing arguments used for modules can be extended to an arbitrary abelian category. Below we indicate a proof of the Serpent Lemma in an arbitrary abelian category to illustrate how this is done. Proposition (i) The pull back of an epic f : Y → Z by any map X → Y is again epic, i.e., X ×Z Y

g

Y

f

f g

X

f epic implies f  epic.

Z

(ii) Dually, the push out of a monic f : A → B by any map A → C is again monic, i.e., g

A

C f

f

B

g

B +A C

f monic implies f  monic.

11 Diagram Chasing in Abelian Categories

59

Proof C is abelian if and only if C op is abelian, so we prove only (ii).  ( f,−g)  We define B + A C = Cok A −−−−→ B ⊕ C . Then by definition of Cok, we have an exact sequence 0

Hom(B + A C, T )

Hom(B ⊕ C, T )

( f,−g)∗

Hom(A, T )

f ∗ −g ∗

Hom(B, T ) ⊕ Hom(C, T ) Thus, a map B + A C → T is the “same” as maps B → T and C → T which agree on A, and so B + A C is the usual push out. ( f,−g)

g

Suppose now that A −→ C is monic. Then we claim A −−−−→ B ⊕ C is monic. Suppose we have T A B ⊕C C so T → C is the zero map 0

and A → C is monic which implies that T → A is zero, proving the claim. We proved before that X → Y epic implies Y = Cok (Ker f → X ), and dually for A → B ⊕ C monic, we have A = Ker {B ⊕ C → B + A C}. We thus have the commutative diagram Ker g  0

unique h

B in1

A

( f,−g)

B ⊕C

−g

g

B +A C

pr 2

C But −g monic and −g ◦ h = 0 implies h = 0, so Ker g  → B is the zero map whence Ker g  = 0, as desired.  To substantiate the general principle above, we indicate a proof of the serpent lemma in an abelian category A. Serpent Lemma Given a commutative diagram

60

11 Diagram Chasing in Abelian Categories

A u

0

u 

u

B

0

A

A

B 

B

Then there exists an exact sequence δ

Ker u  −→ Ker u −→ Ker u  −→ Cok u  −→ Cok u −→ Cok u  . The interesting part of the proof is the construction of δ, which we next discuss. Take the pull back A × A Ker u 

ξ

Ker u 

A

A and by the usual diagram chase, we get a map ∂

A × A Ker u  −→ Cok u  . Now, Ker u  = Coker (Ker ξ → A × A Ker u  ) so by the remark before the general principle above, to get δ : Ker u  → Cok u  it suffices to check that Ker ξ maps to zero under ∂. To see this, let C = Ker (A → A ) and note that a diagram chase gives A × A Ker u  → C. Take the pull back A ×C Ker ξ

Ker ξ

A × A Ker u 

α

A

C

u

B Thus given x ∈ Ker ξ, pull back to y ∈ A ×c Ker ξ. Then u  ◦ α(y) is the coset of ∂(x), i.e., ∂(Ker ξ) = 0, as desired. Exercise Finish the proof of the serpent lemma.



Chapter 12

Fibered and Cofibered Categories

Suppose that f : C → C  is a functor and Y is an object in C  . Then we have already discussed the following categories: (1) the fiber f −1 Y of f over Y is the subcategory of C where arrows lie over idY ; (2) the left fiber f /Y over Y whose objects are (X, u), where X ∈ Ob C and u : v f (X ) → Y , and a map (X, u) → (X  , u  ) is given by maps X −→ X  so that the diagram u Y f (X ) u

f (v)

F(X  ) commutes; (3) the right fiber Y \ f consists of (X, u : Y → f (X )).  G → G ◦ f .  = Funct(C, Sets), for C small, f induces f ∗ : C → C, With C Proposition We have adjoint functors f!

C

f∗

C

f∗

that is, (i) Hom( f ! F, G) = Hom(F, f ∗ G), (ii) Hom(G, f ∗ F) = Hom( f ∗ G, F), given by Kan’s formulae © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_12

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12 Fibered and Cofibered Categories

f ! (F)(Y )

F(X ) ,

lim − →

X, f X →Y ∈ f /Y

f ∗ (F)(Y ) =

F(X ) .

lim ← −

X,Y → f X ∈Y \ f

Proof (i) Start with θ : F → f ∗ G, θ = (θ X : F(X ) → G( f X )) . Let M( f ) = C ×C  Arr C  , where Arr C  are the arrows (that is, the morphisms) in C  , denote the category whose objects are given by pairs (X, u : f X → Y ) and whose a

b

→ Y  so that maps (X, u) → (X  , u  ) are a pair X −→ X  and Y − u

f (X )

Y

f (a)

b u

f (X  )



commutes.

Y

Given (X, u : f (X ) → Y ) in M( f ), we define θX

G(u)

θ(X,u) : F(X ) −−→ G( f X ) −−−→ G(Y ) . Thus θ = (θ X ) gives a family of maps F(X ) → G(Y ), for each (X, u), which are natural with respect to maps in M( f ). Now fix Y . Then the maps θ(X,u) induce a map F(X ) θ(X,u)

G(Y )

lim − →

F(X )

(X,u)∈ f /Y

i.e., induce a map f ! F(Y ) → G(Y ). We check easily that this gives a natural transformation f ! F → G. It is clear that the following gadgets are the same: (1) a natural transformation θ : F → f ∗ G over C, (2) a natural transformation F(X ) → G(Y ) over M( f ), (3) a natural transformation f ! F(Y ) → G(Y ) over C  .

12 Fibered and Cofibered Categories

Therefore,

63

Hom(F, f ∗ G) = Hom( f ! F, G) . 

The proof of (ii) is exactly dual. Fibered Category [due to Grothendieck]

Example Let B be the category of topological spaces. For each X ∈ Ob B, we let E X be the category of spaces over X i.e., pairs E, p : E → X in B. This is denoted B/ X . Now, if u : X → X  is a map in B, we have a pull back functor u ∗ : E X  → E X ⎛

⎞ ⎛ ⎞ E X ×X  E ⎝ ↓ ⎠ −→ ⎝ ↓ ⎠ . X X v

u

Given Z −→ Y −→ X in B, we have a canonical isomorphism Z ×Y (Y × X E) Z × X E given by cu,v : v ∗ u ∗ −→ (uv)∗ .

We have the following (cocycle) condition w∗ v ∗ u ∗

w ∗ (cu,v )

(cv,w ) u ∗

(v w)∗ u ∗

w ∗ (uv)∗ cuv,w

cu,vw

(uvw)∗

where the diagram commutes. First definition of fibered category over a category B: It is a pseudo-functor from B op to the category Cat of categories. Namely, a way of assigning a category E X to each X ∈ Ob B, to each arrow u : Y → X a functors u ∗ : E X → EY v u ∼ and to each pair Z −→ Y −→ X in B an isomorphism of functors cu,v :v ∗ u ∗ −→ (uv)∗ so that the cocycle condition holds. Note that a functor from B to Cat is a pseudofunctor so that cu,v is the identity. New definition of fibered category: Suppose we have a functor p : E → B. If f

X ∈ Ob B, then let E X denote the fiber p −1 X . Let E  −→ E be a morphism in E u lying over X  −→ X , i.e., p(E  ) = X  , p(E) = X and p( f ) = u. We say that f is cartesian provided that for any F ∈ Ob E X  , we have the isomorphism ∼

HomE X  (F, E  ) −→ {g ∈ HomE (F, E) : p(g) = u} α −→ f α . Given p : E → B, we say E is a pre-fibered category over B if given any u : X  → X in B and object E ∈ E X , there exists a cartesian arrow E  → E lying

64

12 Fibered and Cofibered Categories

over u. A fibered category is a pre-fibered category in which the composition of two cartesian arrows is again cartesian. Suppose p : E → B is pre-fibered. Then given any u : X  → X , we get a functor ∗ u : E X → E X  by choosing for each E ∈ E X a cartesian arrow E  → E over u and setting u ∗ E = E  . Then HomE X  ( f, u ∗ E) {g : F → E : p(g) = u} , d

HomE (F, E)u = {g : F → E : p(g) = u} , and then for all F over X  , ∼

HomE X  (F, u ∗ E) −→ HomE (F, E)u . Therefore u ∗ E is unique up to canonical isomorphism. v u If X  −→ X  −→ X , then there is a canonical arrow v ∗ u ∗ −→ (uv)∗ , and we can check it satisfies the co-cycle condition and is an isomorphism E → B in a fibered category. This indicates that the two definitions are the same. Dual Notions p : E → B is pre-cofibered when for every u : X → Y in B we have a functor u ∗ : E X → EY so that HomEY (u ∗ E, F) HomE (E, F)u . Cofibered means pre-cofibered and composition of co-cartesian is co-cartesian. Now go back to f : C → C  with Y and object in C  where f ! (F)(Y ) =

F(X ) .

lim − →

(X,u : f X →Y )∈ f \Y

Claim 12.1 If f is pre-cofibered, then f ! (F)(Y ) = lim F(X ) . − → −1 X∈ f

Y

Moreover, f ∗ (F)(Y ) =



lim F(X ) . ← − → f X ∈Y \ f X,Y − u

12 Fibered and Cofibered Categories

65

Claim 12.2 If f is pre-fibered, then f ∗ (F)(Y ) = lim F(X ) . ← − −1 X∈ f

Exercise Prove the two claims.

Y

Chapter 13

Examples of Fibered Categories

f

In a pre-cofibered category C −→ C, we have maps i : f −1 Y → f /Y   E : E −→ id fE =Y and r : f /Y −→ f −1 Y   X −→ u ∗ X , : u →Y fX − and it is clear that ri = id f −1 Y . Claim That r is left adjoint to i. Proof 

   X E Hom f /Y , u fX − →Y fE =Y = HomC (X, E)u by definition = Hom f −1 Y (u ∗ X, E) by the universal property of u ∗ X.



Exercise Check that f is pre-cofibered if and only if f −1 Y → f /Y has a left adjoint for all Y ∈ C  .

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68

13 Examples of Fibered Categories

Claim 

F(X ) = lim F X . lim − → − →u  X ∈ f −1 Y X, f X − →Y

r

Proof Consider A

B with ri = idA and r a left adjoint to i, so there is a

i

F

natural transformation idB → ir , and suppose B −→ Sets. We show lim F(B) = lim F(i A) . − → − →

B∈ B

A∈A

We have a canonical map lim F(i A) → lim F(B), and using the universal property − → − → A∈A B∈B of lim, we have to show that − → ∼

lim HomSets (F(B), S) −→ lim HomSets (F(i A), S) . ← − ← −

B∈ B

A∈A

Then G(B) = HomSets (F(B), S) is a contravariant functor, and we want to prove ∼

(∗)

lim G(B) −→ lim G(i A) . ← − − →

B∈ B

A∈A

An element of the left-hand side is a family g B ∈ G(B) compatible with the arrows in B. That the map in (∗) is an isomorphism follows from the picture • ir (B)

Bu

ir (u) A

• B • 

i

Example 13.1 For A → B, i has a left adjoint if and only if i(point) = e ∈ B is point

a final object for B. Then the claim says lim F(B) = F(e) for F : B → Sets. − →

13 Examples of Fibered Categories

69

Example 13.2 This is false for contravariant functors, e.g., take a

B :

• •e

•

b

 Then lim F = F(a) F(b) = F(e). − → F(e) Example 13.3 A discrete category has all morphisms identity morphisms, so the p arrows comprise a set. Suppose E −→ B is a fibered category so that the fibers are discrete for each X in B. We get, then, a contravariant functor B op → Sets, B → Ob E X , that is, a pseudo-functor is in fact a functor in a discrete category. Conversely, given any functor F : B op → Sets, we can form the category B/F whose objects are (X, ξ), X ∈ Ob(B), ξ ∈ F(X ), and p : B/F −→ B (X, ξ) −→ X is a fibered category with discrete fibers. The fiber over X is the set F(X ) regarded as a discrete category. Note that HomB/F ((X, ξ), (Y, η)) = { f : X → Y : F( f ) η = ξ} , and in B/F every arrow is cartesian. Thus fibered categories over C with discrete fibers are the same as F : C op → Sets, and cofibered categories over C with discrete fibers are the same as F : C → Sets. p

Example 13.4 Suppose U −→ G is a group homomorphism and consider  → G.  If this is fibered, then p must be epic, and every arrow is cartesian p:U →G  is a fibered category if and only if U → G by unique cancellation. Thus U  is the category K  where K is is epic, and the fiber over the unique object of G Ker{ p : U → G}. In general, given a fibered category p : E → B, we can define for each u : X → Y in B a functor u ∗ : EY → E X , unique up to canonical isomorphism.  → G,  such a choice is the same as a set-theoretic section of U In the case of U over G. Check that the canonical isomorphisms

70

13 Examples of Fibered Categories ∼

(uv)∗ ←− v ∗ u ∗ are the same thing as the 2-cocycle describing U as a group extension. Example 13.5 Let f : X → Y be a simplicial map of simplicial complexes. Take C to be the simplexes in X and C  to be the simplexes in Y as posets and let f : C → C  be the induced map of posets. If σ is a simplex in Y , then what is f /σ? f /σ is the poset of simplices in the subcomplex f −1 σ. Is f : C → C  fibered? We have u∗ T ⊂ T ↧

σ ⊂ σ u

So u ∗ T is the face of T whose vertices lie over vertices σ  . So yes it is fibered but not cofibered. Furthermore, f −1 σ is the poset of all simplexes mapping onto σ, and this is interesting because f −1 (open simplex σ) is homeomorphic to f −1 (ξ) × σ, where ξ ∈ Int σ. So f −1 ξ is not a simplicial complex, but it is a union of contractible pieces whose inclusion relations are described by the poset f −1 σ.

