Derived algebraic geometry 9782856299388

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DERIVED ALGEBRAIC GEOMETRY D. Ben-Zvi, D. Calaque, J. Grivaux, E. Mann, D. Nadler, T. Pantev, M. Robalo, P. Safronov, G. Vezzosi

Panoramas et Synthèses Numéro 55

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

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PANORAMAS ET SYNTHÈSES 55

DERIVED ALGEBRAIC GEOMETRY D. Ben-Zvi, D. Calaque, J. Grivaux, E. Mann, D. Nadler, T. Pantev, M. Robalo, P. Safronov, G. Vezzosi

Société mathématique de France

David Ben-Zvi Department of Mathematics University of Texas 2515 Speedway Stop C1200 Austin TX 78712, USA David Nadler Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA Étienne Mann Université dAngers, Département de mathématiques, Bâtiment I, Faculté des Sciences, 2 Boulevard Lavoisier, F-49045 Angers Cedex 01, France Marco Robalo Sorbonne Université. Université Pierre et Marie Curie, Institut Mathématiques de Jussieu Paris Rive Gauche, CNRS, Case 247, 4, place Jussieu, F-75252 Paris Cedex 05, France Gabriele Vezzosi Dipartimento di Matematica ed Informatica U. Dini, Viale Morgagni, 67/a, 50134 Firenze, Italy Tony Pantev Department of Mathematics, University of Pennsylvania, DRL 209 South 33rd Street, Philadelphia, PA 19104-6395, USA Damien Calaque IMAG, Univ Montpellier, CNRS, Montpellier, France & Institut Universitaire de France Julien Grivaux IMJ-PRG, UMR 7586, Sorbonne Université, Case 247, 4 place Jussieu, F-75252 Paris Cedex 05, France Pavel Safronov School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh UK, EH9 3FD

Classification mathématique par sujets. (2010) — 14-01, 14-06, 18-01, 18-06, 13D03, 13D10 14C40, 14D23, 14F05, 14A20, 14B10, 14D15, 14D20, 14N35, 17B63, 18D50, 18G55, 53D17, 55U40. Mots-clés et phrases. — Algèbres de Poisson, opérades, champs dérivés, invariants de GromovWitten, géométrie algébrique dérivée, catégories supérieures, théorie des déformations, problèmes de modules, algébroïdes de Lie différentiels gradués, dualité de Koszul, traces, homologie de Hochschild, espaces de lacets, théorème de Grothendieck-Riemann- Roch, espaces de modules, géométrie symplectique. Keywords and phrases. — Poisson algebras, operads, derived stacks, Gromov-Witten invariants, derived algebraic geometry, higher categories, deformation theory, moduli problems, differential graded Lie algebroids, Koszul duality, traces, Hochschild homology, loop spaces, GrothendieckRiemann-Roch theorem, moduli spaces, symplectic geometry.

DERIVED ALGEBRAIC GEOMETRY D. Ben-Zvi, D. Calaque, J. Grivaux, E. Mann, D. Nadler, T. Pantev, M. Robalo, P. Safronov, G. Vezzosi

Abstract. — The present volume covers the content of one of the session of “États de la recherche” on derived algebraic geometry which has been held in Toulouse in June 2017. It contains the contributions of David BenZvi and David Nadler, Damien Calaque and Julien Grivaux, Etienne Mann and Marco Robalo, Tony Pantev and Gabriele Vezzosi, and Pavel Safronov, taken from their original lectures. These cover a wide variety of subjects, from foundations of derived algebraic geometry and derived deformation theory, to its applications to enumerative geometry, geometric representation theory and categorification in algebraic geometry. Résumé (Géométrie algébrique dérivée). — Le présent volume est un compte rendu d’une des sessions des « États de la recherche » sur la géométrie algébrique dérivée qui s’est tenue à Toulouse en juin 2017. Il contient les contributions de David BenZvi et David Nadler, Damien Calaque et Julien Grivaux, Etienne Mann et Marco Robalo, Tony Pantev et Gabriele Vezzosi, et de Pavel Safronov, associées à leurs mini-cours respectifs. Elles couvrent une grande variété de sujets du domaine, depuis les fondements de la géométrie algébrique dérivée et de la théorie des déformation, jusqu’à leurs applications à la géométrie énumérative, la théorie géométrique des représentations et à la catégorification dans le contexte de la géométrie algébrique.

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Tony Pantev & Gabriele Vezzosi — Introductory topics in derived algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Affine derived geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Étale topology and derived stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Derived algebraic stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Lurie’s Representability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. DAG “explains” classical deformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Forms and closed forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Shifted symplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 9 11 13 15 21 28 36

David Ben-Zvi & David Nadler — Nonlinear Traces . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Traces in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Traces in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Traces for sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 46 58 68 74 82

Damien Calaque & Julien Grivaux — Formal moduli problems and formal derived stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction to pointed formal moduli problems for commutative algebras 2. General formal moduli problems and Koszul duality . . . . . . . . . . . . . . . . . . . 3. DG-Lie algebroids and formal moduli problems under Spec(A) . . . . . . . . . . 4. Global aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 92 100 119 130 144

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Etienne Mann & Marco Robalo — Gromov-Witten theory with derived algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Moduli space of stable maps, cohomological field theory and operads . . . . 3. Lax algebra structure on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Proof of our main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Comparison with other definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Proof of Theorem 5.3.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Proof of Theorem 5.4.3.(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Alternative proof of Corollary B.4 in the affine case . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 150 158 162 166 177 178 180 183

Pavel Safronov — Lectures on shifted Poisson geometry . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Shifted Poisson algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Coisotropic and Lagrangian structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 190 207 221 228

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RÉSUMÉS DES ARTICLES

Introduction à la géométrie algébrique dérivée Tony Pantev & Gabriele Vezzosi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Nous donnons une introduction à la géométrie algébrique dérivée (DAG) en se concentrant sur les constructions et techniques de base. Nous discutons les schémas dérivés affines, les champs algébriques dérivés et le théorème de représentabilité d’Artin-Lurie. À travers l’exemple des déformations de schémas lisses et propres, nous expliquons comment DAG éclaire la théorie classique de la déformation. Dans les deux dernières sections, nous introduisons les formes différentielles sur le champs dérivées, puis nous nous spécialisons au formes symplectiques décalées, donnant les principaux théorèmes d’existence prouvés dans [17]. Traces non-linéaires David Ben-Zvi & David Nadler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Nous combinons la théorie des traces en algèbre homotopique avec la théorie des faisceaux en géométrie algébrique dérivée afin déduire des formules de caractéres et de points fixes. La notion de dimension (ou homologie de Hochschild) d’un objet dualisable dans le contexte de l’algèbre supérieure fournit un cadre unificateur pour les notions classiques telles que les caractéristiques d’Euler, les caractères de Chern et les caractères des représentations de groupes. De plus, la fonctorialité de ces dimensions clarifient certaines formules célèbres et les étendent à de nouveaux contextes. Nous observons qu’il est avantageux de calculer les dimensions, les traces et leurs fonctorialités directement dans le cadre géométrique non-linéaire des catégories de correspondances, où elles sont respectivement identifiés directement avec (les versions dérivées des) les espaces de lacets, les lieux de points fixes et leurs fonctorialités. Il en résulte des versions universelles non-linéaires des formules de traces de Grothendieck-Riemann-Roch, de la formule de Atiyah-Bott-Lefschetz et des formules de caractères de Frobenius-Weyl. Il est par ailleurs possible de linéariser en appliquant certaines théories de faisceaux, telle que la théorie des faisceaux ind-cohérents et des D-modules construites par Gaitsgory-Rozenblyum [16]. Cela permet de retrouver les formules classiques, valables en familles et sans

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hypothèse de lissité ou de transversalité. D’un autre côté, notre formalisme s’applique également à certains invariants catégoriques supérieurs non présents dans le cadre linéaire, tels que la notion de caractères des actions de groupes sur des catégories. Problèmes de modules formels et champs dérivés formels Damien Calaque & Julien Grivaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Cet article présente un survol des problèmes de modules formels. Nous commençons par une introduction aux problèmes de modules formels pointés, et esquissons la démonstration d’un théorème (dû à Lurie et Pridham) donnant une formulation mathématique précise à la philosophie dite de « déformation dérivée » de Drinfeld. Ce résultat donne une correspondance entre les problèmes de modules formels et les algèbres de Lie différentielles graduées. Dans un second temps, nous présentons la théorie générale des contextes de déformation de Lurie, en insistant sur la notion (plus symétrique) de contexte de dualité de Koszul. Nous appliquons ensuite ce cadre général au cas des problèmes de modules formels non scindés sous un schéma affine dérivé fixé ; cette situation a été étudiée récemment par Nuiten, et nécessite de remplacer les algèbres de Lie différentielle graduée par des a lgébroïdes de Lie différentiels gradués. Dans la dernière partie, nous esquissons la globalisation au cas plus général des épaississements formels de champs dérivés, et suggérons une approche alternative à certains résultats de Gaitsgory et Rozenblyum. Théorie de Gromov-Witten à l’aide de la géométrie algébrique dérivée Etienne Mann & Marco Robalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Dans cet article de synthèse nous ajoutons deux nouveaux résultats qui n’étaient pas dans notre papier [42]. En utilisant l’action des membranes découvertes par Bertrand Toën, nous construisons une action associative relachée de l’opérade des courbes stables de genre zéro sur une variété projective lisse, vue comme une correspondance dans les champs dérivés. Cette action encode la théorie de Gromov-Witten en termes purement géométrique. Conférences sur la géométrie de Poisson décalée Pavel Safronov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Nous introduisons la théorie de structures de Poisson décalées (supérieures) sur les champs algébriques dérivés. Nous discutons de plusieurs résultats fondamentaux, tels que les intersections coisotropes, l’additivité de Poisson, et une comparaison avec la théorie des structures symplectiques décalées. Enfin, nous fournissons plusieurs exemples de structures de Poisson décalées liées aux structures de Poisson-Lie et aux espaces de modules de fibrés.

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ABSTRACTS

Introductory topics in derived algebraic geometry Tony Pantev & Gabriele Vezzosi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

We give a quick introduction to derived algebraic geometry (DAG) sampling basic constructions and techniques. We discuss affine derived schemes, derived algebraic stacks, and the Artin-Lurie representability theorem. Through the example of deformations of smooth and proper schemes, we explain how DAG sheds light on classical deformation theory. In the last two sections, we introduce differential forms on derived stacks, and then specialize to shifted symplectic forms, giving the main existence theorems proved in [17]. Nonlinear Traces David Ben-Zvi & David Nadler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. We observe that it is advantageous to calculate dimensions, traces and their functoriality directly in the nonlinear geometric setting of correspondence categories, where they are directly identified with (derived versions of) loop spaces, fixed point loci and loop maps, respectively. This results in universal nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz trace formulas, and Frobenius-Weyl character formulas. We can then linearize by applying sheaf theories, such as the theories of ind-coherent sheaves and D-modules constructed by Gaitsgory-Rozenblyum [16]. This recovers the familiar classical identities, in families and without any smoothness or transversality assumptions. On the other hand, the formalism also applies to higher categorical settings not captured within a linear framework, such as characters of group actions on categories.

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Formal moduli problems and formal derived stacks Damien Calaque & Julien Grivaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by Lurie and Pridham) which gives a precise mathematical formulation for Drinfeld’s derived deformation theory philosophy. This theorem provides a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie’s general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasizing the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by Joost Nuiten, and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of Gaitsgory and Rozenblyum. Gromov-Witten theory with derived algebraic geometry Etienne Mann & Marco Robalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 In this survey we add two new results that are not in our paper [42]. Using the idea of brane actions discovered by Toën, we construct a lax associative action of the operad of stable curves of genus zero on a smooth variety X seen as an object in correspondences in derived stacks. This action encodes the Gromov-Witten theory of X in purely geometrical terms. Lectures on shifted Poisson geometry Pavel Safronov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 These are expanded notes from lectures given at the États de la Recherche workshop on “Derived algebraic geometry and interactions”. These notes serve as an introduction to the emerging theory of Poisson structures on derived stacks.

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INTRODUCTION

Derived algebraic geometry, often refer to as “DAG,” was formally born in the early 2000s. It proposes a general setting for algebraic geometry particularly well suited to deal with non-generic situations such as non-transversal intersections and non-free group actions. We refer the reader to [13] for an extended historical introduction to the subject, and we restrict ourselves to a shorter overview below. The origin of the main ideas that has led to the present shape of DAG can be tracked back to the 1950s and to the so-called intersection formula of Serre of [12]. In this work, intersection numbers of two subvarieties, in an ambient smooth variety, are defined in terms of the alternating length of the higher Tor between the corresponding structure sheaves. Today, this is interpreted as the length, at the generic points, of the structure sheaf of the derived intersection of the two varieties, which is a special example of a derived scheme. Serre’s intersection formula is probably the very first instance of a derived scheme used in the context of algebraic geometry and is often referred to as the starting point of the whole subject. From a different perspective, another source of ideas came from deformation theory with the derived deformation theory program (DDT for short) initiated by Drinfeld in a letter to Schechtmann (see [6]). This program has been developed further by various authors and can be subsumed by the fact that any reasonable deformation theory problem is governed by a differential graded Lie algebra (dg-Lie algebras for short). When a deformation problem X is associated to a dg-Lie algebra g, the geometric invariants associated to X can be extracted by algebraic construction out of g. For instance, the ring of formal functions on X is given by the 0-th cohomology group H 0 (g, k) of g with coefficients in the trivial module k, or equivalently H 0 (C ∗ (g)) where C ∗ (g) is the Eilenberg-MacLane complex of the dg-Lie algebra g. It turns out that the other cohomologies H i (g, k) are in general non-zero for i > 0, and these groups can be interpreted as coherent sheaves on X which are sometimes called the virtual structure sheaves on X. When X exists as a global moduli space, the typical example here is the moduli of stable maps of Kontsevich, these sheaves exist globally as coherent sheaves on X and encode the so-called virtual fundamental class of X, in

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a very similar fashion that the higher Tor’s of Serre’s intersection formula encode the correct intersection numbers. Both of these streams of ideas reminded above have been eventually formalized by Ciocan-Fontanine and Kapranov into a notion of dg-scheme (see [5]), which is a first approximation of what a derived scheme is. The formalism of dg-schemes has been very useful to prove various results, such as the construction of virtual classes, the virtual Riemann-Roch formula, the existence of dg-enhancements of Hilbert and Quot schemes and so on and so forth. However, the theory of dg-schemes has been recognized too rigid by many aspects, and in particular the theory was missing a reasonable functor of point view point, making it difficult to consider representability theorems, or the notion of stacks as well as more complicated objects such as higher stacks. The modern foundations of DAG are based on the more flexible notion of derived schemes and of derived stacks, far-reaching generalizations of schemes and stacks. The reader will find all the precise definitions in the first contribution of this volume, as well as in [9, 13, 15]. Let us remind here that a derived scheme is a simple data consisting of a space X together with a sheaf of commutative dg-algebras O X (or more generally simplicial commutative rings if one works outside of characteristic zero), satisfying the following two simple conditions. (1) The ringed space (X, H 0 ( O X )) is a scheme. (2) The sheaves H i ( O X ) are quasi-coherent sheaves on (X, H 0 ( O X )). For a derived scheme (X, O X ), the scheme (X, H 0 ( O X )) is called the truncation or the classical part of the derived schemes. The sheaves H i ( O X ) are themselves the virtual structure sheaves mentioned above, and they combine in the derived structure sheaf namely O X itself. Even though derived schemes are perfectly easy to define, it turns out that defining the notion of morphisms between derived schemes requires some rather involved categorical constructions. The reason is that the sheaf O X of commutative dg-algebras must be considered only up to quasi-isomorphism, and the correct manner to deal with this is to use higher category theory, or its avatar Quillen homotopical algebra. Unfortunately, the use of higher categories or model categories is totally unavoidable in order to get a flexible theory of derived schemes for which for instance gluing (also called descent) is possible. Fortunately, this work has been done some years ago and is the content of the foundational texts on DAG [9] and [15]. DAG is today a well funded subject, and is used in many domains of geometry. To mention a few: the geometric Langlands correspondence [7], geometric representation theory [1], quantization of symplectic and Poisson structures [10, 4, 11], enumerative geometry [3], and more recently p-adic geometry [2]. We would like here to insist on two particular directions in which DAG has found applications recently, categorification and formal/infinitesimal geometry, which are the main themes of the contributions to the present volume. On the categorification side, it is important to note that the derived category of a derived scheme is most often better behaved than the derived category of its truncation, this is particularly true when considering

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moduli spaces. One reason for this is that the base change formula is always satisfied in the setting of derived schemes and derived stacks, making the action of natural correspondences on derived categories of certain moduli spaces more regular and less pathological than the induced action on their truncations. This principle has led to many interesting applications of DAG to categorification. In another direction, the infinitesimal and formal study of derived schemes and stacks is also more natural and easier to compute explicitly. The reason here can be already seen at the level of tangent complexes. The tangent complexes of classical (underived) moduli spaces are often impossible to compute explicitly, as these are most of the time unbounded. However, their derived counter-parts have easy modular interpretations in terms of cohomology and thus have explicit descriptions. This has been used recently in order to study the infinitesimal and formal structures of derived schemes and stacks, and has made possible for instance the theory of shifted symplectic and Poisson structures. In the same vein, derived stacks and formal derived stacks have been used in order to integrate dg-Lie algebroids, the same manner than Lie groupoids and differential stacks were used in order to integrate Lie algebroids (see [16]). The present volume covers the content of one of the session of “États de la recherche” on derived algebraic geometry which has been held in Toulouse in June 2017. The purpose of this event was to gather world experts who are using DAG to approach different problems in algebraic geometry, but also in Lie theory and geometric representation theory. The week was organized in several mini-courses by David Ben-Zvi, Damien Calaque, Dominic Joyce, Etienne Mann, Tony Pantev, Marco Robalo, Pavel Safronov and Gabriele Vezzosi, from which the contributions to this volume are taken. Introductory topics in derived algebraic geometry, by Tony Pantev and Gabriele Vezzosi, is a short introduction to the main ideas and concepts of DAG and its applications to the notions of shifted symplectic and Poisson structures. It covers the basic definitions of derived schemes and derived stacks, starting from scratch with the homotopy theory of commutative dg-algberas, cotangent complexes etc. It covers a wide range of fundamental results such as Lurie’s representability theorem, the interpretation of derived deformation theory, and the notion of differential forms and symplectic structures in the derived setting. This contribution serves as an expanded introduction to the subject for the reader who is not familiar to DAG, and contains the details of all the notions we have discussed in this introduction. In Nonlinear traces, by David Ben-Zvi and David Nadler, the authors present a general categorical formalism for traces, with applications to algebraic geometry and Grothendieck-Riemann-Roch’s type formula. One of the leading idea is that the derived category of varieties, or more generally of certain stacks, behaves like finitedimensional vector spaces when suitably considered as dg-categories or stable ∞categories. It is then possible to define the traces of various endomorphisms, including the identity morphism whose trace is called the dimension. By observation, the traces are preserved by symmetric monoidal functors, and when applied to derived categories of sheaves (quasi-coherent, ind-coherent or D-modules) this preservation produces formula of Grothendieck-Riemann-Roch type. The authors present several very nice

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applications, such as Atiyah-Bott-Lefschetz fixed point formula. In this setting, DAG is used in an essential way to insure that correspondence acts functorially on various derived categories, in a compatible manner with compositions. For this to be true, composition of correspondences, which are given by certain fiber products, must be computed in the category of derived schemes and derived stacks. The paper Formal moduli problems and formal derived stacks, by Damien Calaque and Julien Grivaux, contains a fundational work on deformation theory. It is a wellestablished statement (see [14]) that formal moduli problems over a field are classified by dg-Lie algebras. The extension of this important classification result to more general bases is an important question which is addressed by the authors. They study the notion of formal moduli problems over and under a base derived scheme X. These can be considered as families of formal moduli problems parametrized by X in some sense. The main result states that these are classified by dg-Lie algebras over X, and dg-Lie algebroids over X, depending of the fact that one considers split (i.e. pointed) or non-split formal moduli problems. This result can be understood as an integration result, making a correspondence between dg-Lie algebroids and certain derived formal groupoids over X, similar to the well-known Lie correspondence between Lie algebroids and Lie groupoids (see [16]). These results are for instance useful for the study of the formal completions of a scheme X along a closed subscheme Y , by means of a dg-Lie algebroid over Y . The subject Gromov-Witten theory with derived algebraic geometry, by Etienne Mann and Marco Robalo, is a construction of a categorification of the Gromov-Witten invariants. The Gromov-Witten invariants of a variety X can be described by an action of the modular operad on the cohomology of X. In this work the authors categorify this action by showing that it is induced by an action of the family of ¯ derived categories D( M g,n ) on the derived category D(X). For this, it is shown ¯ that the family of categories D( M g,n ) forms a categorical operad. An action of this categorical operad on D(X) is constructed using the derived moduli space of stable ¯ maps R M g,n (X), considered as a correspondence ¯ M g,n+1

×X o

¯ RM g,n+1 (X)

/ X n.

The fact that this action satisfies associativity is the core of the main result, and is proven by using the base change formula in DAG as well as the theory of ∞-operads of [8]. The more traditional Gromov-Witten invariants are then recovered by passing to K-theory or periodic cyclic homology. In Lectures on shifted Poisson geometry, by Pavel Safronov, applications and interactions between DAG and geometric representation theory are presented. The main subject is that of shifted Poisson structures and their quantizations. Shifted Poisson structures are Poisson brackets defined on functions of a derived scheme or a derived stack, with some possibly non-trivial cohomological degree. In the classical setting of smooth varieties they are the usual Poisson structures, but there are lots of natural examples with non-zero cohomological shift, the most fundamental being

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degree 2 Poisson structures on the classifying stack BG of a reductive group G. In this manuscript the author explains how to recover all the standard notions appearing in quantum groups using shifted Poisson structures, starting from this fundamental example of BG. He also presents the comparisons between shifted Poisson structures and the Lagrangian morphisms, a far-reaching generalization of the well known relation between Poisson bracket and symplectic groupoid actions. The paper ends with some application to the FeiginOdesskii Poisson structure on the moduli space of bundles on an elliptic curve as well as the construction of the Poisson structure on the moduli space of monopoles. As a final note, we would like to point out that the lectures and the slides are all available at the following address: https://www.math.univ-toulouse.fr/dagit/index.php?page=video.php Acknowledgments. – We would like to warmly thank all the participants to the session of États de la recherche Derived Algebraic Geometry and Interactions, that was held in Toulouse 12-15 June 2017. A particular thank to all the lecturers and to their contributions to this volume, that, we hope, will be useful to students and colleagues who would like to learn the subject. We also thank very warmly the Société Mathématique de France for their help in the organization of this event. Bertrand Toën and Michel Vaquié Toulouse, November 2019. References [1] D. Ben-Zvi & D. Nadler – « Loop spaces and representations », Duke Math. J. 162 (2013), p. 1587–1619. [2] B. Bhatt, M. Morrow & P. Scholze – « Topological Hochschild homology and integral p-adic Hodge theory », Publ. Math. Inst. Hautes Études Sci. 129 (2019), p. 199– 310. [3] D. Borisov & D. Joyce – « Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds », Geom. Topol. 21 (2017), p. 3231–3311. [4] D. Calaque, T. Pantev, B. Toën, M. Vaquié & G. Vezzosi – « Shifted Poisson structures and deformation quantization », J. Topol. 10 (2017), p. 483–584. [5] I. Ciocan-Fontanine & M. Kapranov – « Derived Quot schemes », Ann. Sci. École Norm. Sup. 34 (2001), p. 403–440. [6] V. Drinfeld – « A letter from Kharkov to Moscow », EMS Surv. Math. Sci. 1 (2014), p. 241–248. [7] D. Gaitsgory – « Progrès récents dans la théorie de Langlands géométrique », Séminaire Bourbaki, vol. 2015/2016, exposé no 1109, Astérisque 390 (2017), p. 139–168. [8] J. Lurie – « Higher algebra », available at http://www.math.harvard.edu/~lurie/ papers/HA.pdf.

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[9]

INTRODUCTION

, « Spectral algebraic geometry », available at http://www.math.harvard.edu/ ~lurie/papers/SAG-rootfile.pdf.

[10] T. Pantev, B. Toën, M. Vaquié & G. Vezzosi – « Shifted symplectic structures », Publ. Math. Inst. Hautes Études Sci. 117 (2013), p. 271–328. [11] J. P. Pridham – « Deformation quantisation for unshifted symplectic structures on derived Artin stacks », Selecta Math. (N.S.) 24 (2018), p. 3027–3059. [12] J-P. Serre – Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Math., vol. 11, Springer, 1965. [13] B. Toën – « Derived algebraic geometry », EMS Surv. Math. Sci. 1 (2014), p. 153–240. [14]

, « Problèmes de modules formels », Séminaire Bourbaki, vol. 2015/2016, exposé no 1111, Astérisque 390 (2017), p. 199–244.

[15] B. Toën & G. Vezzosi – « Homotopical algebraic geometry. II. Geometric stacks and applications », Mem. Amer. Math. Soc. 193 (2008), p. 224. [16] H.-H. Tseng & C. Zhu – « Integrating Lie algebroids via stacks », Compos. Math. 142 (2006), p. 251–270.

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Panoramas & Synthèses 55, 2021, p. 1–37

INTRODUCTORY TOPICS IN DERIVED ALGEBRAIC GEOMETRY by Tony Pantev & Gabriele Vezzosi

Abstract. – We give a quick introduction to derived algebraic geometry (DAG) sampling basic constructions and techniques. We discuss affine derived schemes, derived algebraic stacks, and the Artin-Lurie representability theorem. Through the example of deformations of smooth and proper schemes, we explain how DAG sheds light on classical deformation theory. In the last two sections, we introduce differential forms on derived stacks, and then specialize to shifted symplectic forms, giving the main existence theorems proved in [17].

Résumé (Introduction à la géométrie algébrique dérivée). – Nous donnons une introduction à la géométrie algébrique dérivée (DAG) en se concentrant sur les constructions et techniques de base. Nous discutons les schémas dérivés affines, les champs algébriques dérivés et le théorème de représentabilité d’Artin-Lurie. À travers l’exemple des déformations de schémas lisses et propres, nous expliquons comment DAG éclaire la théorie classique de la déformation. Dans les deux dernières sections, nous introduisons les formes différentielles sur le champs dérivées, puis nous nous spécialisons au formes symplectiques décalées, donnant les principaux théorèmes d’existence prouvés dans [17].

1. Introduction Derived Algebraic Geometry (DAG) starts with the idea of replacing the affine objects of Algebraic Geometry, i.e., commutative rings, by some kind of “derived commutative rings” whose internal homotopy theory is non trivial. This can be achieved over Q by considering commutative differential non-positively graded algebras (cdga’s), while in general one might instead consider simplicial commutative 2010 Mathematics Subject Classification. – 14D23, 14F05. Key words and phrases. – Moduli spaces, derived algebraic geometry, symplectic geometry.

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algebras. (1) For simplicity, we will stick to the case of cdga’s (i.e., we will assume to work over Q). As in classical Algebraic Geometry, the first step is to develop the local or affine theory, i.e to define and study finiteness conditions, flat, smooth, étale properties for morphisms between cdga’s. This is the content of Section 2. Once this is set up, we will define the analog in DAG of being a sheaf or stack with respect to a derived version of a topology on these derived affine objects (Section 3), and then specify which sheaves or stacks are of geometric type (these will be called derived algebraic or derived Artin stacks). This last step will be done by using atlases and a recursion on the “level of algebraicity” in Section 4 where we will also list the main basic properties of derived algebraic stacks. It’s not always easy to decide whether a given derived stack is algebraic by using only the definition, and a powerful criterion is given by J. Lurie’s DAG version of M. Artin’s representability theorem: this is explained in Section 5, where we also give a simplified but often very useful version of Lurie’s representability. In Section 6 we explain, through the example of deformations of a given smooth proper scheme, how DAG fills some conceptual gaps in classical deformation theory, the idea being that once we allow ourselves to consider also deformations over an affine derived base, then deformation theory becomes completely transparent. The second part of this article describes symplectic geometry in DAG. We start by describing differential forms and closed differential forms on a derived stack (Section 7). These forms have two new features with respect to the classical case: first of all they have a degree (this new degree of freedom comes from the fact that in DAG the module of Kähler differentials is replaced by a complex, the so-called cotangent complex), secondly the notion of a closed form in DAG consists of a datum rather than a property. Once these notions are in place, the definition of a derived version of symplectic structure is easy. The final Section 8 reviews some the main examples and existence results in the theory of derived symplectic geometry taken from [17], namely the derived symplectic structure on the mapping derived stack of maps from a O -compact oriented derived stack to a symplectic derived stack, and the derived symplectic structure on a lagrangian intersection. 2. Affine derived geometry 2.1. Homotopical algebra of dg-modules and cdga in characteristic 0. – Let k be a commutative Q-algebra. Definition 2.1. – We will write — dgmod≤0 for the model category of non-positively graded dg-modules over k k (with differential increasing the degree), with weak equivalences W = “quasiisomorphisms” and fibrations Fib = “surjections in deg < 0”, endowed with (1)

Note that DAG based on cdga’s over Q or on simplicial commutative Q-algebras are equivalent theories.

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the usual symmetric ⊗ structure compatible with the model structure (i.e., a symmetric monoidal model category, [13]). — dgmod≤0 for the symmetric monoidal ∞-category obtained by inverting (in the k ∞-categorical sense (2)) quasi-isomorphisms in dgmod≤0 k . ≤0 — cdga≤0 k := CAlg(dgmodk ) for the model category of commutative unital monoids ≤0 in dgmodk , with W = “quasi-isomorphisms” and Fib = “surjections in deg < 0” (here we need that k is a Q-algebra). These are non-positively graded commutative differential graded k-algebras (cdga’s for short). ≤0 — cdga≤0 k (k) := CAlg(dgmodk ) for the ∞-category of non-positively graded commutative differential graded k-algebras (derived k-algebras) that can also be obtained by inverting (in the ∞-categorical sense) quasi-isomorphisms in cdga≤0 k .

— S will denote the ∞-category of spaces. We will use analogous notations for unbounded dg-modules over k, and unbounded cdga’s over k, by simply omitting the (−)≤0 superscript. Same convention for non necessarily commutative algebras (by writing Alg instead of CAlg). There is an ∞-adjunction: dgmod≤0 k o

Symk

/

cdga≤0 k

U

(induced by a Quillen adjunction on the corresponding model categories) where Symk denotes the free cdga functor, and U the forgetful functor. Mapping spaces. – We have the following explicit models for mapping spaces P • in cdga≤0 k . Let Ωn be the algebraic de Rham complex of k[t0 , t1 , . . . , tn ]/( i ti − 1) ≤0 • over k. Consider [n] 7→ Ωn as a simplicial object in cdgak . Thus, if B ∈ cdgak , then the assignment [n] 7→ τ≤0 (Ω•n ⊗k B) defines a simplicial object in cdga≤0 k . ≤0 For any pair (A, B) in cdgak , there is an equivalence of of spaces (simplicial sets) Mapcdga≤0 (A, B) ' ([n] 7→ Homcdga≤0 (Qk A, τ≤0 (Ω•n ⊗k B))) k

k

where Qk A is a cofibrant replacement of A in the model category cdga≤0 k . Cautionary exercises. – The following exercises are meant to make the reader aware of the boundaries of the territory where we will be working. Exercise 2.2. – Let char(k) = p. Show that CAlg(dgmodk ) cannot have a model structure with W=“quasi-isomorphisms” and Fib = “degreewise surjections”. Hint: 1) Construct a map f : A → B in CAlg(dgmodk ) with the following property: ∃i , α ∈ H i (B) such that αp is not in the image of H ip (f ). 2) Prove that no such f can be factored as Fib ◦ (W ∩ Cof ). (2)

See M. Robalo’s paper in this volume.

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Exercise 2.3. – Let char(k) = 0. Show that the obvious ∞-functor CAlg(dgmodk ) → Alg(dgmodk ), though conservative, is not fully faithful. For A ∈ cdga≤0 we consider dgmod(A) the symmetric monoidal ∞-category of k (unbounded) dg-modules over A, and we define CAlg(dgmod(A)) ' A/cdga≤0 k . Given any map f : A → B ∈ cdga≤0 k , there is an induced ∞-adjunction f ∗ =(−)⊗A B

dgmod(B), o

/

dgmod(A)

f∗

which is an equivalence of ∞-categories if f is an equivalence in cdga≤0 k . 2.2. Cotangent complex. – Let f : A → B be a map in cdga≤0 k . For any M ∈ dgmod≤0 (B), let B ⊕ M ∈ cdga≤0 be the trivial square zero extension of B k by M (i.e., B acts on itself and on M in the obvious way, and M · M = 0). B ⊕ M is naturally an A-algebra and it has a natural projection map prB : B ⊕ M → B of A-algebras. Definition 2.4. – The space of derivations from B to M over A is defined as DerA (B, M ) := MapA/cdga≤0 /B (B, B ⊕ M ). k

Equivalently,  DerA (B, M ) = f ib

MapA/cdga≤0 (B, B ⊕ M ) k

prB,∗

/ Map

 ≤0

A/cdgak

(B, B) ; idB .

M → DerA (B, M ) can be constructed as an ∞ functor dgmod≤0 (B) → S, and we have the following derived analog of the classical existence results for Kähler differentials. Proposition 2.5. – The ∞-functor M 7→ DerA (B, M ) is corepresentable, i.e., ∃ LB/A ∈ dgmod≤0 (B) and a canonical equivalence DerA (B, −) ' Mapdgmod(B) (LB/A , −). Proof. – Let QA B ' B be a cofibrant replacement of B in A/cdga≤0 k . Then LB/A := Ω1QA B/A ⊗QA B B does the job. Let us list the most useful properties of the cotangent complex construction (for details, see [23]). (1) Given a commutative square in cdga≤0 k A  B

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u

/ A0  / B0

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there is an induced map LB/A → u∗ LB 0 /A0 ⇔ LB/A ⊗B B 0 → LB 0 /A0 . In particular, we get a canonical map LB/A → LH 0 (B)/H 0 (A) . If B = S, A = R are discrete (3), the canonical map H 0 (LS/R ) → Ω1S/R is an isomorphism. (2) If the commutative diagram in (1) is a pushout square, then the map LB/A ⊗B B 0 → LB 0 /A0 is an equivalence. (3) If C → A → B are maps in cdga≤0 k , then there is an induced cofiber sequence LA/C ⊗A B → LB/C → LB/A in dgmod(B), where the first map is as in (1) via u0 = idC , u : A → B, and the second map is as in (1) with u = idB and u0 = (C → A)). (4) If A  B is a pushout in

cdga≤0 k ,

u0

u

/ A0  / B0

then we have a fiber sequence 0

LA/k ⊗A B → LB/k ⊗B B 0 ⊕ LA0 /k ⊗A B 0 → LB 0 /k in dgmod(B 0 ). Exercise 2.6. – Compute LA/k for A := k[x1 , . . . , xn ]/(f ), for f 6= 0 Hint: Use the associated Koszul resolution. Exercise 2.7. – Compute the cotangent complex of k[ε] over k, where ε2 = 0, ε has cohomological degree −1, and use this to classify explicitly all derivations k[ε] → k. Postnikov towers. – We will describe here the Postnikov tower of a cdga and its relation to the cotangent complex. Definition 2.8. – Let A ∈ cdga≤0 k . A Postnikov tower for A is the data of a sequence {Pn (A)}n≥0 in A/cdga≤0 , and of maps k ···

/ Pn (A)

πn

/ Pn−1 (A)

πn−1

/ ···

/ P1 (A)

π1

/ P0 (A)

in A/cdga≤0 k , satisfying the following properties: (1) Pn (A) is n-truncated (i.e., H −i (Pn (A)) = 0 for i > n); (2) H j (A) → H j (Pn (A)) is an iso, for all 0 ≤ j ≤ n; (3)

0 An object D ∈ cdga≤0 k is discrete if the canonical map D → H (D) is a quasi-isomorphism.

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(3) For any n-truncated B ∈ cdga≤0 k , the map Mapcdga≤0 (Pn (A), B) → Mapcdga≤0 (A, B) k

k

induced by A → Pn (A), is an equivalence of spaces. Proposition 2.9. – A Postnikov tower is unique in the ∞-category Fun∞ (Nop , A/cdga≤0 k ) of ∞-functors. Moreover, the canonical map A → limn≥0 Pn (A) is an equivalence in cdga≤0 k . Exercise 2.10. – Construct a Postnikov tower such that P0 (A) = A0 /d(A−1 )[0] P1 (A) = ( · · · → 0 → A−1 /d(A−2 ) → A0 ) etc. The following result explains how the cotangent complex controls the Postnikov tower of a cdga. Proposition 2.11. – Let ···

/ Pn (A)

πn

/ Pn−1 (A)

πn−1

/ ···

/ P1 (A)

π1

/ P0 (A)

be a Postnikov tower for A. For any n, there exists a map φn : LPn (A)/A → H −n−1 (A)[n + 2] such that the following square is cartesian in A/cdga≤0 k / Pn (A)

Pn+1 (A) πn

 Pn (A)



dφn

d0

/ Pn (A) ⊕ H −n−1 (A)[n + 2],

where dφn is the derivation corresponding to φn via the identification π0 (MapA/cdga≤0 /Pn (A) (Pn (A), Pn (A) ⊕ H −n−1 (A)[n + 2])) k

' HomHo(dgmod(Pn (A)) (LPn (A)/A , H −n−1 (A)[n + 2]). Corollary 2.12. – A map A → B in cdga≤0 is an equivalence iff the following two k properties hold — H 0 (A) → H 0 (B) is an isomorphism (of discrete k-algebras) — LB/A ' 0. The previous corollary is the first of several examples of the following general Principle: derived algebraic geometry = algebraic geometry (of the truncation) () + deformation theory Remark 2.13. – Note that this principle seriously fails for derived geometry over unbounded cdga’s or non-connective commutative ring spectra.

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Definition 2.14. – Let n ∈ N. A non-positively graded dg-module M is said to be n-connective if H −i M = 0 for every 0 ≤ i < n. A morphism f : M → N is said to be n-connected if fib(f ) is n-connective. The following result gives more precise and useful relations between the degree of connectedness of a map between cdga’s and the degree of connectivity of its cotangent complex. Proposition 2.15. – Let f : A → B be a map in cdga≤0 k , and n ∈ N. (1) If f is n-connected, then LB/A is (n + 1)-connective. (2) If LB/A is (n+1)-connective and H 0 (f ) is an isomorphism, then f is n-connected. 2.3. Derived commutative algebra: finiteness, flat, smooth, étale. – In order to motivate the basic definitions in derived commutative algebra, we will give here also the classical definitions in commutative algebra in a form that is appropriate for our generalization. Definition 2.16. – [Classical] A map R → S of discrete commutative k-algebras is finitely presentable if HomR/CAlg(k) (S, −) commutes with filtered colimits. [Derived] A map A → B in cdga≤0 (B, −) k is derived finitely presentable if MapA/cdga≤0 k commutes with (homotopy) filtered colimits. Proposition 2.17. – A map A → B in cdga≤0 k is fp iff — H 0 (A) → H 0 (B) is classically finitely presentable, and — LB/A is a perfect B-dg module (i.e., is a dualizable object in (dgmod(B), ⊗B )). See [23, 2.2.] or [15, Theorem 7.4.3.18] for a proof. Thus the fact that a map between discrete rings is classically finitely presentable does not imply that the map is derived finitely presented. However, “classical finitely presentable + lci” =⇒ “derived finitely presentable”. Definition 2.18. – [Classical] A map R → S of discrete commutative k-algebras is flat if (−) ⊗R S : mod(R) → mod(S) preserves pullbacks (i.e., preserves kernels). ≤0 [Derived] A map A → B in cdga≤0 (A) → k is derived flat if (−) ⊗A B : dgmod ≤0 dgmod (B) preserves pullbacks.

Hence, a map of discrete commutative k-algebras is classically flat iff it is derived flat. Remark 2.19. – Note that a map A → B in cdga≤0 k is derived flat iff for any discrete A-module M , the derived tensor product M ⊗L B is discrete (i.e of zero tor-amplitude). A

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Definition 2.20. – [Classical] A map R → S of discrete commutative k-algebras is formally étale if τ≥−1 LS/R ' 0. [Derived] A map A → B in cdga≤0 k is derived formally étale if LB/A ' 0. [Classical] A map R → S of discrete commutative k-algebras is formally smooth if HomD(S) (τ≥−1 LS/R , M ) = 0 for any M ∈ dgmod≤0 (S) s.t. H 0 (M ) = 0. [Derived] A map A → B in cdga≤0 k is derived formally smooth if HomD≤0 (B) (LB/A , M ) = 0 for any S ∈ dgmod≤0 (B) s.t. H 0 (M ) = 0. Exercise 2.21. – Show that the above definition of classical formally smooth (respectively, étale) coincides with the usual one given by the infinitesimal lifting property ([10]). Hint (for the formally smooth case): show that f : R → S is formally smooth in the sense of [10] iff τ≥−1 Lf ' P [0] with P projective over S. Remark 2.22. – Note that considering τ≥−1 Lf and not all of Lf , in Definition 2.20 is strictly necessary. In fact there is an example ([3, TAG 06E5]) of a map f : k → S, where k is a field, such that f is formally étale (hence formally smooth) but H −2 (Lf ) 6= 0. Note that such an f is necessarily not classically finitely presentable. Definition 2.23. – [Classical] A map R → S of discrete commutative k-algebras is étale (respectively smooth) if it is finitely presentable and formally étale (respectively formally smooth). [Derived] A map A → B in cdga≤0 k is derived étale (respectively derived smooth) if it is derived finitely presentable and derived formally étale (respectively derived formally smooth). [Classical] A map R → S of discrete commutative k-algebras is a Zariski open immersion if it is flat, finitely presentable and the product map S ⊗R S → S is an isomorphism. [Derived] A map A → B in cdga≤0 is a Zariski derived open immersion if it is k flat, finitely presentable, and the product map B ⊗A B → B is an equivalence. There is a characterization of the above derived properties which is very useful in practice. Definition 2.24. – A map A → B in cdga≤0 is strong if the canonical map k 0 i i 0 H (B) ⊗H (A) H (A) → H (B) is an isomorphism for all i. Theorem 2.25. – A map A → B in cdga≤0 k is derived flat (respectively derived smooth, derived étale, a Zariski derived open immersion) iff it is strong and H 0 (A) → H 0 (B) is flat (respectively smooth, étale, a Zariski open immersion). Note that if A → B is derived flat and A is discrete, then B is discrete as well. Exercise 2.26. – Assume Theorem 2.25. Show that f : A → B in cdga≤0 is derived k smooth iff the following three conditions hold:

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— f is derived finitely presentable; — f is derived flat; — B is a perfect B ⊗A B-dg module. Convention. – In the rest of this article, we will omit the adjective “derived” when writing any of the above properties for morphisms of derived algebras, i.e., “derived étale” will be replaced by “étale”, etc. 3. Étale topology and derived stacks Definition 3.1. – A family {A → Ai }i of maps in cdga≤0 k is an étale covering family if (i) each A → Ai is étale; (ii) the family {H 0 (A) → H 0 (Ai )}i is a classical étale covering family (of discrete commutative k-algebras). op This defines a topology on the ∞-category dAff(k) := (cdga≤0 (i.e., by definik ) tion, a Grothendieck topology on Ho(dAff(k))). This topology is called the étale topology on derived rings and denoted by (ét).

— The ∞-category Sh(dAff(k), (ét)) of sheaves of spaces on this ∞-site, is denoted by dSt(k) and called the ∞-category of derived stacks over k. — An ∞-functor F : dAff(k)op → S is a derived stack iff for any A ∈ cdga≤0 k and any étale hypercover A → B • , the induced map F (A) → lim F (B • ) is an equivalence of spaces (in this case, we say that F has étale hyperdescent). The étale topology on derived rings is subcanonical, i.e., for any A ∈ cdga≤0 k , the ∞-functor Spec A : dAff(k) −→ S : B 7−→ Mapcdga≤0 (A, B) k

is a sheaf for (ét) (i.e., it has étale hyperdescent). Any derived stack equivalent to Spec A will be called a derived affine scheme. A consequence of Theorem 2.25 is the following result saying that topologically (étale or Zariski) a derived affine scheme is indistiguishable from its truncation. Proposition 3.2. – H 0 induces an equivalence of étale or Zariski sites of A and of H 0 (A), for any A ∈ cdga≤0 k . This statement is analogous to the equivalence between the small étale or Zariski site of a scheme and of its reduced subscheme. Remark 3.3. – Čech nerves of étale coverings are special étale hypercovers. One may also consider ∞-functors F : dAff(k)op → S having descent just for these special Čech étale hypercovers: we obtain, in general, different categories of derived stacks. However, the full subcategories of truncated stacks (i.e., whose values are truncated homotopy types) are in fact equivalent.

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Next we list a few basic facts concerning the ∞-category of derived stacks. We will denote by St(k) := Sh(Aff(k), ét) the ∞-category of (underived, higher) stacks over k, for the classical étale topology (see also [23, 2.1]). — There is an ∞-adjunction i

St(k) o

/

dSt(k)

t0

between derived and underived stacks (for the étale topology), i being the left adjoint. The truncation functor t0 is determined by the property t0 (Spec A) = Spec H 0 (A), and the fact that it preserves limits (being right adjoint). Note that i is fully faithful. — The ∞-category dSt(k) is cartesian closed, i.e., there are internal Hom’s, denoted as MAPdSt(k) (F, G) ∈ dSt(k), and equivalences of spaces MapdSt(k) (F × G, H) ' MapdSt(k) (F, MAP(G, H)). More generally, we have an equivalence in dSt(k): MAPdSt(k) (F × G, H) ' MAPdSt(k) (F, MAPdSt(k) (G, H)). — The ∞-functor i does not preserve pullbacks nor internal Hom’s (in fact, a lucky feature). Definition 3.4. – If F ∈ dSt(k), we define QCoh(F ) :=

lim

Spec A→F

dgmod(A)

(the limit being taken inside the ∞-category of k-linear stable, symmetric monoidal ∞-categories). This is called the symmetric monoidal ∞-category of quasi-coherent complexes on F . We can globalize the definition of cotangent complex to stacks, as follows. Let F ∈ dSt(k), x : S = Spec A → F , and M ∈ dgmod≤0 (A). We have a projection map pr : A ⊕ M → A. Definition 3.5. – We say that a derived stack F has a cotangent complex at x, if there is a (−n)-connective A-module LF,x (for some n) such that the ∞-functor DerF,x :

dgmod≤0 (A)

/S

M 

  (pr) / fib F (A ⊕ M ) F−→ F (A); x

is equivalent to the functor Mapdgmod(A) (LF,x , −) (restricted to dgmod≤0 (A)). If this is the case, LF,x is called a cotangent complex of F at x (4). (4)

Obviously, any two cotangent complexes of F at x are canonically equivalent.

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— We say that F has a (global) cotangent complex if ∃ LF ∈ QCoh(F ), such that for any x : S = Spec A → F , x∗ LF is a cotangent complex for F at x. 4. Derived algebraic stacks In this section we will single out, among all derived stacks, the “geometric” ones, called derived algebraic stacks. The definition will involve a long induction on n ≥ 0. We start with the case n = 0. Definition 4.1. – A map f : F → G in dSt(k) is defined to be an epimorphism (respectively, a monomorphism) if the induced map π0 (F ) → π0 (G) of sheaves of sets on the usual site (Ho(dAff(k)), (ét)) is an epimorphism (respectively, if the diagonal ∆f : F → F ×G F is an equivalence). Definition 4.2 (Derived schemes). – A map u : F → S = Spec A of derived stacks is a Zariski open immersion if it is a monomorphism ` and there ` is a family {pi : Spec Ai → F } of morphisms in dSt(k) such that p : i i i Spec Ai → Spec F is an epimorphism, and each composite ui : Spec Ai → F → Spec A is a Zariski open immersion of cdga’s (so that we already know what this means). — A morphism F → G of derived stacks is a Zariski open immersion if for any S → G with S affine, the induced map F ×G S → S is a Zariski open immersion (as defined in the previous item). — A derived stack F is a derived scheme if there exists a family {Spec Ai → F }i ` of Zariski open immersions, such that the induced map i Spec Ai → F is an epimorphism. Such a family is called a Zariski atlas for F . Note that if F → Spec A is a Zariski open immersion, then F is automatically a derived scheme. Once derived schemes are defined, we can extend the notion of smooth, flat, étale to maps between them. Definition 4.3. – A morphism of derived schemes f : X → Y is smooth (respectively flat, respectively étale) if there are Zariski atlases {Ui → X}i∈I , {Vj → Y }j∈J and, for any i ∈ I there is j(i) ∈ J and a commutative diagram Ui  X

fi,j(i)

f

/ Vj(i)  /Y

such that fi,j(i) is smooth (resp. flat, resp. étale) between derived rings. Having defined derived schemes, we may give the general inductive definition of derived algebraic n-stacks, for n ≥ 0.

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Definition 4.4 (Derived 0-algebraic stacks). – (equivalent to) a derived scheme.

A derived stack F is 0-algebraic if it is

— A morphism of derived stacks F → G is 0-representable if for any S → G where S is 0-algebraic, the pullback S ×G F is 0-algebraic. — a 0-representable morphism of derived stacks F → G is smooth if for any S → G where S is 0-algebraic, the morphism of derived schemes S ×G F → S is smooth. Definition 4.5 (Derived n-algebraic stacks). – Let n > 0, and suppose we have already defined the notions of derived (n − 1)-algebraic stack, of (n − 1)-representable morphism (between arbitrary derived stacks), and of smooth (n − 1)-representable morphism (between arbitrary derived stacks). Then — A derived stack F is n-algebraic if there exists a smooth (n−1)-algebraic morphism p : U → F of derived stacks, where U is 0-algebraic and p is an epimorphism. Such a p is called a smooth n-atlas for F . — A morphism of derived stacks F → G is n-representable if for any S → G where S is 0-algebraic, the pullback S ×G F is n-algebraic. — An n-representable morphism of derived stacks F → G is smooth (resp. flat, resp. étale) if for any S → G where S is 0-algebraic, there exists a smooth n-atlas U → S ×G F , such that the composite U → S, between 0-algebraic stacks, is smooth (resp. flat, resp. étale). — A derived stack is algebraic if it is m-algebraic for some m ≥ 0. Remark 4.6. – Exactly the same Definitions 4.4 and 4.5, with “derived scheme” replaced by (classical) “scheme” and “affine derived scheme” by (classical) “affine scheme”, give us a notion of underived algebraic (higher) n-stack, for each n ≥ 0. This notion was first proposed by C. Simpson and C. Walter. Exercise 4.7. – If F is n-algebraic, then the diagonal map F → F × F is (n − 1)-representable. We list below some important and useful properties of derived algebraic stacks. — The full sub-∞-category dStalg (k) ⊂ dSt(k) of algebraic stacks is closed under pullbacks and finite disjoint unions. — Representable morphisms are stable under composition and arbitrary pullbacks. — If F → G is a smooth epimorphism of derived stacks, then F is algebraic iff G is algebraic. — A non-derived stack X is algebraic iff the derived stack i( X ) is algebraic. — If F is a derived algebraic stack, then its truncation t0 (F ) is an algebraic (underived) stack.

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— If F is a derived algebraic stack and t0 (F ) is an m-truncated (5) (underived) stack, then for any n-truncated A ∈ cdga≤0 (i.e., H i (A) = 0 for i < −n), the space k F (A) is (n + m)-truncated. — If F → G is a flat morphism of derived algebraic stacks, then: G underived (i.e., ' i( X )) implies that F is underived. — A (n − 1)-representable morphism is n-representable. — A (n−1)-representable smooth (resp. étale, flat, Zariski open immersion) is n-representable smooth (resp. étale, flat, Zariski open immersion). — If F → G is a map of derived stacks, G is n-algebraic and there is a smooth atlas S → G such that S ×G F is n-algebraic, then F is n-algebraic (i.e., being n-algebraic is smooth-local on the target). 0 — A map A → B in cdga≤0 k is quasi finitely presentable (qfp, for short) if H (A) → H 0 (B) is (classically) finitely presentable. Flat qfp covers define a topology (qfpf) on dAff(k). We can replace (ét) by (qfpf), and smooth (atlases) by flat (atlases), and we get another full subcategory dStqfpf, alg (k) of derived stacks which are algebraic for this new pair (qfpf, flat). A deep result of B. Toën [20] says that in fact dStqfpf, alg (k) = dStalg (k). The analogous statement for underived stacks in groupoids was proven by M. Artin [2]).

— Derived algebraic stacks admitting étale atlases are called derived DeligneMumford stacks, and often, by analogy with the classical case, general derived algebraic stacks (i.e., with smooth atlases) are also called derived Artin stacks. — An (underived) algebraic n-stack locally of finite presentation over k (with bounded cotangent complex) has a cotangent complex of amplitude ⊆ [−1, n] (6) On the opposite side, a derived affine scheme always have a cotangent complex with amplitude ⊆ (−∞, 0]. Therefore, for an arbitrary derived algebraic stack F , the negative degrees of LF are often referred to as its “derived” degrees, while the positive ones as its “stacky” degrees. 5. Lurie’s Representability Theorem Let us introduce some fundamental deformation-theoretic properties of a derived stack. Definition 5.1. – A derived stack F is — nilcomplete if for any A ∈ cdga≤0 k , the canonical map F (A) → limn≥0 F (Pn (A)) is an equivalence in S (recall that {Pn (A)}n≥0 is a Postnikov tower for A); (5)

I.e., it sends any discrete commutative k-algebra to an m-truncated space. By a famous result of L. Avramov, a scheme locally of finite type over k either has a perfect cotangent complex with amplitude ⊆ [−1, 0] (and this happens iff the scheme is lci), or has an unbounded cotangent complex.

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— infinitesimally cohesive if the F -image of any cartesian square in cdga≤0 k A0

/A

 B0

 /A

p

q

where H 0 (p) and H 0 (q) are surjective with nilpotent kernels, is cartesian in S; ≤0 — infinitesimally cartesian if for any A ∈ cdga≤0 (A), s.t. k , any M ∈ dgmod 0 H (M ) = 0, and any derivation d from A to M , the F -image of the pullback

A ⊕d ΩM

/A

 A

 / A⊕M

d

triv

is a pullback in S; — discretely integrable (7) if for any classical complete local noetherian k-algebra b → limn F (R/mn ) is an equivalence in S. (R, m), the canonical map F (R) Obviously, being infinitesimally cohesive implies being infinitesimally cartesian, and the notion of infinitesimal cohesiveness is a derived version of the classical Schlessinger condition [3, Tag 06J1]. We are now able to state the following important result that is the most useful known criterion for checking algebraicity of a derived stack. Theorem 5.2 (Lurie’s Representability Theorem). – Let k be a noetherian G-ring (e.g., noetherian and excellent). A derived stack F over k is algebraic iff the following conditions hold: — F is nilcomplete. — F is infinitesimally cohesive. — F is discretely integrable. — F has a cotangent complex Remark 5.3. – There is also a more general version of Theorem 5.2 where the base k is a derived ring ([16, Theorem 18.4.0.1]). Corollary 5.4 (“Easy” Representability Theorem). – Let k be a noetherian commutative ring. A derived stack F over k is n-algebraic k iff the following conditions hold: — The truncation t0 (F ) is an (underived) algebraic n-stack. — F is nilcomplete, infinitesimally cartesian and has a cotangent complex. (7)

This is the same as integrable (as in [16, Definition 17.3.3.1]) if F is already nilcomplete, infinitesimally cohesive and admits a cotangent complex.

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In this corollary, we have separated the first condition which is a global one but only concerns the truncation, and the second condition which is purely deformationtheoretic. This is another instance of Principle (). A proof of Corollary 5.4, independent of Theorem 5.2, can be found in [23, Appendix C]. Remark 5.5. – Note that the G-ring condition in Theorem 5.2 re-appears when we want to further unzip the first condition in Corollary 5.4. Exercise 5.6. – Let X/k be a flat and proper underived scheme and Y /k an underived smooth scheme. Let MAPdSt(k) (X, Y ) : A 7−→ MapdSt(k) (X × Spec A, Y ) the internal mapping derived stack. — Show that t0 (MAPdSt(k) (X, Y )) is the classical scheme of morphisms from X to Y . — Let MAPdSt(k) (X, Y ) o

p

X × MAPdSt(k) (X, Y )

ev

/Y .

Show that T := p∗ ev ∗ (TY /k [0]) is a tangent complex for MAPdSt(k) (X, Y ). — Apply Lurie’s representability theorem (in the easy case, if the reader so wishes) to deduce that MAPdSt(k) (X, Y ) is a derived geometric stack (actually, a derived scheme). 6. DAG “explains” classical deformation theory In this section, we illustrate how derived deformation theory makes classical deformation theory completely transparent, by working out an explicit example of a very classical deformation problem: the infinitesimal deformations of a proper smooth scheme over k = C. We will first describe what classical deformation theory tells us in this special case, point out some weak points in this approach, and finally describe how derived algebraic geometry fixes these issues. The reader will find all of the omitted details in [18]. Recall that the objects of study of classical (formal) deformation theory are reduced functors F : ArtC → Grpd ,→ S Here, ArtC denotes the category of Artin rings over C (i.e., artinian local C-algebras with residue field isomorphic to C, or equivalently, augmented over C), and “reduced” means that F (C) is weakly contractible. For example, if calgC denotes the usual category of (discrete) commutative C-algebras, and F : calgC → Grpd is a classical moduli problem, then the choice of any point ξ ∈ F (C) determines a functor Fbξ := F ×F (C) ∗ : ArtC −→ Grpd , R 7−→ F (A) ×F (C) ∗ by forming the homotopy pullback: the morphism F (A) → F (C) is the F -image of the augmentation a : A → C, while the map ∗ → F (C) is the chosen point ξ). Equivalently, Fbξ (R) = fib(F (a) : F (R) → F (C) ; ξ)

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where fib denotes the homotopy fiber in Grpd. The functor Fbξ is obviously reduced, and is called the formal completion of F at ξ. Let us apply this formal completion construction to our chosen, well known moduli functor F : calgC −→ Grpd ,→ S sending a commutative C-algebra R into the groupoid of proper smooth morphisms Y −→ Spec(R) and isomorphisms between them. In this case, if we fix a proper smooth C-scheme ξ : X0 −→ Spec(C), this is a C-point of F , and the corresponding homotopy base change Fbξ is exactly the usual reduced functor, classically denoted as Def X0 . Definition 6.1. – The groupoid Fbξ (C[t]/tn+1 ) is called the groupoid of n-th order infinitesimal deformations of of F at ξ. The elements of the connected component π0 (Fbξ (C[t]/tn+1 )) are called n-th order infinitesimal deformations of F at ξ. The following properties are classically well known (see e.g., [12]): (1) if ξ1 ∈ Fbξ (C[ε]) is a first order deformation of ξ, then AutFbξ (C[ε]) (ξ1 ) ' H 0 (X0 , TX0 ); (2) π0 (Fbξ (C[ε])) ' H 1 (X0 , TX ); 0

(3) if ξ1 is a first order deformation, then there exists a class obs(ξ1 ) ∈ H 2 (X0 , TX0 ) such that obs(ξ1 ) = 0 if and only if ξ1 extends to a second order deformation. The class obs(ξ1 ) is called an obstruction class for ξ1 . The first two properties are completely satisfactory: they give algebraical interpretations (the rhs’s) of deformation theoretic objects (the lhs’s). Or conversely, as the reader prefers. This is not quite true for the third property, and it raises two natural questions : (A) What is the deformation-theoretic meaning of the entire H 2 (X0 , TX0 ) ? (B) How can we intrinsically identify the space of all obstructions (8) inside H 2 (X0 , TX0 ) ? Derived algebraic geometry gives a more general perspective on the subject, and answers both questions. It allows a natural interpretation of H 2 (X0 , TX0 ) as the group of derived deformations i.e., (isomorphism classes of) deformations over a specific derived ring, and it identifies, consequently, the obstructions space in a natural way. Let’s work these answers out. Define F : cdga≤0 C −→ S It can happen that every obstruction is trivial but H 2 (X0 , TX0 ) 6= 0. An example is given by a smooth projective surface X0 ⊆ P3C of degree ≥ 6.

(8)

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sending a cdga A to the maximal ∞-groupoid of equivalences inside the ∞-category of proper (9) and smooth maps of derived schemes Y −→ Spec A. It is clear that F is a derived stack. Moreover, since a derived scheme that is (derived) smooth over an underived scheme is automatically an underived scheme, we easily conclude that F(R) ' F (R) for any discrete commutative C-algebra R, and more generally, that t0 (F) ' F . Exercise 6.2. – Show that F is infinitesimally cohesive (Definition 5.1). Consider the full ∞-subcategory dArtC of cdga≤0 C of cdga’s A such that — H 0 (A) ∈ ArtC ; — for all i < 0, H i (A) is a module of finite type over H 0 (A); — H i (A) = 0 for i  0. The objects of dArtC will simply be called derived Artin rings. Our choice of a ξ ∈ F (C) ' F(C), identified with a morphism ξ : ∗ → F(C), allows us to consider the formal completion of F at ξ, by taking the pullback: bξ := F ×F(C) ∗ : dArtC −→ S. F Equivalently, bξ (A) = fib(F(a) : F(A) → F(C) ; ξ), dArtC 3 A 7−→ F where α is the derived version of the augmentation, i.e., the composite A

/ H 0 (A)

a

/ C,

and fib denotes the (homotopy) fiber in S. The following result answers to Question A above: the entire H 2 (X0 , TX0 ) can be interpreted as a space of derived deformations. Proposition 6.3. – There is a canonical isomorphism of C-vector spaces bξ (C ⊕ C[1])) ' H 2 (X0 , TX ). π0 (F 0 Proof. – First of all, F has a cotangent complex at ξ in the sense of [23, Definition 1.4.1.5], and it can be shown that TF,ξ ' RΓ(X0 , TX0 [1]) (note that since X0 is smooth TX0 ' TX0 ). Therefore (using e.g., [23, Proposition 1.4.1.6]), we obtain LF,ξ ' T∨ F,ξ ' RΓ(X0 , ΩX0 [−1]). (9)

By definition, a morphism of derived schemes or stacks is proper if it is so on the truncations.

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As a consequence (recall Definition 3.5), we get π0 (DerF,ξ (C[1])) ' π0 (MapdgmodC (LF,ξ , C[1])) ' Ext0 (LF,ξ , C[1]) ' ' Ext0 (LF,ξ [−1], C) ' H 0 (TF,ξ [1]) ' H 0 (RΓ(X0 , TX0 [2])) ' H 2 (X0 , TX0 ). We conclude that bξ (C ⊕ C[1])) ' π0 (fib(F(C ⊕ C[1]) → F(C), ξ)) ' π0 (DerF,ξ (C[1])) ' H 2 (X0 , TX ) π0 ( F 0

Remark 6.4. – Proposition 6.3 also explains why a classical deformation theoretic interpretation of the full H 2 (X0 , TX0 ) is impossible: H 2 (X0 , TX0 ) is the vector space of deformations (of F at ξ) over the base C ⊕ C[1] which is not a classical Artin ring but a derived one. Now that we have answered Question A, i.e., we have a derived deformationtheoretic interpretation of the entire H 2 (X0 , TX0 ) at hand, we can proceed by answering Question B above. We begin by an auxiliary result (for a more general version the reader is invited to consult [18, Thm. 3.1]). Lemma 6.5. – Let /R

J

f

/S

be a square zero extension of (augmented) classical Artin rings over C (i.e., f is surjective, J = ker f , and J 2 = 0). Then, there exists a derived derivation d : R → R ⊕ J[1] and a homotopy cartesian diagram /S

f

R



 C

π◦d

/ C ⊕ J[1]

where π : S ⊕ J[1] → C ⊕ J[1] is the natural map induced by the augmentation S → C. Proof. – Note that S ⊕ J[1] can be represented by the obvious cdga 0

/J

0

/S

/0

where S sits in degree 0. The trivial derivation d0 is then represented by the commutative diagram /0 /S /0 0  0 0  id / /S / 0. 0 J Observe that we may represent S also by the cdga 0

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/J

i

/R

/ 0,

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19

where i denotes the inclusion map. Then, we can define a derived derivation d : S → S ⊕ J[1] by the commutative diagram 0

/J

0

 /J

i id 0

/R  /S

/0

π

/ 0,

and remark that d is a fibration of cdga’s. Since the model category of cdga’s is proper, the ordinary pullback of the zero derivation d0 and of d computes the homotopy pullback S ⊕d J. But the ordinary pullback is given by just /0

0

/R

/0

(i.e., by just R sitting in degree 0). So, we conclude that the square /S

R  S

d



d0

/ S ⊕ J[1]

is a (homotopy) pullback of cdga’s over C. So, we are left to show that S

/C

 S ⊕ J[1]

 / C ⊕ J[1]

d0

is a (homotopy) pullback. However, the map S ⊕ J[1] → C ⊕ J[1] is a fibration, hence it is enough to show that this diagram is a strict pullback, which is a straightforward verification. If, in particular, we take the square-zero extension R = C[t]/(t3 ) → S = C[t]/(t2 ), by Lemma 6.5 we obtain a (homotopy) pullback C[t]/(t3 )

/ C[t]/(t2 )

 C

 / C ⊕ C[1].