Chapter 14

Projective Resolutions

F

Suppose A −→ B is a functor between abelian categories. F is an additive functor if HomA (A, A ) → HomB (F A, F A ) is a homomorphism of abelian groups. Example 14.1 A = Funct (C, Ab), B = Ab, with F = limC , that is, − → F(A) = lim A(X ) . − → X∈ C

Example 14.2 A = G-mod, B = Ab, and F(M) = Z ⊗Z[G] M, i.e., the largest quotient on which G acts trivially. An object P of an abelian category A is projective (respectively injective) if HomA (P, ·) : A → Ab is exact (respectively HomA (·, P) : Aop → Ab is exact). In general, given 0 → A → A → A → 0 exact in A, we know that for any P ∈ A we have 0 −→ Hom(P, A ) −→ Hom(P, A) −→ Hom(P, A ) exact. So P is projective if and only if Hom(P, ·) carries epic maps to epic maps. Similarly, P is injective if and only if HomA (·, P) carries monic maps to epic maps. If P is projective and we have an epic A → P, then Hom(P, A)  Hom(P, P), so there exists α → id P so that P is a direct summand of A and conversely, and therefore P is projective if and only if every A  P has a section. Thus an R-module P is projective in the category of R-modules if and only if it is a direct summand in a free R-module.

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14 Projective Resolutions

Suppose A is an abelian category. A complex in A is a diagram d

d

(∗)

d

d

· · · −→ An+1 −→ An −→ An−1 −→ · · · ,

for n ∈ Z, so that d 2 = 0. The complexes in A form a category in the obvious way called C(A), and C(A) is an abelian category. A· is the notation for a complex, Ker{A· −→ B· }n = Ker{An −→ Bn } and so on. Now (∗) can also be written d

d

d

· · · −→ An −→ An+1 −→ · · · where An = A−n , and we define C≥0 (A) = category of chain complexes C≤0 (A) = C

≥0

= {A· so that An = 0, n < 0} , (A) = category of co-chain complexes ,

C + (A) = category of complexes with An = 0, n > 0 . Homology and cohomology are defined as usual. Suppose we have f, g : A· → B· . Then a homotopy between f and g is h = {h n : An → Bn+1 } so that f − g = dh + hd . Proposition Homotopic maps induce the same maps on homology. The proof is standard and left as an exercise. Homotopy is an equivalence relation on HomC(A) (A· , B· ), so we can form the abelian group of homotopy classes [A· , B· ]. Denote K (A) = category with the same objects as C(A), but in which Hom K (A) (A· , B· ) = [A· , B· ], and define K + , K − as before. Note that the notion of isomorphism in K (A), namely homotopy equivalence, is too strong. A quasi-isomorphism of complexes, or sometimes simply quis for short, is a map A· → B· in C(A) which induces a homology isomorphism.

14 Projective Resolutions

73

Example A homotopy equivalence f : A → B is a quis. Given M ∈ ObA, a (left) resolution of M is an exact sequence of the form . . . −→ A1 −→ A0 −→ M −→ 0 . A projective resolution is a resolution so that each Ai is projective. Key Proposition Any two projective resolutions of M are homotopy equivalent, and this assignment of projective resolution is functorial in M up to homotopy. Again the proof is an exercise. Think of M as being a complex concentrated in degree 0 and a resolution of M as being a chain complex A· equipped with a quasi-isomorphism ···

A2

A1

A0

0

0

···

···

0

0

M

0

0

···

Let P be the full subcategory of A consisting of projectives, where a subcategory B of A is full if for any two of its objects A, B ∈ B, we have HomB (A, B) = HomA (A, B). P is an additive category, so it makes sense to talk about C+ (P), K (P), etc. Define K (P) to be the full subcategory of K (A) consisting of complexes made up of projectives. A (left) resolution of a complex M· is another complex A· together with a quis A· → M· , and a projective resolution is a resolution A· ∈ C(P). Proposition Provided we stay in K + (A), i.e., bounded above, any two projective resolutions of M· are isomorphic in K + (P). Moreover, by choosing for each M a projective resolution P· → M· , we get a well-defined functor K + (A) → K + (P), and this is adjoint to inclusion, where M· → P· . Proof (of the simplest case) Given ···

P2 u2

···

A2

P1 u1

A1

P0 u0

A0

M

0

u

N

0

where successive horizontal arrows compose to zero, the Pi ’s are projective and the bottom sequence is exact, there exist u 0 , u 1 , u 2 , . . . making the diagram commute, and the map of complexes is unique up to homotopy. A diagram chase gives u · : M· → P· as usual. Suppose we extend u also to u · , u · . Then we have and consider u · − u · = 

74

14 Projective Resolutions

P2  u2

A2

P1

P0

 u1

h0

A1

M

 u0

0

0

A0

N

0.

Now  u 0 = d h 0 and d ( u 1 − h 0 d) = 0, so we find h 1 with  u 1 − h 0 d = d h 1 , and so on. This proves the simplest case. Why are two projective resolutions P· → M, Q · → M homotopy equivalent? Use the simplest case to get M

P· v·

u·

id

id

Q·

M

P·

M

Then v· u ·

id

P·

M

so v· u · id M and similarly for u · v· .



At this point, we have a functor well defined up to canonical isomorphism given by A −→ K ≥0 (P) M −→ projective resolution of M. Important We must assume that A has enough projectives, i.e., any M ∈ Ob A is the quotient of some projective P. Then we construct projective resolutions . . . −→ P1 −→ P0 −→ M −→ 0 . Example The category A of finite abelian p-groups does not have enough projectives (or injectives) since A = Z/ pr Z × · · · for A ∈ A is not projective. Note that Z/ pr +1 Z  Z/ pr Z is epic yet there is no section.

Chapter 15

Analogues of Homotopy Liftings

Recall last time we constructed · · · → P1 → P0 → M → 0, which is the same as a quis ···

P2

P1

P0

0

0

···

···

0

0

M

0

0

···

Let A = abelian category with enough projectives, P = full subcategory of projectives. Suppose given a complex A· ∈ C+ A; we want to construct a complex and an epimorphism P· −→ A· which is a quasi-isomorphism with P· ∈ C+ (P). Without loss of generality, A· is so that An = 0 for all n < 0. Choose P0  A0 and consider the diagram

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15 Analogues of Homotopy Liftings

0

0

0

K1

K0

K0

P1

A1 × A0 P0

P0

A1

A0

0

0

A1

=

0

0

Apply the Serpent Lemma to the first two columns to get K 1  K 0 . So far, this gives 0

0

K1

K0

0

P1

P0

0

A1

A0

0

0

0

Put K 10 = Ker{K 1 → K 0 } , P10 = Ker{P1 → P0 } , A10 = Ker{A1 → A0 } . The Serpent Lemma gives the commutative diagram

15 Analogues of Homotopy Liftings

77

0

0

0

K2

K 10

K 10

P2

A2 × A10 P10

P10

A2

A2

A10

0

0

0

with exact verticals. Apply the Serpent Lemma again to get K 2  K 10 . Iterate to get a diagram with exact verticals 0

0

0

···

K2

K1

K0

0

···

P2

P1

P0

0

···

A2

A1

A0

0

0

0

0

so that K · has trivial homology, whence H∗ P·  H∗ A· , so P· → A· is indeed a quis. Another construction: given A· ∈ C+ (A), we can find a P·  A· so that P· ∈ C+ (P) and P· is acyclic

78

15 Analogues of Homotopy Liftings id

P2

P2  id

P1 ⊖

P1

P0

A2

A1

id

P0

A0

0

Incidentally, any P· ∈ C+ P with P· acyclic looks this way. To see this, consider P0

P1

0

so P1 ≈ P10 ⊕ P0 and so on. The next step using either construction is to show that two projective resolutions of A· ∈ C+ (A) are homotopy equivalent and to prove functoriality in A. Recall, Homotopy Extension (HE) Theorem: Suppose given CW complexes A and B, maps A i

E f

B and a homotopy h : A × I → E so that h 0 = f i. Then there exists an extension  h : B × I → E so that  h = h on A × I and  h 0 = f , i.e., a lift −− → making the diagram commute B × {0} ∪ A × I

f∪h

E

 h

B×I or in other words

∗

15 Analogues of Homotopy Liftings

79 h

A

EI

 h

evaluation at 0 f

B

E

Chain Homotopy (CH) Theorem: Suppose given a fiber space E with

X A

E 0

A×I

X

so that EI

E

E are fibrations in the sense of Serre, and

E

X

∗

B × {0} ∪ A × I

A

A

B×I

B

A×I

are inclusions of CW complexes.

Then each of the following is a homotopy equivalence B × {0} ∪ A × I

EI

A

B×I

E

A×I

Chain Homotopy Extension (CHE) Theorem: Given A

E a fibration in the sense of Serre

CW i

B

X

If either i or p is a homotopy equivalence, then a lift −− → exists making the diagram commute.

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15 Analogues of Homotopy Liftings

By analogy, we have Proposition Suppose given in C+ (A) a diagram A·

f

p

i

B·

E·

g

X·

with p epic in each dimension, i monic in each dimension, B· /A· ∈ C+ (P) and either i or p a quis. Then there exists a lift −− → making the diagram commute.

Proof Suppose first that p is quis 0

0

0

A2

A1

A0

0

B2

B1

B0

0

P2

P1

P0

0

0

0

0

Let S K n (B, A) be the complex · · · −→ An+1 −→ An −→ Bn−1 −→ Bn−2 −→ · · · Then A· = S K 0 (B, A) ⊂ S K 1 (B, A) ⊂ · · · ⊂ S K n (B, A) ⊂ · · · ⊂ B· are all subcomplexes. Moreover  S K n+1 (B, A)/S K n (B, A) =

Pn , in degree n, 0, else.

15 Analogues of Homotopy Liftings

81

Thus we have E·

A·

S K0 

S K1 

S K2 

.. . 

B0

X·

and have thus reduced the problem to the case where A· and B· differ in one degree by a projective object. Denote P[n] = complex with P in dimension n, and zeroes elsewhere. Because P is projective, 0 −→ An −→ Bn Pn 0 splits, and so B· = A· ⊕ P[n] with dn (a, p) = da + θ p where θ : Pn → Ker{An−1 → An−2 }. Thus we have E n+1

X n−1 h

An+1

An

An−1 gn

...

An+1

An ⊕ Pn

An−1

E n−1

Pn

fn

...

En

Xn

X n−1

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15 Analogues of Homotopy Liftings

Problem: Define h : Pn → E n so that pn h = gn on P with dh = f n−1 d : An ⊕ P → E n−1

En

P (∗)

this is epic because p is epic quis

(gn , f n−1 d)

X n × Ker{E n−1 → E n−2 } Ker{X n−1 →X n−2 }

and h exists because (∗) is epic and P is projective.



Chapter 16

The Mapping Cylinder and Mapping Cone

Recall the lifting result from last time which we continue discussing: Given f

A i

E

h

B

p

in C+ (A)

X

g

with i monic, Coker i ∈ C+ (P), p epic and either i or p a quis, then there exists a lifting h making diagram commute. We did the case where p is a quis last time. Now suppose i is a quis. Then Coker i = (B/A)· = P· , and the long exact sequence gives H∗ P· = 0. Since P· ∈ C+ (P), we showed P· is a direct sum of complexes of the form . . . 0 −→ 0 −→ P −→ deg n

id

P

deg n−1

−→ 0 −→ 0 −→ . . . .

We have the commutative diagram

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16 The Mapping Cylinder and Mapping Cone

B3

B2

B1

B0

0

P2

P1

P0

0

P0

id

P0

 id

Q1

Q1



Q2 Consider the following special case f

{0}

E p

i

  id . . . → 0 → P −→ P → 0 → . . . n

X

g

n−1

Then the dotted arrow exists because    id 0 −→ P −→ P −→ 0 , K · = Hom(P, K n ), HomC(A) n

n−1

and we have En

P

Xn

  id since P is projective. We denote Dn (P) = 0 −→ P −→ P −→ 0 . n

n−1

16 The Mapping Cylinder and Mapping Cone

85

Returning to the general case, we have A i

B ∗

 B/A = Dn (Q n ) n≥0

where Q n ∈ P. By the special case, we can lift each Dn (Q n ) so that ∗ exists and so B  A⊕



Dn (Q n ) .

n≥0

Finally we have f

A

E

h

A⊕



Dn (Q n )

X

where we construct h by lifting each Dn (Q n ) by the special case.



Recall that in homotopy theory we have the mapping cylinder construction, namely, f

given A −→ B, we construct M( f ) = (A × I )



B

A×1

and have then a factorization of f as follows A

i

M( f )

p

B

with i an embedding and p a homotopy equivalence. The analogue for complexes: let A· ∈ C(A), and define A I as follows (A I )n = An ⊕ An−1 ⊕ An . Notice that if we were working with modules, then we would like to denote an element of (A I )n by (0) ⊗ a + (0, 1) ⊗ b + (1) ⊗ c, for a, c ∈ An and b ∈ An−1 , and then d is given by

86

16 The Mapping Cylinder and Mapping Cone

d((0) ⊗ a + (0, 1) ⊗ b + (1) ⊗ c) = (0) ⊗ da − (0) ⊗ b + (1) ⊗ b − (0, 1) ⊗ db + (1) ⊗ dc . Back to complexes, the differential d : (A I )n → (A I )n−1 is defined as follows 

An d

−id

An−1





An−1

id

−d



An−2

An d

An−1

i.e., d(a, b, c) = (da − b, −db, b + dc) . We check d 2 (a, b, c) = d(da − b, −db, b + dc) = (d(da − b) − (−db), −d(−db), −db + d(b + dc)) =0 Proposition To give a map of complexes A I → B· amounts to giving two maps f, g : A· → B· and a homotopy between them, i.e., giving h : f → g so that g − f = dh + hd. Proof An ⊕ An−1 ⊕ An −→ Bn a ⊕ b ⊕ c −→ f (a) + h(b) + g(c) . Check this is a chain map (a, b, c)

f (a) + h(b) + g(c)

d f (a) + dh(b) + dg(c)

(da − b, −db, b + dc)

f (da − b) − hdb + gb + gdc 

Now given f : A· → B· , a map in C(A), we define M( f ) by push out

16 The Mapping Cylinder and Mapping Cone

87 f

A

B

a ↧ i1 (0, 0, a)

M( f )

AI Therefore

(M( f ))n = An ⊕ An−1 ⊕ Bn and

d(a, a , b) = (da − a , −da , f a + db) .