Exercise 6.6. – Construct the rightmost vertical map in the above diagram as in the very first part of the proof of Lemma 6.5, and prove directly (i.e., without using Lemma 6.5) that the above square is cartesian. Observe that both maps C → C ⊕ C[1] and C[t]/(t2 ) → C ⊕ C[1] are surjective on H 0 , with nilpotent kernels. Since F is infinitesimally cohesive (Exercise 6.2), the

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diagram F(C[t]/t3 ) ' F (C[t]/t3 )

/ F(C[t]/t2 ) ' F (C[t]/t2 )

 F(C) ' F (C)

 / F(C ⊕ C[1])

is cartesian in S. Via the augmentation maps, this whole diagram maps to F (C), and taking fibers at ξ gives us a diagram bξ (C[t]/t3 ) ' Fbξ (C[t]/t3 ) F

bξ (C[t]/t2 ) / Fbξ (C[t]/t2 ) ' F

 *

 bξ (C ⊕ C[1]), /F

which is, obviously, again cartesian in S∗ (the ∞-category of pointed spaces or simplicial sets). In other words, we obtain a fiber sequence of pointed spaces bξ (C[t]/(t3 )) −→ F bξ (C[t]/(t2 )) −→ F bξ (C ⊕ C[1]), F and therefore a corresponding exact sequence on π0 ’s (*) bξ (C[t]/(t2 ))) obs / π0 (F bξ (C ⊕ C[1])) ' H 2 (X0 , TX ) bξ (C[t]/(t3 ))) / π0 ( F π0 ( F 0 of pointed sets. As a consequence, we see that bξ (C[t]/(t2 ))) = π0 (Fbξ (C[t]/(t2 )))) — a first order deformation (i.e., an element in π0 (F bξ (C[t]/(t3 ))) = extends to a second order deformation (i.e to an element in π0 (F π0 (Fbξ (C[t]/(t3 )))), if and only if its image via obs vanishes. In other words, the set Obs2 (F ; ξ) of all obstructions to extending a first order deformation to a second order one is given by the image of the obstruction map bξ (C ⊕ C[1])) ' H 2 (X0 , TX ). obs : π0 (Fbξ (C[t]/(t2 ))) −→ π0 (F 0 This is a complete answer to Question B. Remark 6.7. – Although strictly speaking this is not necessary, one can furthermore observe that the middle and the rightmost pointed sets in sequence (∗) are actually C-vector spaces, pointed at 0, and that the obs is a morphism of vector spaces. Exercise 6.8. – Prove the statements in Remark 6.7. Exercise 6.9. – Extend the previous arguments and results to higher order infinitesimal deformations and obstructions.

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7. Forms and closed forms In this section we describe the theory of differential forms on derived algebraic stacks and its implications for the local geometry of derived stacks in general and for derived moduli stacks in particular. 7.1. The case of affine derived schemes. – As explained in Sections 2.2 and 3, given A ∈ cdga≤0 k , and a cofibrant (quasi-free) replacement QA → A of A, the affine derived scheme Spec A ∈ dSt(k) has a cotangent complex:  1  ΩQA ⊗QA A, where Ω1QA is the module LSpec A = LA = of Kähler differentials of QA Remark 7.1. – We will often use implicitly the equivalence of ∞-category dgmod(QA) ' dgmod(A), and consequently identify Ω1QA with Ω1QA ⊗QA A = LA . This definition globalizes to general (algebraic) derived stacks X as in Definition 3.5. Moreover, X is locally of finite presentation if and only if LX , which is a priori an object in QCoh(X), is in fact perfect. In this case the intrinsically defined tangent complex of X can be computed as TX = L∨ X = Hom O X (LX , O X ). In the affine case the complex of p-forms on Spec A is defined to be ∧pQA LA = ΩpQA . p We will write Ωp,i QA for the terms of the complex ΩQA . The condition that QA is quasi-free means that if we ignore the differential, then as a graded algebra QA is a polynomial algebra in a finite number of variables {ti } of various grading degrees. By definition the module Ω1QA of Kähler differentials for QA is the graded module over the graded algebra QA which is freely generated by formal generators dti where the grading degree of dti is set to be equal to the grading degree 1,a+1 which of ti . The differential on Ω1QA is the unique k-linear map d : Ω1,a QA → ΩQA 1 makes ΩQA a dg module over QA and such that the de Rham differential dDR : QA → Ω1QA assigning the formal derivative to each polynomial is a map of dg modules over QA. As usual dDR : QA → Ω1QA extends to a de Rham differential on p-forms dDR : L p p → Ωp+1 QA by the graded Leibnitz rule. Thus the sum ⊕p≥0 ∧QA LA = p≥0 ΩQA is a fourth quadrant bicomplex with

ΩpQA

p,i+1 vertical differential: d : Ωp,i induced by dQA , and QA → ΩQA p+1,i horizontal differential: dDR : Ωp,i given by the de Rham differential. QA → ΩQA

In analogy with the underived setting, the differential forms on an affine derived L q scheme admit a Hodge filtration with steps F p (A) := q≥p ΩQA , each of which is still a fourth quadrant bicomplex. This Hodge filtration turns out to be the correct vessel for defining closed p-forms and working with the closedness condition in the derived setting.

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i h p+1 p dDR Motivation: If X is a smooth scheme/k, then the sheaf Ωp,cl X = ker ΩX −→ ΩX of closed p-forms on X is naturally quasi-isomorphic to the stupid truncation of the algebraic de Rham complex, i.e.,   ≥p ∼ Ωp,cl Ω [p], d = DR . X X Up to a shift this truncation is precisely the p-the step of the Hodge filtration of the complex of algebraic forms, and so we can use the hypercohomology of the complex F p Ω•X [p] = (Ω≥p X [p], dDR ) as a model for closed p forms in general. Thus we have the following natural Definition 7.2. – If Spec A is an affine derived scheme, the complex of closed p-forms Q Q on Spec A is the complex Ap,cl (A) := tot (F p (A))[p], where tot (F p (A)) denotes the completed (i.e., product) totalization of the double complex F p (A). Furthermore, in the derived setting all these notions admit natural refinements which account for the freedom of performing a shift in the internal homological grading of the cotangent complex. This leads to the notions of shifted forms and shifted closed forms. Definition 7.3. – For an affine derived scheme Spec A define — the complex of n-shifted p-forms on Spec A: Ap (A; n) :=

Vp

LA [n] = ΩpQA [n] Q

— the complex of n-shifted closed p-forms on Spec A: Ap,cl (A; n) := tot (F p (A))[n + p] — the Hodge tower of Spec A: · · · → Ap,cl (A)[−p] → Ap−1,cl (A)[1 − p] → · · · → A0,cl (A) Explicitly an n-shifted closed p-form ω on Spec A is an infinite collection p+i,n−i ω i ∈ ΩA ,

ω = {ωi }i≥0 ,

satisfying

dDR ωi = −dωi+1 .

Notation. We write |E| for Mapdgmod≤0 (k, τ≤0 E), i.e., the Dold-Kan construction apk plied to the ≤ 0-truncation of a complex E. Remark 7.4. – (i): It is clear from this description that the notion of an n-shifted closed p-form is much more flexible than the naive notion of an n-shifted strictly closed p-form, i.e., an element ω0 ∈ Ωp,n A satisfying dDR ω0 = 0. Given an n-shifted closed p-form ω = {ωi }i≥0 we will call the collection {ωi }i≥1 the key closing ω. (ii): By definition we have an underlying p-form map Ap,cl (A; n) → ∧p LA [n] which induces a map on cohomology H 0 (Ap,cl (A)[n]) → H n (X, ∧p LA ). (iii): The homotopy fiber of the underlying p-form map is the complex of keys for a given n-shifted p-form and can be very complicated. Thus for a p-form in derived geometry being closed is not a property but rather a list of coherent data.

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23

(iv): The complex A0,cl (A) of closed 0-forms on X = Spec A is exactly Illusie’s derived de Rham complex of A [14, ch. VIII]. (v): If A ∈ cdga≤0 k is quasi-free, then   Y p+1 ΩA [n − i], d + dDR Ap,cl (A; n) = i≥0 = totΠ (F p (A))[n] = |N C w (A)(p)[n + p]| , where N C w (A) denotes the weighted negative cyclic complex for A. Hence n−p π0 Ap,cl (A; n) = HC− (A)(p).

7.2. Functoriality and gluing. – To globalize the definitions of forms and closed forms we consider: — The ∞-functor of n-shifted p-forms A

p

(−; n) : cdga≤0 k → S,

A 7→ |Ap (A; n) | .

— The ∞-functor of n-shifted closed p-forms A

p,cl

(−; n) : cdga≤0 k → S,

A 7→ Ap,cl (A)[n] .

One can check [17] that the functors A p (−; n) and A p,cl (−; n) are derived stacks for the étale topology and that the assignment of an underlying p-form A

p,cl

(−; n) → A p (−; n)

is a map of derived stacks. With this in mind one can now give the following general Definition 7.5. – Let X ∈ dSt(k) be a derived algebraic stack locally of finite presentation. Then we define: — A p (X) := MapdSt(k) (X, A p (−)) to be the space of p-forms on X; — A p,cl (X) := MapdSt(k) (X, A p,cl (−)) to be the space of closed p-forms on X; — the corresponding n-shifted versions : A p (X; n) := MapdSt(k) (X, A p (−; n)) A

p,cl

(X; n) := MapdSt(k) (X, A p,cl (−; n))

— an n-shifted (respectively closed) p-form on X is an element in π0 A p (X; n) (respectively in π0 A p,cl (X; n)) Remark 7.6. – The definition has some straighforward consequences: (1) If X is a smooth underived scheme, then there are no negatively shifted forms. (2) When X = Spec A is an affine derived scheme, then there are no positively shifted forms. For a general derived stack X shifted differential forms might exist for any n ∈ Z.

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As in the affine case the underlying p-form map of simplicial sets A

p,cl

(X; n) → A p (X; n)

will not typically be a monomorphism. Its homotopy fiber at a given p-form ω0 is the space of keys of ω0 . However, if X is a smooth and proper scheme then this map is indeed a monomorphism, i.e., its homotopy fibers are either empty or contractible [17]. Thus we have no new phenomena in this case. In the general case the following theorem provides a concrete and expected algebraic model for global forms.

Theorem 7.7 (Proposition 1.14 in [17]). – For a derived algebraic stack X the n-shifted p-forms satisfy smooth descent, i.e., A

p

(X; n) ' MapQCoh(X) ( O X , (∧p LX )[n]).

In particular an n-shifted p-form on X is an element in H n (X, ∧p LX ).

Guided by the classical case we can also define algebraic de Rham cohomology for derived stacks.

Definition 7.8. – Given a derived algebraic stack X the n-th algebraic de Rham cohon mology of X is defined to be HDR (X) = π0 A 0,cl (X; n). Remark 7.9. – schemes.

This notion agrees with Illusie’s definition in the case of affine

— if X is a algebraic derived stack locally of finite presentation, then [21, Proposi• • (X) ∼ (t0 X) = algebraic de Rham cohomology of the underived tion 5.2] HDR = HDR higher stack t0 X defined by the standard Hartshorne’s completion formalism [11].

The previous remark combined with the canonical resolution of singularities in characteristic zero and smooth and proper descent for algebraic de Rham cohomology have the follwoing important consequence

Corollary 7.10 (Corollary 5.3 in [21]). – Let X be a algebraic derived stack which is locally of finite presentation. Suppose ω is an n-shifted closed p-form on X with n < 0. n+p Then ω is exact, i.e., [ω] = 0 ∈ HDR (X).

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25

7.3. Examples. – In this section we describe several natural examples where forms and closed forms can be computed exactly. (1) If X = Spec A is an usual (underived) smooth affine scheme, then A

p,cl

(X; n) = (τ≤n ( ΩpA 0

and hence

dDR

/ Ωp+1 A

dDR

/ · · · ))[n],

1

  n 0. DR

n+p−i (2) If X is a smooth and proper scheme, then πi A p,cl (X; n) = F p HDR (X).

(3) If X is a (underived, higher) algebraic stack, and X• → X is a smooth affine simplicial groupoid (10) presenting X, then π0 A p (X; n) = H n (Ωp (X• ), δ) with δ = Čech differential. In particular if G is a complex reductive group, then ( 0, n 6= p p π0 A (BG; n) = p ∨ G (Sym g ) , n = p. (4) Similarly Y
G p,cl A (BG; n) = Symp+i g∨ [n + p − 2i] , i≥0 and so

( π0 A

p,cl

(BG; n) =

0, if n is odd G (Symp g∨ ) , if n is even.

(5) Derived schemes naturally arise as derived intersections of ordinary schemes. A dg scheme ([9]) (over k) is a scheme X equipped with a sheaf A •X of non-positively graded quasi-coherent k-dg algebras such that A 0X = O X . There is an obvious notion of morphism between dg schemes, and the equivalences are defined as those morphisms inducing quasi-isomorphisms between the corresponding sheaves of cdga’s. If we localize the ∞-category of dg schemes with respect to such equivalences, we get an ∞-category dgSchk admitting a functor Θ to the ∞-category of derived schemes: for an affine dg scheme (X = Spec R, A X ), A X is given by a k-cdga AX , and the functor sends (X, A X ) to Spec AX . For more general, non affine dg schemes (X, A X ), the functor is defined using an affine Zariski covering of X (see [22] for more details). The functor Θ is conservative but neither full, nor faithful, nor essentially surjective. The reason for not being essentially surjective is that, for a dg scheme (X, A X ), (10)

E.g., the nerve of a smooth affine atlas.

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Θ(X, A X ) is always an globally embeddable derived scheme, i.e., there exists an underived scheme X and a closed immersion of derived schemes Θ(X, A X ) → X. Not all derived schemes have this property. However, for the purpose of this example, we will stick to dg schemes, and tacitly identify them as derived schemes by using (though not writing) the functor Θ. In a typical setup one considers a smooth variety M over k and two smooth subvarieties L1 , L2 ⊂ M . The derived intersection X of L1 and L2 is defined as a dg scheme, i.e., a space equipped with a sheaf of non-positively graded dg algebras:   L h O X := L1 L2 = L1 ∩ L2 , O L1 O L2  .

× M

OM

The tangent complex of X is a a perfect complex concentrated in degrees 0 and 1 and is explicitly given by / TM,x ],

[ TL1 ,x ⊕ TL2 ,x

TX,x =

0

1.

In particular we have — H 0 (TX,x ) = TL1 ∩L2 ,x , and — H 1 (TX,x ) = failure of transversality of the intersection L1 ∩ L2 . An important special case of a derived intersection is the derived zero locus of a section of a vector bundle. Let L be a smooth variety over k, E → L an algebraic vector bundle on L, and s ∈ H 0 (L, E) and a section in E. The derived zero locus X of s is defined as the derived intersection of s with the zero section of E inside M = tot(E):   h • ∨ [ X := Rzero(s) = L L = Z, i−1 L (Sym (E [1]), s ) ,

×

s,M,o

where: — Z = t0 X = zero(s) is the scheme theoretic zero locus of s, — iL : Z → L is the natural inclusion, and — s[ is the contraction with s. In particular TX =

[ i∗L TL ⊕ i∗L TL

∗ i∗ L do+iL ds

/ i∗ TM ], M

0

1

— M = tot(E), and — iM , o, and s are the natural maps

?L

iL

Z

iM

iL

PANORAMAS & SYNTHÈSES 55



L

o

! / M. =

s

where

27

INTRODUCTORY TOPICS IN DERIVED ALGEBRAIC GEOMETRY

Exercise 7.11. – Let ∇ : E → E ⊗ Ω1L be an algebraic connection on E. Show that there is a natural quasi-isomorphism " TX = i∗L TL

(∇s)[

/ i∗ E

#

L

" = TL

(∇s)[

/E

# . |Z

The algebraic connection ∇ : E → E ⊗ Ω1L may exists only locally on L and is not unique. Check that (∇s)|Z is well defined of the choice of ∇.  ˚ globally and independent 

iL E

E

Suppose again X = Rzero(s) for s ∈ H 0 (L, E) on a smooth L, then ∨ Ω1X = E|Z

(∇s)[

/ Ω1 . L|Z

−1

0

 Ω1 , L|Z ∇ : E → E ⊗ Ω1 which is also Assume we have chosen (11) an algebraic connection L 2 flat, i.e., ∇ = 0. Using such a ∇ we can explicate Ω1X as a module over the Koszul dga of s:   [ [ s[ • ∨ [ 2 ∨ s ∨ s (Sym (E [1]), s ) = · · · → ∧ E → E → O L .

In other words using ∇ we can resolve Ω1X by a double complex of vector bundles on L so that this double complex is on-the-nose a module over (Sym• (E ∨ [1]), s[ ):

s5

 ^ 2 E _ b Ω1 L

¨¨¨ ¨¨¨

 ^2 E _ b E _

s5

 E _ b Ω1 L r∇,s5 s

 E_ b E_

s5

 Ω1 L

 Ω1 L|Z

s5

r∇,s5 s

0

 E_

p∇sq5

 E_

|Z

´1

||

Ω1X In the same way we can describe Ω2X as a module over the Koszul dga:

(11)

  E _ b Ω2 L Such a connection always exists Zariski locally on L. 

 Ω2

 Ω2 L|Z

L



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 E _ b Ω1 L 



28



 Ω1 L



 Ω1 L|Z



TONY PANTEV & GABRIELE VEZZOSI

¨¨¨

 ^ 2 E _ b Ω2 L

 E _ b Ω2

 Ω2 L

 Ω2 L|Z

0

¨¨¨

 ^ 2 E _ b E _ b Ω1 L

 E _ b E _ b Ω1 L

 E _ b Ω1 L

 pE _ b Ω1 q|Z L

´1

¨¨¨

 ^2 E _ b S 2 E _

 E_ b S2E_

 S 2E _

 S 2E _ |Z

´2

L

||

Ω2X

In particular the −1 shifted 2-forms on the derived critical locus X will be given by pairs of sections α ∈ E ∨ ⊗ Ω2L , and φ ∈ E ∨ ⊗ Ω1L . Note also that in terms of these resolutions, the de Rham differential dDR : Ω1X → 2 ΩX is the map that term-by-term is given by the sum dDR = ∇ + κ. Here κ is the Koszul contraction κ : ∧a E ∨ ⊗ S b E ∨ → ∧a−1 E ∨ ⊗ S b+1 E ∨ , i.e., the contraction with idE ∈ E ⊗ E ∨ followed by the multiplication E ∨ ⊗ S b E ∨ → S b+1 E ∨ . The case when X = Rzero(s) is the derived zero locus of an algebraic one form s ∈ Ω1L plays a special role in Donaldson-Thomas theory [4]. The explicit Koszul model of forms on such an X leads to the following important description of the −1 shifted 2-forms due to Behrend: Remark 7.12 (K. Behrend). – If E = Ω1L and so s is a 1-form on L, then a (−1)-shifted 2-form on X = Rzero(s) corresponds to a pair of elements α ∈ (Ω1L )∨ ⊗ Ω2L and φ ∈ (Ω1L )∨ ⊗ Ω1L such that [∇, s[ ](φ) = s[ (α). Take φ = id ∈ (Ω1L )∨ ⊗ Ω1L . Suppose the local ∇ is chosen so that ∇(id) = 0 (i.e., ∇ is a torsion free connection). Then [∇, s[ ](id) = ds. In other words the pair (α, id) gives a (−1)-shifted 2-form on X if and only if ds = s[ (α) = −s ∧ α. Equivalently (α, id) gives a −1-shifted 2-form on X when ds|Z = 0, i.e., if and only if s is an almost closed 1-form on L. Exercise 7.13. – Suppose s is an almost closed one form on L and let (α, id) be an associated (−1)-shifted 2-form. Describe the complex of keys for (α, id). 8. Shifted symplectic geometry To illustrate the power of the general theory we will review a geometric concept that is inherently derived in nature - the notion of a shifted symplectic structure. Here we only sketch the highlights of the theory. Full details can be found in [17]. Recall that for an ordinary smooth scheme X over k a symplectic structure on X is a non-degenerate closed algebraic 2-form. In other words a symplectic strucure is an

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29

[ 1 element ω ∈ H 0 (X, Ω2,cl X ) such that its adjoint ω : TX → ΩX is a sheaf isomorphism. This straightforward definition does not work when X/k is a derived stack for at least two reasons

— The tangent complex TX of X need not have amplitude which is symmetric around zero. When the amplitude of TX is not symmetric, the tangent and cotangent complex of X will not have the same amplitude so they can not be quasi-isomorphic. — A form being closed is not just a condition but rather an extra structure implemented by the key closing the form. Taking these comments into account we arrive at the following natural Definition 8.1. – Let X be a derived algebraic stack/k locally of finite presentation (so that LX is perfect). — A n-shifted 2-form ω : O X → LX ∧ LX [n] - i.e., ω ∈ π0 ( A 2 (X; n)) - is nondegenerate if its adjoint ω [ : TX → LX [n] is an isomorphism in QCoh(X). — The space of n-shifted symplectic forms Sympl(X; n) on X/k is the subspace of A 2,cl (X; n) of closed 2-forms whose underlying 2-forms are nondegenerate i.e., we have a homotopy cartesian diagram of spaces Sympl(X, n)

/ A 2,cl (X, n)

 (X, n)nd

 / A 2 (X, n).

A

2

Remark 8.2. – The nondegeneracy condition in the definition of shifted symplectic structure should be viewed as a duality between the stacky (positive degree) and the derived (negative degree) parts of LX (see the end of §4). — Suppose the perfect complex LX has amplitude [−m, n] for m, n ≥ 0. Then at best X can admit an (m − n)-shifted symplectic structure. — If G = GLn , then BG has a canonical 2-shifted symplectic form (see Example (4) Section 7.3) : k → (LBG ∧ LBG )[2] ' (g∨ [−1] ∧ g∨ [−1])[2] = Sym2 g∨ given by the dual of the trace map (A, B) 7→ tr(AB). — Similarly, for a reductive G/k choosing a non-degenerate G-invariant symmetric bilinear form on g gives rise (see Example (4) Section 7.3) to a 2-shifted symplectic form on BG. — For a smooth variety the n-shifted cotangent bundle T ∨ X[n] := SpecX (Sym(TX [−n])) has a canonical n-shifted symplectic form. The same holds for the shifted cotangent bundle of a general derived algebraic stack locally of finite presentation over k [8].

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Beyond these elementary examples shifted symplectic structures frequently arise on derived moduli stacks. A common method for producing such structures comes from a algebraic version of the AKSZ formalism in quantum field theory [1]. Before we can explain this we need to recall some basic facts about O -orientations and push-forwards in derived geometry. Definition 8.3. – Let X be a derived stack/k and let A ∈ cdga≤0 k . Let XA denote the derived A-stack X × Spec A. We will say that X is O -compact over k if for every A ∈ cdga≤0 we have that O XA is a compact object in QCoh(XA ) and for any perfect k complex E ∈ QCoh(XA ), the cochain module C(XA , E) = RHom( O , E) is a perfect A-dg module. Suppose X is derived stack over k. For every A ∈ cdga≤0 k , the cup product ∪ on C(XA , O ) turns this cochain complex into a commutative dga over A. Given any morphism η : C(X, O ) → k[−d] of k-cdga, we get a morphism C(XA , O ) C(X, O ) ⊗k A

/ A[−d]

ηA

η⊗idA

/ k[−d] ⊗k A

of A-cdga and an induced morphism (1)

C(XA , O ) ⊗A C(XA , O )

∪

/ C(XA , O )

ηA

/ A[−d].

If X is also O -compact over k, then writing C(XA , O )∨ = RHom(C(XA , O ), A) for the dual of C(XA , O ) over A then (1) can be rewritten as the ajoint map: C(XA , O )

−∪η

/ C(XA , O )∨ [−d] .

More generally, for any perfect complex E ∈ QCoh(XA ) we can compose the natural pairing C(XA , E) ⊗A C(XA , E ∨ ) → C(XA , O ) with ηA to obtain a morphism C(XA , E)

−∪ηA

/ C(XA , E ∨ )∨ [−d] .

With this notation we now have the following Definition 8.4. – Let X be an O -compact derived stack/k and let d be an integer. An O -orientation of X of degree d is a morphism of cdga [X] : C(X, O ) → k[−d] such that for any A ∈ dalg and any perfect complex on XA , the natural map − ∪ [X]A : C(XA , E) → C(XA , E ∨ )∨ [−d]. is a quasi-isomorphism of A-dg modules. Example 8.5. – (a): Let X be a smooth proper Deligne-Mumford stack of dimension d over k, and let u be a Calabi-Yau structure on X. In other words u is an isomorphism between the structure sheaf of X and the canonical line bundle ωX = ΩdX .

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Since X is smooth and proper it is O -compact when viewed as a derived stack/k. The isomorphism u composed with the trace map gives an isomorphism H d (X, O )

/ H d (X, ωX )

u

tr

/k .

This in turn gives a natural map of complexes [X] : C(X, O ) → k[−d] which by Serre duality is an O -orientation on X of degree d. (b): Let M be an oriented connected and compact topological manifold of dim M = d. Let X = S(M ) be the simplicial set of singular simplices in M viewed as a constant derived stack/k. The category QCoh(X) is naturally identified with the ∞-category of complexes of k-modules on M with locally constant cohomology sheaves. The perfect complexes on X correspond to complexes of k-modules on M that are locally quasi-isomorphic to finite complexes of constant sheaves of projective k-modules of finite type. I particular X is O -compact. Furthermore, the fundamental class [M ] ∈ Hd (M, k) on M given by the orientation defines a morphism of complexes [X] : C(M, k) = C(X, O ) → k[−d]. Finally Poincaré duality on M implies that [X] is an O -orientation on X of degree d. Exercise 8.6. – Let D = Spec k[t]/(t2 ) be the spectrum of the dual numbers. Show that D admits a natural O -orientation of degree 0. The main utility of O -orientations is that they give natural integration maps on negative cyclic complexes and thus induce natural push-forward maps on shifted closed forms. Specifically let X be a derived O -compact stack/k with an O -orientation [X] : C(X, O ) → k[−d] of degree d. Suppose that X is a algebraic stack locally of finite presentation and let S be another algebraic derived stack locally of finite presentation. Then the O -orientation on X induces a natural integration map of mixed graded complexes Z : N C w (X × S) → N C w (S)[−d]

[X]

between the weighted negative cyclic complex on X × S and the weighted negative cyclic complex on X. R We will not spell out the definition of [X] but direct the reader to [17, Section 2.1] R which contains a detailed construction of this map. Here we only point out [X] induces a natural map between spaces of shifted closed forms. Indeed, recall (see Remark 7.4 (v)) that an n-shifted closed p-form α is nothing but a map α : k[n − p](p) → N C w of mixed graded complexes. Thus, givenR α ∈ A p,cl (X × S; n) we can compose the map α : k[n − p](p) → N C w (X × S) with [X] to obtan a map of mixed graded complexes k[n − p](p)

α

/ N C w (X × S) R [X]

R [X]

/ N C w (S)[−d]. 2

α

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TONY PANTEV & GABRIELE VEZZOSI

The assignment α → 7 closed p-forms: Z

R [X]

α can therefore be viewed as a map of spaces of shifted

: A p,cl (X × S; n) → A p,cl (S; n − d).

[X]

After these preliminaries we are now ready to state the algebraic version of the AKSZ formalism: Theorem 8.7 (Theorem 5.2 in [17]). – Let F be a derived algebraic stack equipped with an n-shifted symplectic form ω ∈ Symp(F, n). Let X be an O -compact derived stack equipped with an O -orientation [X] : C(X, O X ) → k[−d] of degree d. Assume that the derived mapping stack MAP(X, F ) is itself a derived algebraic stack locally of finite presentation over k. Then, MAP(X, F ) carries a canonical (n − d)-shifted symplectic structure. The shifted symplectic form on MAP(X, F ) is explicitly constructed from the symplectic form ω and the O -orientation [X]. Indeed, consider the natural evaluation map ev /F X × MAP(X, F ) / f (x). (x, f )  Since ω ∈ Symp(F, n) → A 2,cl (X; n) can be viewed as an n-shifted 2-form on F we can pull it back to X × MAP(X, F ) and integrate it against [X] to obtain a closed (n − d)-shifted 2-form Z ev∗ ω ∈ A 2,cl (MAP(X, F ); n − d). [X]

It can be cheked directly [17, Theorem 5.2] that the underlying shifted 2-form is R non-degenerate, which shows that [X] ω is symplectic. Example 8.8. – (1): Let X/C be a smooth and proper Calabi-Yau variety of dimension d and let G be complex reductive group. Then the derived moduli stack BunX (G) = MAP(X, BG) of algebraic G-bundles on X is (2 − d)-shifted symplectic. (2): Let M be an oriented connected compact topological manifold of dimension d, and let G be a complex reductive group. If X = S(M ) is the simplicial set of singular simplices in M viewed as a constant derived stack, then the derived moduli of flat G-bundles on M , can be computed as the derived moduli BunX (G) = MAP(X, BG) of algebraic G-bundles on X. Since X admits an O -orientation of degree d, it again follows that BunX (G) is (2 − d)-shifted symplectic. Exercise 8.9. – (a) Suppose (M, ω) is an algebraic symplectic manifold over C, and let D = Spec C[t]/(t2 ) be the spectrum of the dual numbers. Then by the above theorem MAP(D, M ) is a symplectic manifold. But MAP(D, M ) is the total space of the tangent bundle TM of M . Check that the symplectic structure on TM is the pullback of the ∨ ∨ tautological symplectic structure on TM via the isomorphism ω [ : TM → TM .

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(b) Suppose (F, ω) is an n-shifted symplectic derived scheme over C, and let D = Spec C[t]/(t2 ) be the spectrum of the dual numbers. The theorem gives an n-shifted symplectic structure on the total space TF = MAP(D, F ) of the tangent stack of F . How does this symplectic structure depend on the key of ω? Another source of examples of shifted symplectic structures is the derived intersection of Lagrangians. Before we can formulate the relevant construction we will need to discuss the notions of isotropic and Lagrangian structures on shifted symplectic geometry. Definition 8.10. – Let (Y, ω) be a n-shifted symplectic derived stack/k and let f : X → Y be a map from a algebraic derived stack X locally of finite presentation/k. — An isotropic structure on f : X → Y is a path γ in the space A 2,cl (X; n) connecting f ∗ ω to 0 — A Lagrangian structure on f : X → Y is an isotropic structure γ which is nondegenerate in the sense that the induced map γ [ : Tf → LX [n−1] is an equivalence. Here Tf denotes the relative tangent complex of the map f : X → Y , and the map γ [ is constructed as follows. By definition γ gives a homotopy between the map f ∗ (ω [ ) : f ∗ TY → f ∗ LY [n] and the zero map of complexes. Pre-composing and postcomposing f ∗ (ω [ ) with the differential and the codifferential of f respectively we get a map of complexes df ∨ ◦f ∗ (ω [ )◦df

TX

/ LX [n] O df ∨

df

 f ∗ TY

f ∗ (ω [ )

/ f ∗ LY [n]

and hence pre-composing and post-composing γ the differential and the codifferential of f we will get a homotopy between df ∨ ◦ f ∗ (ω [ ) ◦ df : TX → LX [n] and the zero map of complexes. If we write ι : Tf → TX for the natural map from the relative tangent complex of f to the tangent complex of X, then we will get a homotopy between the map df ∨ ◦ f ∗ (ω [ ) ◦ df ◦ ι : Tf → LX [n] and the zero map of complexes. On the other hand by definition we have an exact triangle of complexes Tf

ι

/ TX

df

/ f ∗ TX

/ Tf [1]

so we have an intrinsic homotopy between df ◦ ι : Tf → f ∗ TY and the zero map of complexes. In other words we get two homotopies between df ∨ ◦ f ∗ (ω [ ) ◦ df ◦ ι

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and zero: one coming from the isotropic structure γ and the other coming from the definition of the relative tangent complex. Composing these two homotopies we get a self-homotopy of the zero map of complexes 0 : Tf → LX [n], i.e., an element γ [ in π1 (MapQCoh(X) (Tf , LX [n]); 0) = π0 (MapQCoh(X) (Tf , LX [n − 1]); 0) = HomQCoh(X) (Tf , LX [n − 1]).

Remark 8.11. – Any smooth Lagrangian L ,→ (Y, ω) where (Y, ω) is a smooth (0)symplectic scheme has a natural Lagrangian structure in the derived sense. Moreover in this case the space of Lagrangian structures is contractible, so that this natural Lagrangian structure is essentially unique. — For any n the point Spec k has a natural (n + 1)-shifted symplectic form ωn+1 , namely the zero form. As first observed by D. Calaque, it is straightforward to check that the Lagrangian structures on the canonical map X → (Spec k, ωn+1 ) are the same thing as n-shifted symplectic structures on X.

The relevance of Lagrangian structures for constructing new shifted symplectic structures is captured by the following

Theorem 8.12 (Theorem 2.9 in [17]). – Let (F, ω) be n-shifted symplectic derived stack, and let Li → F , i = 1, 2 be maps of derived stacks equipped with a Lagrangian structures. Then the derived intersection L1 ×hF L2 of L1 and L2 is canonically a (n − 1)-shifted symplectic derived stack.

Example 8.13. – (1) An important special case of this construction is the (−1)-shifted symplectic structure on the derived critical locus of a function. Suppose L/C is a smooth variety and let w : L → C be a regular function. By definition the derived critical locus Rctit(w) of w is the derived zero scheme Rzero(dw) of the one form dw ∈ H 0 (L, Ω1L ). Consider TL∨ with its standard (0-shifted) symplectic structure. Since dw is closed, the map dw : L → TL∨ is Lagrangian. But the zero section o : L → TL∨ is Lagrangian as well, and so h

Rcrit(w) = L

× L dw,T ,o ∨ L

is equipped with a natural (−1)-shifted symplectic structure. In terms of the local description of Remark 7.12 the underlying (−1)-shifted 2-form is given by the pair(dw, id).

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Variant. – By the same construction the derived critical locus of a shifted function w : L → A1 [n] (i.e., an element w ∈ H n (L, O ) is equipped with a natural (n−1)-shifted symplectic structure. Note that if w is locally constant, then dw = 0 and so Rcrit(w) = TL∨ [−1] is the (−1)-shifted cotangent stack with its canonical symplectic structure. By the same token if w ∈ H n (L, O ) is in the image of H n (L, C), then Rcrit(w) = TL∨ [n − 1] is the (n − 1)-shifted cotangent stack with its canonical symplectic structure. In particular if L is smooth and projective over C the Hodge theorem implies that H n (L, C) → H n (L, O ) is surjective for all n and therefore the derived critical locus of any shifted function on L is a shifted cotangent bundle. (2) Let (X, ω) be an algebraic symplectic manifold over C equipped with a Hamiltonian action of a complex reductive group G. Let g = Lie(G) and let µ : X → g∨ be a G-equivariant moment map. Consider the stack quotient [g∨ /G] where G acts by the ∨ ∨ [1] is a 1-shifted = [g[−1]/G] it follows that [g∨ /G] = TBG coadjoint action. Since TBG symplectic stack. The equivariant map µ induces a map of stacks µ : [X/G] → [g∨ /G] and the moment map condition translates into the statement that the symplectic form ω induces a Lagrangian structure on this map. Additionally, for any coadjoint orbit O ⊂ g∨ the Kostant-Kirillov symplectic form on O induces a Lagrangian structure on the map of stacks [O/G] → [g∨ /G]. In particular the homotopy fiber product h

[X/G]

× [O/G] [g /G] ∨

will have a natural 0-shifted symplectic structure. Note that this fiber product is simply the stack quotient [Rµ−1 (O)/G], where Rµ−1 (O) = X ×hg∨ O is the derived preimage of the coadjoint orbit O under the moment map µ. Thus the 0-shifted symplectic structure on [Rµ−1 (O)/G] can be viewed as an extension of the Marsden-Weinstein symplectic reduction: at the expense of adding a stacky and a derived structure on the reduction we obtain a symplectic structure that makes sense at all points along the preimage of the moment map. More details on the statements in this example can be found in [7, 19]. We conclude this section with a brief discussion of the local structure of shifted symplectic derived stacks. Recall that In classical symplectic geometry the local structure of a symplectic manifold is described by the Darboux-Weinstein theorem: a symplectic structure is locally (in the C ∞ or analytic setting) or formally (in the algebraic setting) isomorphic to the standard symplectic structure on a cotangent bundle.

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By analogy one might expect that in the derived stacky context shifted symplectic structures will be modeled locally by the standard symplectic structures on shifted cotangent bundles. This expectation is too naive. Exercise 8.14. – Exhibit a derived critical locus (defined in Example 8.13 (1)) whose derived structure is not locally formal. However, as we already noted in Example 8.13, the shifted cotangent bundles are symplectically derived critical loci of shifted constant functions. It turns out that the derived critical loci of arbitrary shifted functions have enough flexibility to provide local models. This leads to a remarkable shifted version of the Darboux theorem due to Brav-Bussi-Joyce: Theorem 8.15 ([6]). – Let X be a derived scheme, and let ω be an n-shifted symplectic structure on
X, with n < 0. Then Zariski locally (X, ω) is isomorphic to Rcrit(w), ωRcrit(w) for some shifted function w : M → A1 [n + 1] on a derived scheme M . Remark 8.16. – A more general result holds for locally finitely presented derived algebraic stacks (see [5]). Acknowledgements. – We are very grateful to the French Mathematical Society and the Labex CIMI at the University Paul Sabatier (Toulouse) for organizing the DAGIT session of États de la Recherche and for encouraging us to write up our introductory lectures. We would like to thank our friends and co-authors D. Calaque, B. Toën, and M. Vaquié for the interesting mathematics we did together. We also thank M. Porta for his detailed and thoughtful comments on the first version of this manuscript. Tony Pantev was partially supported by NSF research grant DMS-1601438 and by grant # 347070 from the Simons Foundation. Gabriele Vezzosi is a member of the GNSAGA-INDAM group (Italy).

References [1] M. Alexandrov, A. Schwarz, O. Zaboronsky & M. Kontsevich – “The geometry of the master equation and topological quantum field theory”, Internat. J. Modern Phys. A 12 (1997), p. 1405–1429. [2] M. Artin – “Versal deformations and algebraic stacks”, Invent. math. 27 (1974), p. 165– 189. [3] Authors of The Stacks Project – “Stacks project”, online book project available at http://stacks.math.columbia.edu, 2008–today. [4] K. Behrend – “Donaldson-Thomas type invariants via microlocal geometry”, Ann. of Math. 170 (2009), p. 1307–1338.

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[5] O. Ben-Bassat, C. Brav, V. Bussi & D. Joyce – “A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications”, Geom. Topol. 19 (2015), p. 1287–1359. [6] C. Brav, V. Bussi & D. Joyce – “A Darboux theorem for derived schemes with shifted symplectic structure”, J. Amer. Math. Soc. 32 (2019), p. 399–443. [7] D. Calaque – “Lagrangian structures on mapping stacks and semi-classical TFTs”, in Stacks and categories in geometry, topology, and algebra, Contemp. Math., vol. 643, Amer. Math. Soc., 2015, p. 1–23. [8] , “Shifted cotangent stacks are shifted symplectic”, Ann. Fac. Sci. Toulouse Math. 28 (2019), p. 67–90. [9] I. Ciocan-Fontanine & M. Kapranov – “Derived Quot schemes”, Ann. Sci. École Norm. Sup. 34 (2001), p. 403–440. [10] A. Grothendieck – “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III”, Inst. Hautes Études Sci. Publ. Math. 28 (1966), p. 255. [11] R. Hartshorne – “On the De Rham cohomology of algebraic varieties”, Inst. Hautes Études Sci. Publ. Math. 45 (1975), p. 5–99. [12] , Deformation theory, Graduate Texts in Math., vol. 257, Springer, 2010. [13] M. Hovey – Model categories, Mathematical Surveys and Monographs, vol. 63, Amer. Math. Soc., 1999. [14] L. Illusie – Complexe cotangent et déformations. II, Lecture Notes in Math., vol. 283, Springer, 1972. [15] J. Lurie – “Higher algebra”, book in progress, available at http://www.math.harvard. edu/~lurie/papers/HA.pdf, 2017. , “Spectral algebraic geometry”, book in progress, available at http://www.math. [16] harvard.edu/~lurie/papers/SAG-rootfile.pdf, 2018. [17] T. Pantev, B. Toën, M. Vaquié & G. Vezzosi – “Shifted symplectic structures”, Publ. Math. Inst. Hautes Études Sci. 117 (2013), p. 271–328. [18] M. Porta & G. Vezzosi – “Infinitesimal and square-zero extensions of simplicial algebras”, preprint arXiv:1310.3573. [19] P. Safronov – “Quasi-Hamiltonian reduction via classical Chern-Simons theory”, Adv. Math. 287 (2016), p. 733–773. [20] B. Toën – “Descente fidèlement plate pour les n-champs d’Artin”, Compos. Math. 147 (2011), p. 1382–1412. [21] , “Derived algebraic geometry”, EMS Surv. Math. Sci. 1 (2014), p. 153–240. [22] B. Toën & G. Vezzosi – “From HAG to DAG: derived moduli stacks”, in Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., 2004, p. 173–216. [23] , Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc., vol. 193, 2008.

Tony Pantev, Department of Mathematics, University of Pennsylvania, USA Gabriele Vezzosi, DIMAI, Università di Firenze, Italy

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Panoramas & Synthèses 55, 2021, p. 39–84

NONLINEAR TRACES by David Ben-Zvi & David Nadler

Abstract. – We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. We observe that it is advantageous to calculate dimensions, traces and their functoriality directly in the nonlinear geometric setting of correspondence categories, where they are directly identified with (derived versions of) loop spaces, fixed point loci and loop maps, respectively. This results in universal nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz trace formulas, and Frobenius-Weyl character formulas. We can then linearize by applying sheaf theories, such as the theories of ind-coherent sheaves and D-modules constructed by Gaitsgory-Rozenblyum [16]. This recovers the familiar classical identities, in families and without any smoothness or transversality assumptions. On the other hand, the formalism also applies to higher categorical settings not captured within a linear framework, such as characters of group actions on categories. Résumé (Traces non-linéaires). – Nous combinons la théorie des traces en algèbre homotopique avec la théorie des faisceaux en géométrie algébrique dérivée afin déduire des formules de caractéres et de points fixes. La notion de dimension (ou homologie de Hochschild) d’un objet dualisable dans le contexte de l’algèbre supérieure fournit un cadre unificateur pour les notions classiques telles que les caractéristiques d’Euler, les caractères de Chern et les caractères des représentations de groupes. De plus, la fonctorialité de ces dimensions clarifient certaines formules célèbres et les étendent à de nouveaux contextes. Nous observons qu’il est avantageux de calculer les dimensions, les traces et leurs fonctorialités directement dans le cadre géométrique non-linéaire des catégories de correspondances, où elles sont respectivement identifiés directement avec (les versions dérivées des) les espaces de lacets, les lieux de points fixes et leurs fonctorialités. Il en 2010 Mathematics Subject Classification. – 14C40, 13D03. Key words and phrases. – Traces, Hochschild homology, loop spaces, Grothendieck-Riemann-Roch theorem.

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résulte des versions universelles non-linéaires des formules de traces de GrothendieckRiemann-Roch, de la formule de Atiyah-Bott-Lefschetz et des formules de caractères de Frobenius-Weyl. Il est par ailleurs possible de linéariser en appliquant certaines théories de faisceaux, telle que la théorie des faisceaux ind-cohérents et des D-modules construites par Gaitsgory-Rozenblyum [GR2]. Cela permet de retrouver les formules classiques, valables en familles et sans hypothèse de lissité ou de transversalité. D’un autre côté, notre formalisme s’applique également à certains invariants catégoriques supérieurs non présents dans le cadre linéaire, tels que la notion de caractères des actions de groupes sur des catégories.

1. Introduction This paper is devoted to traces and characters in homotopical algebra and their application to algebraic geometry and representation theory. We observe that many geometric fixed point and trace formulas can be expressed as linearizations of fundamental nonlinear identities, describing dimensions and traces directly in the setting of correspondence categories of varieties or stacks. This gives a simple uniform perspective on (and useful generalizations of) geometric character and fixed point formulas of Grothendieck-Riemann-Roch and Atiyah-Bott-Lefschetz type. In addition, one can also specialize the universal geometric formulas to higher categorical settings not captured within a linear framework, such as characters of group actions on categories. The paper is organized as follows: after a brief summary in Section 1.1, we give a detailed overview in Section 2, in three sections: first, the abstract functoriality of traces in higher category theory; second, their calculation in correspondence categories in derived algebraic geometry; and third, their specialization via sheaf theories. The rest of the paper follows the same structure with more details provided. We emphasize the formal nature and appealing simplicity of the constructions in any sufficiently derived setting. For example, in the second part, we work within derived algebraic geometry, but the statements and proofs should hold in any setting (for example, derived manifolds) with a suitable notion of fiber product to handle non-transversal intersections. The main objects appearing in trace formulas are the derived loop space (the selfintersection of the diagonal in its role as the nonlinear trace of the identity map) and more general derived fixed point loci. The importance of a derived setting also appears prominently in the third part, where the sheaf theories we apply must have good functorial properties with respect to fiber products. As a result, the theory of characters in Hochschild and cyclic homology is expressed directly by the geometry, resulting in simpler formulations. For example, the Todd genus in Grothendieck-Riemann-Roch and the denominators in the classical Atiyah-Bott formula arise naturally from derived calculations.

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1.1. Summary. – We now describe the main theorem, extending classical trace and dimension formulas to a very general setting in derived algebraic geometry (including equivariance for arbitrary Lie algebroids or affine algebraic groups) without any smoothness or transversality assumptions, while emphasizing that the main contribution of the paper is the simple geometric formalism underlying these formulas. For our general and formal nonlinear results, we need not assume anything about what classes of derived stacks and morphisms we work with. For applications, we need to be in a setting in which the powerful mechanism of sheaf theory is fully developed [13, 9, 16]. Setting 1.1. – Throughout this paper, we work over a field k of characteristic zero. A stack X connotes either (1) a QCA derived stack in the sense of [9]. In other words X is quasicompact with affine diagonal and the underived inertia of X is finite presentation over the underlying underived stack Xcl . (2) an ind-inf-scheme, in the sense of [16]. These include (derived) schemes of finite (or locally almost of finite) type and ind-schemes built out of unions of the former along closed embeddings, as well as their quotients by arbitrary Lie algebroids, or equivalently formal groupoids. By a proper map we indicate a proper and schematic map, while for ind-proper indicates ind-proper and ind-inf-schematic. All appearances of proper maps in this paper may be replaced by ind-proper ones. Thus the class of spaces we consider includes all k-derived schemes of finite type and their quotients by either Lie algebroids or finite type affine group schemes. Given a derived stack πX : X → Spec k, we denote by π L X : L X = Map(S 1 , X) → Spec k its derived loop space. In general, the derived loop space is a derived thickening of the inertia stack. For a map f : X → Y , we will denote by L f : L X → L Y the induced map on loops. Example 1.2. – For many applications, the following two special cases are noteworthy. When X is a smooth scheme, L X ' TX [−1] is the total space of the shifted tangent space by the HKR theorem. The same holds for an arbitrary scheme, if we replace the tangent space by the tangent complex, see for example [3]. For f : X → Y a map of schemes, L f : TX [−1] → TY [−1] is (the shift of) the usual tangent map. When Y = BG is a classifying stack, L Y ' G/G is the adjoint quotient. For X a G-scheme, and f : X/G → BG the corresponding classifying map, L f : L (X/G) → L BG ' G/G is the universal family of derived fixed point loci. More precisely, for any element g ∈ G, the derived fixed point locus X g ⊂ X is precisely the derived fiber X g ' L (X/G) ×G/G {g} We will measure stacks X by differential graded (dg) enhancements of derived categories of sheaves. The most familiar is the assignment X 7→ Q(X) of the (unbounded) category of quasicoherent sheaves. However, we will make essential use of Grothendieck-Serre

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duality, in the guise of an adjunction (f∗ , f ! ) between push-forward and extraordinary pullback for proper maps X. This duality is most naturally expressed in the setting of ind-coherent sheaves X 7→ Q! (X) as developed in [13, 16]. Ind-coherent sheaves agree with quasicoherent sheaves for smooth schemes but differ on singular schemes, where [bounded complexes of] coherent sheaves (the compact objects of Q! ) differ from perfect complexes (the compact objects of Q). In other words, ind-coherent sheaves are to coherent sheaves and G-theory (the setting of Grothendieck-Riemann-Roch theorems) as quasicoherent sheaves are to perfect complexes and K-theory. Another sheaf theory to which the general formalism developed in [16] applies is the theory of D-modules X → D(X), which for smooth schemes agrees with the classical notion of quasicoherent complexes of modules for the sheaf of differential operators DX (i.e., with compact objects given by bounded coherent complexes of DX -modules). In general [15, 16] D(X) is defined as the category of crystals, i.e., as ind-coherent complexes on the de Rham space of X D(X)

:= Q! (XdR ).

The compact objects in D(X) for X a scheme are the coherent D-modules, while for X a stack they form a smaller class, the safe D-modules of [9]. The book [16] develops the theories Q! and D in particular as functors out of 2categories of correspondences of schemes and stacks, with 1-morphisms from X to Y given by correspondences representable over Y and 2-morphisms given by ind-proper ind-schematic morphisms of correspondences. This theory encodes a huge amount of structure, including in particular pullback and pushforward functors f ! and f∗ satisfying base change, as well as the (f∗ , f ! ) adjunction for proper (or even ind-proper) maps. They also establish symmetric monoidal properties of the sheaf theory. (1) (See Sections 2.3 and 5 for more background and precise statements.) Let S = Q! or S = D denote either of these sheaf theories. ! We let ωX = πX O Spec k ∈ S (X) denote the appropriate dualizing sheaf. Thus for ind-coherent sheaves, ωX ∈ Q! (X) is the algebraic dualizing sheaf, and for D-modules, ωX ∈ D(X) is the Verdier dualizing sheaf. Let ω(X) = πX∗ ωX denote the corresponding complex of global volume forms: for ind-coherent sheaves, ω(X) ∈ k-mod consists of algebraic volume forms, and for D-modules, ω(X) ∈ k-mod consists of locally constant distributions (Borel-Moore chains) for X a scheme. For X a stack we use the continuous “renormalized” pushforward functor on D-modules of [9], which roughly replaces equivariant cohomology (derived invariants) by a shift of equivariant homology (derived coinvariants, see [9, Example 9.1.6]), so that ω(Y /G) for a G-variety Y is given by a shift of G-coinvariants on Borel-Moore chains on Y . (Note that even for X = BG this differs from the standard (1)

We will also need from [9] the construction of continuous pushforward functors for all maps of QCA stacks (such as the non-representable projection πX : X → pt from a stack) and their base-change property.

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definition of equivariant Borel-Moore homology, which is identified in this case with equivariant cohomology.) ForR a proper (or ind-proper) map f : X → Y , adjunction provides an integration map f : ω(X) → ω(Y ). Example 1.3. – Let us continue with the special cases of Example 1.2, and focus in particular on algebraic distributions ω L X ∈ Q! ( L X) on the loop space. When X is a smooth scheme, L X ' TX [−1] is naturally Calabi-Yau, and its global volume forms are identified with differential forms ω(TX [−1]) ' O (TX [−1]) ' Sym• (ΩX [1]). The canonical “volume form” on L X is given by the Todd genus (as explained by Markarian [29]): the resulting integration of functions on L X differs from the integration of differential forms on X by the Todd genus. When Y = BG is a classifying stack, L Y ' G/G is naturally Calabi-Yau, and its global volume forms are invariant functions ω(G/G) ' O (G/G) ' O (G)G . If G is reductive with Cartan subgroup T ⊂ G and Weyl group W , the naive invariants O (G)G ' O (T )W are equivalent to the derived invariants, but in general there may be higher cohomology. Theorem 1.4. – Let S = Q! or S = D denote either the theory of ind-coherent sheaves or D-modules. Recall our conventions for stacks and morphisms, Setting 1.1. • For a stack X, there is a canonical identification HH∗ ( S (X)) ' ω( L X) of the Hochschild homology of sheaves on X with distributions (or renormalized Borel-Moore chains) on the loop space. • For a proper (or ind-proper) map of stacks f : X → Y the induced map HH∗ ( S (X)) → HH∗ ( S (Y )) is given by integration along the loop map L f : L X → LY . • Grothendieck-Riemann-Roch. In particular, for any compact object M ∈ S (X) (coherent sheaf or safe coherent D-module) with character [M ] ∈ HH∗ ( S (X)) ' ω( L X), there is a canonical identification Z [f∗ M ] ' [M ] ∈ HH∗ ( S (Y )) ' ω( L Y ). Lf

In other words, the character of a pushforward along a proper map is the integral of the character along the induced loop map. • Atiyah-Bott-Lefschetz. Let G be an affine group, and X a proper stack with G-action, or equivalently, a proper map f : X/G → BG. Then for any compact object M ∈ S (X/G) (G-equivariant coherent sheaf or safely equivariant coherent D-module on X), and element g ∈ G, there is a canonical identification Z [f∗ M ]| ' [M ]| g . g

Lf

X

In other words, under the identification of invariant functions and volume forms on the group, the value of the character of an induced representation at a group element

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is given by the integral of the original character along the corresponding fixed point locus of the group element. • Extension to traces. The trace Tr( S (Z)) of the endofunctor of S (X) given by a self-correspondence (e.g., a self-map) X ← Z → X is given by distributions on the fixed points ω(Z | ). ∆

• For a map f : (X, Z) → (Y, W ) of stacks with self-correspondences (2), the induced map Tr( S (Z)) → Tr( S (W )) is given by integration along fixed points Z | → W ∆Y . ∆X

Example 1.5 (Frobenius-Weyl Character Formula). – Here is a reminder of a wellknown application of the Atiyah-Bott-Lefschetz formula in representation theory. • If G is a finite group, and X = G/K is a homogeneous set, and M = k[G/K] the ring of functions, one recovers the Frobenius character formula for the induced representation k[G/K]. • If G is a reductive group, X = G/B is the flag variety, X/G = pt/B → pt/G. The loop map e L (X/G) = B/B ' G/G → G/G is the (group) Grothendieck-Springer simultaneous resolution, with fibers giving fixed point loci on the flag variety. For M = L an equivariant line bundle on G/B, and g ∈ G runs over a maximal torus, one recovers the Weyl character formula for the induced representation H ∗ (G/B, L ). Remark 1.6. – The reader will note no explicit appearance of the Todd genus in the above formulas. In other words as for K-theory, pushforward of sheaves naturally agrees with the pushforward in Hochschild homology. The Todd genus arises in comparing these natural pushforwards with the pushforward in cohomology, i.e., integraR tion of forms. It arises when one unwinds the integration map L f : ω( L X) → ω( L Y ), given by Grothendieck duality, in terms of functions (or differential forms) using the Hochschild-Kostant-Rosenberg theorem. In particular, the familiar denominators in the Atiyah-Bott formula are implicit in the integration measure on the fixed point locus. For instance, as mentioned above, when X is a smooth scheme, a geometric version of the HKR theorem asserts that the loop space is the total space of the shifted tangent complex L X ' TX [−1], and global volume forms are canonically functions ω( L X) ' O (TX [−1]) ' Sym• (ΩX [1]). Under this identification (as explained by Markarian [29]), the resulting integration of functions on L X differs from the integration of differential forms on X by the Todd genus. Remark 1.7. – The paper [22] carries out the program described in this paper (i.e., recovering classical identities from non-linear ones) by calculating explicitly the derived contributions in the case of the Atiyah-Bott formula. (2)

I.e., we lift f to an identification Z ' X ×Y W of correspondences from X to Y .

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45

Remark 1.8. – The main contribution of this paper is hidden in the statement of this theorem: we establish nonlinear versions of character formulas in the setting of derived stacks, and deduce classical formulas and new higher categorical analogues formally by applying suitable sheaf theories. Thanks to the great generality of sheaf theory in derived algebraic geometry [16], the resulting applications hold with remarkably few assumptions. We are particularly interested in the higher categorical variants where one considers sheaves of categories, in particular Frobenius-Weyl character formulas for group actions on categories. Since the requisite foundations are not yet fully developed, we postpone details of this to future works. Applications include an identification of the character of the category of D-modules on the flag variety with the GrothendieckSpringer sheaf, and of the trace of a Hecke functor on the category of D-modules on the moduli of bundles on a curve with the cohomology of a Hitchin space. 1.2. Inspirations and motivations. – This work has many inspirations. It is heavily reliant on the ∞-categorical foundations of higher algebra, derived algebraic geometry and sheaf theory due to Lurie [27, 28] and Gaitsgory-Rozenblyum [16]. It is also inspired by Lurie’s cobordism hypothesis with singularities [26], which provides a powerful unifying tool for higher algebra. Already in the setting of one-dimensional field theory, this result can be viewed as a vast generalization of the classical theory Hochschild and cyclic homology and characters therein [23], (in particular the natural cyclic symmetry of Hochschild homology is generalized to a circle action on the dimensions of arbitrary dualizable objects). In particular, the formal properties of traces we use are simple instances of the cobordism hypothesis with singularities on marked intervals and cylinders. The work of Toën and Vezzosi [40] on traces and higher Chern characters of sheaves of categories (and in particular the role of the cobordism hypothesis therein) has also profoundly influenced our thinking. Another important inspiration is the categorical theory of strong duality, dimensions and traces introduced by Dold and Puppe in [8] (see [30, 33] for more recent developments) with the express purpose of proving Lefschetz-type formulas. In [8], dualizability of a space is achieved by linearization (passing to suspension spectra), while our approach is to pass to categories of correspondences (or spans) instead. We were also inspired by the preprint [29] and the subsequent work [7, 6, 35, 36, 37]. There have been many recent papers [31, 24, 32, 5] building on related ideas to prove Riemann-Roch and Lefschetz-type theorems in the noncommutative context of differential graded categories and Fourier-Mukai transforms; our work instead places these results in the context of the general formalism of traces in ∞-categories, and generalizes them to commutative but nonlinear settings. The Grothendieck-Riemann-Roch type applications in this paper concern the character map taking coherent sheaves to classes in Hochschild homology (or in a more refined version, to cyclic homology). This is significantly coarser than the well established theory of Lefschetz-Riemann-Roch theorems valued in Chow groups (see the seminal [38], the more recent [20] and many references therein). Thus for schemes,

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the quantities compared are Dolbeault (or de Rham) cohomology classes rather than algebraic cycles or K-(or rather G-)theory classes. Our primary motivation is the development of foundations for “homotopical harmonic analysis” of group actions on categories, aimed at decomposing derived categories of sheaves (rather than classical function spaces) under the actions of natural operators. This undertaking follows the groundbreaking path of Beilinson-Drinfeld within the geometric Langlands program and is consonant with general themes in geometric representation theory. The pursuit of a geometric analogue of the ArthurSelberg trace formula by Frenkel and Ngô [12] has also been a source of inspiration and applications. Remark 1.9. – A companion paper [4] presents an alternative approach to AtiyahBott-Lefschetz formulas (and in particular a conjecture of Frenkel-Ngô) as a special case of the “secondary trace formula” identifying trace invariants associated to two commuting endomorphisms of a sufficiently dualizable object. This is also applied to establish the symmetry of the 2-class functions on a group constructed as the 2-characters of categorical representations. 1.3. Acknowledgements. – We would like to thank Dennis Gaitsgory and Jacob Lurie for providing both the foundations and the inspiration for this work, as well as helpful comments and specifically D.G. for discussions of [16]. We would also like to thank Toly Preygel for many discussions about derived algebraic geometry. We gratefully acknowledge the support of NSF grants DMS-1103525 (D.BZ.) and DMS-1319287 (D.N.).

2. Overview 2.1. Traces in category theory. – We highlight structures arising in the general theory of dualizable objects in symmetric monoidal higher categories (see also [8, 30, 33]). For legibility, we suppress all ∞-categorical notations and complications from the introduction. We rely on [16] for the theory of symmetric monoidal (∞, 2)-categories, though only the formal outline of the theory is in fact needed for this paper. See [40, 19] for thorough treatments of the theory of traces in higher category theory. The basic notion in the theory is that of dimension of a dualizable object of a symmetric monoidal category A . By definition, for such an object A there exists another A∨ together with a coevaluation map ηA and evaluation map A satisfying standard identities. By definition, the dimension of A is the endomorphism of the the unit 1 A given by the composition 1A

ηA

/ A ⊗ A∨ dim(A)

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A

/ 7 1A

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Example 2.1. – For V a vector space, V ∨ = Homk (V, k) is the vector space of functionals, V : V ⊗ V ∨ → k is the usual evaluation of functionals, ηV : k → End(V ) ' V ⊗ V ∨ is the identity map (which exists only for V finite-dimensional), and dim(V ) can be regarded as an element of the ground field (by evaluating it on the multiplicative unit). Remark 2.2 (Duality and naivëté in ∞-categories). – It is a useful technical observation that the notion of dualizability in the setting of ∞-categories is a “naive” one: it is a property of an object that can be checked in the underlying homotopy category. As a result, all of the categorical and 2-categorical calculations in this paper are similarly naive and explicit (and analogous to familiar unenriched categorical assertions), involving only small amounts of data that can be checked by hand. The notion of dimension is a special case of the trace of an endomorphism Φ of a dualizable object A. By definition, the trace of Φ is the endomorphism of the unit 1 A given by the composition 1A

ηA

/ A ⊗ A∨

Φ⊗idA∨

/ A ⊗ A∨

A

/6 1 A

Tr(Φ)

which recovers the dimension for Φ = idA . A key feature of dimensions and traces is their cyclicity, which at the coarsest level is expressed by a canonical equivalence m(Φ, Ψ) : Tr(Φ ◦ Ψ)

∼

/ Tr(Ψ ◦ Φ),

see Proposition 3.15. At a much deeper level, an important corollary of the cobordism hypothesis [26] is the existence of an S 1 -action on dim(A) for any dualizable object A (and an analogous structure for general traces, see Remark 3.28). Remark 2.3 (Dimensions and traces are local). – It is useful for applications to note that the notion of dualizability and the definition of dimension and are local in the category A . Namely, they only require knowledge of the objects 1 A , A, A∨ , A ⊗ A∨ , the morphisms ηA , A , and standard tensor product and composition identities among them. Likewise, the notion of trace only requires the additional endomorphism Φ along with a handful of additional identities. 2.1.1. Functoriality of traces. – Now suppose the ambient symmetric monoidal category A underlies a 2-category, so there is the possibility of noninvertible 2-morphisms. This allows for the notion of left and right adjoints to morphisms. Let us say a morphism A → B is continuous, or right dualizable, if it has a right adjoint. (The terminology derives from the setting of presentable categories, where the adjoint functor theorem guarantees the existence of right adjoints for colimit preserving functors.) Here are natural functoriality properties of dimensions and traces. Proposition 2.4. – Let A, B denote dualizable objects of A and f∗ : A → B a continuous morphism with right adjoint f ! .

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(1) There is a canonical map on dimensions dim(A)

=

/ Tr(IdA )

/ Tr(f ! f∗ )

/ Tr(f∗ f ! )

∼

/ Tr(IdB )

=

/ dim(B) 3

dim(f∗ )

compatible with compositions of continuous morphisms. (2) Given endomorphisms Φ ∈ End(A), Ψ ∈ End(B), and a commuting structure α : f∗ ◦ Φ

∼

/ Ψ ◦ f∗

there is a canonical map on traces / Tr(Ψ)

Tr(f∗ , α) : Tr(Φ)

compatible with compositions of continuous morphisms with commuting structures. We refer to the compatibility with compositions stated in the proposition as abstract Grothendieck-Riemann-Roch. To see its import more concretely, let us restrict the generality and focus on an object of A in the sense of a morphism V : 1 A → A. Corollary 2.5. – Let A, B denote dualizable objects of A and f∗ : A → B a continuous morphism. For V : 1 A → A an object of A, we obtain a map on dimensions dim(V ) : 1 A ' dim(1 A )

/ dim(A)

called the character of V and alternatively denoted by [V ]. It satisfies abstract Grothendieck-Riemann-Roch in the sense that the following diagram commutes 1A

[V ]

/ dim(A)

dim(f∗ )

/ dim(B). 6

[f∗ V ]

Remark 2.6 (Functoriality of dimensions and traces is local). – As in Remark 2.3, it is useful to note that the functoriality of dimension is local, depending only on a handful of objects, morphisms and identities, along with the additional adjunction data (f∗ , f ! ). A similar observation applies to the functoriality of traces. Remark 2.7. – It follows from the cobordism hypothesis with singularities [26] (see [40]) that the morphism dim(f∗ ) is S 1 -equivariant, and hence the character [V ] is S 1 -invariant, though we will not elaborate on this structure here. We refer to [19] for a thorough study of the functoriality and cyclicity of traces. Example 2.8. – Let dgCatk denote the symmetric monoidal ∞-category of presentable k-linear differential graded categories (or alternatively, stable presentable k-linear ∞-categories), see e.g., [16]. In this setting, any compactly generated category A is dualizable, and its dimension is the Hochschild chain complex dim(A) = HH∗ (A). The S 1 -action on dim(A) corresponds to Connes’ cyclic structure on HH∗ (A), so that in particular, the localized S 1 -invariants of dim(A) form the periodic cyclic homology of A.