We have a canonical factorization p

i

→ M( f ) −→ B· , A· − that is, An −→ An ⊕ An−1 ⊕ Bn −→ Bn given by i : a → (a, 0, 0) and p : (a, a , b) → f (a) + b, where i is monic, p is epic and Exercise p : M( f ) → B is a homotopy equivalence. We can also define A I so that a map B· → A I is the same as a pair f, g : B· → A· of maps and a homotopy h : f → g. In fact, (A I )n = An × An+1 × An . Exercise Find the differential. i

p

Then we can factor f : B· → A· into B· − → B· × A· A I −→ A· , where i is monic, p is epic and i is homotopy equivalence. The mapping cone of f : A → B is given by the quotient C( f ) = M( f )/Image of A. Proposition Let E · → X · be a quis in C+ (A) and let P· ∈ C+ (P). Then ∼

[P· , E · ] −→ [P· , X · ] is an isomorphism, where [·, ·] denotes homotopy classes. · −→ X · , where p is epic and i is a homotopy →E Proof We factor E · → X · into E · − equivalence as well as being monic. Since E · → X · is a quis, it follows that p is also a quis, and moreover, · ] , i ∗ : [P· , E · ]  [P· , E i

p

88

16 The Mapping Cylinder and Mapping Cone p

so without loss of generality we assume E · −− X · is an epic quis. We must show [P· , E · ] → [P· , X · ] is epic, and to this end, given E

0

P

g

X

we use the lifting theorem. To show it is monic, suppose f, f : P· → E · and given a homotopy h : p f → p f where p : E · → X · , we have the diagram P⊕P

( f , f )

E

(i 0 ,i 1 )

PI

p

h

X

where (i 0 , i 1 ) is monic with cokernel in C+ (P). Use the lifting theorem to conclude  that f and f are homotopic. Note that this Proposition gives uniqueness up to canonical homotopy equivalence of projective resolutions. . . more next time.

Chapter 17

Derived Categories

From last time, we saw Proposition If P· ∈ C+ (P) and E · → X · is a quis in C+ (A), then ∼ =

[P· , E · ] −→ [P· , X ] is a natural isomorphism. Remark Check from the proof that this holds without assuming E · , X · are in C+ (A) but just in C(A). Now, recall that we have K + (P), which has the same objects as C+ (P), but the morphisms are homotopy classes of maps of complexes. We have of course i

r

K + (P) → K + (A) and claim that i has a right adjoint functor K + (P) ← K + (A) defined by choosing for each A· ∈ K + (A) a quis P· → A· with P· ∈ C+ (P) and setting r (A· ) = P· . Claim that Hom K + (P) (r (Q · ), r (A· ))

∼ =

=

[Q · , P· ]

Hom K + (A) (i Q · , A· ) =

∼ =

[Q · , A· ]

via the proposition above.

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_17

89

90

17 Derived Categories

These two functors of A· are isomorphic. Thus it follows that r is, in fact, a functor. So we have K + (P)

i r

K + (A) with ri = id .

For any A· , the canonical map ir (A· ) → A· is the quis P· → A· which we have chosen. Let D+ (A) be the category with same objects as C+ (A), but with morphisms ∼

Hom D+ (A) (A· , B· ) = [r A· , r B· ] −→ [r A· , B· ] . Then we notice Proposition We have K + (A) −→ D+ (A) [A· , B· ] −→ [r A· , r B· ] , and this canonical functor is universal with respect to the property that it sends quasi-isomorphisms into isomorphisms K + (A)

C

D+ (A) Proof K + (P)

K + (A)

F

C

 F

equivalence

D+ (A) as F carries quasi-isomorphisms to isomorphisms and where the equivalence comes from the fact that the composition gives a clear equivalence of categories. We want to show that given F : K + (A) → C inverting the quis, there is a unique way of defining F∗ : Hom D+ (A) (A· , B· )

[r A· , r B· ]

F∗

Hom(F(A· ), F(B· ))

Hom(Fr A· , Fr B· )

17 Derived Categories

91

∼

∼

for F(r A· ) −→ F(A· ) and F(r B· ) −→ F(B· ) since r is quis. Clearly, this will work.  D+ (A) is called the derived category of complexes in A bounded below. D(A) is defined by an analogous universal property from K (A). However, its existence is proved by a process of localization. The idea is to define a map A· → B· in D(A) to be represented by a diagram C· quis

f s

A·

B·

where C· is projective. Think of this as a fraction of s and then proceed as in the construction of localization for a multiplicative system. Consider a projective resolution P·

M[0]

−→ · · · −→ P1 −→ P0

−→ · · · −→ 0 −→ M −→ 0 .

Let F : A → B be an additive functor between abelian categories. Then given M in A, assuming A has enough projectives, choose a projective resolution · · · −→ P2 −→ P1 −→ P0 −→ M −→ 0 −→ · · · , which is unique up to homotopy. Form F(P· ) : · · · −→ F(P2 ) −→ F(P1 ) −→ F(P0 ) −→ 0 −→ · · · and take the homology groups of this chain complex (L q F)(M) = Hq (F(P· )) . Since P· is unique up to homotopy equivalence and F is additive, the homotopy type of F(P· ), and hence its homology, depends only on M. L q F is called the q th left-derived functor of F. Generalization F induces C(A) −→ C(B) , K (A) −→ K (B) . Namely, given A· ∈ K + (A), we have r A· ∈ K + (P), and we can define

92

17 Derived Categories

(L F) : K + A → D+ B by (L F)(A· ) = F(r A· ) . This L F has the property that it inverts quasi-isomorphisms. To see this, note that if A· quis

quis

A·

quis

quis

r A·

r A·

then r A· , r A· are homotopy equivalent complexes, so L F(A· ) → L F(A· ) is a homotopy equivalence, hence an isomorphism in D+ (B). Thus L F can be viewed as a functor L F : D+ (A) → D+ (B). By definition, if M ∈ Ob A, then L F(M[0]) ∈ D+ (B) is an object which is well defined up to canonical isomorphism with Hq (L F(M[0])) = (L q F)(M) . Properties of {L q F} 1: Long exact sequence: Given 0 → M  → M → M  → 0 exact in A, we get a long exact sequence of derived functors −→ (L 1 F) M  −→ (L 0 F)(M  ) −→ (L 0 F)(M) −→ (L 0 F)(M  ) −→ 0 . 2: (L q F)(P) = 0 for q > 0 if P is projective. 3: There is a canonical map F → L · F which is an isomorphism if and only if F is right exact, that is, 0 → M  → M → M  → 0 exact implies that F(M  ) → F(M) → F(M  ) → 0 is also exact. Proof of 2 If P is projective, then a projective resolution is id

−→ 0 −→ 0 −→ P −→ P −→ 0 2

1

0



so (L q F)(P) = Hq (F(P[0])) =

F(P) q = 0 , 0 q = 0 .

Note that Property 3 implies that Property 1 implies that L 0 F is right exact, for if · · · → P1 → P0 → M → 0 is exact, then · · · → F(P1 ) → F(P0 ) → F(M) → 0 is not necessarily exact, but L 0 F(M) = H0 (F(P· )) → F(M). Lemma The following are equivalent conditions on F (a) 0 → M  → M → M  → 0 exact implies F(M  ) → F(M) → F(M  ) → 0 exact;

17 Derived Categories

93

(b) M  → M → M  → 0 exact implies F(M  ) → F(M) → F(M  ) → 0 exact. u

Proof (b) obviously implies (a). Conversely, given (a) and N  −→ N → N  → 0 exact, break it into 0 → Im u → N → N  → 0 exact, so F(Im u) → F(N ) → F(N  ) → 0 exact, and 0 → Ker u → N  → Im u → 0 exact, so F(Ker u) → F(N  ) → F(Im u) → 0 exact, and putting them back together, F(N  ) → F(N ) → F(N  ) → 0 is also exact. 

Chapter 18

The First Homotopy Property

Given functors C

f

g

C

between small categories and a suitable natural transformation h : f → g, we want to show that f and g induce the same map on homology. This is the first homotopy property. Suppose F is a complex of functors C → Ab. Then we have H∗ (C, F), and for f : C → C  a functor, we have H∗ (C, F) = H∗ (C  , L f ! (F)) . α

Thus, if we give L f ! (F) −→ F  , we get an induced homomorphism H∗ (C, F) −→ H∗ (C  , F  ) associated to ( f, α). For simplicity we assume that F and F  are just functors (instead of complexes). Then H0 (L f ! (F)) = f ! (F), so α is the same as a map f ! F → F  which is in turn the same as F → f ∗ F  . Thus, given f : C → C  , F, F  and α : F → f ∗ F  , we get an induced map on homology Hn (C, F) −→ Hn (C  , F  ) , and in particular, we always have Hn (C, f ∗ F  ) −→ Hn (C  , F  ) © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_18

95

96

18 The First Homotopy Property

for any F  : C  → Ab. Claim The pair f, g : C → C  together with h : f → g is the same thing as a functor H

C × I −→ C  , where I = {0 < 1} is regarded as a category, with H i 0 = f , with H i 1 = g and with C

i0

i1

C × I the obvious maps.

There are the three kinds of maps in C × I :

(u,id)

(i) (X, 0)

(X  , 0)

H

f (u)

f (X )

f (X  ) ,

f (X  ) f (u)

(ii) (X, 0)

(u, 0. It follows that − → I

18 The First Homotopy Property

97

L n p! ( p ∗ F)(X ) =



0, n = 0, F(X ), n = 0,

so L p! ( p ∗ F) is quis to F. Therefore Hn (C × I, p ∗ F) = Hn (C, F). Exercise 1 Compute Hn (C × I, G) for any G on C × I . One of the embeddings i 0 , i 1 has p as adjoint of the sort that (by last time) L p! is i ∗j . Quillen is not sure that this will work. Proposition (First Homotopy Property) Suppose given C

f

g 

C  with h : f → g

and F  : C  → Ab, such that h induces an isomorphism f ∗ F → g ∗ F  . Recall F  (h X )

( f ∗ F  )(X ) = F  ( f (X )) −−−−→ F  (g(X )) = (g ∗ F  )(X ) , and we claim Hn (C, f ∗ F  ) f∗

Hn (C  , F  )

∼ = g∗ ∗



Hn (C, g F ) commutes, where f ∗ here means maps induced on homology, not the adjoint of f ∗ and likewise for g∗ . Example If F  is the constant functor with value A ∈ Ob Ab, then the proposition says that the following diagram commutes. Hn (C, A) f∗

Hn (C, A) g∗

Hn (C, A) Proof of Proposition We have

98

18 The First Homotopy Property f

i0

C

i1

C×I

H

C

p

C g

and given F  , we know F  ( f X )

∼ F  (h X )

F  (g X ) . But H ∗ (F  ) is constant on

the fibers of p, and so H ∗ (F  ) ∼ = p ∗ f ∗ (F  ) = p ∗ G . d

Thus we have the commutative diagram Hn (C, f ∗ (F  )) i 0∗

Hn (C × I, H ∗ (F  ))

Hn (C  , F  )

p∗

i 1∗ ∗

H∗



Hn (C, g (F ))

Hn (C, G) 

where i 0∗ , i 1∗ and p∗ are isomorphisms.

We noted in (∗) that if C has a final object, then lim C is exact so Hn (C, F) = 0, for − → all n = 0. However, the conclusion is not true if C has just an initial object. Example a

b ≡C c

F(a)

F(b) F(c)

and Hn (C) fits into a Mayer–Vietoris type sequence 0 −→ H1 (C, F) −→ F(a) −→ F(b) ⊕ F(c) −→ H0 (C, F) −→ 0, with all Hn (C, F) = 0 for n = 0, 1. This example was abandoned apologetically. We know, for i b : point → C with image b, that

18 The First Homotopy Property

99

Hn (C, i b! A) = Hn (C, L i b! (A)) since i b! is exact = Hn (point, A)  A, n = 0, = 0, else. Claim For a constant functor A and a category C with an initial object that  Hn (C, A) =

A, n = 0, 0, else, f

using the homotopy property, i.e., that if C Hn (C, A)

f∗

g

C  and h : f → g, then

Hn (C  , A) coincide.

g∗

f

Thus, if we have a pair of adjoint functors C f∗

Hn (C, A)

g∗

C  , then

g

Hn (C  , A) ,

and so f ∗ and g∗ are inverses. iX

Proof point −−→ C has a right adjoint Y → ∗, and HomC (i X ∗, Y ) = Hompoint (∗, ∗), when X is initial in C. Thus Hn (point, A) = Hn (C, A) when C has an initial object.  Claim We have Hn (C, A) ∼ = Hn (C op , A). Proof We form over C op × C the cofibered category E belonging to the functor f

(X, Y ) → HomC (X, Y ). The objects of E are arrows X −→ Y , and a morphism is a diagram f

X

Y

in C op

in C f

X

Y

We will show that the two obvious functors E p

C

p

C

100

18 The First Homotopy Property

induce isomorphisms on homology. To this end, the functor p : E −→ C f

: (X −→ Y ) −→ Y is the composite of cofibered hence cofibered (exercise) with p −1 Y = (C/Y )op . But C/Y has a final object, which implies that (C/Y )op has an initial one. Similarly,

p  : E −→ C  f

: (X −→ Y ) −→ X is cofibered with ( p  )−1 (X ) = X \C, which also has an initial object. Now (†)

Hn (E, A) = Hn (C, L p! (A))

and L p! (A) has homology groups given by L q p! (A)(Y ) = Hq ( p −1 Y, A) since p cofibered  A, q = 0, = since p −1 Y has an initial object. 0, else, Therefore L p! A = A, so (†) follows, and similarly for p  .