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More generally, the trace of an endofunctor Φ : A → A is the Hochschild homology Tr(Φ) = HH∗ (A, Φ). For example, if A = R-mod for a dg algebra R, then Φ is represented by an R-bimodule M , and we recover the Hochschild homology HH∗ (R, M ). Any compact object M ∈ A defines a continuous functor /A

M

1dgCatk = dgVectk whose character is a vector

dim(M ) ∈ HH∗ (A) in Hochschild homology (with refinement in cyclic homology). The abstract Grothendieck-Riemann-Roch theorem expresses the natural functoriality of characters in Hochschild homology (or their refinement in cyclic homology). In fact, the construction of characters factors through the canonical Dennis trace map / K(A)

Acpt

/ HH∗ (A)

from the space Acpt of compact objects of A. 2.2. Traces in geometry. – To apply the preceding formalism to geometry, it is useful to organize spaces and maps within a suitable categorical framework. We then arrive at loop spaces and fixed point loci as nonlinear expressions of dimensions and traces. This simple observation provides the core of the paper. Throughout the discussion, we continue to suppress all ∞-categorical notations and complications. Our reference for the correspondence 2-category of stacks is Section V of [16] (see also [18]). To begin, consider the general setup of the symmetric monoidal category Corr( C ) of correspondences, or spans, in a category C such as stacks (or formally a symmetric monoidal ∞-category with finite limits; see [1, 18]). Here the objects X ∈ Corr( C ) are the objects of C , the morphisms Corr C (X, Y ) are arbitrary spans in C , Z

X

~ 

Y

(more generally one can require the left and right legs to live in specified subcategories of C as in [16]). The composition of morphisms Z ∈ Corr C (X, Y ) and W ∈ Corr C (Y, U ) is given by fiber product Z ×Y W

Z

X

~

$ z $

Y

z

W

U

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and the symmetric monoidal structure is given in terms of that on C (Cartesian product in the case of stacks). For the purpose of calculating dimensions and traces, we need not require any further properties of the spaces of Corr( C ), since we need only the modest local data discussed in Remarks 2.3 and 2.6. (See [26] and [11], where the higher categories Famn of iterated correspondences of manifolds are constructed and applied.) With applications in mind, we will specialize to the correspondence category Corrk = Corr(Stk ) of derived stacks over k. It would also be interesting to work with smooth manifolds instead, for example through the theory of C ∞ -stacks [21] (see Remark 2.20). It is natural to enhance Corr( C ) to a 2-category Corr( C ) by allowing non-invertible maps, or more generally correspondences, between correspondences (see Section V of [16] or [18] for full details), so that maps from X to Y form the category of objects over X × Y . Our constructions naturally fit into the 2-category Corrprop with nonk invertible 2-morphisms restricted to be proper (or more generally ind-proper) maps of correspondences. Remark 2.9 (Correspondences are bimodules). – It is useful to view the correspondence category Corr within the framework of coalgebras in symmetric monoidal categories. The diagonal map X → X × X makes any space or stack into a cocommutative coalgebra object with respect to the Cartesian product monoidal structure (or commutative coalgebra in the opposite category). Moreover, a map Z → X is equivalent to an X-comodule structure on Z. Thus correspondences from X to Y may be interpreted as X-Y -bicomodules, with composition of correspondences given by tensor product of bicomodules. Furthermore, it is natural to enhance Corr to a 2-category by allowing noninvertible maps between correspondences. This can be viewed as a special case of the Morita category of coalgebras in a symmetric monoidal category. The 2-category Corr of spaces, correspondences, and maps of correspondences is the Morita category on spaces regarded as coalgebra objects. (In particular, the cocommutativity of the coalgebra objects implies they are canonically self-dual, and the transpose of a correspondence is the same correspondence read backwards.) If we further keep track of the En -coalgebra structure of spaces and consider the corresponding Morita (n + 1)-category, we recover the (n + 1)-category of iterated correspondences of correspondences. (See [18, 17] for a thorough treatment of categories of spans and Morita categories of En algebras; see also for example the category Famn of [26] and [11] in the topological setting.) 2.2.1. Geometric dimensions and loop spaces. – A crucial feature of the category Corrk is that any object X ∈ Corrk is dualizable (in fact, canonically self-dual),

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thanks to the diagonal correspondence. (3) Note for this it is crucial that we allow all maps, including the map πX : X → pt for any X, as possible legs in a span. We have the following calculations of dimensions and their functoriality. Note that the point pt = Spec k is the unit of Corrk . We keep track of properness of maps of correspondences for the later application of sheaf theory. Proposition 2.10. – Let Corrk be the category of derived stacks and correspondences, and Corrprop the 2-category of derived stacks, correspondences, and proper maps of k correspondences. (1) Any derived stack X is dualizable as an object of Corrk , and its dimension dim(X) is identified with the loop space LX

1

= X S ' X ×X×X X

regarded as a self-correspondence of pt = Spec k. (2) A map f : X → Y regarded as a correspondence from X to Y is continuous if and only if f is proper. Given a proper map f : X → Y , its induced in Corrprop k map / dim(Y ) dim(f ) : dim(X) is identified with the loop map Lf

: LX

/ L Y.

Remark 2.11. – All of the objects and maps of the proposition have natural S 1 -actions, on the one hand coming from loop rotation, on the other hand coming from the cyclic symmetry of dimensions. One can check that the identifications of the proposition are S 1 -equivariant (see Remark 4.2). Remark 2.12. – Recall [39, 3] that for a derived scheme X, the loop space L X ' TX [−1] is the total space of the shifted tangent complex. The action map of the S 1 -rotation action is encoded by the de Rham differential. For an underived stack X, the loop space is a derived enhancement of the inertia stack IX = {x ∈ X, γ ∈ Aut(x)}. The action map of the S 1 -rotation action is manifested by the “universal automorphism” of any sheaf on L X. Example 2.13. – Let G denote an algebraic group and BG = pt/G its classifying space. There is a canonical identification L BG ' G/G of the loop space and adjoint quotient. Suppose we are given a G-derived stack X, or equivalently a morphism π : X/G → BG, from which one recovers X ' X/G ×BG pt. (Note that if we want π proper we should take X itself proper.) (3) Likewise, if we wish to make a space n-dualizable for any n we may simply consider it as an object of a higher correspondence category as in Remark 2.9, since En -(co)algebras are n + 1-dualizable objects of the corresponding Morita category. In other words, a space X defines a topological field theory of any dimension valued in the appropriate correspondence category.

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Let us explain how the loop map L π : L (X/G) → L (BG) captures the fixed points of G acting on X. For any self-map g : X → X, let us write X g for the derived fixed point locus given by the derived intersection X g = Γg ×X×X X of the graph Γg ⊂ X × X with the diagonal. Then L π map fits into a commutative square

∼

/ L (BG)

Lπ

L (X/G)

∼



{g ∈ G, x ∈ X g }/G

p

 / G/G,

where p projects to the group element. In particular, fix a group element g ∈ G, with conjugacy class Og ⊂ G, and centralizer Z G (g) ⊂ G, so that Og /G ' B Z G (g) ∈ G/G. Then the corresponding g fiber of L π is the equivariant fixed point locus XG = X g / Z G (g), or in other words we have a fiber diagram / Og /G

g XG

 L (X/G)

Lπ

 / L (BG).

Let us specialize to the case of a subgroup K ⊂ G, and the quotient X = G/K, so that we have a map of classifying stacks π : BK ' G\(G/K) → BG. Here the loop map L π realizes the familiar geometry of the Frobenius character formula L (BK) ∼

/ L (BG)

Lπ

∼



K/K ' {g ∈ G, x ∈ (G/K)g }/G

p

 / G/G.

The equivariant fixed point loci express the equivariant inclusion of conjugacy classes. Specializing further, for G a reductive group, B ⊂ G a Borel subgroup, and X = G/B the flag variety, we recover the group-theoretic Grothendieck-Springer resolution L (BB)

/ L (BG)

Lπ

∼

 B/B ' {g ∈ G, x ∈ (G/B)g }/G

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∼

p

 / G/G.

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2.2.2. Geometric traces of correspondences. – More generally, we have the following calculations of traces and their functoriality. Proposition 2.14. – Let Corrk be the category of derived stacks and correspondences, and Corrprop the 2-category of derived stacks, correspondences, and proper maps of k correspondences. (1) The trace of a self-correspondence Z ∈ Corrprop (X, X) is its fiber product with k the diagonal Tr(Z) ' Z | = Z ×X×X X ' Z ×X L X. ∆

In particular, for the graph Γf → X × X of a self-map f : X → X, its trace is the fixed point locus of the map Tr(Γg ) ' X f = Γf ×X×X X. (2) Given a proper map f : X → Y regarded as a correspondence from X to Y , and self-correspondences Z ∈ Corrprop (X, X) and W ∈ Corrprop (Y, Y ), together with k k an identification ∼ / α:Z X ×Y W of correspondences from X to Y , the induced abstract trace map Tr(f, α) : Tr(Z)

/ Tr(W )

is equivalent to the induced geometric map τ (f, α) : Z |

∆X

/ W| . ∆Y

2.3. Trace formulas via sheaf theories. – Given any sufficiently functorial method of measuring derived stacks, the preceding calculations of geometric dimensions, traces and their functoriality immediately lead to trace and character formulas. To formalize the functoriality needed, we will use the language of sheaf theories. Broadly speaking, a sheaf theory is a representation (symmetric monoidal functor out) of a correspondence category in the way a topological field theory is a representation of a cobordism category. (4) It provides an approach to encoding the standard operations on coherent sheaves and D-modules, developed by Gaitsgory and Rozenblyum in the book [16] (following a suggestion of Lurie and previous versions in [10, 13, 9, 15]). We will take the target of our sheaf theories to be the linear setting of the symmetric monoidal (∞, 2)-category dgCatk of presentable k-linear differential graded categories with continuous functors and natural transformations. (Recall from Setting 1.1 that for applications all stacks are assumed to be QCA or ind-inf-schemes. Also all proper maps can be replaced by ind-proper ones at no cost.) (4)

Indeed a typical mechanism to construct “Lagrangian” field theories is as the composition of a sheaf theory with a “classical field theory” as in [11, 18], a symmetric monoidal functor from a cobordism category to a correspondence category.

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Definition 2.15. – A sheaf theory is a symmetric monoidal functor of (∞, 2)-categories S

: Corrprop k

/ dgCatk

from correspondences of stacks (with 2-morphisms given by proper maps of correspondences) to dg categories. We denote by S

: Corrk

/ dgCatk

the underlying 1-categorical sheaf theory, i.e., the symmetric monoidal functor on (∞, 1)-categories obtained by forgetting noninvertible morphisms. Let us first spell out some of the structure encoded in a 1-categorical sheaf theory S . The graph of a map of stacks f : X → Y provides a correspondence from X to Y and a correspondence from Y to X. We denote the respective induced maps by f∗ : S (X) → S (Y ) and f ! : S (Y ) → S (X). For π : X → pt = Spec k, we denote by ωX = π ! k ∈ S (X) the S -analogue of the dualizing sheaf, and by ω(X) = π∗ ωX ∈ S (pt) = dgVectk the S -analogue of “global volume forms”. We adopt traditional notations whenever possible, for example writing Γ(X, F ) = π∗ ( F ), for F ∈ S (X) The functoriality of S concisely encodes base change for f∗ and f ! . Its symmetric monoidal structure provides equivalences S (X

× Y ) ' S (X) ⊗ S (Y ),

as well as a symmetric monoidal structure on S (X) for any X (using pullback along diagonal maps). The 2-categorical extension S further encodes an identification of f ! with the right adjoint of f∗ for f proper. Since a sheaf theory S is symmetric monoidal, it is automatically compatible with dimensions and traces: for any X ∈ Corrk , and any endomorphism Z ∈ Corrk (X, X), we have dim( S (X)) ' S (dim(X)) Tr( S (Z)) ' S (Tr(Z)) Let us combine this with the calculation of the right hand sides and highlight specific examples of interest. Proposition 2.16. – Fix a sheaf theory S : Corrk → dgCatk . (1) The S -dimension dim( S (X)) = HH∗ ( S (X)) of any X ∈ Corrk is S 1 -equivariantly equivalent with S -global volume forms on the loop space dim( S (X)) ' ω( L X). In particular, for G an affine algebraic group, characters of S -valued G-representations are adjoint-equivariant S -global volume forms dim( S (BG)) ' ω(G/G). (2) The S -trace of any endomorphism Z ∈ Corrk (X, X) is equivalent to S -global volume forms on the restriction to the diagonal Tr( S (Z)) ' ω(Z | ). ∆

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55

In particular, the S -trace of a self-map f : X → X is equivalent to S -volume forms on the f -fixed point locus Tr(f∗ ) ' ω(X f ). Remark 2.17 (Local sheaf theory). – To apply this proposition, far less structure than a full sheaf theory is required. We only need the data of the functor S on the handful of objects and morphisms involved in the construction of dimensions and traces as in Remark 2.3. In particular, we only need base change isomorphisms for pullback and pushforward along specific diagrams, rather than the general base change provided by a functor out of Corrk . This is often easy to verify in practice, in particular for the examples Q, Q! and D (see for example [2] for the quasicoherent setting). 2.3.1. Examples of sheaf theories. – As we explain in Section 5, the work [16] (combined with essential results from [9]) construct two sheaf theories Q! and D: • Theory Q! : the theory of ind-coherent sheaves Q! (X). This is the “large” version Q (X) = Ind Coh(X) of the category of coherent sheaves, which by definition are the compact objects in Q! (X). (For smooth X, ind-coherent and quasicoherent sheaves are equivalent.) Maps are given by the standard pushforward f∗ and exceptional pullback f ! . The Q! -dualizing sheaf is the usual dualizing complex ωX , and (for X proper) the Q! -global volume forms are its sections RΓ(X, ωX ) = RΓ(X, O X )∗ . The K-theory of Q! (X) is algebraic G-theory G(X), the homological version of algebraic K-theory for potentially singular spaces suited to Grothendieck-Riemann-Roch theorems. !

• Theory D: the theory of D-modules D(X) with the standard functors f∗ and f ! . The compact objects are necessarily coherent D-modules (this suffices for X a scheme; see [9] for a characterization in the case of a stack). The D-dualizing sheaf is the Verdier dualizing complex ωX , and the D-global volume forms (for X smooth) are the Borel-Moore homology RΓ(XdR , ωX ) = HdR (X)∗ . Remark 2.18. – More precisely, [16] construct the sheaf theories Q! and D as lax symmetric monoidal functors on a much broader class of stacks, with pullbacks allowed for arbitrary maps but pushforward only for schematic morphisms. The strictness follows from results of [9], as we explain in Section 5.3.1, as does the definition of pushforwards for arbitrary maps of QCA stacks (without the functorial apparatus of [16] but sufficient for all the “local” constructions we need, in the sense of Remarks 2.3, 2.6 and 2.17). Remark 2.19 (Quasicoherent sheaves). – The theory of quasicoherent sheaves X 7→ Q(X) behaves similarly with respect to 1-categorical properties. It also defines a symmetric monoidal functor out of the (∞, 1)−category of correspondences of stacks, using standard pullback f ∗ and pushforward f∗ functors. Assuming X is perfect (in the sense of [2]), the compact objects of Q(X) form the subcategory of perfect complexes Perf (X), and we have Q(X) = Ind Perf (X). The analog of the dualizing sheaf is

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the structure sheaf O X , and the Q-“global volume forms” are the global functions RΓ(X, O X ). The K-theory of Q(X) is the usual algebraic K-theory K(X). However while we have (f ∗ , f∗ ) adjunction and f ∗ preserves perfection for arbitrary morphisms, proper pushforward does not preserve perfection and we do not have proper adjunction of the form (f∗ , f ∗ ) (unless we add smoothness and twisting by relative dualizing sheaves). In other words, Q and K-theory are better adapted to pullback, while Q! and G-theory are better adapted to integration and character formulas. Remark 2.20 (Sheaf theories in differential topology and elliptic operators) It is tempting to think of sheaf theories in algebraic geometry as analogues of elliptic operators or complexes in differential topology. In particular, the theory Q! (X) for a smooth variety X is a natural setting for the study of the Dolbeault ∂-operator coupled to vector bundles, while the theory D(X) is similarly a natural setting for the study of the de Rham operator d coupled to vector bundles. The pushforward operation is the analogue of the index. In this direction, it would be interesting to develop sheaf theories on derived manifolds, for example C ∞ -schemes and stacks. Quasicoherent sheaves in the sense of Joyce [21] are a natural candidate. Another interesting setting is categories of elliptic complexes on manifolds. The general results below would then provide an approach to generalizations of the classical Atiyah-Singer and Atiyah-Bott theorems. Let us spell out the main ingredients of Proposition 2.16 for our examples. Recall that for X a smooth scheme, L X ' SpecX Sym• (ΩX [1]), and for BG a classifying stack, L (BG) ' G/G. • Theory Q: For X a smooth scheme, we have the HKR identification of functions on the loop space (or the Hochschild chain complex) with differential forms, dim( Q(X)) ' Γ(X, Sym• (ΩX [1])), or more generally, Q-global volume forms on X f are the coherent cohomology O (X f ). For BG a classifying stack, Q-global volume forms on L (BG) are the coherent cohomology O (G/G), which for G reductive are the underived invariants O (T )W . • Theory Q! : For X smooth, we have Q(X) ' Q! (X), and so we recover the above descriptions. For X proper, Q-global volume forms on L X are the dual of the Hochschild chain complex (see [34]). • Theory D: For X a smooth scheme, D-global volume forms on L X are the de ∗ Rham cochains dim( D(X)) ' CdR (X), or more generally, D-global volume forms f ∗ on X are the de Rham cochains CdR (X f ), or equivalently those of the underlying unf derived scheme of X . For BG a classifying stack, D-global volume forms on L (BG) are the Borel-Moore homology of G/G.

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2.3.2. Integration formulas for traces. – Now let us turn to the functoriality of dimensions and traces, which is reflected in integration of volume forms along proper maps. For a sheaf theory prop S : Corrk → dgCatk , the counit of the (f∗ , f ! ) adjunction for a proper map f : X → Y gives rise to a canonical integration map R / ω(Y ). : ω(X) f Theorem 2.21. – Fix a sheaf theory S : Corrk → dgCatk . (1) For any proper map f : X → Y , the induced map on dimensions / dim( S (Y ))

dim(f∗ ) : dim( S (X))

is identified (S 1 -equivariantly) with integration along the loop map R / ω( L Y ). dim(f∗ ) ' L f : ω( L X) (2) Given a proper map f : X → Y regarded as a correspondence from X to Y , and self-correspondences Z ∈ Corrk (X, X) and W ∈ Corrk (Y, Y ), together with an identification ∼ / α:Z X ×Y W of correspondences from X to Y , the induced trace map is identified with integration along the natural map R / ω(W | ). Tr(f∗ , α) ' τ (f,s) : ω(Z | ) ∆X

∆Y

Remark 2.22. – Similarly, in the case of the theory Q of quasicoherent sheaves, the standard adjunction (f ∗ , f∗ ) leads to the evident contravariant functoriality of dimensions under arbitrary maps, given by pullback of functions on loop spaces. Remark 2.23 (Categorified version). – For applications to categorical representation theory, in particular the geometric Langlands program, it is interesting to have character formulas for group actions on categories. Such formulas would follow from a good formalism of “stack theories,” the higher unstable analogs of sheaf theories, such as the assignment X → ShvCatk (X). Such stack theories could be formulated as symmetric monoidal functors S : Corrk → A out of a correspondence (∞, 2) (or more naturally (∞, 3)) category with values in a category A such as that of module categories for dgCatk . Namely, we are interested in categorified analogues of D and Q, taking values in the ∞-category PrL of presentable ∞-categories, in which we assign to a scheme or stack X the ∞-category of quasicoherent sheaves of module categories over D or Q. Since such theories have not been fully constructed yet, we will only briefly sketch the idea.

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For any stack X and sheaf theory S , the category of sheaves S (X) is naturally symmetric monoidal, and so we may consider its ∞-category of (presentable, stable) module categories S (X)-mod. To obtain a more meaningful geometric theory we should sheafify this construction. For example, strong or Harish-Chandra G-categories (in other words, module categories over D(G) with convolution) are identified with sheaves of categories over the de Rham stack of BG. However, in the quasicoherent case, the “1-affineness” theorem of Gaitsgory [14] identifies Q(X)-modules with sheaves of categories on X for a large class of stacks (specifically, for X an eventually coconnective quasi-compact algebraic stack of finite type with an affine diagonal over a field of characteristic 0). In particular, Q(BG)-modules are identified with algebraic G-categories. In the quasicoherent case, the general formalism of this paper should provide an S 1 -equivariant equivalence dim( Q(X)-mod) = Q( L X), identifying the class [ Q(X)] of the structure stack with the structure sheaf O ( L X). In particular, the characters of quasicoherent G-categories are given by Q(G/G). The induced map on dimensions dim(f∗ ) : dim( Q(X)-mod) → dim( Q(Y )-mod) is identified S 1 -equivariantly with the morphism given by pushforward along the loop map dim(f∗ ) = L f∗ : Q( L X)

/ Q( L Y ).

In particular, for an algebraic group G and G-space X with π : X/G → BG, the character of the G-category Q(X/G) is given by the pushforward L π∗ O ( L X/G) ∈ Q (G/G). Analogous results are expected for strong or Harish-Chandra G-categories (module categories for D(G) with convolution) using the sheafification of the theory of D(X)-module categories. We hope to return to these applications in future works. 3. Traces in category theory 3.1. Preliminaries. – Our working setting is the higher category theory and algebra developed by J. Lurie [25, 27, 28], see Chapter I.1 of [16] for an excellent overview. Throughout what follows, we will fix once and for all a symmetric monoidal (∞, 2)-category A with unit object 1 A . By forgetting non-invertible 2-morphisms we obtain a symmetric monoidal (∞, 1)-category f ( A ), which we will abusively refer to as A whenever only invertible higher morphisms are involved. Conversely, given a symmetric monoidal (∞, 1)-category C , we can always regard it as a symmetric monoidal (∞, 2)-category i( C ) with all 2-morphisms invertible. (5) Thus developments for higher ∞-categories equally well apply to the more familiar (∞, 1)-categories. In what follows, noninvertible 2-morphisms only play a significant role starting with Section 3.4. We will use ⊗ to denote the symmetric monoidal structure of A . We will write Ω A = End A (1 A ) for the “based loops” in A , or in other words, the symmetric (5)

One can understand the above two operations as forming an adjoint pair (i, f ).

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monoidal (∞, 1)-category of endomorphisms of the monoidal unit 1 A . Note that the monoidal unit 1Ω A is nothing more than the identity id1 A of the monoidal unit 1 A . Example 3.1 (Algebras). – Fix a symmetric monoidal (∞, 1)-category C , and let A = Alg( C ) denote the Morita (∞, 2)-category of algebras, bimodules, and intertwiners of bimodules within C . The forgetful map A = Alg( C ) → C is symmetric monoidal, and in particular, the monoidal unit 1 A is the monoidal unit 1 C equipped with its natural algebra structure. Finally, we have Ω A ' C . For a specific example, one could take C = k-mod = dgVectk the (∞, 1)-category of complexes of k-modules (with quasi-isomorphisms inverted). Then A = Alg( C ) is the (∞, 2)-category of k-algebras, bimodules, and intertwiners of bimodules. Example 3.2 (Categories). – A natural source of (∞, 2)-categories is given by various theories of (∞, 1)-categories. For example, one could consider dgCatk , the (∞, 2)-category of k-linear stable presentable ∞-categories (or k-linear presentable dg categories), k-linear continuous functors, and natural transformations. Observe that Alg(k-mod) is a full subcategory of dgCatk , via the functor assigning to a k-algebra its stable presentable ∞-category of modules. The essential image consists of dg categories admitting a compact generator. 3.2. Dualizability Definition 3.3. – An object A of the symmetric monoidal (∞, 2)-category A is said to be dualizable (equivalently, A is dualizable in the (∞, 1)-category f ( A )) if it admits a monoidal dual: there is a dual object A∨ ∈ A and evaluation and coevaluation morphisms / 1A / A ⊗ A∨ A : A∨ ⊗ A ηA : 1 A such that the usual compositions are naturally equivalent to the identity morphism A A∨

ηA ⊗idA

/ A ⊗ A∨ ⊗ A

idA ⊗A

/ A,

idA∨ ⊗ηA

/ A∨ ⊗ A ⊗ A∨

A ⊗idA∨

/ A∨ .

Example 3.4. – Any algebra object A ∈ Alg( C ) is dualizable with dual the opposite algebra Aop ∈ Alg( C ). The evaluation morphism A : Aop ⊗ A

/ 1C

is given by A itself regarded as an A-bimodule. The coevaluation morphism ηA : 1 C

/ A ⊗ Aop

is also given by A itself regarded as an A-bimodule.

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3.2.1. Dualizable morphisms. – Consider two objects A, B ∈ A , and a morphism / B.

Φ:A

Example 3.5. – If A = Alg( C ), then Φ is simply an Aop ⊗ B-module. If B is dualizable with dual B ∨ , we can package Φ in the equivalent form of the morphism eΦ : B ∨ ⊗ A → 1 A defined by / 1A :

eΦ

B∨ ⊗ A B

idB ∨ ⊗Φ

 B ⊗ B∨.

If A is dualizable with dual A∨ , we can package Φ in the equivalent form of the morphism uΦ : 1 A → B ⊗ A∨ defined by A: ⊗ A∨ ηA

1A

uΦ



Φ⊗idA∨

/ B ⊗ A∨ .

If both A and B are dualizable, we can also encode Φ by its dual morphism Φ∨ : B ∨

/ A∨

defined by B∨

idB ∨ ⊗ηA

/ B ∨ ⊗ A ⊗ A∨

idB ∨ ⊗Φ⊗idA∨

/ B ∨ ⊗ B ⊗ A∨

B ∨ ⊗idA∨

/ 4 A∨ .

Φ∨

There is a natural composition identity (ΦΨ)∨ ' Ψ∨ Φ∨ . Note that for fixed A, B, the construction Φ 7→ Φ∨ naturally defines a covariant map (−)∨ : Hom(A, B)

/ Hom(B ∨ , A∨ )

∨ and in particular a morphism Φ1 → Φ2 induces a natural morphism Φ∨ 1 → Φ2 .

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Let us record the canonical equivalences encoded by the following commutative diagrams A ; ⊗ A∨

(3.1) ηA

1A

uΦ

ηB

A∨ ⊗ O A Φ∨ ⊗idA

Φ⊗idA∨

 / B ⊗ A∨ O

B∨ ⊗ A

idB ⊗Φ∨

idB ∨ ⊗Φ

# B ⊗ B∨



A∨ eΦ

$ / 1A :

B ∨

B⊗B .

Example 3.6. – In the setting of algebras, bimodules and intertwiners, the morphisms Φ, uΦ , eΦ and Φ∨ are all different manifestations of the same bimodule Φ, making their various compatibilities particularly evident. Definition 3.7. – (1) A morphism Φ : A → B is said to be left dualizable if it admits a left adjoint: there is a morphism Φ` : B → A and unit and counit morphisms / Φ ◦ Φ`

ηΦ : idB

/ idA

Φ : Φ` ◦ Φ

satisfying the usual identities. (2) A morphism Φ : A → B is said to be right dualizable if it admits a right adjoint: there is a morphism Φr : B → A and unit and counit morphisms / Φr ◦ Φ

ηΦ : idA

/ idB

Φ : Φ ◦ Φ r

satisfying the usual identities. Remark 3.8. – If A and B are dualizable, and Φ : A → B is left (resp. right) dualizable, then Φ∨ : B ∨ → A∨ is right (resp. left) dualizable with right adjoint (Φ` )∨ : A∨ → B ∨ (resp. left adjoint (Φr )∨ : A∨ → B ∨ ). 3.3. Traces and dimensions. – (We continue to refer to [40, 19] for thorough treatments of the theory of traces in higher category theory.) Let A ∈ A be a dualizable object with dual A∨ . Consider an endomorphism Φ:A

/ A.

Since A is dualizable, Φ has a trace defined as follows. Definition 3.9. – (1) The trace of Φ : A → A is the object Tr(Φ) ∈ Ω A defined by 1A

ηA

/ A ⊗ A∨

Φ⊗idA

/ A ⊗ A∨

A

/ 1A .

Tr(Φ)

Given a natural transformation ϕ : Φ → Ψ, we define the induced morphism Tr(ϕ) : Tr(Φ)

/ Tr(Φ0 )

by applying ϕ ⊗ idA∨ to the middle arrow above.

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(2) The dimension (or Hochschild homology) of A is the trace of the identity dim(A) = Tr(idA ) ∈ Ω A or in other words, the object defined by ηA

1A

/ A ⊗ A∨

/ 1A.

A

dim(A)

Remark 3.10. – Equivalently, we can describe the trace as the composition Φ

1A

/ End(A)

∼

/ A ⊗ A∨

A

/ 1A

where the middle arrow is the identification deduced from the dualizability of A. Remark 3.11. – Observe that for fixed dualizable A ∈ A , taking traces gives a functor / ΩA.

Tr : End(A)

Remark 3.12. – Observe that for any dualizable endomorphism Φ, the standard identities encoded by Diagrams 3.1 give rise to an identification Tr(Φ) ' Tr(Φ∨ ). Example 3.13. – When A = 1 A is the monoidal unit, and Φ : 1 A → 1 A is an endomorphism, we have an evident equivalence of endomorphisms Tr(Φ) ' Φ. Theorem 3.14 ([26]). – There is a canonical S 1 -action on the dimension dim(A) of any dualizable object A of a symmetric monoidal ∞-category A . 3.3.1. Cyclic symmetry Proposition 3.15. – Given two morphisms Ao

Φ

/B

Ψ

between dualizable objects A, B ∈ A , there is a canonical equivalence m(Φ, Ψ) : Tr(Φ ◦ Ψ) functorial in morphisms of both Φ and Ψ.

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∼

/ Tr(Ψ ◦ Φ)

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Proof. – We construct m(Φ, Ψ) following the commutative diagram below:

ηA

A : ⊗ A∨

Φ⊗idA∨

/ B ⊗ A∨ Ψ⊗idA∨/ A ⊗ A∨ A

ηB

$ B ⊗ B∨

$

idB ⊗Ψ∨

idA ⊗Ψ∨

1A

 / A ⊗ B∨

Ψ⊗idB ∨

: 1A

 / B ⊗ B∨.

B

Φ⊗idB ∨

Following the top edge, we find the definition of Tr(Ψ ◦ Φ). Following the bottom edge, we find the definition of Tr(Φ ◦ Ψ). The identifications filling the left and right diamonds arise from the standard identities encoded by Diagrams 3.1. The identification filling the central square results from the symmetric monoidal structure. The construction is evidently functorial for morphisms Φ → Φ0 . The functoriality for morphisms Ψ → Ψ0 is similar, once one recalls that the construction Ψ 7→ Ψ∨ is covariantly functorial in morphisms of Ψ. Example 3.16. – Taking Φ = idA yields a canonical equivalence γ 0 : idTr(Φ0 )

/ m(idA , Φ0 ) A

∼

and likewise, taking Φ0 = idA yields a canonical equivalence γ : idTr(Φ)

/ m(ΦA , idA ).

∼

Thus taking Φ = Φ0 = idA yields an automorphism of the identity of the Hochschild homology ∼ / (γ 0 )−1 ◦ γ : idTr(id ) idTr(id ) A

A

called the BV homotopy. Remark 3.17. – The proposition is only the initial part of the full cyclic symmetry of trace (see Remark 3.28), and the example is the lowest level structure of the S 1 -action on Hochschild homology (see Theorem 3.14) defining cyclic homology. Lemma 3.18. – Given morphisms A

Φ

/B

Ψ

/C

Υ

/A

between dualizable objects A, B, C ∈ A , there is a canonical commutative diagram Tr(ΨΦΥ)

m(Ψ,ΦΥ)

/ Tr(ΦΥΨ) m(Φ,ΥΨ)

m(ΨΦ,Υ)

)  Tr(ΥΨΦ).

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Proof. – We construct the desired equivalence from the following diagram: C; ⊗ C∨

/ A ⊗ C∨

Υ ∨

∨

Ψ

/ B ⊗ B∨

1A

Ψ

% / C ⊗ B∨

Υ

Φ

Ψ

C

Ψ

% / A ⊗ B∨

Φ∨

% / B ⊗ A∨

/ C ⊗ C∨

Ψ ∨

Ψ

Φ∨

# A ⊗ A∨

/ B ⊗ C∨

Φ

% / B ⊗ B∨

Φ Φ∨

% / C ⊗ A∨

Υ

A

$ / 1A :

% / A ⊗ A∨ .

The natural transformations m(Ψ, ΦΥ) and m(Φ, ΥΨ) describe passage from the top row to the middle row and from the middle to the bottom, respectively. The transformation m(ΨΦ, Υ) can then be identified with the transformation from the top row to the bottom given by inserting the diagonal morphisms id ⊗Φ∨ ◦ Ψ∨ and using standard composition identities. 3.4. Functoriality of dimension. – Let A cont ⊂ A denote the (∞, 2)-subcategory of dualizable objects and continuous or right dualizable morphisms (morphisms that are left duals). Definition 3.19. – Let Ψ : A → B denote a morphism in A cont with right adjoint Ψr : B → A. We define the induced morphism of dimensions dim(Ψ) : dim(A)

/ dim(B)

to be the composition Tr(idA )

ηΨ

/ Tr(Ψr ◦ Ψ)

m(Ψr ,Ψ)

/ Tr(Ψ ◦ Ψr )

Ψ

/ Tr(idB ).

Remark 3.20. – In other words, the morphism dim(Ψ) is defined by the following diagram A < ⊗ O A∨ dim(A) ηA

1A

uΨ

ηB

Ψ⊗idA∨

Ψr ⊗idA

 / B ⊗ A∨ O idB ⊗Ψ∨

A

" / 1A

cΨ

idB ∨ ⊗Ψr∨ B


Y  W.

In other words, the morphisms Corrk (X, Y ) now form the ∞-category Stacks/X×Y of stacks over X × Y with arbitrary morphisms rather than isomorphisms as in Corrk (X, Y ).

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We will also have need to restrict the class of morphisms of correspondences to some subcategory of Stacks/X×Y . In particular, we will consider the subcategory Corrprop k in which we only allow proper maps of correspondences.

4.2. Traces of correspondences. – Given a map Z → X, it is convenient to introduce the symmetric presentation of the based loop space LZX

= Z ×Z×X Z.

Note the two natural identification with the traditional based loop space LX

×X Z ' X ×X×X Z o

∼

Z ×Z×X Z

∼

/ Z ×X×X X ' Z ×X L X.

There is a natural rotational equivalence L X ×X Z ' Z ×X L X that makes the above two identifications coincide. (It does not preserve base points and is not given by swapping the factors.) Thus we can unambiguously identify all of the above versions of the based loop space. Proposition 4.1. – (1) Any derived stack X is dualizable as an object of Corrk , with dual X ∨ identified with X itself, and dimension dim(X) identified with the loop space LX

1

= X S ' X ×X×X X

regarded as a self-correspondence of pt = Spec k. (2) The transpose of any correspondence X ← Z → Y is identified with the reverse correspondence Y ← Z → X. The trace of a self-correspondence X ← Z → X is identified with the based loop space Tr(Z) ' Z |

∆X

= Z ×X×X X ' L Z X

regarded as a self-correspondence of pt = Spec k. In particular, the trace of the graph Γf → X × X of a self-map f : X → X is identified with the fixed point locus Tr(f ) ' Γf |

∆X

= Γf ×X×X X ' X f .

Proof. – The evaluation and coevaluation presenting the self-duality of X are both given by X itself as a correspondence between pt = Spec k and X ×X via the diagonal map. The standard identities follow from the calculation of the fiber product of the two diagonal maps X ∆12 ×X×X×X

∆23 X

' X.

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Thus the dimension of X is the loop space LX

X

pt

{ #

X

# { X ×X ~

pt.

By definition, the transpose of a correspondence X ← Z → Y is identified with Y ← Z → X by checking the definition

Z

Y ×Z

v

x Y ×X

Y

'

Z ×X

' w Y ×Z ×X & w Y ×X ×X

{

& Y ×X ' x Y ×Y ×X #

X.

The trace of a self-correspondence X ← Z → X is then calculated by the composition Z ×Z×X Z

Z

X

pt



x &

Z

& x Z ×X { # x X ×X

# & { X ×X

X

pt.

Finally, the case of the graph Z = Γf of a self-map gives the fixed point locus by definition. Remark 4.2 (Cyclic version). – The identification dim(X) ' L X above is naturally S 1 -equivariant for the standard loop rotation on L X and the cyclic symmetry of dim(X) provided by the cobordism hypothesis. To see this it is useful to consider X as an E∞ -algebra object in Stacksop k via the diagonal map (or as an En -object for any n). In other words, for n = 1 we identify stacks and correspondences with objects and morphisms in the Morita category Alg(Stacksop k ). It follows from the

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properties of topological chiral homology [27, Theorem 5.3.3.8] that for a (constant) commutative algebra A its topological chiral homology over R a manifold is given by the tensoring of commutative algebras over simplicial sets M A = M ⊗ A. In particular R 1 (passing back from the opposite category to stacks) we have S 1 X = X S = L X. Moreover this identification holds not just for a fixed circle but over the moduli space of circles BDiff(S 1 ) ∼ BS 1 , i.e., equivariantly for rotation. We also know from [27, Example 5.3.3.14] or [26, Example 4.2.2] that the S 1 -action on the dimension of an associative algebra A (given classically by the cyclic structure on the Hochschild 1 Rchain complex) is given by the rotation S -action on the topological chiral homology A, i.e., the family of topological chiral homologies over the moduli space of circles. S1 In our case this recovers the rotation action on the loop space. 4.3. Geometric functoriality of dimension Proposition 4.3. – The graph X ← Γf → Y of any proper morphism f : X → Y gives a continuous morphism F : X → Y in Corrprop , with right adjoint F r : Y → X k identified with the opposite correspondence Y ← Γf → X. Proof. – We construct the unit and counit of the adjunction as follows. Consider the composition F r F : X → X of correspondences X ×Y X

Γf

X

{ $

~ $

Y

Γf

z

X.

r

The unit ηf : idX = X → F F ' X ×Y X is given by the relative diagonal map. Consider the opposite composition of correspondences X ×X X

Γf

Y

z

~ $

$

X

z

Γf

Y.

r

The counit f : F F ' X → idY = Y is given by f itself. The standard identities are easily verified by identifying the resulting composite map / Γf ×Y Γf ×X Γf / Γf Γf of correspondences with the identity.

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Lemma 4.4. – Let FZ : X → Y and FW : Y → X be morphisms in Corrk given by respective correspondences X ← Z → Y and Y ← W → X. Then the canonical equivalence ∼ / Tr(FZ ◦ FW ) m(FW , FZ ) : Tr(FW ◦ FZ ) is given by the composition of evident geometric identifications / W ×X×Y Z

∼

(Z ×Y W ) ×X×X X

/ Z ×Y ×X W

∼

/ (W ×X Z) ×Y ×Y Y.

∼

Proof. – Returning to the definition and using our previous identifications, observe that m(FZ , FW ) is calculated by commutativity of the diagram of correspondences X; × X

Z×X

/ Y ×X

W ×X

/ X ×X

X

X

Y

# Y ×Y

W ×Y

#

Y ×W

X×W

pt

 / X ×Y

X×Y

 / Y × Y.

; pt

Y

Following the top edge, we see Tr(FW ◦ FZ ) ' (Z ×Y W ) ×X×X X. Following the bottom edge, we see Tr(FZ ◦ FW ) ' (W ×X Z) ×Y ×Y Y . Moving from the top to bottom edge via the successive equivalences of the three commuting squares, one finds the three successive equivalences in the assertion of the lemma. Proposition 4.5. – Suppose f : X → Y is a proper morphism, and F : X → Y denotes the induced morphism in Corrprop given by the graph X ← Γf → Y . Then dim(F ) : k dim(X) → dim(Y ) is canonically identified with the S 1 -equivariant morphism L f : LX → LY . Proof. – Denote by F r : Y → X the right adjoint to F . We must calculate dim(X)

/ Tr(F r F )

m(F r ,F )

/ Tr(F F r )

/ dim(Y ).

We have seen that the first and third morphisms correspond to the natural geometric maps / (X ×Y X) ×X×X X L X ' X ×X×X X X ×Y ×Y Y

/ Y ×Y ×Y Y ' L Y,

induced by the relative diagonal X → X ×Y X and given map f : X → Y respectively. Furthermore, by Lemma 4.4, the middle map is the natural geometric identification (X ×Y X) ×X×X X

∼

/ X ×Y ×Y Y.

Altogether, the composition is easily identified with the loop map L f : L X → L Y .

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Remark 4.6. – It follows from the proposition that the loop map L f : L X → L Y must be proper when the given map f : X → Y is proper. Let us note why this is true geometrically from the factorization L X → L X Y → L Y appearing in the proof. First, the natural morphism L X → L X Y is the restriction along the diagonal X → X × X of the relative diagonal X → X ×Y X. The relative diagonal is a closed embedding since f is proper, and hence the natural morphism L X → L X Y is as well. Second, the natural morphism L X Y → L Y is the restriction along the diagonal Y → Y ×Y of the proper morphism f : X → Y and thus is proper as well. Altogether, we see that L f : L X → L Y is itself proper. Remark 4.7. – One can invoke the cobordism hypothesis with singularities to endow the morphism dim(F ) : dim(X) → dim(Y ) with a canonical S 1 -equivariant structure, and it will agree with the canonical geometric S 1 -equivariant structure on the map L f : L X → L Y under the identification of the proposition. 4.4. Geometric functoriality of trace. – Consider a proper morphism f : X → Y and endomorphisms FZ : X → X and FW : Y → Y in Corrk given by respective selfcorrespondences X ← Z → X and Y ← W → Y . By an f -morphism from the pair (X, FZ ) to the pair (Y, FW ), we mean an identification ∼ / s:Z X ×Y W of correspondences from X to Y . This in turn induces an identification of what might be called relative traces ∼ / X ×Y ×Y W Z ×Y ×Y Y generalizing the relative loop space L X Y from the case of the identity correspondences Z = X, W = Y . We thus obtain a map of traces τ (f, s) : Z |

∆X

= Z ×X×X X

/ Z ×Y ×Y Y

∼/

/ Y ×Y ×Y W = W |

X ×Y ×Y W

∆Y

.

Proposition 4.8. – With the preceding setup, the trace map Tr(f, s) : Tr(FZ ) → Tr(FW ) is canonically identified with the geometric map τ (f, s) : Z |

∆X

/ W| . ∆Y

Proof. – Denote by F : X → Y the morphism given by the graph X ← Γf → Y , and by F r : Y → X its right adjoint. We must calculate Tr(FZ )

/ Tr(F r F FZ )

s

/ Tr(F r FW Fm(F )

r

,FW F )

/ Tr(FW F F r )

/ Tr(FW ).

We have seen that the first and fourth morphisms correspond to the natural geometric maps / Z ×X×X (X ×Y X) Z| = Z ×X×X X ∆X

X ×Y ×Y W

/ Y ×Y ×Y W = W |

∆Y

.

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induced by the relative diagonal X → X ×Y X and given map f : X → Y respectively. Using associativity, the second map, induced by s, is the natural geometric identification ∼ / Z ×X×X (X ×Y X) ' Z ×Y ×Y Y W ×Y ×Y X. By Lemma 4.4, the third map, given by the cyclic symmetry, is nothing more than the natural identification W ×Y ×Y X

∼

/ X ×Y ×Y W.

Thus assembling the above maps we arrive at the composition defining τ (f, s).

5. Traces for sheaves In this section, we spell out how to apply the abstract formalism of traces of Section 3 and its geometric incarnation of Section 4 to categories of sheaves. Recall Definition 2.15: Definition 5.1. – A sheaf theory is a symmetric monoidal functor of (∞, 2)-categories S

: Corrprop k

/ dgCat k

from correspondences of stacks (with 2-morphisms given by proper maps of correspondences) to dg categories. We denote by S

: Corrk

/ dgCatk

the underlying 1-categorical sheaf theory, i.e., the symmetric monoidal functor on (∞, 1)-categories obtained by forgetting noninvertible morphisms. Applying a sheaf theory to the geometric descriptions of traces of correspondences, one immediately deduces trace formulas for dg categories. We first spell out the consequences of the 1-categorial structure of a sheaf theory S , then the trace formulae arising from its 2-categorical enhancement S , and finally in Section 5.3 explain how to use the results of [9, 16] to deduce applications of this formalism. 5.1. Dimensions and traces of sheaf categories: 1-categorical consequences. – The graph of a map of derived stacks f : X → Y provides a correspondence from X to Y and a correspondence from Y to X. We denote the respective induced maps by f∗ : S (X) → S (Y ) and f ! : S (Y ) → S (X). The functoriality of S concisely encodes base change for f∗ and f ! . For π : X → pt = Spec k, we denote by ωX = π ! k ∈ S (X) the S -analogue of the dualizing sheaf, and by ω(X) = π∗ ωX ∈ S (pt) = dgVectk the S -analogue of “global volume forms”. Next we will record formal consequences of our prior calculations deduced from the fact that a sheaf theory is symmetric monoidal.

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Proposition 5.2. – Fix a sheaf theory S : Corrk → dgCatk , and X, Y ∈ Corrk . (1) S (X) ∈ dgCatk is canonically self-dual, and for any f : X → Y , f ! : S (Y ) → S (X) and f∗ : S (X) → S (Y ) are canonically transposes of each other. (2) S (X) is canonically symmetric monoidal with tensor product F

⊗! G = ∆! (π1! F ⊗ π2! G )

F

, G ∈ S (X).

(3) For any f : X → Y , the projection formula holds: f∗ F ⊗! G ' f∗ ( F ⊗! f ! G )

∈ S (X), G ∈ S (Y ).

F

(4) There is a canonical equivalence of functors and integral kernels HomdgCatk ( S (X), S (Y )) ' S (X × Y ). (5) The functor q∗ p! : S (X) → S (Y ) associated to a correspondence Xo

p

Z

/Y

q

is represented by the integral kernel (p × q)∗ ωZ ∈ S (X × Y ). Proof. – (1) Follows immediately from Proposition 4.1. (2) Follows immediately from the commutative algebra structure on X ∈ Corrk (in fact commutative coalgebra structure on X ∈ Stacksk ) provided by the diagonal map. (3) Follows from base change for the diagram f

X id×f

 X ×Y

/Y ∆

f ×id

 / Y × Y.

(4) Since S is monoidal, we have S (X) ⊗ S (Y )

' S (X × Y ).

The self-duality of S (X) provides HomdgCatk ( S (X), S (Y )) ' S (X)∨ ⊗ S (Y ) ' S (X) ⊗ S (Y ). By construction, the composite identification assigns the functor FK ( F ) = π2∗ (π1! F ⊗! K)

K ∈ S (X × Y ).

(5) Follows from the projection formula: consider the diagram Z q

p Π

{ Xo

π1

 X ×Y

π2

#

/ Y,

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where Π = p × q. Then we have q∗ p! (−) ' π2∗ Π∗ Π! π1! (−) ' π2∗ Π∗ (ωZ ⊗! Π! π1! (−)) ' π2∗ (Π∗ ωZ ⊗! π1! (−)) Proposition 5.3. – Fix a sheaf theory S : Corrk → dgCatk . (1) The S -dimension dim( S (X)) = HH∗ ( S (X)) of any X ∈ Corrk is S 1 -equivariantly equivalent with S -global volume forms on the loop space dim( S (X)) ' ω( L X). In particular, for G an affine algebraic group, characters of S -valued G-representations are adjoint-equivariant S -global volume forms dim( S (BG)) ' ω(G/G). (2) The S -trace of any endomorphism Z ∈ Corrk (X, X) is equivalent to S -global volume forms on the restriction to the diagonal Tr( S (Z)) ' ω(Z | ). ∆

In particular, the S -trace of a self-map f : X → X is equivalent to S -global volume forms on the f -fixed point locus Tr(f∗ ) ' ω(X f ). Proof. – (1) Follows immediately from Proposition 4.1(1). To spell this out, using the previous proposition and base change, dim( S (X)) results from applying the composition / dgVectk π∗ ∆! ∆∗ π ! ' π∗ p2∗ p!1 π ! ' L π∗ L π ! : dgVectk to the unit 1dgVectk = k. Here π : X → pt and L π : L x → pt are the maps to the point, and p1 , p2 : L X ' X ×X×X X → X are the two natural projections. Thus we find dim( S (X)) ' L π∗ L π ! (k) ' ω( L X). Furthermore, the S 1 -equivariance results from the one-dimensional cobordism hypothesis: the one-dimensional topological field theory defined by the dualizable object S (X) ∈ dgCatk factors through that defined by the dualizable object X ∈ Corrk . Moreover, we identified the S 1 -action on the dimension L X with loop rotation. (6) (2) Similarly follows immediately from Proposition 4.1(2). (6)

One can also check directly that the cyclic structure on the cyclic bar construction of the dg category S (X) is induced by the cyclic structure of the loop space L X under the identification ω( L X) ' dim( S (X)).

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5.2. Integration formulas for traces: 2-categorical consequences. – Now we turn to the functoriality of dimensions and traces. For this we require the 2-categorical enhanced version S of a sheaf theory, so as to take advantage of the resulting functorial adjunction S (X)

o

f∗ f!

/ S (Y )

for f : X → Y proper. In particular, applying the functor S on two-morphisms we find that • A proper map f : Z → W of correspondences Z pZ

X`

~

qZ

>Y

f

pW

 W

qW

induces a canonical integration morphism of integral transforms R / qW ∗ p! : qZ∗ p!Z W f • In particular, when X = Y = pt, it induces a map of global volume forms R / ω(W ). : ω(Z) f • There is a canonical composition identity R R R ◦ f ' g◦f . g Proposition 5.4. – For f : X → Y proper, the unit and counit of the (f∗ , f ! ) adjunction are given respectively by integration along the proper maps of self-correspondences ∆/Y : X → X ×Y X of X and f/Y : X → Y of Y : X

X f

z

Xd

$ :X

∆f p1

 X ×Y X

p2

Y_

~

f

f

?Y

 Y.

Proof. – The assertion follows immediately from the geometric description of the unit and counit in the correspondence category, Proposition 4.3, upon applying the functor S .

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Proposition 5.5. – For any proper map f : X → Y , the induced map on dimensions / dim( S (Y ))

dim(f∗ ) : dim( S (X))

is identified (S 1 -equivariantly) with integration along the loop map R / ω( L Y ). dim(f∗ ) ' L f : ω( L X) Proof. – According to Definition 3.19, we must calculate the composition dim(f∗ ) : Tr(id S (X) )

Tr(ηf )

/ Tr(f ! f∗ )

∼

/ Tr(f∗ f ! )

Tr(f )

/ Tr(id S (Y ) ).

The equivalence of the middle arrow is given by the canonical identifications Tr(f ! f∗ ) ' ω((X ×Y X) ×X×X X) ' ω(X ×Y ×Y Y ) ' Tr(f∗ f ! ). By Proposition 5.4, the unit ηf : id S (X) → f ! f∗ is given by the integration morphism R / ωX× X : ∆f ∗ ωX Y ∆f and hence its trace Tr(ηf ) : Tr(id S (X) ) → Tr(f ! f∗ ) is given by the induced integration map R / ω((X ×Y X) ×X×X X). : ω( L X) ∆f

Likewise, the counit f : f∗ f ! → id S (Y ) is given by by the integration morphism R / ωY : f∗ ωX f and hence its trace Tr(f ) : Tr(f∗ f ! ) → Tr(id S (Y ) ) is given by the induced integration map R / ω( L Y ). : ω(X ×Y ×Y Y ) f Finally, by functoriality, their composition is given by the integration map R / ω( L Y ). : ω( L X) Lf Finally, we have the functoriality of traces in parallel with the previous theorem on the functoriality of dimensions. Let us recall the relevant setup. Consider a proper morphism f : X → Y and endomorphisms FZ : X → X and FW : Y → Y in Corrk given by respective self-correspondences X ← Z → X and Y ← W → Y . By an f -morphism from the pair (X, FZ ) to the pair (Y, FW ), we mean an identification ∼ / s:Z X ×Y W of correspondences from X to Y . This in turn induces an identification of what might be called relative traces Z ×Y ×Y Y

PANORAMAS & SYNTHÈSES 55

∼

/ X ×Y ×Y W

79

NONLINEAR TRACES

generalizing the relative loop space L X Y from the case of the identity correspondences Z = X, W = Y . We thus obtain a map of traces τ (f, s) : Z |

∆X

= Z ×X×X X

/ Z ×Y ×Y Y

∼

/ X ×Y ×Y W

/ Y ×Y ×Y W = W |

∆Y

.

Proposition 5.6. – With the preceding setup, the trace map Tr(f∗ , s) : Tr(FX∗ ) → Tr(FY ∗ ) is canonically identified with the integration map R / ω(W | ). : ω(Z | ) τ (f,s) ∆X ∆Y Proof. – The argument is parallel to the proof of Proposition 5.5. One calculates Tr(f∗ , α) from Definition 3.24 using Proposition 4.8 and the compatibility of Proposition 5.2 and the integration morphism for integral transforms from Section 2.3.2. 5.3. Ind-coherent sheaves and D-modules. – We now apply the results of GaitsgoryRozenblyum [16] and Drinfeld-Gaitsgory [9] establishing functoriality properties of categories of ind-coherent sheaves and D-modules. We first state a fundamental result of Gaitsgory-Rozenblyum [16]: Theorem 5.7 ([16, Theorem III.3.5.4.3, III.3.6.3]). – There is a uniquely defined rightlax symmetric monoidal functor Q! from the (∞, 2)-category whose objects are laft (locally almost of finite type) prestacks, morphisms are correspondences with vertical arrow ind-inf-schematic, and 2-morphisms are ind-proper and ind-inf-schematic, to the (∞, 2) category dgCatk of k-linear presentable dg categories with continuous morphisms. The functor Q! is strictly symmetric monoidal on the full subcategory of laft ind-inf-schemes.

The theorem encodes a tremendous amount of structure in great generality. Let us highlight some salient features useful in practice. The theorem assigns a symmetric monoidal dg category Q! (X) to any stack satisfying a reasonable finite type assumption. The symmetric monoidal structure, the !-tensor product, is induced by !-pullback along diagonal maps. For an arbitrary morphism p : X → Y there is a continuous symmetric monoidal pullback functor f ! : Q! (Y ) → Q! (X), while for f schematic (or ind-schematic) there is a continuous pushforward f∗ : Q! (X) → Q! (Y ), which satisfies base change with respect to !-pullbacks. Moreover for f proper (or ind-proper), (f∗ , f ! ) form an adjoint pair. The theorem goes much further through the powerful formalism of inf-schemes: prototypical inf-schemes are quotients of schemes by infinitesimal equivalence relation. Thus one can treat on an equal footing ind-coherent sheaves that are equivariant for any formal groupoid. The most important example is the de Rham space XdR of a scheme, and one recovers D-modules on X as ind-coherent sheaves on the de Rham functor of X, D(X) = Q! (XdR ). Thus by first applying the functor (−)dR the theorem encodes the theory of D-modules, as a functor out of the correspondence 2-category

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of stacks (or laft prestacks) with pullbacks for arbitrary maps and pushforward for (ind-)schematic maps. Corollary 5.8. – The functors Q! and D define sheaf theories on laft ind-inf-schemes, i.e., define symmetric monoidal functors Q

!

, D : Corr(ind-inf-Schk )ind-prop −→ dgCatk .

Thus the conclusions of Theorem 1.4 apply in this setting (in particular the Grothendieck-Riemann-Roch theorem for ind-proper maps of ind-inf-schemes). 5.3.1. The QCA setting.– We are mostly interested in applications of sheaf theory on stacks, e.g., in an equivariant setting. That requires two features of the theories Q! and D that are not encoded in Theorem 5.7. Theorem 5.7 produces in general a right-lax symmetric monoidal functor – in other words, we have the natural map Q

!

(X) ⊗ Q! (Y ) −→ Q! (X × Y )

satisfying the expected coherences, but it is not an equivalence in general (though it is for schemes). Also, while the theorem encodes arbitrary pullbacks, it does not encode a continuous pushforward functor p∗ : Q! (X) → Q! (Y ) for non-schematic morphisms (though a generalization to include QCA morphisms has been announced by the authors). This precludes an immediate application of our formalism to traces on stacks. However, since the full structure of a sheaf theory is far stronger than is needed for the “local” statements we discuss in this paper, we can get around this issue, by taking advantage of the following compilation of results of Drinfeld and Gaitsgory (specifically, see Section 3.6.1, Corollarys 3.7.14, 4.2.3, and 4.4.7, Proposition 4.4.11, Corollarys 8.3.4 and 8.4.3, Definition 9.3.2 and Proposition 9.3.12). Theorem 5.9 ([9]). – (1) For a QCA stack X, the categories Q! (X) and D(X) are dualizable and canonically self-dual. (1) The canonical functors define equivalences Q

!

(X) ⊗ Q! (Y ) ' Q! (X × Y ),

D(X) ⊗ D(Y )

' D(X × Y ).

(2) For a morphism f : X → Y of QCA stack, the “renormalized pushforwards” (7) f• : Q! (X) → Q! (Y ),

f• : D(X) → D(Y )

defined as the transpose of f ! (by the self-duality of (1)) are continuous functors satisfying base-change and the projection formula with respect to pullback. We note that for Q! the “renormalized” pushforward is the standard pushforward functor, while for D it differs for non-safe objects from the more familiar, but discontinuous, de Rham pushforward.

(7)

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Let us now briefly indicate how the results of the previous two sections carry over to QCA stacks in the absence of a fully fledged sheaf theory, where we use the renormalized pushforward functors f• provided by Theorem 5.9 to carry out nonrepresentable pushforwards. In particular, for X a general QCA stack, so that piX : X → pt is not representable, this means that the notation ω(X) has to be taken in a renormalized fashion, ! ω(X) := πX,• πX k = πX,• ωX .

For the theory of ind-coherent sheaves this produces the usual notion of derived global sections of the dualizing complex, but for the theory of D-modules this will differ in general from the nonrenormalized version, namely Borel-Moore chains on X: ω(X)non-renorm = RΓdR (ωX ) = C∗BM (X). In Proposition 5.2, the self-duality in assertion (1) for QCA stacks is the content of Theorem 5.9(1), and f• is defined so as to make it the transpose of f ! . Assertion (2) follows from the sheaf theory construction (i.e., is independent of non-representable morphisms), the projection formula is asserted in item (3) of the theorem and the last two assertions are deduced from the first three. Proposition 5.3 is also deduced directly from Proposition 5.2. R The general trace construction f discussed in Section 5.2 depends only on the functoriality of proper adjunction (which is part of the [16] formalism) and the definition of pullback and (renormalized) pushforward. The identities needed to verify Propositions 5.4, 5.5 and 5.6 only depend on base change, which is guaranteed by Theorem 5.9. We therefore get as a payoff that the conclusions of Theorem 1.4 hold for QCA stacks, in particular: Theorem 5.10. – Let S = Q! or S = D denote either ind-coherent sheaves or D-modules. Let f : X → Y denote a proper morphism of QCA stacks. • For any compact object M ∈ S (X) (coherent sheaf or safe coherent D-module) with character [M ] ∈ HH∗ ( S (X)) ' ω( L X), there is a canonical identification Z [f∗ M ] ' [M ] ∈ HH∗ ( S (Y )) ' ω( L Y ). Lf

• Assume Y = BG for an affine group and X = Z/G for Z a proper QCA stack. Then for any compact object M ∈ S (Z/G) (G-equivariant coherent sheaf or safely equivariant coherent D-module on X), and element g ∈ G, there is a canonical identification Z [f∗ M ]|g ' [M ]| g . Lf

X

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• For a map f : (X, Z) → (Y, W ) of QCA stacks with self-correspondences, the induced map Tr( S (Z)) → Tr( S (W )) is given by integration along fixed points Z| → W ∆Y . ∆X References [1] C. Barwick – arXiv:1301.4725.

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[29] N. Markarian – “The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem”, J. Lond. Math. Soc. 79 (2009), p. 129–143. [30] J. P. May – “Picard groups, Grothendieck rings, and Burnside rings of categories”, Adv. Math. 163 (2001), p. 1–16. [31] F. Petit – “A Riemann-Roch theorem for DG algebras”, Bull. Soc. Math. France 141 (2013), p. 197–223. [32] A. Polishchuk – “Lefschetz type formulas for dg-categories”, Selecta Math. (N.S.) 20 (2014), p. 885–928. [33] K. Ponto & M. Shulman – “Traces in symmetric monoidal categories”, Expo. Math. 32 (2014), p. 248–273. [34] A. Preygel – Thom-Sebastiani and Duality for Matrix Factorizations, and Results on the Higher Structures of the Hochschild Invariants, ProQuest LLC, Ann Arbor, MI, 2012, Thesis (Ph.D.)–Massachusetts Institute of Technology. [35] A. C. Ramadoss – “The relative Riemann-Roch theorem from Hochschild homology”, New York J. Math. 14 (2008), p. 643–717. [36]

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David Ben-Zvi, Department of Mathematics, University of Texas, Austin, TX 78712-0257, USA E-mail : [email protected] David Nadler, Department of Mathematics, University of California, Berkeley, CA 947203840, USA • E-mail : [email protected]

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Panoramas & Synthèses 55, 2021, p. 85–145

FORMAL MODULI PROBLEMS AND FORMAL DERIVED STACKS by Damien Calaque & Julien Grivaux

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. G. H. Hardy – A Mathematician’s Apology To the memory of Jean-Louis Koszul

Abstract. – This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by Lurie and Pridham) which gives a precise mathematical formulation for Drinfeld’s derived deformation theory philosophy. This theorem provides a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie’s general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasizing the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by Joost Nuiten, and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of Gaitsgory and Rozenblyum. Résumé (Problèmes de modules formels et champs dérivés formels). – Cet article présente un survol des problèmes de modules formels. Nous commençons par une introduction aux problèmes de modules formels pointés, et esquissons la démonstration d’un théorème (dû à Lurie et Pridham) donnant une formulation mathématique précise à la philosophie dite de « déformation dérivée » de Drinfeld. Ce résultat donne une correspondance entre les problèmes de modules formels et les algèbres de Lie différentielles graduées. Dans un second temps, nous présentons la théorie générale des contextes de déformation de Lurie, en insistant sur la notion (plus symétrique) de contexte de dualité de Koszul. Nous appliquons ensuite ce cadre général au cas 2010 Mathematics Subject Classification. – 13D10, 14A20, 14B10, 14D15, 14D20, 18G55. Key words and phrases. – Deformation theory, moduli problems, differential graded Lie algebroids, Koszul duality contexts, derived stacks, formal thickenings.

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des problèmes de modules formels non scindés sous un schéma affine dérivé fixé; cette situation a été étudiée récemment par Nuiten, et nécessite de remplacer les algèbres de Lie différentielle graduée par des a lgébroïdes de Lie différentiels gradués. Dans la dernière partie, nous esquissons la globalisation au cas plus général des épaississements formels de champs dérivés, et suggérons une approche alternative à certains résultats de Gaitsgory et Rozenblyum.

Introduction The aim of these lecture notes is to present a recent work, due independently to Lurie and Pridham, concerning an equivalence of infinity-categories between formal moduli problems and differential graded Lie algebras. The link between deformation theory and dg-Lie algebras has a long history, which is as old as deformation theory itself. One of the first occurrence of a relation between these appears in KodairaSpencer’s theory of deformations of complex compact manifolds (see [23]): if X is such a manifold, then infinitesimal deformations of X over a local artinian C-algebra A with maximal ideal mA correspond to Maurer-Cartan elements of the dgLie algebra gX ⊗C mA modulo gauge equivalence, where gX is the dg-Lie algebra Γ(X, AX0,• ⊗ TX ), the differential being the ∂ operator. This very concrete example illustrates the following general principle: to any sufficiently nice dg-Lie algebra it is possible to attach a deformation functor (see e.g., [24, §3.2]), which is defined on local artinian algebras as follows: Def g (A) = MC(g ⊗ mA )/ gauge equivalence This correspondence was carried out by many people, including Quillen, Deligne and Drinfeld. In a letter to Schechtman [7], Drinfeld introduced in 1988 the Derived Deformation Theory (DDT) philosophy: Every (dg/derived) deformation problem is controlled by a dg-Lie (or L∞ -)algebra. Since then, there has been a lot of work confirming this philosophy. We can refer the interested reader to the expository paper [26] for more details. Let us give now a few examples, which are related to derived algebraic geometry. In 1997 Kapranov [21] studied deformations of local systems: given an affine algebraic group G and a G-local system E on a manifold S, deformations of E are encoded by a formal dg-scheme RDef(E) whose tangent complex T[E] RDef(E) at the closed point [E] is RΓ(S, ad(E))[1]. The next result is that T[E] RDef(E)[−1] is naturally an L∞ -algebra, the corresponding Lie algebra structure on the cohomology groups Hi (S, ad(E)) being induced by the natural Lie structure on ad(E).

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This last observation is not at all a hazard, and reflects a far more general phenomenon: in loc. cit., Kapranov proved (1) that for any smooth dg-scheme X, the shifted tangent complex Tx X[−1] at a point x : ∗ → X carries an L∞ -algebra structure that determines the formal geometry of X around x. A modern rephrasing of Kapranov’s result can be presented as follows: for any derived affine scheme X = Spec(A) with a k-point x = Spec(A → k), we have that Tx X[−1] ' TA ⊗A k[−1] ' Tk/A ' RDerA (k, k) and thus Tx X[−1] is a derived Lie algebra. Let us explain another folklore calculation (2) confirming this result in another concrete example. Let X = [G0 /G1 ] be a 1-stack presented by a smooth groupoid s G1 −→ −→ G0 , where G0 and G1 are smooth affine algebraic varieties over a field k. If x is t

a k-point of G0 , then the tangent complex T[x] X of X at the k-point [x] is a 2-step s complex V → W where V = Tid G1 sits in degree −1, W = Tx G0 sits in degree 0, x and the arrow is given by the differential of the target map t at idx . According to Kapranov’s result, there should exist a natural L∞ -structure on T[x] X[−1]. Let us sketch the construction of such a structure. Observe that L := (Ts G1 )|G0 has the structure of a (k, A) Lie algebroid, where G0 = Spec(A). In particular we have a k-linear Lie bracket Λ2k L → L and an anchor map L → T G0 = Derk (A). Very roughly, Taylor components at x of the anchor give us maps S n (W ) ⊗ V → W , and Taylor components of the bracket at x give us maps S n (W ) ⊗ Λ2 (V ) → V . These are the only possible structure maps for an L∞ -algebra concentrated in degrees 0 and 1. Equations for the L∞ -structure are guaranteed from the axioms of a Lie algebroid. As a particular but illuminating case, we can consider the stack BG for some affine algebraic group G. Here G0 = {∗} and G1 = G. Then T∗ BG[−1] ' g where g is the Lie algebra of G, and the L∞ -structure is simply the Lie structure of g. Let us explain how all this fits in the DDT philosophy. Any derived affine scheme Spec(A) endowed with a k-point x defines a representable deformation functor B → Hom(A, B), the Hom functor being taken in the category of k-augmented commutative differential graded algebras (cdgas). Hence it should be associated to a dg-Lie (or indifferently a L∞ -) algebra. This dg-Lie algebra turns out to be exactly Tx Spec(A)[−1], endowed with Kapranov derived Lie structure. In this way, we get a clear picture of the DDT philosophy for representable deformation functors. However, the work of Lurie and Pridham goes way beyond that; what they prove is the following statement: Theorem ([25, 29]). – Over a base field of characteristic zero, there is an equivalence of ∞-categories between formal moduli problems and dg-Lie algebras. (1) (2)

This idea is also present in Hinich [16]. We don’t know a precise reference for that, but it seems to be known among experts.