Chapter 19

Group Completions and Grothendieck Groups

We want to define the group completion of an abelian monoid S, (being a monoid means that S has a commutative and associative operation + and the unit 0 exists). More precisely: u

Problem Construct an abelian group G and a monoid morphism S −→ G which is universal. G is then called the group completion of S. Solution Take G = (S × S)/ ∼ where ∼ is generated by (s1 , s2 ) ∼ (s + s1 , s + s2 ). Check this works and yields an abelian group with −(s1 , s2 ) = (s2 , s1 ) and u : s → (s, 0), so (s1 , s2 ) = u(s1 ) − u(s2 ). This is Grothendieck’s construction. Example S = N. N × N −→ Z (n 1 , n 2 ) −→ n 1 − n 2 . We get G = (N × N)/ ∼ → Z, and this map is clearly a bijection. More generally, if S is a sub-monoid of an abelian group A so that A = S + (−S), then A ≈ group completion of S. Notice that we can do this for a category leading to a groupoid, but this loses most structure, e.g., higher homotopy. If X is a set on which S acts, construct S −1 X = (X × S)/ ∼ where ∼ is generated by (x, s) ∼ (s  x, s  + s). S −1 X is a set with S action s  (x, s) = (s  x, s), and each s  acts invertibly on S −1 X . We also have a map X → S −1 X which is universal for maps from X to S-sets on which S acts invertibly. Example Localization for S a multiplicative system in a commutative ring A and X an A-module. Note that the group completion of S is S −1 S. © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_19

101

102

19 Group Completions and Grothendieck Groups

Exercise Calculate S −1 S where S = Z under multiplication. Let A be a ring and P A the category of finitely generated projective A-modules P, so Hom (P, ·) is exact and in particular id : P → P comes from P → An . Take S to be the set of isomorphism classes of P A and addition on S induced by (P, Q) → P ⊕ Q. The Grothendieck group of P A is the group completion of S, denoted K 0 A. Example 1 If F is a field, then isomorphism classes of P F ∼ = N by dimension whence K 0 F = Z. More generally, if every P ∈ P A is free, then K 0 A ∼ = Z. For instance (1) local commutative rings by Nakayama, (2) principal ideal domains, (3) [1976] (Serre problem) A = k[X 1 , . . . , X n ] for k a principal ideal domain. Example 2 Consider a Dedekind domain A, e.g., algebraic integers in an algebraic number field. Any finitely generated torsion free A-module is projective and is a direct sum P = A1 ⊕ . . . ⊕ An , where the Ai are ideals. Actually P = An−1 ⊕ A where A is an ideal. In fact, the exterior algebra n P ∼ = n−1 An−1 ⊗ 1 A ∼ = A, and the class of A is the ideal class group Pic A, an invariant of P. Moreover, it depends additively on the projective module. The isomorphism classes in Pic A are therefore S ⊂ N × Pic A in the obvious way and K 0 A = Z ⊕ Pic A. Example 3 [Serre]: Consider a compact Hausdorff space X and let A be the ring of continuous C-valued functions on X . Serre and Swan showed there is an equivalence between P A and the category of complex vector bundles E over X as follows. If E is a vector bundle, let (X, E) denote the continuous global sections of E. (X, E) is a finitely generated projective A-module: Any E is a direct summand of a trivial vector bundle, and we get a surjection X × Cn  E which splits. Thus (X, E) is a direct summand of (X, X × Cn ) = An . Conversely, given P ∈ P A , express P as a summand of An . This gives an idempotent n × n matrix M over A. Then M can be viewed as an endomorphism of X × Cn which splits the bundle by idempotence. To see locally triviality, note that Im M = Ker{1 − M} again by idempotence. Now, from algebraic topology for X connected, we have isomorphism classes of n-dimensional vector bundles over X corresponding to [X, BUn ]. Instead of P A , we can use any category C having a set of isomorphism classes ⊥

together with a functor C × C −→ C, where X × Y −→ X ⊥ Y , so that X ⊥Y Y ⊥ X, (X ⊥ Y ) ⊥ Z X ⊥ (Y ⊥ Z ) , there is 0 ∈ C so that 0 ⊥ X X . Given all this, we can group complete the isomorphism classes in C to get a Grothendieck group K 0 C.

19 Group Completions and Grothendieck Groups

103

Exercise For G a finite group, C the category of finite G-sets and X ⊥ Y the disjoint union of G-sets, describe K 0 , called the Burnside ring of G. Grothendieck’s original: Let M be a full subcategory of an abelian category A. Assume 0 ∈ M and M is closed under extensions, i.e., if 0 → M  → A → M → 0 is exact with M, M  ∈ M, then A ∈ M as well. The Grothendieck group K 0 M is an abelian group together with a map M −→ [M] from isomorphism classes of M to K 0 M so that for any exact sequence 0 → M  → M → M  → 0, we have [M] = [M  ] + [M  ], and moreover, this map is universal. Remark K 0 (M, ⊕)  K 0 M. Example For M the category of finite abelian groups, we get K 0 (M, ⊕) =



Z,

primes p r ∈ Z+

K 0 (M) =



primes p

Z.

Chapter 20

Devissage and Resolution Theorems

Suppose that M is a full subcategory of an abelian category A where M is closed under finite direct sums, contains 0 and whose isomorphism classes form a set. Recall that K 0 M is an abelian group together with a universal map Ob M −→ K 0 M so that

0 −→ M  −→ M −→ M  −→ 0

exact in M implies that [M] = [M  ] + [M  ]. Remark If M = P A , then exact sequences split, so K 0 (M, ⊕) = K 0 M. In general, K 0 M is a quotient of K 0 (M, ⊕). Example Let M be the category of finite abelian p-groups. Recall that Krull– Schmidt says every M ∈ M is a direct sum of indecomposables in a unique way up to isomorphism. Thus isomorphism classes of M are given by the free abelian monoid generated by indecomposable isomorphism classes, i.e., Z/ pr , r = 1, . . .. Thus K 0 (M, ⊕) is the free abelian group generated by Z/ pr Z , r ≥ 1 . Whilst in general if we have a filtration 0 = F−1 M ⊂ F0 M ⊂ . . . ⊂ Fn M = M of M ∈ M so that each F p M and F p M/F p−1 M ∈ M, then in K 0 M, we have n  (†) M = [F p M/F p−1 M]. p=0

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_20

105

106

20 Devissage and Resolution Theorems

Thus in our example, every M has a filtration with quotients Z/ p Z so K 0 M = Z. Namely we have the map assigning the integral filtration length which is a homomorphism K 0 M → Z by universality. More generally, suppose M is an abelian category where every object has the descending chain condition and the ascending chain condition on its sub-objects. Then Jordan–Hölder applies, so we can speak of the multiplicity m σ (M) of the simple object σ in any composition series for M. We have M −→ {m σ (M)} in the free ≈  monoid generated by simple objects and get a map K 0 M −→ Z. σ

Devissage Theorem (Grothendieck) Let A be an abelian category and A ⊂ A full so that also A is abelian. Assume every A in A has a finite filtration 0 = F−1 ⊂ · · · ⊂ Fn = A ∼

so that F p /F p−1 ∈ A . Then K 0 A −→ K 0 A. Example 1 Take A to be finite abelian p-groups and A to be finitely generated Z/ p-modules. Example 2 Suppose A is a noetherian ring, M = Modf(A), the finitely generated A-modules, and I ⊂ A is a nilpotent ideal. Take M = Modf(A/I ) where 0 = I n M ⊂ . . . ⊂ I M ⊂ M. Thus K 0 Mod f (A/I ) = K 0 Modf(A). The proof of the Devissage Theorem is based on the Schreier refinement lemma. Lemma (Schreier) Suppose M has two filtrations · · · ⊂ F p M ⊂ F p+1 M ⊂ · · · and  M ⊂ · · · , neither necessarily terminating. Then F  induces a · · · ⊂ Fq M ⊂ Fq+1  F F p M/F p−1 M via Fq (F p M) = Fq M ∩ F p M, i.e., filtration on gr M = p

Fq (F p M/F p−1 M) = Image {Fq ∩ F p −→ F p /F p−1 } . 

 Similarly F induces a filtration on gr F M = ⊕ Fq /Fq−1 . Then we have 



gr F (gr F (M)) = gr F (gr F (M)) . Proof We have the formulae Fq (F p /F p−1 ) = Fq ∩ F p + F p−1 /F p−1 , 

grqF (F p /F p−1 ) =

Fq ∩ F p + F p−1  Fq−1 ∩ F p + F p−1

=

Fq ∩ F p  (Fq−1 ∩ F p + F p−1 ) ∩ (Fq ∩ F p )

20 Devissage and Resolution Theorems

but

107

  ∩ F p ) + F p−1 ) ∩ (Fq ∩ F p ) = Fq−1 ∩ F p + F p−1 ∩ Fq . ((Fq−1

 Remark The refinement lemma fails for 3 filtrations since the modular law fails. Proof of Devissage Given M ∈ A, choose a filtration F p M exhausting M with quotients gr Fp (M) ∈ A and consider 

[gr Fp M] ∈ K 0 A .

p

Claim This does not depend on the choice of {F p M}. This follows easily from the previous lemma and (†). Thus we get a map γ : Ob A → K 0 A . Then 0 → M  → M → M  → 0 exact in A implies that γ(M) = γ(M  ) + γ(M  ). By universality, we get γ : K 0 A → K 0 A , which is inverse to the map induced by inclusion.  Example Take A = Modf(Z), so PZ is the category of free finitely generated abelian groups, and we have K 0 PZ ) = Z. Now, any finitely generated abelian group has a resolution 0 −→ Zq −→ Z p −→ M −→ 0 so [M] = [Z p ] − [Zq ] = ( p − q)[Z] , whence K 0 PZ  K 0 (Mod f (Z)) is in fact an isomorphism. Resolution Theorem (Grothendieck) Suppose A is a noetherian ring so that every finitely generated A-module M has a finite resolution of projectives (∗)

0 −→ Pn −→ . . . −→ P0 −→ M −→ 0

A is said to be regular in this case. Then

∼

K 0 (P A ) −→ K 0 Modf(A) . The idea is to define K 0 Modf(A) −→ K 0 P A n  [M] −→ (−1)i [Pi ] . i=0

108

20 Devissage and Resolution Theorems

If this is well defined, then it is an inverse to K 0 P A → K 0 Modf(A). To see this, decompose (∗) into 0 −→ R1 −→ P0 −→ M −→ 0 0 −→ R2 −→ P1 −→ R1 −→ 0 .. . 0 −→ 0 −→ Pn −→ Rn −→ 0 . Thus in K 0 Modf(A), we have [M] =

n  (−1)i [Pi ] , i=0

and it remains as an exercise only to show it is well defined.



Chapter 21

Exact Sequences of Homotopy Classes

Suppose that F : A → B is an additive functor between abelian categories and A has enough projectives. For M in A, choose a projective resolution, i.e., a quis P· → M[0]. Then F(P· ) = LF(M[0]) is unique up to canonical isomorphism in D+ (B), and we define L i F(M) = Hi LF(M[0]) = Hi F(P· ).

Claim Given 0 → M  → M → M  → 0 exact in A, then we have a natural long exact sequence (†) · · · −→ L 1 F M  −→ L 1 F M −→ L 1 F M  −→ L 0 F M  −→ L 0 F M −→ L 0 F M  −→ 0 .

We shall prove this more generally for M· in C+ (A) and P· → M· , and then the sequence (†) continues on to the right: L 0 F M  → L −1 F M  → · · · . Proof Start with projective resolutions of M  and M P

(∗)

f

quis p

0

M

P p quis

i

M

M 

0 ∼

and note that the induced P  → P comes from [P  , P] −→ [P  , M]. Thus we get f : P  → P and a homotopy h : i p  → p f , i.e., dh + hd = p f −  ip .

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_21

109

110

21 Exact Sequences of Homotopy Classes

Note that given i

A

B g

f

X i0

with a homotopy h from f to g, we can form A −→ A × I ∪i B and get a map M(i) → X , where M(•) is the mapping cylinder of •. satisfying but

M(i)

A

A

M(i) f



B

X

X

where the right-hand diagram and the bottom triangle of the left-hand diagram each commutes and the diagonal morphism of the left-hand diagram is a homotopy equivalence.   ⊕ Recall we have M( f : P  → P) = PI P as M( f : P  → P)n = Pn ⊕ Pn−1 P

Pn with the appropriate differential. Thus we can replace P in (∗) by M( f : P  → P) to get Pn

(id,0,0)

 Pn ⊕ Pn−1 ⊕ Pn

p

(i p ,±h, p)

Mn

Mn

i

where we can check that with an appropriate choice of sign in ±h that (i p  , h, p) is a chain map. We get (∗∗)

0

P

  f M P  −→ P

quis

0

M

quis

M

Cone ( f )

0

quis

M 

0

where Cone(•) is the mapping cone of • and the leftmost square commutes, proving the rightmost map to be a quis by a long exact and the 5-lemma. It is also clear that Cone ( f ) is projective. Now apply F to get a sequence

21 Exact Sequences of Homotopy Classes

(‡)

111

0 −→ F(P  ) −→ F M( f ) −→ F Cone ( f ) −→ 0 ,

which is exact because the sequence (∗∗) splits since Cone ( f ) ∈ C+ (P). In fact F Cone ( f ) = Cone Ff for F additive. Now take the long exact homology sequence of (‡). Naturality follows easily.  Suppose given a module exact sequence and two projective resolutions, so we have

P1

etc.