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A very nice exposition of the above theorem, with several examples and perspectives, is in Toën [32]. Remark. – Lurie’s work [25] extends to En -deformation problems. Pridham’s work [29] has some extension to the positive characteristic setting. In a recent paper, Brantner and Mathew [2] actually generalize the above result over any field, proving that there is an equivalence of ∞-categories between formal moduli problems and so-called partition Lie algebras. In these lectures, we will work in characteristic 0, and our goal is to provide a version of Lurie-Pridham Theorem in families. In other words, we are aiming at first at a statement “over a cdga A”, and build an extension from the affine case X = Spec(A) to an arbitrary derived Artin stack X. One shall be very careful as there are different meanings to “over a base”. We will provide two variants. The second one, which is the one we are interested in, will actually rather be named “under a base”. Split families of formal moduli problems. – In the same year 1997, Kapranov [20] considered the family of all formal neighborhood of points in a smooth algebraic ˆ X of the diagonal variety X, which is nothing but the formal neighborhood X × in X × X. He showed that the sheaf TX [−1] ' TX/X×X is a Lie algebra object ˆ X. in Db (X), whose Chevalley-Eilenberg cdga gives back the structure sheaf of X × ˆ X) that It is important to observe here that we have a kind of formal stack (X × lives both over X and under X. This example is a prototype for split families of formal moduli problems: a generalization of Lurie-Pridham Theorem has recently been proven by Benjamin Hennion [13] in 2013, in the following form: Theorem ([13]). – Let A be a noetherian cdga concentrated in nonpositive degrees, and let X be a derived Artin stack of finite presentation. – A-pointed A-linear formal moduli problems are equivalent (as an ∞-category) to A-dg-Lie algebras. – X-pointed X-families of formal moduli problems are equivalent to Lie algebra objects in QCoh(X). Formal moduli problems under a base. – In 2013, Căldăraru, Tu and the first author [5] looked at the formal neighborhood Yˆ of a smooth closed subvariety X into a smooth algebraic variety Y . They prove that the relative tangent complex TX/Y is a dg-Lie algebroid whose Chevalley-Eilenberg cdga gives back the structure sheaf of Yˆ . Similar results had been previously proven by Bhargav Bhatt in a slightly different formulation (see [1]). We also refer to the work of Shilin Yu [35], who obtained parallel results in the complex analytic context. Note that this time, the derived scheme Yˆ doesn’t live anymore over X (although it still lives under X). Keeping this example in mind, we expect the following generalization of Hennion’s result:

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– A-pointed k-linear formal moduli problems are equivalent to dg-Lie algebroids over A, which has recently been proven by Joost Nuiten in [28]. This appears as Theorem 3.9 of the present survey. – X-pointed formal moduli problems are equivalent to dg-Lie algebroids over X. A weaker version of this appears as Theorem 4.12 in the present paper. Remark. – These results are somehow contained in the recent book [8, 9] of GaitsgoryRozenblyum, though in a slightly different formulation (3). We finally observe that it would be very interesting to understand the results from [11], describing the derived geometry of locally split first order thickenings X ,→ S of a smooth scheme X, in terms of Lie algebroids on X. (4) Description of the paper §1 We present the basic notions necessary to understand Lurie-Pridham result, relate them to more classical constructions in deformation theory, and provide some examples. We finally give a glimpse of Lurie’s approach for the proof. §2 A general framework for abstract formal moduli problems is given in details. In §2.1, we recall Lurie’s deformation contexts. In §2.2, we deal with dual deformation contexts. In §2.3, we introduce the useful notion of Koszul duality context, which is a nice interplay between a deformation context and a dual deformation context. In §2.4 we talk about morphisms between these, which is a rather delicate notion. In §2.5 we restate some results of Lurie using the notion of Koszul duality context, making his approach a bit more systematic. In §2.6 we discuss tangent complexes. §3 We extend former results from dg-Lie algebras to dg-Lie algebroids. We prove Hennion’s result [13] in §3.1. Most of the material in §3.2, §3.3 and §3.4 happens to be already contained in the recent preprints [27, 28] of Joost Nuiten. §3.5 presents some kind of base change functor that will be useful for functoriality in the next section. §4 We explain how to globalize the results of §3. In §4.1 we introduce formal (pre)stacks and formal thickenings, mainly following [8, 9] and [6], and compare these with formal moduli problems. In §4.2 we state a consequence, for Lie algebroids, of the previous §, and we sketch an alternative proof that is based on Koszul duality contexts and morphisms thereof. In §4.3 we show that formal thickenings of X fully faithfully embed in Lie algebroids on X, and we conjecture that this actually is an equivalence. (3) Actually, Gaitsgory and Rozenblyum define Lie algebroids as formal groupoids. Our results, as well as Nuiten’s one, thus somehow indirectly give a formal exponentiation result for derived Lie algebroids. (4) This question is now solved [12].

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Acknowledgements. – The first author thanks Mathieu Anel, Benjamin Hennion, Pavel Safronov, and Bertrand Toën for numerous discussions on this topic. This survey paper grew out of lecture notes, taken by the second author, of a 3 hours mini-course given by the first author at the session DAGIT of the États de la Recherche, that took place in Toulouse in June 2017. We both would like to thank the organizers for a wonderful meeting, as well as the participants to the lectures for their enthusiasm. Damien Calaque acknowledges the financial support of the Institut Universitaire de France, and of the ANR grant “SAT” ANR-14-CE25-0008. Julien Grivaux acknowledges the financial support of the ANR grant “MicroLocal” ANR-15-CE40-0007, and ANR Grant “HodgeFun” ANR-16-CE40-0011. Credits – §1 actually doesn’t contain more than the beginning of [25], and §2 is somehow a nice re-packaging of the general construction presented in loc.cit. However, we believe the formalism of §2 is way more user-friendly for the reader wanting to apply [25] in concrete situations. – Even though we thought they were new at the time we gave these lectures, Joost Nuiten independently obtained results that contain and encompass the material that is covered in §3 (see the two very nice papers [27, 28], that appeared while we were writing this survey). These results must be attributed to him. – Global results presented in §4 rely on the general theory of formal derived (pre)stacks from [9] (see also [6]). Some of these results seem to be new. Notation. – Below are the notation and conventions we use in this paper. Categories, model categories and ∞-categories – We make use all along of the language of ∞-categories. We will only use (∞, 1)-categories, that is categories where all k-morphisms for k ≥ 2 are invertible. By ∞-categories, we will always mean (∞, 1)-categories. – Abusing notation, we will denote by the same letter an ordinary (i.e., discrete) category and its associated ∞-category. – If M is a given model category, we write W M for its subcategory of weak equivalences, and M := M [ W −1 M ] for the associated ∞-category (unless otherwise specified, localization is always understood as the ∞-categorical localization, that is Dwyer-Kan simplicial localization). – Going from M to M is harmless regarding (co)limits: homotopy (co)limits in M correspond to ∞-categorical (co)limits in M. – Conversely, any presentable ∞-category can be strictified to a model category (i.e., is of the form M for some model category M ). In the whole paper, we will only deal with presentable ∞-categories.

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– Let cat∞ be the ∞-category of (small) ∞-categories. It can be obtained as the ∞-category associated with the model category qcat of (small) quasi-categories. – We denote by sSet the model category of simplicial sets (endowed with Quillen’s model structure). The associated ∞-category is denoted by sSet, it is equivalent to ∞Grpd. We denote by spa its stabilization (see §2.1), it is the category of spectra. – When writing an adjunction horizontally (resp. vertically), we always write the left adjoint above (resp. on the left) and the right adjoint below (resp. on the right). This means that when we write an adjunction as F : C −→ ←− D : G, F is the left adjoint. Complexes – The letter k will refer to a fixed field of characteristic zero. – The category modk is the model category of unbounded (cochain) complexes of k-modules from [15] (it is known as the projective model structure), and modk is its associated ∞-category. For this model structure, weak equivalences are quasiisomorphisms, and fibrations are componentwise surjective morphisms. is the model category of complexes of k-modules sitting – The category mod≤0 k in nonpositive degrees, and mod≤0 is its associated ∞-category. For this model k structure, fibrations are componentwise surjective morphisms in degree ≤ −1. – In the sequel, we will consider categories of complexes with an additional algebraic structure (like commutative differential graded algebras, or differential graded Lie algebras). They carry model structures for which fibrations and weak equivalences are exactly the same as the ones for complexes. For these examples, the “freeforget” adjunction with modk (or, mod≤0 k ) is a Quillen adjunction. We refer the reader to the paper [15] for more details. Differential graded algebras – The category cdgak denotes the model category of (unbounded) unital commutative differential graded k-algebras (that is commutative monoids in modk ), and cdgak denotes its associated ∞-category. – A (unbounded) unital commutative differential graded k-algebra will be called a “cdga”. – The category Calgaug is the slice category cdgak/k , i.e., the category of augmented k k-algebras. It is equivalent to the category Calgnu k of non-unital commutative differential graded algebras. This equivalence is actually a Quillen equivalence. – The category cdga≤0 k denotes the model category of cdgas over k sitting in nonpositive degree. The associated ∞-category is denoted by cdga≤0 k . – For any A ∈ cdgak we write modA for the model category of left A-modules, and modA for its associated ∞-category.

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– Note that modA is a stable ∞-category. This can be deduced from the fact that modA is a triangulated dg category. The looping and delooping functors Ω∗ and Σ∗ are then simply given by degree shifting (∗)[−1] and (∗)[1], respectively. – We define cdgaA as the model category of A-algebras (commutative monoids in modA ), and cdgaA is the corresponding ∞-category. – All these definitions have relative counterparts: if A is a cdga and if B is an object of cdgaA , then cdgaA/B is the slice category (cdgaA )/B of relative A-algebras over B. Differential graded Lie algebras and L∞ -algebras – We denote by Liek the model category of (unbounded) diffential graded Lie algebras over k (i.e., Lie algebra objects in modk ). The associated ∞-category is denoted by Liek . – A differential graded Lie algebra over k will be called a “dgla”. – Remark that Liek is equivalent to the localization of the category of L∞ -algebras, with morphisms being ∞-morphisms, with respect to ∞-quasi-isomorphisms (see e.g., [33]). 1. Introduction to pointed formal moduli problems for commutative algebras 1.1. Small augmented algebras. – For any dg-algebra A and any A-module M , we can form the square zero extension of A by M ; we denote it by A ⊕ M where there is no possible ambiguity. We set first some crucial definitions for the rest of the paper: Definition 1.1. – Recall that Calgaug is the ∞-category of augmented cdgas. k – A morphism in Calgaug is called elementary if it is a pull-back (5) of k → k ⊕ k[n] k for some n ≥ 1, where k → k ⊕ k[n] is the square zero extension of k by k[n]. – A morphism in Calgaug is called small if it is a finite composition of elementary k morphisms. – An object in Calgaug is called small if the augmentation morphism  : A → k is k small. (6) We denote by Calgsm sub-∞-category of small objects of Calgaug k the full k . Small objects admits various concrete equivalent algebraic characterizations:

Proposition 1.2 ([25, Proposition 1.1.11 and Lemma 1.1.20]). – An object A of Calgaug k is small if and only if the three following conditions hold: – H n (A) = {0} for n positive and for n sufficiently negative. – All cohomology groups H n (A) are finite dimensional over k. (5) (6)

Since we are working in the ∞-categorical framework, pullback means homotopy pullback. Hence we allow all morphisms in the category Calgsm k , not only small ones.

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– H 0 (A) is a local ring with maximal ideal m, and the morphism H 0 (A)/m → k is an isomorphism. Moreover, a morphism A → B between small objects is small if and only if H 0 (A) → H 0 (B) is surjective. Remark 1.3. – Let us make a few comments on this statement, in order to explain its meaning and link it to classical results in commutative algebra and deformation theory. – Observe that small algebras are nothing but dg-artinian algebras concentrated (cohomologically) in nonpositive degree. – To get a practical grasp to the definitions of elementary and small morphisms, it is necessary to be able to compute homotopy pullbacks in the model category Calgaug k . This is a tractable problem since the model structure on cdgak is fairly explicit, and Calgaug is a slice category of cdgak . k – If a small object A is concentrated in degree zero, the theorem says that A is small if and only if A is a local artinian algebra with residue field k. Let us explain concretely why this holds (the argument is the same as in the general case). If A is a local artinian algebra, then A can be obtained from the residue field as a finite sequence of (classical) small extensions, that is extensions of the form 0 → (t) → R2 → R1 → 0 where R1 , R2 are local artinian with residue field k, and (t) is the ideal generated by a single element t annihilated by the maximal ideal of R2 (hence it is isomorphic to the residue field k). In this way we get a cartesian diagram (7) R2

/k

 cone ((t) → R2 )

 / k ⊕ k[1]

in Calgaug k . Since the bottom horizontal map is surjective in each degree, it is in particular a fibration. Therefore this diagram is also cartesian in Calgaug k , and is isomorphic to a cartesian diagram of the form R2

/k

 R1

 / k ⊕ k[1]

in Calgaug k . Hence the morphism R2 → R1 is elementary. (7)

The cdga structure we put on the cone is the natural one: we require that (t) has square zero.

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– An important part of classical deformation theory in algebraic geometry is devoted to formal deformations of algebraic schemes. For a complete account, we refer the reader to the book [31]. Following the beginning of [25], we will explain quickly how small morphisms fit in this framework. Given a algebraic scheme Z over k (that will be assumed to be smooth for simplicity), the formal deformation theory of Z deals with equivalence classes of cartesian diagrams Z

/Z

 Spec(k)

 / Spec(A)

where A is a local artinian algebra with residue field k. This construction defines a deformation functor Def Z from the category of local artinian algebras to sets. It is possible to refine the functor Def Z to a groupoid-valued functor that associates to any A the groupoid of such diagrams. Since any groupoid defines a homotopy type (by taking the nerve), we get an enriched functor Def Z from local artinian algebras to the homotopy category of topological spaces, such that π0 ( Def Z ) = Def Z . The first important case happens when A = k[t]/t2 . In this case, KodairaSpencer theory gives a bijection between isomorphism classes of deformations of X over Spec(k[t]/t2 ) and the cohomology group H1 (Z, TZ ). In other words, H1 (Z, TZ ) is the tangent space to the deformation functor Def Z . The next problem of the theory is the following: when can an infinitesimal deformation θ in H1 (Z, TZ ) be lifted to Spec(k[t]/t3 )? The answer is: exactly when [θ, θ] vanishes in H2 (Z, TZ ). This can be interpreted in the framework of derived algebraic geometry as follows: the extension 0 → (t2 ) → k[t]/t3 → k[t]/t2 → 0 yields a cartesian diagram k[t]/t3

/k

 k[t]/t2

 / k ⊕ k[1]

in Calgaug k . This gives a fiber sequence of homotopy types Def Z (k[t]/t

3

) → Def Z (k[t]/t2 ) → Def Z (k ⊕ k[1])

hence a long exact sequence · · · → π0 ( Def Z (k[t]/t3 )) → π0 ( Def Z (k[t]/t2 )) → π0 ( Def Z (k ⊕ k[1]).

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It turns out that the set π0 ( Def Z (k ⊕ k[1]) of equivalence classes of deformations of Z over the derived scheme Spec(k ⊕ k[1]) is isomorphic to H2 (Z, TZ ). Hence the obstruction class morphism Def Z (k[t]/t2 ) ∼ = H1 (Z, TZ ) −→ 7−→

θ

H2 (Z, TZ ) [θ, θ]

can be entirely understood by writing k[t]/t3 → k[t]/t2 as an elementary morphism. 1.2. The ∞-category of formal moduli problems. – We start by introducing formal moduli problems in the case of cdgas: Definition 1.4. – A formal moduli problem (we write fmp) is an ∞-functor X : Calgsm k → sSet satisfying the following two properties: – X(k) is contractible. – X preserves pull-backs along small morphisms. The second condition means that given a cartesian diagram N

/A

 M

 /B

in Calgsm k where A → B is small, then X(N )

/ X(A)

 X(M )

 / X(B)

is cartesian. Remark 1.5. – Observe that the second condition is stable under composition and pullback. Hence it is equivalent to replace in this condition small morphisms with elementary morphisms. We claim that we can even replace elementary morphisms with the particular morphisms k → k ⊕ k[n] for every n ≥ 1. Indeed, consider a cartesian diagram /A N  M

f

 /B

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where f is elementary, that is given by a cartesian diagram A

/k

 B

 / k ⊕ k[n].

f

If we look at the diagram X(N )

/ X(A)

/ X(k)

 X(M )

 / X(B)

 / X(k ⊕ k[n])

and assume that X preserves pullbacks along the morphisms k → k ⊕ k[n], then the right square is cartesian, so the left square is cartesian if and only if the big square is cartesian (which is the case). Corollary 1.6. – A functor X : Calgsm k → sSet is a fmp if and only if X(k) is contractible and preserves pull-backs whenever morphisms in the diagram are surjective on H 0 . Proof. – Assume that X preserves pull-backs whenever morphisms in the diagram are surjective on H 0 . Consider a cartesian diagram of the following type, with n ≥ 1: N

/k

 M

 / k ⊕ k[n].

Since M is an augmented k-algebra, the map M → k⊕k[n] is surjective on H 0 . Hence X(N )

/ X(k)

 X(M )

 / X(k ⊕ k[n])

is cartesian. According to the preceding remark, this implies that X is a fmp. The reverse implication is obvious thanks to the last sentence of Proposition 1.2.

We write FMPk for the full sub-∞-category of Fun(Calgsm k , sSet) consisting of formal moduli problems.

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1.3. A glimpse at the description of FMPk . – In this section, we explain some heuristical aspects of the proof of the following theorem: Theorem 1.7 ([25], [29]). – There is an equivalence of ∞-categories Liek → FMPk . Remark 1.8. – Again, we discuss various points in this theorem related to more classical material. – Let us first give a naive idea about how the ∞-functor can be defined on a “sufficiently nice” dgla. The procedure is rather classical; we refer the reader to [17] and [18] for further details. If g is a dgla of over k, we can consider its (discrete) Maurer-Cartan set MC(g) = {x ∈ g1 such that dx + [x, x]g = 0}. The “good” object attached to g is the quotient of MC(g) under gauge equivalence. This can be better formulated using a simplicial enrichment as follows: for any n ≥ 0, let Ω• (∆n ) be the cdga of polynomial differential forms on the n-simplex ∆n . The collection of the Ω• (∆n ) defines a simplicial cdga. Then we define the simplicial set M C (g) as follows (8): M C (g)n = MC(g ⊗ Ω• (∆n )). This being done, we can attach to g a deformation functor Def g : Calgsm → sSet defined by Def g (A) = M C (g ⊗k IA ), where IA is the augmentation ideal of A. This defines (again in good cases) a formal moduli problem. – The main problem of the Maurer-Cartan construction is that the functor g → Def g does not always preserve weak equivalences. For dglas that satisfy some extra conditions (like for instance nipoltence conditions), Def g will be exactly the fmp we are seeking for. We will see very soon how Lurie circumvents this problem using the Chevalley-Eilenberg complex. – To illustrate an example where the Maurer-Cartan construction appears, let us come back to deformation theory of algebraic schemes in a slightly more differential-geometric context: instead of algebraic schemes we deform compact complex manifolds. We can attach to a complex compact manifold Z the Dolbeault complex of the holomorphic tangent bundle TZ , which is the complex ∂

∂

0 → C ∞ (TZ ) − → A 0,1 (TZ ) − → ··· We see this complex as a dgla over C, the Lie structure being given by the classical Lie bracket of vector fields and the wedge product on forms. Then it is well known (see e.g., [19, Lemma 6.1.2]) that deformations of Z over an artinian algebra A yield IA -points in the set-valued deformation functor associated with the dgla (A 0,• (TZ ), ∂). (8)

The π0 of M C (g) is exactly the quotient of MC(g) under gauge equivalence.

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Let us now explain the construction of the equivalence from Liek to FMPk . First we start by recalling the following standard definitions: Definition 1.9. – For any dgla g, we define the homological and cohomological Chevalley-Eilenberg complexes CE• (g) and CE• (g) as follows: – As a graded vector space, CE• (g) = S (g[1]) is the (graded) symmetric algebra of g[1]. The differential is obtained by extending, as a degree 1 graded coderivation, the sum ot the differential g[1] → g[2] with the Lie bracket S2 (g[1]) → g[2]. Jacobi identity and Leibniz rule ensure that this coderivation squares to zero. The complex CE• (g) is actually a (coaugmented, counital, and conilpotent) cocommutative coalgebra object in the category of complexes. – CE• (g) is the linear dual of CE• (g), it is an augmented cdga. Remark 1.10. – Observe that the above definition still makes sense for an L∞ -algebra g. Indeed, an L∞ -algebra structure on g is defined as a degree 1 graded codifferential that makes S (g[1]) a coaugmented counital cocommutative differential graded coalgebra. L

– It is possible to prove that CE• (g) ' k ⊗U(g) k and CE• (g) ' RHomU(g) (k, k). – Let V be an object of modk , and let free (V ) be the free dgla generated by V . Then CE• (free (V )) and CE• (free (V )) are quasi-isomorphic to the square zero extensions k ⊕ V [1] and k ⊕ V ∗ [−1] respectively. – For any cdga A and any dgla g of finite dimension over k, there is a map (1)

Homcdgak (CE• (g), A) → MC(g ⊗ IA ) which is in good cases an isomorphism (where IA is the augmentation ideal of A). To see how the map is constructed, let ϕ be an element in Homcdgak (CE• (g), A). Forgetting the differential, it defines an algebra morphism from the completed algebra b S (g∗ [−1]) to A, and in fact to IA (since the morphism is pointed). In particular, we have a map φ : g∗ [−1] → IA , and since g is finite-dimensional, we can see the morphism φ as a map k[−1] → g⊗IA , hence an element x of (g⊗IA )1 . Now we have a commutative diagram g∗ [−1]

φ

/A

/A O ?A

[. , . ]∗

 g∗ [−1] ⊗k g∗ [−1]

dA

φ⊗φ

/ A ⊗k A.

Unwrapping what it means expressing φ with x, we end up exactly with the Maurer-Cartan equation dx + [x, x] = 0. The reason why the map (1) is not always bijective is that CE• (g) is not the symmetric algebra of g∗ [−1], but it is the completed symmetric algebra.

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– There is a simplicially enrichement of (1), given by Homcdgak (CE• (g), A) → M C (g ⊗ IA ). Apart from the completion issue that we have already discussed, this morphism may not be an equivalence as CE• (g) may not be a cofibrant object in cdgak . Sketch of the construction of the equivalence. – The Chevalley-Eilenberg construction preserves weak equivalences, hence defining an ∞-functor aug CE• : Lieop k → Calgk .

The functor (CE• )op commutes with small colimits (see [25, Proposition 2.2.17]), so since Liek is presentable, CE• admits a left adjoint. We call this adjoint D. Hence we have an adjunction op • −→ D : Calgaug k ←− Liek : CE

that can be seen as some version of Koszul duality. The main point in this step is that the Chevalley-Eilenberg functor does only commute with small homotopy limits, not usual small limits. Hence the adjoint functor is only defined in the ∞-categorical setting (i.e., it does not come from a Quillen adjunction). – We define an ∞-functor from Liek to Fun(Calgaug k , sSet) as follows:

∆(g) = HomLieop g, D(−) = HomLiek D(−), g . k The functor ∆ will define the equivalence we are seeking for. – Let us explain why ∆ factors through FMPk . We introduce the notion of good object: a dg-Lie algebra L is good if there exists a finite chain 0 = L 0 → L 1 → . . . → L n = L such that each of these morphisms appears in a pushout diagram free k[−ni − 1]

/ Li

 {0}

 / L i+1

in Liek , or equivalently a pullback diagram L i+1

 Li

/ {0}  / free k[−ni − 1]

gd op in Lieop k . We denote by Liek the full subcategory of Liek consisting of good objects. We see that good objects are formally the same as small ones in Lieop k , using the sequence of objects free k[−n − 1] instead of k ⊕ k[n]. This will be formalized using the various notions of deformation contexts developed in the next section.

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– The next step consists of proving that if g is good, the counit morphism DCE• (g) → g in Lieop k is an equivalence. This is the crucial technical input and will be proved in Proposition 2.30. By purely formal arguments (see Proposi• tion 2.22), this implies that the adjunction D −→ ←− CE defines an equivalence of gd categories between Calgsm k and Liek .

– Using this, if we have a cartesian diagram N

/k

 M

 / k ⊕ k[n]

where N and M are small, then D(N )

/ {0}

 D(M )

 / D(k ⊕ k[n])

is cartesian in Lieop k , and therefore ∆(g)(N )

/∗

 ∆(g)(M )

 / ∆(g)(k ⊕ k[n])

is also cartesian in sSet. This implies that ∆ is an object of FMPk . Hence ∆ factors through the category FMPk .

2. General formal moduli problems and Koszul duality 2.1. Deformation contexts and small objects. – In this section, we will explain how the notions of small and elementary morphism make sense in a broader categorical setting. We start by some general facts on ∞-categories. – If C is an ∞-category with finite limits (9), its stabilization Stab(C) can be described as the ∞-category of spectrum objects (also called infinite loop objects) in C. An object of Stab(C) is a sequence E = (En )n∈Z of pointed objects (10) together with weak equivalences En → ΩEn+1 , where Ω := Ω∗ denotes the based loop functor: Ω(c) = ∗ × ∗. We often write En = Ω∞−n E. c

(9)

This hypothesis will be always implicit in the sequel. It implies among other things that C has a terminal object ∗. (10) The ∞-category of pointed objects is the coslice ∞-category ∗/C under the terminal object.

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– If C is stable, then Stab(C) is naturally equivalent to C via the map sending (En )n to E0 . – If C is an ∞-category with finite limits and c is an object of C, then the stabilization Stab(c/C) of its coslice ∞-category c/C is equivalent to Stab(C). Indeed, the ∞-categories of pointed objects in C and c/C are themselves equivalent. – If C is an ∞-category with finite limits and c is an object of C, then the stabilization Stab(C/c ) of its slice ∞-category C/c is the category of spectrum objects in the ∞-category idc /(C/c ) of sequences c → d → c such that the composition is the identity. – If C = sSet, then Stab(C) is the ∞-category spa of spectra (that is spectrum objects in spaces). Definition 2.1. – A pair (C, E), where C is a presentable ∞-category with finite limits and E is an object of Stab(C), is called a deformation context. Given a deformation context (C, E): – A morphism in C is elementary if it is a pull-back of ∗ → Ω∞−n E for n ≥ 1 (where ∗ is a terminal object in the category C). – A morphism in C is small if it can be written as a finite sequence of elementary morphisms. – An object c is small if the morphism c → ∗ is small. We let (C, E)sm be the full subcategory of C spanned by the small objects. When it is clear from the context, we may abuse notation and write Csm := (C, E)sm . Let us give two examples of deformation contexts: Example 2.2. – If C = modk , which is already stable, then we have an equivalence modk

−→ ˜

Stab(C)

M

7−→

(M [n])n .

In this context we will mainly consider the spectrum object E = (k[n + 1])n∈Z . Remark 2.3. – Instead of working over the ground field k, we can work over an arbitrary cdga A. Then we can take C = modA and E = (A[n + 1])n∈Z in Stab(C). Example 2.4. – The category Stab(Calgaug k ) is equivalent to modk . Indeed, we have nu an equivalence Calgaug ' Calg , and the based loop functor Ω0 in Calgnu k k (0 is the k terminal nonunital cdga) sends a non-unital algebra R to R[−1] equipped with the trivial product; hence the equivalence modk

−→ ˜

Stab(Calgaug k )

M

7−→

(k ⊕ M [n])n∈Z .

In this context we will mainly consider the spectrum object E = (k ⊕ k[n])n∈Z .

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Remark 2.5. – One can prove in a similar way that the stabilization Stab(cdgaA/A ) of the ∞-category cdgaA/A of A-augmented A-algebras, where A is in cdgak , is equivalent to modA . Moreover, if cdgak/A is the ∞-category of A-augmented k-algebras, then Stab (cdgak/A ) ' Stab (cdgaA/A ) ' modA . In this case a natural spectrum object to consider is (A ⊕ A[n])n∈Z . We have the following obvious (though very useful) lemma, which allows to transfer deformation contexts along adjunctions: Lemma 2.6. – Suppose (C, E) is a deformation context. If we are given an adjunction 0 T 0 : C0 −→ ←− C : T , then T preserves small limits, so that (C , T (E)) is a deformation context whenever C0 is presentable. Besides, in this case T induces a functor from Csm to (C0 )sm . Let us give four examples of this transfer principle: (11) ` Example 2.7. – Given an A-module L, the push-out functor − L 0 : L/modA → modA along the zero map L → 0 admits a right adjoint, being the functor sending 0 an A-module M to the zero map L → M . Hence the deformation context from Example 2.2 and Remark 2.3 can be transferred to L/modA . Example 2.8. – The relative cotangent complex functor LA/− : Calgaug A → modA admits a right adjoint, which is the split square zero extension functor M 7→ A ⊕ M . Hence the deformation context from Example 2.4 and Remark 2.5 can be obtained by transfer from the one given in Example 2.2 and Remark 2.3. Example 2.9. – The relative cotangent complex functor LA/− : cdgak/A →

LA/k /

modA

d

admits a right adjoint, being the functor sending a morphism LA/k → M in modA to the (non-necessarily split) square zero extension A ⊕ M [−1] that it classifies. d

Hence the deformation context from Example 2.7 (with L = LA/k ) can be transferred to cdgak/A . Example 2.10. – The forgetful functor Calgaug A → cdgak/A is a right adjoint: its left adjoint is A⊗−. Hence the deformation context from Example 2.9 can also be obtained by transfer from the one given in Example 2.8. The main observation is that formal moduli problems make sense with Calgaug k being replaced by any deformation context (C, E). We write FMP(C, E) for the ∞-category of formal moduli problems associated with it.
Then FMP Calgaug k , (k ⊕ k[n])n = FMPk . (11)

In all these examples, the use of transfer is not strictly necessary. Indeed, all the categories involved have a stabilization that is equivalent to modA .

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2.2. Dual deformation contexts and good objects. – We introduce the dual concept of a deformation context: Definition 2.11. – A pair (D, F ), where D is a presentable ∞-category and F is an object of Stab(Dop ), is called a dual deformation context. If (D, F ) is a dual deformation context, (Dop , F ) is not in general a deformation context because Dop is almost never presentable. Example 2.12. – The first easy example is for D = modA . Nevertheless, this example is of high interest (see Lemma 2.16 below). Since modA is stable, so is modop A , and the looping and delooping functors are swapped when passing to the opposite category. Hence spectrum objects in modop A
are of the form (M [−n])n for some object M . In particular, modA , (A[−n − 1])n is a dual deformation context. Just like deformation contexts, dual deformation contexts can be transported using adjunctions: assume to be given a dual deformation context (D, F ) as well as an 0 0 adjunction T : D −→ ←− D : ι. Then it is easy to see that (D , T (F )) is a dual deformation context on D0 whenever D0 is presentable, since T preserves colimits (hence limits in the opposite category). Using this, one can build a lot of dual deformation contexts starting from modA . Example 2.13. – For instance, the adjunction free : modA −→ ←− LieA : forget
yields the dual deformation context LieA , free(A[−n − 1])n . Example 2.14. – Let T be an A-module. Then the pull-back functor − ×T 0 : modA/T → modA along the zero morphism 0 → T admits a left adjoint: it is the 0

functor sending an A-module S to the zero morphism S → T . This yields the dual
0 deformation context modA/T , (A[−n − 1] → T )n . Definition 2.15. – Given a dual deformation context (D, F ), an object (resp. morphism) of D is good if it is small when considered as an object (resp. morphism) of Dop . More explicitely, if ∅ denotes the initial element of D, an object b of D is good if there is a finite sequence of morphisms fm f2 f1 ∅ = bm −−→ · · · −→ b1 −→ b0 ∼ =b

that are pushouts along Fn → ∅ in D: i.e., each fi fits into a pushout square Fn  ∅

/ bi 

fi

/ bi−1

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in D for some n ≥ 1. We let (Dop , F )gd be the full subcategory of Dop spanned by the good objects, which we might simply denote Dgd if there is no ambiguity. (12) Let us give a somehow nontrivial example. Let A
be a k-algebra , and consider the dual deformation context modA , (A[−n − 1])n .

Lemma 2.16. – An element of modA is good for the dual deformation context
modA , (A[−n − 1])n if and only if it is perfect and cohomologically concentrated in positive degrees. Proof. – Recall that a perfect complex is the same as a dualizable object, which means a complex quasi-isomorphic to a finite complex consisting of projective A-modules of finite type. Let K be a perfect complex concentrated in positive degree and prove, by induction on the amplitude, that K is good. If the amplitude is 0 then K is quasiisomorphic to P [−n], for n ≥ 1, with Ar = P ⊕ Aq . We have the following push-out square in modA : /0 Aq [−n − 2]  / Aq [−n − 1].

 0

Hence Aq [−n − 1] is good. We then also have another push-out square: Ar [−n − 1]

/ Aq [−n − 1]

 0

 / P [−n].

Hence the morphism Aq [−n − 1] → P [−n] is good, and since Aq [−n − 1] is good, then P [−n] is good as well. Performing the induction step is now easy: let K be a positively graded complex of finitely generated projective A-modules of some finite amplitude d > 0, let n > 0 be the index where K starts and let P = Kn . We have a push-out square:

(12)

A is concentrated in degree 0.

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P [−n − 1]

/ τ >n K

 0

 / K,

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105

where τ >n is the stupid truncation functor, and using again that Ar = P ⊕ Aq we get another push-out square: Ar [−n − 1]

/ Aq [−n − 1] ⊕ τ >n K

 0

 / K.