P1

0

P0

P0 ⊕ P0

P0

0

0

M

M

M 

0

0

0

0

This does not work for complexes because projective in each degree does not imply projective. Exercise Describe projective objects in C+ (A); the answer is not all of C+ (P). Puppe Sequences Fix an additive category A and suppose we have a map f : A → B of complexes. Then we get, for any complex N , a long exact sequence · · · ←− [S −1 Cone f, N ] ←− [A, N ] ←− [B, N ] ←− [Cone f, N ] ←− ←− [S A, N ] ←− [S B, N ] ←− [S Cone f, N ] ←− · · · where S A = Cone A → 0 so (S A)n = An−1 and d : (S A)n → (S A)n−1 is given by −d : An−1 → An−2 . Lemma Suppose 0 → A → A → A → 0 is an exact sequence of complexes which splits in each dimension. Then for any N , we have exact sequences [A , N ] ←− [A, N ] ←− [A , N ] , [N , A ] −→ [N , A] −→ [N , A ] .

112

21 Exact Sequences of Homotopy Classes

Proof We introduce the complex Hom· (A, N ), where Hom p (A, N ) =



Hom(An , Nn+ p )

n

with d f = d ◦ f − (−1) p f ◦ d. Remark 0-cycles in Hom· (A, N ) are f = ( f : An → Nn ) so that d ◦ f = f ◦ d, i.e., maps of complexes A → N . A zero-boundary is a map f = ( f : An → Nn ) of the form f = d ◦ h + h ◦ d for some h = (h : An → Nn+1 ) i.e., those maps homotopic to zero. Thus H0 Hom· (A, N ) = [A, N ] . Now, if 0 → A → A → A → 0 splits in each dimension, then 0 −→ Hom· (N , A ) −→ Hom· (N , A) −→ Hom· (N , A ) −→ 0 is exact as a product of exact sequences. Take the homology.



Remark For complexes of abelian groups for example, we have K · ⊗ L · with d(x ⊗ y) = d x ⊗ y + (−1)|x| x ⊗ dy. Then the differential in Hom· (A, N ) is the one making the evaluation map Hom· (A· , N· ) ⊗ N· −→ N· a map of complexes.

Chapter 22

Spectral Sequences

Recall the Lemma Suppose 0 → A → A → A → 0 is exact in C(A) and splits in each dimension. Then [A , X ] ←− [A, X ] ←− [A , X ] is exact.  f Recall also that given A −→ B, we can construct M( f ) = A I A B. Thus, we have the commutative diagram f

A

B homotopy equivalence

i0

M( f ) and set C( f ) = Cok i 0 . The Lemma applies to i0

0 −→ A −→ M( f ) −→ C( f ) −→ 0 because in dimension n we have 0

An

An ⊕ An−1 ⊕ Bn

An−1 ⊕ Bn

0,

and so we get an exact

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_22

113

114

22 Spectral Sequences

[A, X ]

[M( f ), X ]

[C( f ), X ]

f∗

j∗

[B, X ] where j : B → C( f ). Now construct A

f

j

B

j

C( f )

C( j  )

C( j)

···

and get exact sequences of homotopy classes [A, X ] ←− [B, X ] ←− [C( f ), X ] ←− [C( j), X ] ←− [C( j  ), X ] . Note that in general C( f )/B = S(A) and C( j)/C( f ) = S(B). (†) Lemma: If A canonical map

f

B is injective and splits in each dimension, then the C( f ) −→ B/A

is a homotopy equivalence. homotopy

homotopy

Thus C( f )/B ←−−−−− C( j) and C( j)/C( f ) ←−−−−− C( j  ),  S(A)

equivalence

 S(B)

equivalence

and so we get the exact sequence · · · ←− [A, X ] ←− [B, X ] ←− [C( f ), X ] ←− [S A, X ] ←− [S B, X ] ←− · · · Point of Lemma (†): We have 0

A

0

A

f

B

B/A

0

M( f )

C( f )

0

The first lemma gives the following exact

[A, X ]

[M( f ), X ]

[B/A, X ]

H1 Hom· (A, X )

•

[C( f ), X ]

H1 Hom· (A, X )

H1 Hom· C( f )

∼

[B, X ] ∼

[A, X ]

∼

so the 5-lemma implies [B/A, X ] −→ [C( f ), X ] and by Yoneda C( f ) → B/A is an isomorphism in K A, as desired. 

22 Spectral Sequences

115

Spectral Sequences Situation: Suppose K is a complex provided with an increasing filtration by subcomplexes 0 ⊂ · · · ⊂ F p−1 K ⊂ F p K ⊂ F p+1 K ⊂ · · · ⊂ K and suppose T∗ = {Tn } is a “∂-functor on complexes”, that is, it associates naturally to a short exact sequence of complexes 0 → K  → K → K  → 0 a long exact sequence ∂

∂

· · · −→ Tn+1 K  −→ Tn K  −→ Tn K −→ Tn K  −→ Tn−1 K  −→ · · · in an abelian category B. Problem To relate T∗ K with T∗ gr ∗ K where gr p K = F p K /F p−1 K . Useful diagram: Given A

u

B

a

Cok u

0

Cok v

0

b

C

v

D

Cok a

Cok b

0

0

where the top-left square commutes, then we have Cok v/Cok u

Cok b/Cok a = D/(Im b ⊕ Im v) . 

Proof Diagram chase.

Consider the induced filtration on T∗ (K ) and let F p T∗ K = Im{T∗ F p K → T∗ K }. Can we see F p T K /F p−1 T K inside T K p /K p−1 ? Use the exact sequence ∂

Tn+1 (K /F p K ) −→ Tn F p K −→ Tn K −→ Tn K /F p K

116

22 Spectral Sequences

to get ∂

Cok{Tn+1 K /F p K −→ Tn F p K } = F p T K . Then we find Tn+1 (K /F p−1 K )

∂

Tn (F p−1 K )

F p−1 (Tn K )

0

Tn+1 (K /F p K )

∂

Tn (F p K )

F p (Tn K )

0

∂

Tn (F p K /F p−1 K )

Tn (F p K /F p−1 K )

long exact sequence of F p /F p−1 → K /F p−1 → K /F p

long exact sequence of F p−1 → F p → F p /F p−1

Thus by the useful diagram, F p Tn K /F p−1 Tn K is isomorphic to a sub-quotient of Tn F p K /F p−1 K . Precisely, if we put F p Tn K /F p−1 Tn K ∼ =

Im{Tn F p K → Tn F p /F p−1 } , Im{Tn+1 K /F p K → Tn F p /F p−1 }

then we derive Corollary If the filtration is finite and exhausts K and if Tn (F p /F p−1 ) = 0, then Tn K = 0. Define Tn F p K /Fq K = Tn ( p, q) and set Z rp,q = Im{T p+q ( p, p − r ) −→ T p+q ( p, p − 1)} = Im{T p+q F p /F p−r −→ T p+q F p /F p−1 } . Then we find · · · ⊂ Z 3pq ⊂ Z 2pq ⊂ Z 1pq = T p+q ( p, p − 1) , and if we interpret F−∞ K = 0 then Z∞ pq = Im{T p+q (F p ) −→ T p+q F p /F p−1 } . Putting B rpq = Im{T p+q+1 ( p + r − 1, p) −→ T p+q ( p, p − 1)} , we have 0 = B 1pq ⊂ B 2pq ⊂ B 3pq ⊂ · · · ,

22 Spectral Sequences

117

and if we interpret F∞ K = K , then we have B∞ pq = Im {T p+q+1 K /F p K −→ T p+q F p /F p−1 } . Thus we have 0 = B 1 ⊂ B 2 ⊂ · · · ⊂ B ∞ ⊂ Z ∞ ⊂ · · · ⊂ Z 2 ⊂ Z 1 = T p+q F p /F p−1 . Now exactness of Tn ( p − 1, p − r ) −→ Tn ( p, p − r ) −→ Tn ( p, p − 1) implies Z rpq = Cok{T p+q ( p − 1, p − r ) −→ T p+q ( p, p − r )} and

Z rpq+1 = Cok{T p+q ( p − 1, p − r − 1) −→ T p+q ( p, p − r − 1)} .

Similarly ∂

Tn+1 ( p + r − 1, p − 1) −→ Tn+1 ( p + r − 1, p) −→ Tn ( p, p − 1) gives B rpq = Cok{T p+q+1 ( p + r − 1, p − 1) −→ T p+q+1 ( p + r − 1, p)} and B rp−r,q+r −1 = Cok{T p+q ( p − 1, p − r − 1) −→ T p+q ( p − 1, p − r )}, and we find T p+q ( p − 1, p − r − 1)

T p+q ( p, p − r − 1)

Z rpq+1

0

T p+q ( p − 1, p − r )

T p+q ( p, p − r )

Z rpq

0

B rp−r,q+r −1

+1 B rp−r,q+r −1

0

0

118

22 Spectral Sequences

If we set E rpq = Z rpq /B rpq and define dr : E rpq

E rp−r,q+r −1

Z rpq /B rpq

Z rpq /Z rpq+1 = B rpq+1 /B rpq ,

r r r +1 then dr2 = 0 on E ∗∗ and has homology H (E ∗∗ , dr ) = E ∗∗ .

Chapter 23

Spectral Sequences Continued

Recall that Tn ( p, p  ) = Tn (F p K /F p K ) , E 1p,q = T p+q ( p, p − 1) . We define ∞ 2 1 1 0 = B 1pq ⊂ B 2pq ⊂ . . . ⊂ B ∞ pq ⊂ Z pq ⊂ . . . ⊂ Z pq ⊂ Z pq = E pq ,

  ∂ Z rpq = Ker T p+q ( p, p − 1) −→ T p+q−1 ( p − 1, p − r )   = Im T p+q ( p, p − r ) −→ T p+q−1 ( p − 1, p − r )   = Cok T p+q ( p − 1, p − r ) −→ T p+q ( p, p − r ) , Z rpq+1 = Cok{T p+q ( p − 1, p − r − 1) −→ T p+q ( p, p − r − 1)} ,   ∂ B rpq = Im T p+q+1 ( p + r − 1, p) −→ T p+q ( p, p − 1)   = Cok T p+q+1 ( p + r − 1, p − 1) −→ T p+q+1 ( p + r − 1, p) ,   B rp−r,q+r −1 = Cok T p+q ( p − 1, p − r − 1) −→ T p+q ( p − 1, p − r ) ,   +1 B rp−r,q+r −1 = Cok T p+q ( p, p − r − 1) −→ T p+q ( p, p − r ) .

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_23

119

120

23 Spectral Sequences Continued

We thus have T p+q ( p − 1, p − r − 1)

T p+q ( p, p − r − 1)

Z rpq+1

0

∩

T p+q ( p − 1, p − r )

B rp−r,q+r −1

T p+q ( p, p − r )

Z rpq

0

+1 B rp−r,q+r −1

⊂

0

0

Thus, +1 r Z rpq /Z rpq+1 ≈ B rp−r,q+r −1 /B p−r,q+r −1

by the previous useful diagram and lemma from last time. Put E rpq = Z rpq /B rpq and consider dr

E rp+r,q−r +1

E rpq

dr

E rp−r,q+r −1

Z rp+r,q−r +1

Z qr

Z rp−r,q+r −1

∪

∪

∪

+1 Z rp+r,q−r +1

Z rpq+1

+1 Z rp−r,q+r −1

∪

∪

∪

+1 B rp+r,q−r +1

∼ =

B rpq+1

∼ =

+1 B rp−r,q+r −1

∪

∪

∪

B rp+r,q−r +1

B rpq

B rp−r,q+r −1

where the boxed result above gives an isomorphism between the quotients of top two terms with the quotients of the adjacent bottom two terms.

23 Spectral Sequences Continued

121

Define dr : E rp,q → E rp−r,q+r −1 to be the composition Z rpq /Z rpq+1

E rpq

∼ =

+1 r B rp−r,q+r −1 /B p−r,q+r −1 ∩

E rp−r,q+r −1 Clear that dr2 = 0 and E rpq+1 =

Ker{dr : E rpq −→ E rp−r,q+r −1 } Im{dr : E rp+r,q−r +1 −→ E rpq }

.

We get the spectral sequence r = {E rpq } , E ∗∗

r = 1, 2, . . . ,

and on each E r there is dr : E rpq → E rp−r,q+r −1 so that dr2 = 0, and E r +1 = H (Er , dr ). The beginning of the spectral sequence is E 1pq = T p+q ( p, p − 1) = T p+q (F p K /F p−1 K ) and the end is E∞ pq =

Im{T p+q F p K −→ T p+q K } . Im{T p+q F p−1 K −→ T p+q K }

T∗ K is called the abutment. Convergence Question: In what sense is E ∞ = lim E r ?   ∂ → T p+q−1 (F p−1 K ) , Z∞ pq = Ker T p+q ( p, p − 1) − and   B∞ pq = Im T p+q+1 (F p K ) −→ T p+q ( p, p − 1) . We say the spectral sequence converges strongly if for a given p and q we have r ∞ r Z∞ pq = Z pq and B pq = Z pq for r large enough .

122

23 Spectral Sequences Continued

Exercise Consider a ∂-functor {Tn }, where the ∂’s lower degree by 1, and a complex with a decreasing filtration . . . ⊃ Fq K ⊃ Fq+1 K ⊃ . . .. Show then that we have a spectral sequence E rpq , for r ≥ 2, starting with E 2pq = T p+q (Fq K /Fq+1 K ) and with T p+q (Fq /Fq+1 )

T p+q−1 (Fq+1 /Fq+2 ) abutting to T∗ K . Reference: Cartan Seminars. Examples 2 step filtration: 0 = F−1 K ⊂ F0 K ⊂ F1 K = K so E 1pq = T p+q F p K /F p−1 K = 0 for p = 0, 1 . with (∗)

0

F0 /F−1

F1 /F−1

F1 /F0

K

K

K 

and we suppose Tn = 0 for all n < 0.

q

T3 K  T2 K  T1 K  T0 K  T−1 K 

T3 K  T2 K  T1 K  T0 K  p

d1 : E 1pq

T p+q (F p /F p−1 )

E 1p−1,q ∂

T p+q−1 (F p−1 /F p−2 ) ,

and it can be shown that d1 = 0 for the long exact sequence. 2 Now d2 lowers p by 2 and hence is zero, whence E ∞ pq = E pq .

0

23 Spectral Sequences Continued

123

Thus Tn K has a 2-step filtration ∞ 2 = Im{Tn K  → Tn K } = E 0,n E 0,n = Cok{Tn+1 (K  ) → Tn K  } ∞ E 1,n−1 = Tn K /Im{Tn K  → Tn K } 2 = E 1,n−1

= Ker{Tn K  → Tn−1 K  } . This gives the same information as the long exact sequence of (∗). Another picture of the spectral sequence: d3 ∂

T p+q+1 ( p, p − 1)

∂

T p+q ( p − 2)

T p+q ( p − 2, p − 3)

∂

T p+q−1 ( p − 3)

T p+q ( p − 1)

T p+q ( p − 1, p − 2)

∂

T p+q−1 ( p − 2)

d2

T p+q ( p)

T p+q ( p, p − 1)

∂

T p+q−1 ( p − 1)

T p+q−1 ( p − 1, p − 2)

d1

T p+q−1 ( p)

E1

Tn F∗ K

E1

Tn F∗ K

A complex K has two canonical filtrations by dimension: (1) the increasing one: · · · → K n+1 → K n → · · · 

0, n > p, K n , n ≤ p, F p K /F p−1 K = K p [ p] (F p K )n =

(2) the decreasing one (the Postnikov filtration): ···

K n+2

K n+1

Zn

0

0

· · · Fn K

···

K n+2

K n+1

Kn

K n−1

K n−2

··· K

with

≈

Hi (Fn K ) −→ Hi K for i ≥ n = 0 else

124

23 Spectral Sequences Continued

and Fn K /Fn+1 K is given by 0 −→ K n+1 /Z n+1 −→ Z n −→ 0 , so Fn K /Fn+1 K is quis to Hn (K )[n].

Chapter 24

Hyper-Homology Spectral Sequences

General Remark 24.1 Let A be a complex. Then we have the exact 0

A

M(A → 0)

SA

{An }

{An ⊕ An−1 ⊕ 0}

{An−1 }

0

where M(A → 0) is the mapping cylinder of A → 0 and so is homotopy equivalent to 0, and S A is the suspension. Thus if T is a ∂-functor on K A, then we have the exact ···

Tn A

Tn M

Tn S A

Tn−1 A

0

Tn−1 M

···

0

so Tn S A ≈ Tn−1 A canonically. General Remark 24.2 Suppose T is so that Tn A = 0 for all n < 0. Then Tn A/F p A = Tn−( p+1) (S − p−1 A/F p A) = 0 if n − ( p + 1) < 0 . Thus Tn A/F p A = 0 if n ≤ p, and we have the exact · · · −→ Tn+1 A/F p A −→ Tn F p A −→ Tn A −→ Tn A/F p A −→ · · · whence Im{Tn F p A −→ Tn A} = Tn A for p > n . © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_24

125

126

24 Hyper-Homology Spectral Sequences

If moreover F 0. Remark If I is a poset and J is the poset of layers in I , then I and J are homotopy equivalent. More generally if C is a small category, we can form a category A whose objects are u

u

arrows in C and in which a morphism (X  −→ Y  ) −→ (X −→ Y ) is a diagram u

X

Y

u

X

Y

A is the cofibered category over C op × C belonging to the functor X, Y −→ HomC (X, Y ) , p

u

and the functor A −→ C given by (X −→ Y ) −→ Y is cofibered with fiber over Y equal to (C/Y )op , which is contractible since it has an initial object. q

Similarly A −→ C op is cofibered with contractible fiber over Y = X/C. It follows that p and q are homotopy equivalences by Theorem A plus the fact that for a cofibered functor f : C → C  the left fiber C/Y is homotopy equivalent to the fiber f −1 (Y ), that is, we have f −1 Y ⊂ C/Y ,  u ∗ X ←−

X u f X −→ Y

 adjoint to inclusion.

In fact BA is a kind of subdivision of BC where X → Y subdivides to id

X −→ X

X →Y

id

Y −→ Y

•−−−−−−−−−−−−−−•−−−−−−−−−−−−−−• and X

Y

Z sss

subdivides to

•

•

•

•

• •

33 A Filtration of Q(P A )

177

It follows that Y + is homotopy equivalent to the poset of all subspaces contained in but not equal to V , whence Y + ∼ = ∗. Similarly Y − ∼ = ∗, and Y + ∩ Y − is homotopy equivalent to the poset of all proper subspaces of V . Thus Y (V ) =

 (Y + ∩ Y − )

= Fn−1 Q(P A )/P , where



denotes the suspension.

Example The Tits building T (V ) is the simplicial complex associated to the poset of proper subspaces of V . If G = Aut V , then the parabolic subgroups of G which are distinct from G are in one-to-one correspondence with the flags in V . We have the Theorem (Tits) T (V ) is homotopy equivalent to a bouquet of (n − 2) spheres, where n is the dimension of V . Example For n = 2, T (V ) is the zero-dimensional complex with vertices given by lines in V , and for n = 3, T (V ) is a connected graph. n−2 (T (V ), Z). The Steinberg module is H

Chapter 34

Bi-simplicial Sets and Dold–Thom

Theorem → C  is a homotopy equivalence if Y \ f ∼ = ∗ for all Y , where A f : C X u Y \ f = Y −→ f X is the cofibered category corresponding to the map X → HomC  (Y, f X ). Let S( f ) be the cofibered category over C × (C  )op associated to (X, Y ) → HomC  (Y, f X ), so S( f ) has objects {(X, Y, u : Y → f X )} and morphisms given by diagrams u fX Y

Y

u

f X

Let T ( f ) be a bi-simplicial set, that is, [ p], [q] −→ sets of pairs (Y p −→ Y p−1 −→ · · · −→ Y0 −→ f X 0 , X 0 −→ · · · −→ X q ) of diagrams, the first in C  and the second in C, and call this T ( f ) p,q , i.e., T ( f ) p,q =



N p (C  / f X 0 )

X 0 →···→X q ∈N (C)

=



Nq (Y0 \ f ) .

Y p →···→Y0 ∈N (C  )

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34 Bi-simplicial Sets and Dold–Thom

Proposition Given a bi-simplicial set p, q → T p,q , we have homeomorphisms      [ p] −→ [q] −→ T pq   ≀≀   [ p] −→ T pp  ≀≀       [ p] −→ T pq   [q]  − →  Example T pq = X p × Yq for X and Y simplicial sets, whence   [q] −→ X p × Yq  = X p × |Y | so

     [ p] −→ [q] −→ X p × Yq  = |X | × |Y | .

Milnor’s Theorem says |X × Y | = |X | × |Y |, where X × Y : [ p] −→ X p × Y p . Thus the Proposition works here. Proof of Proposition First, by general nonsense, we only have to exhibit these homeomorphisms for the following generators on the category of bi-simplicial sets h ([m],[n]) : [ p], [q] −→ Hom´ ([ p], [m]) × Hom´ ([q], [n]) since for any F : C → Sets, we have lim h X = F . − →

(X,ξ )∈C F

Second, |[ p] → Hom´ ([ p], [m])| is the m-simplex ´m , and the combinatorial part is that ´m × ´n = |[ p] → Hom´ ([ p], [m]) × Hom´ ([ p], [n])| , proving the second assertion and hence the proposition. Claim T ( f ) pp = N p (S( f )). The proposition thus gives homeomorphisms            B(Y0 / f ) ,  p −→ |q −→ T pq |≈  p −→   Y p →Y0        q −→ | p −→ T pq |≈ q −→ 

 X 0 →···X q

    B(C / f X 0 ) . 



34 Bi-simplicial Sets and Dold–Thom

181

By the hypothesis of Theorem A B(C  / f X 0 ) ∼ = ∗ and B(Y0 / f ) ∼ = ∗. We still need the Lemma If X → B X is a functor from C to topological spaces so that B X ∼ = ∗, for all X in C, then               ∼   = BC . [ p] −→  [ p]  − → B point X0  =        X 0 →···→X p Y0 →···→X p    To see this, consider [ p] → 

   B X 0  and conclude that  X 0 →···→X p 

       [ p] −→ B(Y0 / f ) −→ B C  ,    Y p →···Y0    [q] −→  

 X 0 →···→X q

Thus we find B(S( f ))

   B(C  / f X 0 ) −→ B C . 

BC 



BC and compare this with the same thing for idC  : C  → C  : C

S( f )

C f

C

S(idC )

C

where the right-hand square of the diagram commutes. The argument above is natural in f thus proving Theorem A.  f

Recall that if E −→ B is a map of spaces and b ∈ B, then the homotopy fiber of f over b is

182

34 Bi-simplicial Sets and Dold–Thom

E( f ; b) = E × B B I × B {b} = {(e, λ) : λ is a path from f (e) to b} . This is the actual fiber over b of the Serre construction E × B B I = {(e, λ) : λ a path with λ(0) = e}

endpoint map

E f

B We get of course e ∈ f −1 (b) and the homotopy exact sequence e) −→ πn (E, e) −→ πn (B, b) −→ πn−1 −→ · · · . · · · −→ πn+1 −→ πn (E( f ; b), Note that

f −1 b ⊂ E( f, b)

via e = (e, constant path) We call f a quasi-fibration (in the sense of Dold–Thom) when the inclusion f −1 b ⊂ E( f, b) is a weak homotopy equivalence for all b ∈ B. Dold–Thom Example Let A be a “nice” subspace of a “nice” space X , where A is connected and has a base point, and let S P(X ) = ∪ X n / n be the infinite symmetric n

product. We have also S P(X/A), and the map S P(X ) −→ S P(X/A) is a quasi-fibration (this is a theorem), where the fiber over the base point is S P(A). We conclude that there is an exact sequence · · · −→ πn S P A −→ πn S P X −→ πn S P X/A −→ · · ·

We also have the n (X, Z). Theorem πn S P X = H A good paradigm for a quasi-fibration which is not a fibration is given by the natural f

mapping from the mapping cylinder of X −→ Y to the unit interval, which is not a

34 Bi-simplicial Sets and Dold–Thom

183

fibration unless f is the identity map but is a quasi-fibration provided f is a homotopy equivalence. u

Theorem B Suppose f : C → C  is so that for all Y  −→ Y in C  , the functor u ∗ : Y \ f −→ Y  \ f is a homotopy equivalence. Then B(Y \ f ) is homotopy equivalent to the homotopy fiber of B f over Y . Proof We have

       B(Y0 \ f ) [ p] −→   Y p →···→Y0        point  [ p] −→   Y p →···→Y0

This is a quasi-fibration from Dold–Thom theory because all specialization maps are homotopy equivalences by assumption. As in Theorem A, we have     [ p] −→ 

Y P →···→Y0

and

     B(Y0 \ f ) = [q] −→ B(C  / X 0 ) BC 

       [ p] −→ point  BC  .    Y p →···→Y0



Chapter 35

Homology of Q(P A) and the Tits Complex

We begin with the Example Take f

SPX −→ SPX/A and let [y1 , · · · , yn ] ∈ S P(X/A), where without loss of generality yi ∈ X \A. Then f −1 [y1 , · · · , yn ] is the union of [y1 , · · · , yn ] and any set in S P(A). Specialization maps are multiplication by elements of S P(A). S P(A) is a connected monoid, so these are homotopy equivalences. Let A be a Dedekind domain and F its field of fractions. We defined Fn Q(P A ) to be the full subcategory comprised of those P with rank at most n. Then 0 = F0 ⊂ F1 ⊂ · · · ⊂



Fn = Q(P A ) .

n

Problem Compare the homology of Fn and Fn−1 . To simplify what follows, assume Pic (A) = 0, so the only projective module P of rank n is An and Hom Q(P A ) (P, P) = Aut (P) = GLn (A) . If we regard Aut An as a category, then we have inclusions Fn−1

i

Fn

j

Aut An

of categories. Recall from last term that for a functor f : C → C  and a functor F : C → Ab, we have f ! (F)(Y ) = colim F(X ) . (X, f X →Y )

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35 Homology of Q(P A ) and the Tits Complex

186

More generally, if F· is a complex in Funct (C, Ab), then we have a complex L f ! (F· ) in Funct (C  , Ab) unique up to quasi-isomorphism. There is an isomorphism H p (C  , L f ! (F· )) ∼ = H p (C, F· ) , and there is a spectral sequence with E 2pq = H p (C  , L q f ! (F· )) converging to Hn (C  , L f ! (F· )) with L q f ! (P)(Y ) = Hq ( f /Y, F pulled back to f /Y ) . Setting Γ = GLn A, if we put f = j : Γ → Fn , then ⎧ ⎨ ∅, if P ∈ Fn−1 , j/ p = the category whose objects are elements of Hom(An , P) and whose ⎩ maps are given by composition with elements of Γ , if rank P = n. So if M is any Γ -module, then ⎧ ⎨ 0 , if P ∈ Fn−1 , L q j! (M)(P) = 0 , if P ∼ = An and q > 0, ⎩ M , if P ∼ = An and q = 0, so H∗ (Fn , j! (M)) = H∗ (Γ, M) . Note that j! (M) is a typical functor on Fn with support {An }. Now we consider i : Fn−1 → Fn . Then for P ∈ Fn−1 , i/ p is a category with final object, hence Hq (i/ p, F) = 0, for q > 0, and any F : i/ p → Ab. i/An is equivalent to the poset of proper layers in F n , which in turn is equivalent to the suspension of T (F n ), namely the poset of proper subspaces of F n . Thus k−1 (T (F n ), Z) k (i/An , Z) = H H  0 , k − 1 = n − 2, = I (F n ) , k − 1 = n − 2, where I (F n ) is the Steinberg module of F n . Now we have H∗ (Fn−1 , Z) = H∗ (Fn , Li ! (Z)) ,  L q i ! (Z)( p) =

0 , q = 0 , p ∈ Fn−1 , Z , q = 0 , p ∈ Fn−1 ,

35 Homology of Q(P A ) and the Tits Complex

and L q i ! (Z)(An ) = Hq (i/An , Z) =

187

⎧ ⎨

0 , q = 0, n − 1, Z , q = 0, ⎩ I (F n ) , q = n − 1,

provided n = 1. A formula valid even for n = 1 is an exact sequence 0 ←− Z ←− L i ! (Z) ←− j! I (F n )[n − 1] ←− 0 , and from this we get exact homology sequences ···

Hq (Fn , Z)

Hq (Fn , L i ! (Z))

Hq (Fn , j! I (F n )[n − 1])

···

···

Hq (Fn , Z)

Hq (Fn−1 , Z)

Hq−n+1 (GLn (A), I (F n ))

···

The Tits complex For a vector space V , |T (V )| is the simplicial complex associated to the poset T (V ) of proper subspaces of V . Choose a line L in V and consider the map W → W + L. If this were a well-defined map T (V ) → T (V ), then T (V ) would be conically contractible since we would have W ≤W +L ≥ L. But the map fails to be well-defined on H L = {hyperplanes H with H + L = V } , and the above argument shows that the poset T (V ) − H L is contractible. Now for a vertex v in a simplicial complex K , we define Link v = the subcomplex of simplices τ with v ∈ / τ such that v ∪ τ ∈ K , Star v = the subcomplex of simplices σ such that v ∈ σ, Star v = the subcomplex of simplices σ with v ∪ τ ∈ K , so we have Star v = Star v ∪ Link v. Then for any v, we have K = (K − Star v)

 Link v

Star v,

35 Homology of Q(P A ) and the Tits Complex

188

so |T (v)| =



|T (v) − H L | ∪

H ∈H L



cone on Link (H )

Link H

since no two H in H L form a simplex. Then since |T (v) − H L | ∼ = ∗, we find |T (v)| 



S Link (H ) .

H ∈H L

But Link (H ) = |T (H )|, so by induction |T (v)| is a bouquet of spheres, and #spheres in |T (P n )| = (# H in H L ) · (#spheres in |T (P n−1 )|). Moreover, for F a finite field of characteristic q, we have # H in H L = q n . It follows from this argument that once a flag 0 < F1 < F2 < · · · < Fn−1 < V is chosen, then I (V ) has a basis indexed in a natural way by flags V > H1 > H2 > · · · > Hn−1 > 0 with Hi ⊕ Fi = V for each i. As a module over the unipotent radical of the Borel subgroup B n of Aut (V ) = GLn (F) fixing {Fi }, I (V ) is free, i.e., I (V )  Z [B n ]. For F = Fq , B n is a Sylow p-subgroup and I (V ) is projective over Sylow subgroups. It follows that I (K ) p is projective over Z p [GLn (F)].

Chapter 36

Long Exact Sequences of K -Groups

Recall Theorem A: f : C → C  is a homotopy equivalence provided any of the following hold for all Y in C  (i) f /Y  ∗, (i’) f is pre-cofibered and f −1 Y  ∗, (ii) Y \ f  ∗, (ii’) f is pre-fibered and f −1 Y  ∗. Pre-cofibered means “there are cobase change operations,” i.e., given u : Y → Y  in C  , we have a cobase change functor u ∗ : f −1 (Y ) → f −1 (Y  ), that is, f −1 Y → f /Y u has an adjoint, namely, (X, f X −→ Y ) −→ u ∗ (X )). Suppose M is an exact category. Let E(M) denote the category of short exact sequences M  → M  M  in M. E(M) is itself an exact category, and we have three exact functors s, t, q from E(M) to M: s(M  → M  M  ) = sub-object = M  , t (M  → M  M  ) = total object = M, q(M  → M  M  ) = quotient object = M  . Remark Exact functors induce maps on Q-categories and hence on K -groups. Theorem (s, q) : E(M) → M × M induces a homotopy equivalence on Q-categories, i.e., Q(E(M))  QM × QM. (†) Corollary: Given an exact category N and an exact sequence of exact functors 0 → F  → F → F  → 0 from N into M. Then on K -groups, we have F∗ = F∗ + F∗ : K i (N ) → K i (M). Proof We have the diagram © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5_36

189

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36 Long Exact Sequences of K -Groups

Q(E(M))

t

QM

(s,q)

QM × QM (s,q)

and a section η of E(M) −−−→ M × M by (M  , M  ) → (M  → M  ⊕ M   M  ) . 

Thus adding Q(η) to the diagram, we get the result.

Proof of Theorem It is sufficient by Theorem A to show that for any (M  , M  ) ∈ QM × QM, we have Q(E(M))(M  , M  )  ∗. The typical element is M

•

E

ξ

• •

M  η

E

E 

By pulling back with respect to η and pushing forward by ξ, we retract Q(E(M))/ (M  , M  ) into the full subcategory of diagrams M

E

M 

E

E 

But this full subcategory has the initial object M

E

E

E 

E

E

0

0

0

0

M



36 Long Exact Sequences of K -Groups

191

Theorem B: Let f : C → C  and suppose that for any Y → Y  in C  , the induced functor f /Y → f /Y is a homotopy equivalence. Then the homotopy fiber of B f : BC → BC  over Y is homotopy equivalent to B( f /Y ). Geometrical Example Let g : K → L be a map of simplicial complexes. Take C to be the poset of simplices in K , C  to be the poset of simplices in L and suppose y is a simplex in L. Then f /y = {simplices x in K : f x ⊂ y} = {simplices in f −1 y} , and f −1 y   f −1 y for all y  ≤ y implies that f −1 y  homotopy fiber. Localization Theorem Suppose A is an abelian category and B a Serre subcategory with A/B the quotient abelian category. Then B QB  homotopy fiber of (B QA −→ B Q(A/B)) . Proof From Theorem B plus lots of work, we have a short exact sequence QB −→ QA −→ Q(A/B) . 

See Quillen’s paper. Corollary There is an exact sequence of K -groups . . . −→ K i B −→ K i A −→ K i A/B −→ K i−1 B −→ . . .

 Remark Suppose V ∈ A/B. We have S B Aut V −→ B Q(A/B) , so B Aut V −→  B Q(A/B) , and we get B Q(A/B)

B QB

B QA

B Q(A/B) .

B Aut V We exhibit this map. Given V ∈ A/B, take A = Modf A, B = S-torsion and A/B = Modf(S −1 A), where A is a noetherian ring.

192

36 Long Exact Sequences of K -Groups

Construction: Take the poset J V of all finitely generated A-submodules in M ⊂ V so that V = S −1 (M). J (V )  ∗ because we can take suprema. Let JV = poset of layers in J V . We have a functor JV −→ QB (M0 , M1 ) −→ M1 /M0 . Now form the cofibered category JV Aut V over Aut V with fiber JV . The objects are still (M0 , M1 ) but now morphisms (M0 , M1 ) −→ (M0 , M1 ) are automorphisms θ of V so that (θM0 , θM1 ) ≤ (M0 , M1 ). We have a functor JV Aut V

QB

homotopy equivalence by Theorem A

Aut V Example 1 Recall if A is regular noetherian, then K0 A ∼ = K 0 A[T ] . This generalizes with a proof as before, so Kn A ∼ = K n A[T ] . Example 2 For A regular, consider K ∗ A[T, T −1 ]. We have K ∗ (P A ) = K ∗ (Modf A) for A and also for A[T ] and consider Modf A[T ] −→ Modf A([T ]) −→ Modf A[T, T −1 ] . T -torsion A/B A B By Devissage, B has the same K -groups as A[T ]-modules killed by T , and so K∗ B = K∗ A . We get the long exact sequence

36 Long Exact Sequences of K -Groups

···

Ki A

α

K i A[T ]

193

K i A[T, T −1 ]

K i−1 A

··· .

Ki A To compute α, take M ∈ Modf A ⊂ B. We have the exact sequence F

                       ×T 0 −→ A[T ]⊗ A M −−→ A[T ] ⊗ A M −→ M −→ 0 of exact functors from Modf A to Modf A[T ]. By Corollary (†), we conclude that α = 0, i.e., F∗ = F∗ + α, and so finally K i A[T, T −1 ] ∼ = K i A ⊕ K i−1 A .

Chapter 37

Localization

Let X = algebraic variety over a field k , M X = (abelian) category of coherent sheaves on X , P X = category of locally free sheaves on X . (“vector-bundles”) If X = Spec A, for A a finitely generated k-algebra , M X = finitely generated modules over A , P X = finitely generated projective modules over A , d

then K i X = K i P X is a contravariant functor in X because f : X → Y induces an exact functor f ∗ : PY → P X . d Let us also define K i X = K i M X , a covariant functor in X for proper maps. This holds because of Grothendieck’s argument for i = 0: If f : X → Y and F is a coherent sheaf on X , then R q f ∗ F are coherent sheaves on Y and vanish for q > dim X . So we can form  (−1)q [R q f ∗ F] ∈ K 0 MY . q≥0

This gives a map M X → K 0 MY which is additive for short exact sequences, namely, the Euler characteristic, and so induces K 0 M X → K 0 MY . For higher K -groups, assume that X is quasi-projective, i.e., X can be embedded in projective space, so a proper morphism f : X → Y is projective, i.e., factors as a  p composition X → Y × P n → Y with i a closed embedding and p the projection.

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37 Localization

Then the map ∗ : K i (X ) → K i (Y × P n ) is induced by the embedding M X → MY ×P n . The map p∗ : K i (Y × P n ) → K i (Y ) can be defined using Serre’s theorem R q p∗ (F(n)) = 0 , for q > 0 where

F(n) = F ⊗ O(1)⊗n .

This shows that large chunks of MY ×P n (and thus of M X ) map, by R 0 p∗ (resp. by R 0 f ∗ ) into MY , in a way preserving short exact sequences, and that is sufficient. For a more detailed and sophisticated explanation see §6.2 of Quillen’s paper [10]. f

Suppose A −→ B is a map between noetherian rings. This induces an exact functor P A → P B , where P → P ⊗ A B, and hence a map K i A → K i B. However in general, the map Modf A → Modf B given by M → B ⊗ A M is not exact. But assume that B A has finite Tor dimension, i.e., the ith left derived functor Tori (B, M) = 0 for all sufficiently large i. Then we can define a map K 0 Modf A −→ K 0 Modf B  M −→ (−1)q [ToriA (B, M)] . q≥0

For higher K-groups, introduce a filtration F0 M A ⊂ · · · ⊂ F p M A ⊂ · · · ⊂ Modf A = M A , where M ∈ F p M A if and only if ToriA (B, M) = 0 for i > p. Then M → Tor 0 = B ⊗ A M is exact from F0 M A to M B and so induces a map K i F0 M A −→ K 0 M B . Now, we use the Resolution Theorem Suppose we have a category closed under extensions 0 −→ M  −→ M −→ M  −→ 0 . Then we have an induced exact sequence · · · −→ Tor Ap (B, M  ) −→ Tor Ap (B, M) −→ Tor Ap (B, M  ) −→ Tor Ap−1 (B, M  ) −→ · · · .

37 Localization

197

Moreover, if M in a subcategory implies that M  is also in the subcategory, then we find K i F p−1 M A ∼ = K i Fp M A and get an induced map in higher K -groups. Example of Localization Theorem Let Q be the quotient field of Z. Then we have the exact sequence T = finitely generated torsion Z-modules

MZ

MQ ,

and K ∗ MZ = K ∗ Z since Z is regular by the Resolution Theorem and similarly for K ∗ Q. Now T =



Tp ,

p

where the product is over primes p and T p denotes the p-torsion finite abelian groups, whence  K∗ T p . K∗ T = p

Use Devissage to get K ∗ Modf(Z/ p Z) ≈ K ∗ T p , whence from localization   K i Z/ p −→ K i Z −→ K i Q −→ K i−1 Z/ p −→ · · · . · · · −→ p

p

There is an obvious generalization · · · −→



K i A/m −→ K i A −→ K i F −→

m



K i−1 A/m −→ · · ·

m

to a Dedekind domain A with quotient field F. Facts: ⎧ ⎪ ⎪ ⎨

[Quillen]

Z, Fq∗ ∼ = Z/(q − 1), K i (Fq ) = 0, ⎪ ⎪ ⎩ Z/(q j − 1), i = 2 j

i = 0, i = 1, i even , − 1 , for j ≥ 1 .

198

37 Localization

Suppose A is the ring of integers in a number field F with r1 real places and r2 complex places. Then ⎧ 1, i = 0, ⎪ ⎪ ⎪ ⎪ i = 1, ⎨r1 + r2 − 1 , 0 , i = even , [Borel] rank K i A = ⎪ ⎪ , i = 3, 7, 11, 15, . . . , r ⎪ 2 ⎪ ⎩ i = 5, 9, 13, . . . . r1 + r2 , A general fact [Bass] is that K 1 A = A∗ for integers in number fields. From [Quillen], we get ⎧ ⎪ ⎪ ⎪ ⎨

Z, i = 0, 0 , i ≡ 0(2) ,  K i (F p ) = Q /Z , i odd , ⎪ ⎪ ⎪ ⎩  = p prime

and in fact Q /Z =

n≥0

Z/n Z .

Example Suppose k = F p and let X be a complete non singular curve over k. We have k[T −1 ] ⊂ A− ∩ ∩ k(T ) ⊂ F ∪ ∪ k[T ] ⊂ A+ where A± is the integral closure of k[T ±1 ] in F, and we set B = A+ A− ⊂ F. Then X = Spec A+ ∪ Spec A− is a finite ramified cover of P1 (k), and we have P X = (P+ , P− , α), for P± ∈ P A± , where α an isomorphism of P+ ⊗ B with P− ⊗ B. This is like a Dedekind domain where we have the following “localization” · · · −→



K i k(x) −→ K i X −→ K i F −→ · · · ,

closed points x in X

and note that k(x) ∼ = F p because k is algebraically closed. Take D = the group of divisors on curve = free abelian group on closed points, whence

 x∈X

K i (k) = D

Z

K i (k) .

37 Localization

199

This gives · · · −→ D ⊗ K i k −→ K i X −→ K i F −→ D ⊗ K i−1 k −→ · · · , ending K1 F

D ⊗ K0 k

F∗

D

K0 X

K0 F

0

with this bottom map associating to F its divisor D=Z



D0

of zeros and poles, where D0 are divisors with total degree zero and the summand Z captures the degree. Let J be the Jacobian of the curve. Then it is well-known that D0 /Im F ∗ = J , and so   K0 X = Z Z J. rank

Note that J is divisible.

degree

Chapter 38

The Plus Construction, K 1 and K 2

An acyclic map f : X → Y is one so that either of the following two equivalent conditions holds (i) the homotopy fiber of f is acyclic, (ii) for all local coefficient systems L, we have H∗ (X, f −1 L)  H∗ (Y, L). Theorem Given X and a perfect normal subgroup N of π1 X , there is a unique acyclic map f : X → Y with kernel N . Also, there is the following universal property in the homotopy category f

X

Y

g h

Z with a unique h making the diagram commute if and only if N < Ker π1 g. Recall that the set of acyclic maps is closed under push out and pull back. Take X = BG L(A) and so π1 X = G L(A) with πq X = 0 for all q ≥ 2. Take N = E(A), which is perfect and the commutator subgroup by Whitehead. Denote the unique acyclic map f : X → Y with Ker π1 f = E(A) by f : BG L(A) −→ BG L(A)+ .

Review of K 1 A [Bass] and K 2 A [Milnor], see Milnor’s book: According to Bass, K 1 is K 1 A = G L(A)ab = G L(A)/E(A) = π1 (BG L A+ ) . d

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38 The Plus Construction, K 1 and K 2

One way of defining Milnor’s K 2 is d

K 2 A = H2 (E(A), Z) = H2 (B E A, Z) . Meanwhile we have the pull back B E A = BG L A

acyclic

 BG L A+ K1

BG L A+

BG L A and

 π2 BG L A+ = π2 BG L A+  = H2 BG L A+ , by Hurewicz = H2 BG L A = H2 B E A . Note that B Q P A = K 0 A × BG L A+ . The Schur multiplier and the universal central extension of a perfect group Recall from day one that H 2 (G, M) is the collection of extensions ∗ −→ M −→ E −→ G −→ ∗ of G by M, and the extension is central if and only if G acts trivially on M. If M is a trivial G-module, then we have a Universal Coefficient Theorem 0 −→ Ext 1 (H1 G, M) −→ H 2 (G, M) −→ Hom(H2 G, M) −→ 0 , and G perfect implies Ext = 0, so H 2 (G, M) = Hom(H2 G, M) . Translation: there exists a canonical central extension ξ ∈ H 2 (G, H2 G) so that any other extension is induced by a unique map from H2 G to M: ξ:

∗

H2 G

 G

G

∗

∗

M

E

G

∗

38 The Plus Construction, K 1 and K 2

203

For G perfect,  is the universal central extension of G, and G H2 G is the Schur multiplier of G, and in particular for E A,  E A is the Steinberg group, and E A −→ E A} = K 2 A. H2 E A = Ker {  E A is generated by I + a E i j = ei j (a), and the commutators are [ei j a, e jk b] = eik (ab) , for i, j, k pairwise distinct. In fact, Milnor uses this to show K 2 (Fq ) = 0. Exercise π3 (BG L A+ ) = H3 ( E A, Z). BG L A is not an H -space since π1 is not abelian. Put N = {1, 2, . . .} and let θ : N → N be any embedding. Then θ induces a map G L A → G L A given by  αi j →

δi j , if i or j ∈ / Im θ , α θ−1 (i) θ−1 ( j) , else, 

which for instance simply shifts α diagonally α →

 10 in case θ(n) = n + 1. 0α

We get BG L A

BG L A+

BG L A

θ

BG L A+

and have Lemma θ  id. Proof Step 1: Show the homotopy equivalence. We have G Ln A

GL A θ

GL A and there exists an α ∈ E A so that conjugation by α and the map θ have the same effect on G L n A. Thus the diagram

204

38 The Plus Construction, K 1 and K 2

H∗ G L n (A)

H∗ G L A θ∗

H∗ G L A commutes because conjugation acts trivially. So θ∗ acts identically with the same argument showing that θ∗ = id on H∗ (G L A, M) for M any (G L A/E A)-module, and meanwhile H∗ (G L A, M) = H∗ (BG L A, M) = H∗ (BG L A+ , M) . It follows that θ induces an isomorphism on BG L A+ for all kinds of homology, and so it is a homotopy equivalence. Step 2 (Exercise) Any self-homomorphism of the monoid M of all embeddings θ : N → N is trivial. 

This proves the lemma.

Proposition BG L A+ is a homotopy commutative associative H -space.  Proof Choose N N → N. This gives a homomorphism G L A × G L A → G L A and hence a map B(G L A × G L A)+ → BG L A+ . But B(G L A × G L A)+ = (BG L A × BG L A)+ = BG L A+ × BG L A+ , whichgives a product. The previous lemma shows independence of the choice of N N → N. Also, the restriction of the product to both of ∗ × BG L A+ and BG L A+ × ∗ are homotopic to the identity by the lemma again. We thus have an H -space.  Example For F = F p , compute π∗ BG L F + = K ∗ F. To this end, following [R. Brauer], construct a map BG → BU using character theory starting from a representation of a finite group G over F. Namely, suppose ρ given G −→ G L n F. Then choose an embedding F ∗ ⊂ {roots of unity} ⊂ C , where F ∗ =

  = p

Q /Z . Define the Brauer character χ to be

χ(ρ)(g) = the sum of the eigenvalues of ρ(g) in C∗ . Theorem (Brauer) χ(ρ) is the character of a virtual representation of G over C, that is χ(ρ) = χ V1 − χ V2 .

38 The Plus Construction, K 1 and K 2

205

We get BG L F → BU hence BG L F + → BU since BU is simply connected, and this is almost a homology equivalence. ************************* As noted in the Foreword, the class and the notes ended abruptly and embarrassedly here with this vague remark on homology equivalence owing to the end of class time, which Quillen always meticulously respected. The precise statement proved in Quillen’s 1972 paper [9] is that this map lifts to the homotopy fiber of Ψ q − 1, where Ψ is an Adams operation, and this lift is almost a homotopy equivalence.

Afterword by Mikhail Karpranov

This volume contains the notes of a course given by Daniel Quillen at MIT in 1979/80, almost 40 years ago. The goal of the course was to provide an introduction to Quillen’s fundamental work on higher algebraic K-theory. It begins from scratch, making very minimal requirements on the mathematical background of the participants (only an elementary knowledge of groups, rings etc. is assumed), and in the space of 38 lectures brings them to the subject matter of Quillen’s original work. Among the subjects covered are: • • • • • •

Group cohomology. Homological algebra. Derived categories. Simplicial sets and simplicial homotopy theory. Homotopy theory of categories via classifying spaces. Higher K-theory via Quillen’s Q-construction. Sketches of relations to other techniques: – Tits complexes (used in Quillen’s proof of the finite generation of K-groups for algebraic integers and curves over finite fields). – The definition of higher K-groups via the Plus construction.

Most of this material is available in modern textbooks and in Quillen’s original and excellently written papers [9–11]. For further study the reader will definitely need to master these as well. Among the standard modern references on homological algebra one can name the books of Gelfand–Manin [2] and Weibel [15] as well as the classic [1] of Cartan–Eilenberg. The basics of simplicial homotopy theory can also be found in Gelfand–Manin [2] and a more systematic treatment in the book of Goerss–Jardine [3]. The books of Srinivas [13] and Weibel [16] provide systematic treatments of various approaches to higher K-theory. The present volume gives a shortened overview of the subject which may be useful and less intimidating for the beginner than a more fundamental course of reading which can come later. Comparison of modern treatments shows how difficult it is to improve on Quillen. The sheer perfection of his work is almost frightening. As far as the foundations of © Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5

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208

Afterword by Mikhail Karpranov

K-theory go, one important later development is the S-construction of Segal and Waldhausen [14, 16] which can be viewed as a somewhat more flexible analog of the Q-construction, applicable to a wider class of categories. Quillen famously expressed hope [10] that the techniques around his theorems A and B will some day be incorporated into a general homotopy theory for toposes. One can see Lurie’s theory of ∞-categories and ∞-toposes [7] as a confirmation of this hope. These notes, with their informal style, provide a direct window into the way of operation of a great mind. Their preservation, owing to the effort of Robert Penner, will be very much appreciated by the readers.

References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

H. Cartan, S. Eilenberg, Homological Algebra (Princeton University Press, Princeton, 1956) S.I. Gelfand, Y.I. Manin, Methods of Homological Algebra (Springer, Berlin, 1997) P.G. Goerss, J.F. Jardine, Simplicial Homotopy Theory (Birkhäuser, Basel, 1999) D. Grayson, Higher algebraic K-theory II (after Daniel Quillen). Lecture Notes in Mathematics, vol. 551 (Springer, Berlin, 1976), pp. 217–240 D. Grayson, Finite generation of K-groups of a curve over a finite field (after Daniel Quillen), vol. 966, Lecture Notes in Mathematics (Springer, Berlin, 1982), pp. 69–90 A. Grothendieck, J.-L. Verdier, Préfaisceaux (SGA4 Éxp. 1), Lecture Notes in Mathematics, vol. 269 (Springer, Berlin, 1972), pp. 17–233 J. Lurie, Higher Topos Theory (Princeton University Press, Princeton, 2009) J. Milnor, Introduction to Algebraic K-Theory, Annals of Mathematics Studies, vol. 72 (Princeton University Press, Princeton, 1971) D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field. Ann. Math. 96, 552–586 (1972) D. Quillen, Algebraic K-theory I, vol. 341, Lecture Notes in Mathematics (Springer, Berlin, 1973), pp. 85–147 D. Quillen, Finite generation of the groups K i of rings of algebraic integers, vol. 341, Lecture Notes in Mathematics (Springer, Berlin, 1973), pp. 179–198 J.-P. Serre, Local Fields (Springer, Berlin, 1979) V. Srinivas, Algebraic K-Theory (Birkäuser, Basel, 1996) F. Waldhausen, Algebraic K-theory of generalized free products. Ann. Math. 108, 135–204 (1978) C.A. Weibel, Introduction to Homological Algebra (Cambridge University Press, Cambridge, 1994) C.A. Weibel, The K-book: An Introduction to Algebraic K-theory (American Mathematical Society Publishing, Providence, 2013)

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5

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Index

A Abelian category, 51, 55 Acyclic map, 201 Additive category, 54 Additive functor, 23 Additive non-abelian category, 57 Adjoint functors, 43 Adjunction formula, 127 Admissible layer, 172 Admissible morphism, 144 Arrow, 6 Axioms for simplicial objects, 10

B Bi-simplicial set, 179 Burnside ring, 103

C Cartesian morphism, 63 Category, 6 Category object, 156 Classifying space, 156 1-coboundary, 6 2-cocycle, 3 Co-chain complex, 4 Cofibered category, 64 Cogroup structure, 52 Cohomology of cyclic groups, 28 Cohomology of free groups, 22 Cokernel, 54 Complex in abelian category, 72 Conical contractibility, 17 Conical contraction, 170 Connecting homomorphism, 23

Contractibility, 169 Crossed homomorphism, 5

D δ-functor, 23 Derivation, 5, 22 Derived category, 91 Devissage Theorem, 106 Direct limit, 42 Direct sum, 42 Discrete category, 47, 69 Dold–Thom Theorem, 182

E Effaceable functor, 21 Equalizer, 41 Exact category, 143 Exact sequence of K -groups, 191

F Fiber, 47 Fibered category, 63 Final object, 40 First homotopy property, 95 Full subcategory, 73 Functor, 7

G Generalized Kan formula, 134 Geometric realization, 155 Grothendieck group, 102 Group completion, 101

© Springer Nature Switzerland AG 2020 R. Penner, Topology and K-Theory, Lecture Notes in Mathematics 2262, https://doi.org/10.1007/978-3-030-43996-5

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212 Group extension, 1 Groupoid, 7 G-set, 41

H Higher K -groups, 161 Hochschild–Serre spectral sequence, 140 Homology groups with coefficients in a module, 13 Homology of category with values in a functor, 12 Homology of cyclic groups, 27 Homology of simplicial complex, 16 Homotopy between functors, 72 Homotopy equivalence, 169 Hyper-homology spectral sequence, 126

I Increasing filtration, 123 Initial object, 40 Injective object, 71

K Kan formula, 47 Kernel, 55

L Layer, 170 Left-derived functor, 91 Left-fiber, 47 Leray spectral sequence, 133 Localization, 101 Localization Theorem, 191

M Mackey formula, 35 Mapping cone, 87 Mapping cylinder, 85 Modules, 2 Monoid, 6 Morphisms, 6

N Nerve of a category, 7 Nerve of topological category, 156 Normalization Theorem, 15

Index O Objects, 6

P Plus construction, 201 Poset, 6 Postnikov filtration, 123 Pre-cofibered category, 64 Pre-fibered category, 63 Projective object, 71 Projective resolution, 73 Puppe sequence, 111

Q Quasi-fibration, 182 Quasi-isomorphism, 72 Quis, 72

R Representable functor, 40 Resolution of a complex, 73 Resolution Theorem, 107, 144, 196 Right fiber, 47

S Schreier Lemma, 106 Schur multiplier, 203 Schur–Zassenhaus Theorem, 36 Second cohomology group, 4 Section, 3 Semi-simplicial space, 155 Serpent Lemma, 59 Simplex in a category, 7 Simplicial abelian group, 10 Simplicial object, 9 Simplicial set, 7 Small category, 6 Spectral sequence, 115 Split extension, 22 s.s., 155 Steinberg group, 203 Steinberg module, 177

T Theorem A, 189 Theorem B, 191 Thick subcategory, 147 Tits building, 177 Tits complex, 187

Index Topological category, 156 Transfer map, 33 U Universal central extension, 203

213 Y Yoneda Lemma, 39