Hence the morphism Aq [−n − 1] ⊕ τ >n K → K is good. But τ >n K is good (by induction on the amplitude) and Aq [−n − 1] is good as well, thus so is K. This finishes the induction step. For the converse statement, it suffices to observe that, given a push-out square A[−n − 1]

/K

 0

 /L

in modA , where n ≥ 1, and K is perfect and concentrated in positive degrees, then so is L. . Similarly, one
can prove that small objects for the deformation context modA , (A[n + 1])n are perfect complexes of A-modules cohomologicaly concentrated in negative degrees. We leave it as an exercise to the reader. Remark 2.17. – If we replace the k-algebra A by a bounded cdga concentrated in non-positive degrees, then one can still prove that good objects are perfect A-modules that are cohomologically generated in positive degree. In other words, a good object is quasi-isomorphic, as an A-module, to an A-module P having the following property: as a graded A-module, P is a direct summand of A⊗V , where V is a finite dimensional positively graded k-module. 2.3. Koszul duality contexts. – We now introduce the main notion that is needed to state Lurie’s theorem on formal moduli problems in full generality: Definition 2.18. – (1) A weak Koszul duality context is the data of : – a deformation context (C, E), – a dual deformation context (D, F ), op 0 – an adjoint pair D : C −→ ←− D : D ,

such that for every n ≥ 0 there is an equivalence En ' D0 Fn . (2) A Koszul duality context is a weak Kozsul duality context satisfying the two additional properties: – For every good object d of D, the counit morphism DD0 d → d is an equivalence.

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– The functor Θ : HomD (Fn , −) : D → Stab(sSet) = spa is conservative and preserves small sifted colimits. With a slight abuse of notation, we will denote the whole package of a (weak) Koszul duality context by op 0 D : (C, E) −→ ←− (D , F ) : D .

Remark 2.19. – The name “Koszul duality context” has been chosen in agreement with the first non-trivial example we will deal with (see proposition 2.30), as it reflects the well-known Koszul duality between commutative and Lie algebras. Similarly, the characterization of associative formal moduli problems [25, §3], resp. En formal moduli problems [25, §4], can be proven with the help of a Koszul duality context that reflects the Koszul duality for associative algebras, resp. En -algebras. We actually expect that a pair ( O , P ) of Koszul dual operads always lead to a Koszul duality context between O -algebras and P -algebras, allowing to show that formal moduli problems for O -algebras are P -algebras. (13) Observe that there may be more objects of D for which the counit morphism is an equivalence, than just good objects. We call them reflexive objects. Example 2.20. – The only elementary Koszul duality context we can give at this stage is the following: if A is a bounded cdga concentrated in nonpositive degrees,

op ∨ (−)∨ = HomA (−, A) : modA , (A[n + 1])n −→ ←− modA , (A[−n − 1])n : HomA (−, A) = (−) . To prove that it is indeed a Koszul duality context, we use Lemma 2.16 and Remark 2.17: a good object in modop A is a perfect A-module (generated in non-negative degrees), so it is isomorphic to its bidual. Lastly, the functor Θ is simply the forgetful functor modA → modk ' Stab(modk ) → Stab(sSet) ' spa, which is conservative. Note that there are strictly more reflexive objects than good ones: for instance, ⊕n≥0 A[−n] is reflexive, but not good (it is an example of an almost finite cellular object in the terminology of [13]). There is a peculiar refinement of the above example, that will be useful for later purposes.

(13)

This is indeed the case, see [4].

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Example 2.21. – Let A be a bounded cdga concentrated in nonpositive degrees, and let L be an A-module. We have a weak Koszul duality context (14)

L/ 0 0 op ∨ ∨ modA , (L → A[n + 1])n −→ (−)∨ : ←− (modA/L∨ ) , (A[−n − 1] → L )n : (−) , k∨

k

where the right adjoint sends K −→ L∨ to the composed morphism L → (L∨ )∨ −→ K ∨ . We claim that this is actually a Koszul duality context. Indeed, one first observes that good objects are given by morphisms K → L∨ where K is a good object in modA . Hence they are again isomorphic to their biduals. Lastly, the functor Θ is the composition of the pull-back functor − × 0 : (modA/L∨ )op → modA along the L∨

zero morphism 0 → L∨ , with the forgetful functor modA → spa from the previous example. They are both conservative, hence Θ is. There are a few properties that can be deduced from the definition, which are absolutely crucial. Proposition 2.22 ([25, Proposition 1.3.5]). – Assume that we are given a Koszul duality context op 0 D : (C, E) −→ ←− (D , F ) : D (A) For every n ≥ 0, DEn ' Fn . (B) For every small object M in C, the unit map M → D0 D( M ) is an equivalence. (C) There is an equivalence of categories (C, E)sm o

D D0

/

(Dop , F )gd .

(D) Consider a pullback diagram N

/A

 M

 / B,

f

where f is small and M is small. Then the image of this diagram by D is still a pullback diagram. (14)

Observe that, contrary to what one could think, it is not required that L is perfect, or even just `

k

reflexive. Indeed, for L → M and K → L∨ , the following commuting diagrams completely determine each other: K k

 |

L∨

f∨

/ M∨ `∨

and

/ (L∨ )∨

L `



M

f



k∨

/ K∨.

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Proof. – The proof is clever but completely formal, and doesn’t require any extra input. (A) This is straightforward: DEn ' DD0 Fn ' Fn since Fn is good. (B) We proceed in two steps. First we prove it if M is for the form D0 ( N ) for some good object N . This is easy: the composition ∼ id ◦ D0 ⇒ (D0 ◦ D) ◦ D0 ∼ D0 = = D0 ◦ (D ◦ D0 ) ⇒ D0 ◦ id ∼ = D0 is an equivalence (it is homotopic to the identity). This means that the composition D0 D(D0 N ) → D0 N is obtained up to equivalence by applying D0 to the equivalence DD0 ( N ) → N , hence it is also an equivalence. Now we prove that every small object can be written as D0 (X) where X is good. This will in particular imply that D maps small objects to good objects (15). Every small morphism can be written as a composition of a finite number i of elementary morphisms, we will argue by induction on the number i. If i = 0, then since D0 is a right adjoint it preserves limits. Hence D0 (∅) = ∗, where ∅ is the initial element of D. Since ∅ is good, we are done. We now deal with the induction step. Let us consider a cartesian diagram /∗ M i+1  Mi

 / En

in C such that M i → ∗ is obtained by composition of at most i elementary morphisms. By induction, M i+1 = D0 ( N ) for some good object N . This implies that D( M i ) is good. Let X be the pullback the object of D making the diagram X

/∗

 D( M i )

 / D(En )

cartesian in Dop . Since D( M i ) is good and D(En ) ' Fn , X is good. Now we apply D0 , which preserves limits. We get a cartesian diagram isomorphic to D0 (X)

/∗



 / En

Mi

This proves that M i+1 is isomorphic to D0 (X). (C) This is a direct consequence of (B). (15)

This is nontrivial: D preserves colimits, and small objects are defined using limits.

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(D) We can reduce to diagrams of the form /∗ N  M

 / En .

Then the property follows from (C). 2.4. Morphisms of (weak) Koszul duality contexts. – We will now introduce the notion of morphisms between weak Koszul duality contexts. It will be extremely useful in the sequel. Definition 2.23. – Let op 0 D1 : (C1 , E1 ) −→ ←− (D1 , F1 ) : D1

and op 0 D2 : (C2 , E2 ) −→ ←− (D2 , F2 ) : D2

be two weak Koszul duality contexts. A weak morphism between these two duality contexts consists of two additional pairs of adjoint functors appearing (vertically) in the diagram

S

/

D1

C1O o

D01

T

Dop 1O

Y

 C2 o

Z

 / op D2

D2 D02

such that: (A) The diagram consisting of right adjoints C1 o O

D01

commutes. Dop 1O

T

Z

C2 o

Dop 2

D02

(B) We have equivalences Z(F2,n ) ' F1,n . Example 2.24. – Let L be an A-module. We then have the following weak morphism from the Koszul duality context of Example 2.21 to the one of Example 2.20: (−)∨

o mod O A

L/

cofib

(−)∨

/

(modA/L∨ )op O

0

M 7→(L→M )

 modA o

(−)∨ (−)∨

/

fib

0

K7→(K →L∨ )

 modop A ,

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where cofib = 0

`

− and fib = 0 × −. L∨

L

Notice that the natural equivalence θ

from condition (A) above has a mate

which we define pictorially as the composition

where id and id are used to depict units and counits of vertical adjunctions. In other words, the mate θ is the composition SD01 ⇒ SD01 ZY ∼ = ST D02 Y ⇒ D02 Y . Remark 2.25. – The commutativity of the square of right adjoints implies the commutativity of the square of left adjoints. Hence there is a natural equivalence D2 S ∼ = Y D1 that pictorially reads , and the mate θ can also be identified with the following composition: SD01 ⇒ D02 D2 SD1 ∼ = D02 Y D1 D01 ⇒ D02 Y

or, pictorially,

.

Definition 2.26. – One says that the above commuting square of right ajoints satisfies the Beck-Chevalley condition locally at d, where d is an object of D1 , if the mate θd : SD01 (d) → D02 Y (d) is an equivalence. Definition 2.27. – A morphism of weak Koszul duality contexts is a weak morphism such that, borrowing the above notation: (C) The functor Y is conservative, preserves small sifted limits, and sends good objects to reflexive objects. (D) The commuting square of right adjoints satisfies the Beck-Chevalley condition locally at good objects. The main feature of this definition is a result allowing to transfer Koszul duality contexts along morphisms: Proposition 2.28 (Transfer theorem). – Assume to be given a morphism between two weak Koszul duality contexts. If the target deformation context is a Koszul duality context, then the source is also a Koszul duality context. Proof. – We take the notation of Definition 2.23. We have two functors Θ1 and Θ2 , defined on D1 and D2 respectively and with values in spa, defined by ( Θ1 (d)n = HomD1 (F1,n , d) Θ2 (d)n = HomD2 (F2,n , d). For any object d in D1 , we have Θ1 (d)n = HomD1 (F1,n , d) ' HomDop (d, Z(F2,n )) 1 ' HomDop (Y (d), F2,n ) ' HomD2 (F2,n , Y op (d)) 2 ' Θ2 (Y op (d))n .

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So Θ1 = Θ2 ◦ Y op , and thus Θ1 is conservative (because Y and Θ2 are). Moreover, Y preserves small sifted limits, so Y op preserves small sifted colimits, and hence so does Θ1 (because Θ2 does too). Lastly, we consider the following diagram in Dop 2 , where d is still an object in D1 : Y D1 D01 KS

+3 Y KS

Y ◦ co-unit

co-unit ◦Y

∼

 D2 SD01

D2 ◦θ

+3 D2 D0 Y. 2

We claim that it commutes. Indeed, writing the mate explicitely gives a diagram Y D1 D01 KS ai

+3 Y KS

Y ◦ co-unit

co-unit ◦Y

∼

 D2 SD01

co-unit ◦Y D1 D01

D2 D02 Y ◦ co-unit

D2 ◦ unit ◦SD01

 D2 D02 D2 SD01 ks

D2 D02 Y KS

∼

+3 D2 D0 Y D1 D0 , 2 1

where the two triangles commute. Given now a good object d in D1 , we have that Y (d) is reflexive (after (C) in Definition 2.27): thus the co-unit morphism of the adjunction (D2 , D02 ) is an equivalence on Y (d). Besides, θd is also an equivalence (thanks to (D) in Definition 2.27). Therefore Y ◦ co-unit(d) is an equivalence and thus, by conservativity of Y , the co-unit morphism D1 D01 (d) → d is an equivalence. Example 2.29. – Going back to Example 2.24, one sees that the weak morphism is a morphism if and only if L is reflexive (in which case, L∨ is so as well). Indeed: – The functor fib is conservative. – If K → L∨ is good then K itself is good, thus perfect, in modA , and thus fib(K → L∨ ) is reflexive as soon as L∨ is so. – the mate θK→L∨ is the natural morphism cofib(L → K ∨ ) → fib(K → L∨ )∨ , which is an equivalence if and only if the unit morphism L → (L∨ )∨ is an equivalence. Hence the Koszul duality context from Example 2.21 is in general not obtained by transfer from the one of 2.20, but it is in the case when L is reflexive. Proposition 2.30. – The adjunction op • −→ D : (Calgaug k , k ⊕ k[n]) ←− (Liek , free k[−n − 1]) : CE

defines a Koszul duality context.

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Proof. – We consider the diagram (see Remark 1.10): CE•

o Calgaug O k

Lieop Ok

V 7→k⊕V [−1]

free Hom(−,k)

modk o

modop k .

We can fill it with left adjoints everywhere. This gives the following (nice!) diagram:

L

CE•

V 7→k⊕V [−1]

 modk o

/

D

o Calgaug Ok

Lieop Ok

forget

Hom(−,k)

/

Hom(−,k)

free

 modop k ,

where L(R) = Lk/R ' LR/k ⊗R k[1]. We claim that these four adjunctions define a morphism of weak Koszul duality contexts. Properties (A) and (B) are true, so that we have a weak morphism. We will now prove that it is actually a morphism, so that we get the result, using Proposition 2.28 and the fact that the bottom adjunction is a Koszul duality context (Example 2.20). The functor forget is conservative. Let us now prove that, if g is good, then it is equivalent to a very good dgla: a good dgla with underlying graded Lie algebra being generated by a finite dimensional graded vector space sitting in positive degrees. Lemma 2.31. – Any good dgla g is quasi-isomorphic to a very good one. Proof of the lemma. – We first observe that 0 is very good. We then proceed by induction: assume that g is very good, and consider a dgla g0 obtained by a pushout free k[−n − 1]

/g

 0

 / g0

in Liek . The first trick is that free k[−n − 1] is cofibrant and g is fibrant (every object in Liek is). Hence the morphism free k[−n − 1] in the ∞-category Liek can be represented by a honest morphism in Liek . The next step consists in picking a cofibrant replacement of the left vertical arrow. A cofibrant replacement is given by the morphism n o id free k[−n − 1] → free cone (k −→ k)[−n − 1] .

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Hence our pushout in Liek is represented by the following honest non-derived pushout in Liek : /g free k[−n − 1]  n o id free cone (k −→ k)[−n − 1]

 / g0

n o id This pushout is obtained by making the free product of g and free cone (k −→ k)[−n − 1] , and then by taking the quotient by the image of the ideal generated by free k[−n − 1]. This completes the induction step: g0 is still very good. This in particular shows that forget sends good objects to reflexive objects for the dual deformation context (modk , k[−n−1]). Lastly, thanks to [25, Proposition 2.1.16], op the forgetful functor from Lieop k to modk preserves small sifted limits. Thus (C) holds. It remains to prove (D), which is the main delicate point of the proof. Recall for that purpose that, for a dgla g, θg is defined as the following composition, where we omit the forget functor (its appearance being obvious):

L CE• (g) → L CE• (free g) → L(k ⊕ g∗ [−1]) → g∗ . At this point it is important to make a rather elementary observation: the composed cdga morphism CE• (g) → CE• (free g) → k ⊕ g∗ [−1] is nothing but the projection onto the quotient by the square I 2 of the augmentation ideal I = ker(CE• (g) → k) whenever g is very good (16). • f (g) as a sub-cdga We now introduce the uncompleted Chevalley-Eilenberg cdga CE S(g∗ [−1]) ⊂ CE• (g) (it is obviously a graded subalgebra, and it can be easily checked as an exercise that it is stable under the differential). The quotient by the square of its augmentation ideal is still k ⊕ g∗ [−1], and we have the following commuting diagram and its image through the left-most vertical adjunction of our square:
• • f (g) f (g) CE L CE ιg

 CE• (g)

% / k ⊕ g∗ [−1]

θeg

L(ιg )


L CE• (g)

θg

# / g∗ .

In order to prove that θg is an equivalence when g is good, we will prove that both θeg and L(ιg ) are. Let us start with the following lemma: Lemma 2.32. – Let A be an augmented cdga, with augmentation ideal J , that is cofibrant as a cdga. Then the morphism L(A) → J / J 2 [1] associated with the projection A → k ⊕ J / J 2 is an equivalence. (16) Indeed, in this case the natural map S k (g∗ [−1]) → S k (g[1])∗ is an isomorphism. Thus, forgetting ˆ ∗ [−1]). the differential, CE• (g) = S(g

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Proof. – First of all, the projection A → k ⊕ J / J 2 is an actual morphism in the category cdgak , so that we have a factorization L(A) ' LA/k ⊗A k[1] → Ω1A/k ⊗A k[1] → J / J 2 [1], where: – the morphism LA/k ⊗A k → Ω1A/k ⊗A k is an equivalence because A is cofibrant; – the second morphism is an isomorphism in modk . The lemma is proved. We then observe that when g is very good (which we can always assume without • f (g) is cofibrant and thus loss of generality when dealing with good dglas), then CE θeg is an equivalence (17). • f (g), which implies that the Lastly, it can be shown that CE• (g) is flat over CE natural map • LCE f • (g) CE (g) −→ LCE• (g)/k f • (g)/k ⊗CE

is also an equivalence, which shows (after applying − ⊗CE• (g) k) that L(ιg ) is an equivalence. 2.5. Description of general formal moduli problems. – Assume that we are given a Koszul duality context op 0 D : (C, E) −→ ←− (D , F ) : D . We can define a functor Ψ : D → Fun(C, sSet) by the composition Y

◦D

Ψ : D −→ Fun(Dop , sSet) −−→ Fun(C, sSet). where Y is the Yoneda functor d → HomDop (d, −). Theorem 2.33 (Lurie [25]). – Given a Koszul duality context op 0 D : (C, E) −→ ←− (D , F ) : D ,

the functor Ψ factors through FMP(C, E) and the induced functor Ψ : D → FMP(C, E) is an equivalence. Sketch of proof. – The first point is a direct consequence of Proposition 2.22 (D). The proof that Ψ is an equivalence proceeds on several steps. (17)

Here is another approach, avoiding the use of very good models. If g is a dgla then there is an L∞ -structure on H ∗ (g) that makes it equivalent to g in Liek . If moreover g is good then H ∗ (g) is finite dimensional and concentrated in positive degree, so that the L∞ -structure has only finitely
f • H ∗ (g) can many non-trivial structure maps. Thus the uncompleted Chevalley-Eilenberg cdga CE f • (g). be defined, is cofibrant, and is equivalent to CE

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– The first step consists of proving that Ψ is conservative. This follows almost immediately from the hypotheses. Indeed, let f : X → Y an arrow in D inducing isomorphic formal moduli problems. We have Ψ(X) = HomD (D(∗), X) and similarly for Y . Since all En are small and D(En ) ' Fn , f induces an equivalence of spectra HomD (Fn , X) ' HomD (Fn , Y ). Since HomD (Fn , −) : D → spa is conservative, f is an equivalence. – The next step consists in proving that Ψ commutes with limits and accessible colimits, so it has a left adjoint. This part is elementary, and uses the same techniques as in Proposition 2.22. We denote this left adjoint by Φ. – The functor Ψ being conservative, it suffices to prove that the unit idFMP(C,E) ⇒ Ψ ◦ Φ is an equivalence. This is the most technical part in the proof: it involves hypercoverings to reduce to pro-representable moduli problems; this is where the condition on sifted colimits plays a role. Here pro-representable means small limit of fmp representables by small objects. – In the representable case, we can explain what happens: we have an adjunction diagram o D`

Φ

/ FMP(C, E) 9

Ψ

Dop

Y op

C , where Y is the Yoneda functor c → HomC (c, −). We claim that Φ ◦ Y ' Dop on small objects, that is on (Csm )op . Indeed, for any objects c and d of Csm and D respectively, HomD (Φ(Yc ), d) ' HomFMP(C,E) (Yc , Ψ(d)) = HomFun(Csm ,E) (Yc , Ψ(d)) = ψ(d)(c) = HomDop (d, D(c)) = HomD (Dop (c), d). Using this, the unit map of Yc is given by the unit map of the adjunction between D and D0 via the natural equivalence Yc → Ψ ◦ Φ(Yc ) ' Ψ(Dop (c)) = HomDop (D(c), D(−)) ' HomC (c, D0 D(−)). Using Proposition 2.22 (B), we see that this map is an equivalence of formal moduli problems.

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2.6. The tangent complex. – In this section, we introduce the tangent complex associated to a formal moduli problem. We start with a very general definition. Definition 2.34. – Let (C, E) be a deformation context. For any fmp X in FMP(C, E), its tangent complex TX is the spectrum X(E) = X(En )n . Remark 2.35. – There is a slight subtelty in the definition of TX , as X(En ) is only defined for n ≥ 0. However, this suffices to define it uniquely as a spectra, by putting
(TX )−m := Ωm ∗ (TX )0 . Assume now to be given a Koszul duality context op 0 D : (C, E) −→ ←− (D , F ) : D .

Then we have the following result: Proposition 2.36. – The following diagram commutes: / FMP(C, E)

Ψ

D Θ

spa.

y

T

Proof. – This is straightforward: given an object d in D, Ψ(d) = HomDop (d, D(∗)), so TΨ(d) = HomDop (d, D(En )) = HomD (Fn , d) = Θ(d). Remark 2.37. – If D is k-linear (resp. A-linear), then Θ actually lifts to modk (resp. modA ): indeed, replacing HomD (Fn , −) by its enriched version HOMD (Fn , −) gives us the lift of Θ. Example 2.38. – Going back to Example 2.20 we get that, for an A-module M , Θ(M ) = HommodA (A[−n − 1], M )n . In the modA -enriched version, we have Θ(M ) = (M [n+1])n which gives Θ(M ) ' M [1] via the canonical identification Stab(modA ) ' modA . We now deal with functoriality: Proposition 2.39. – Assume to be given a weak morphism C1O o S

D01

T

 C2 o

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/

Dop 1O

Y D2 D02

Z

 / op D2

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between two Koszul duality contexts. Then there is an induced commuting diagram D1

∼

/ FMP (C1 , E1 ) T

Y op

 D2

( 6 spa

◦T

∼

 / FMP (C2 , E2 ).

T

In particular, the functor FMP (C1 , E1 ) → FMP (C2 , E2 ) is conservative. Proof. – We begin by proving that T ∗ := − ◦ T : Fun(C1 , sSet) → Fun(C2 , sSet) indeed defines a functor FMP (C1 , E1 ) → FMP (C2 , E2 ): (1) First, for every n ≥ 0: T (En,2 ) ' T D02 (Fn,2 ) ' D01 Z(Fn,2 ) ' D01 (Fn,1 ) ' En,1 . (2) Then, T being a right adjoint it preserves in particular pull-backs along ∗ → En,2 , and thus sends them to pull-backs along ∗ → En,1 whenever n ≥ 0. Hence it sends small objects to small objects. (3) Finally, let F be an fmp for (C1 , E1 ). Then F ◦ T (∗) ' F (∗) ' ∗ (because T is a right adjoint and F is an fmp), and F ◦ T preserves pull-back along ∗ → En,2 (thanks to the second point and that F is an fmp). Note that Z also sends good objects to good objects (the proof is the same as for the second point above). We now come to the proof of the commutativity of the square, which is essentially based on the following lemma: Lemma 2.40. – There is a natural transformation D1 T ⇒ ZD2 that is an equivalence on small objects. Proof of the Lemma. – The commutativity of the square of right adjoints tells us there is a natural equivalence T D02 ∼ = D01 Z. Hence we have a mate D1 T ⇒ ZD2 defined as the composition D1 T ⇒ D1 T D02 D2 ∼ = D1 D01 ZD2 ⇒ ZD2 , that can also be depicted as

. We then observe that

– on small objects, the unit id ⇒ D02 D2 is an equivalence; – ZD2 sends small objects to good objects (D2 realizes an equivalence between smalls and goods, and Z preserves the goods); – on good objects, the co-unit D1 D01 ⇒ id is an equivalence. Hence the mate D1 T ⇒ ZD2 is an equivalence on small objects.

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The commutativity of the square then reads as follows (recall that we are reasoning on the category of small objects):
T ∗ Ψ1 = HomDop −, D1 T (−) (by definition) 1
' HomDop −, ZD2 (−) (using the mate) 1
' HomDop Y (−), D2 (−) (by adjunction) 2 Ψ2 Y op

=

(by definition).

Finally, we have to prove that the triangle commutes. This is obvious: X ◦ T (En,2 ) ' X(En,1 ). Example 2.41. – Let us see what it implies for our preferred morphism of Koszul duality context: D / o Calgaug Lieop k O Ok CE• V 7→k⊕V [−1]

L

 modk o

forget

Hom(−,k) Hom(−,k)

/

free

 modop k .

Note that we are in the k-linear situation, hence viewing the tangent complex as an object in modk . Let ϑ denote the functor V 7→ k ⊕ V [−1]. If X is in FMPk , then X ◦ ϑ
belongs to FMP modk , (k[n + 1])n , and thus TX = TX◦ϑ . Now: TX◦ϑ = Θ2 Φ2 (X ◦ ϑ) = Φ2 (X ◦ ϑ)[1] = forget Φ1 (X)[1]. Hence we obtain the following beautiful result, explaining one of the phenomenon mentionned in the introduction: The underlying complex of the dgla gX := Φ1 (X) attached to X is TX [−1]. The following proposition describes the tangent complex of a representable moduli problem: Proposition 2.42. – Let A be a representable element in FMPk . Then its tangent complex TA , considered in modk , is equivalent to TA ⊗A k, with TA := (LA/k )∨ . Proof. – It is a simple calculation: (TA )n := HomCalgaug (A, k ⊕ k[n]) ' Hommodk (Lk/A , k[n + 1]) k

' Hommodk (LA/k ⊗A k[1], k[n + 1]) ' TA ⊗A k[n]. Hence, using the canonical identification Stab(modk ) ' modk we get that TA ' TA ⊗A k. As a consequence, we get that Tk/A ' TA ⊗A k[−1] carries a dgla structure.

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3. DG-Lie algebroids and formal moduli problems under Spec(A) In this part, A will denote a fixed cdga over k that is concentrated in non-positive degree and cohomologically bounded. In what follows the boundedness hypothesis is important; we will explain precisely where it has to be used. 3.1. Split formal moduli problems under Spec(A). – One of the main purpose of the work [13] (in the local case) is to prove that the equivalence provided by Theorem 2.22 can be extended when replacing the ground field k by A. We have (see Lemma 2.4) Stab (cdgaA/A ) ' modA .
We consider the deformation context cdgaA/A , (A ⊕ A[n])n . Then Hennion’s result runs as follows: Theorem 3.1 (Hennion [13]). – If A is a cohomologically bounded cdga concentrated in non-positive degree, there is an isomorphism

where FMPA/A

FMPA/A ' LieA ,
:= FMP cdgaA/A , (A ⊕ A[n])n .

Remark 3.2. – In [13], there is the additional assumption that H 0 (A) is noetherian, but it does not appear to be necessary. Hints of proof. – The strategy is to produce a Koszul duality context op (cdgaA/A , A ⊕ A[n]) −→ ←− (LieA , free A[−n − 1])

and then to apply Theorem 2.33. The Chevalley-Eilenberg construction can be performed over A, but we must take its derived version (because the functor M → SA (M ) does not respect weak equivalences). Concretely, assuming that A is cofibrant, the Chevalley Eilenberg is defined in the usual way on A-dg-Lie algebras that are projective as A-modules. Since every element in LieA is isomorphic to a dg-Lie algebra whose underlying A-module is projective, we get a functor CE•A : Lieop A → cdgaA/A , and it admits a left adjoint DA . We consider again a diagram of the form

LA

CE• A

V →A⊕V [−1]

 modA o

/

DA

cdgaA/A o O

HomA (−,A) HomA (−,A)

Lieop OA

forget

/

free

 modop A ,

where LA (R) = LA/R and try to construct a morphism between weak Koszul duality contexts. This works exactly as in the proof of Proposition 2.30, but some aspects have to be taken care of in condition (C) of morphisms between weak Koszul duality contexts: the fact that the forgetful functor commutes with with sifted limits is explained

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in the proof of [13, prop. 1.2.2]. On the other hand, the cohomological boundedness of A is crucial, otherwise the forgetful functor (although conservative) wouldn’t send good objects to reflexive objects. More precisely, even if Lemma 2.31 still holds in this context (18), very good dglas over A are not necessarily reflexive as A-modules (for instance if A is not a bounded k-algebra, then free A[−2] is not quasi-isomorphic to its double dual as a complex of A-modules). But they are whenever A is bounded. All results of the previous section remain true if one replaces k with a bounded A. For instance, we have the following analog of Proposition 2.42: Proposition 3.3. – Let B be a representable element in FMPA/A . Then its tangent complex TB , considered in modA , is equivalent to TB/A ⊗B A, with TB/A := (LB/A )∨ . In particular if B = A ⊗k A, with A-augmentation being given by the product, then LA⊗k A/A ⊗A⊗k A A ' LA and thus TA⊗k A ' TA . Corollary 3.4. – The A-module TA [−1] is a Lie algebra object in modA . In [13], Hennion proves global versions of the above results, and shows in particular that if X is an algebraic derived stack locally of finite presentation, then TX [−1] is a Lie algebra object in QCoh(X). This again explains (and generalizes) a phenomenon that we mentioned in the introduction. 3.2. DG-Lie algebroids. – In this section, we introduce the notion of dg-Lie algebroids, which is the dg-enriched version of Lie algebroids. This will be used to construct a Koszul duality context for A-augmented k-algebras in the next section. Informally, a (dg)-Lie algebroid is a Lie algebra L over k such that L is an A-module, A is a L-module, both structures being compatible. More precisely: Definition 3.5. – A (k, A)-dg-Lie algebroid is the data of a dgla L over k endowed with an A-module structure, as well as an action of L on A, satisfying the following conditions: – L acts on A by derivations, meaning that the action is given by a A-linear morphism of k-dglas ρ : L → Derk (A), called the anchor map. – The following Leibniz type rule holds for any a ∈ A and any `1 , `2 ∈ L: [`1 , a`2 ] = (−1)|a|×|`1 | a[`1 , `2 ] + ρ(`1 )(a)`2 . Morphisms of (k, A)-dg-Lie algebroids are A-linear morphisms of dglas over k commuting with the anchor map. (18) A very good dgla over A is a good dgla over A such that (1) the underlying A-module is projective (2) the underlying graded Lie algebra is freely generated over A by finitely many generators in positive degree.

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Remark 3.6. – Every A-linear dgla defines a dg-Lie algebroid: it suffices to keep the same underlying object and to set the anchor map to zero. On the other hand, if L is a (k, A)-dg-Lie algebroid, the kernel of the anchor map is a true A-linear dgla. These two constructions are adjoint. – Given a pair (k, A), it is possible (see [22]) to attach to any object V of modA/Derk (A) a free (k, A)-dg-Lie algebroid, denoted by free (V ). The functor V → free (V ) is the left adjoint to the forgetful functor from (k, A)-dg-Lie algebroids to A-modules lying over Derk (A). – The Chevalley-Eilenberg construction can be performed for (k, A)-dg-Lie algebroids. If L is a (k, A)-dg-Lie algebroid, then the A-module underlying CE•k/A (L) is the A-dual of SA (L[1]) and the differential reads as follows (omitting signs): n X
dCE ω(`0 , . . . , `n ) = ± ω `0 , . . . , dL (`i ), . . . , `n i=0

±

n X


ρ(`i ) ω(`0 , . . . `ˆi , . . . , `n )

i=0

±

X

ω([`i , `j ], `0 , . . . `ˆi , . . . , `ˆj , . . . , `n ).

0≤i 0, are “corrections terms”. 5.2. About the associativity. – The most important property of these two products is the associativity. It is proved by Kontsevich-Manin [34] (See also [21]) that the quantum product in cohomology is associative. The associativity of this product follows from the so-called WDVV equation (see for example [14, p.240]) that we do not recall here. The geometrical fact that proves this WDVV equation, is given in Theorem 5.3.4 which states that virtual classes behave with respect to the morphisms α’s and the gluing morphisms. Recall that the morphisms α’s are the one that appear in the lax action (3.4.3). Later, when Givental and Lee (See [35]) try to define a quantum product in G0 -theory they want an associative product. If one put the same kind of formula as in (5.1.2), the product is not associative. Hence the key observation of Givental and Lee is Theorem 5.3.9 which is the analogue of Theorem 5.3.4 in G0 -theory that is how the virtual sheaves behave with respect to the morphisms α’s and the gluing morphisms. This will imply a version of WDVV equation in G0 -theory. We refer to [35, §5.1] for this WDVV equation. In the literature, the gluing properties of the virtual classes or virtual sheaves is called the “splitting axiom”. Our contribution to this question is Theorem 5.3.11 which is the geometric explanation of the splitting axiom. Notice that Givental-Lee packed the complicated formula of 5.1.3 in a very clever way. Notice that M 0,2 (X, β) = M 0,2 ×X is empty if β = 0. As before put M 0,2 = pt. Then we put X vir OM (5.2.1) := O X + Qβ O vir ∈ G0 (X) ⊗ Λ. M (X,β) 0,2

0,2

β∈NE(X)

β6=0

Let invert the formula above formally in G0 (X) ⊗ Λ. The terms in front of Qβ is X X (5.2.2) (−1)r O vir ⊗ O vir · · · ⊗ O vir . M (X,β0 ) M (X,β1 ) M (X,βr ) 0,2

0,2

0,2

r∈N (βP 0 ,...,βr )| βi =β

The Formula (5.2.1) and (5.2.2) are the reason of the “metric” (See Formula (16) in [35] for more details) because one can express in a compact form the Formula (5.1.3) using the inverse of the metric. 5.3. Key diagram. – Let us consider the following homotopical fiber product. Let n1 , n2 ∈ N≥2 . Put n = n1 + n2 . (5.3.1)

/ R M 0,n (X, β)

Zβ  M 0,n1 +1 × M 0,n2 +1

g



p

/ M 0,n .

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σ◦τ

The fiber over a point (σ, τ ) is denoted by M (X, β) in § 3.4 that is stable maps where the curve stabilize to σ ◦ τ . In Figure 6, we have an example of a fiber over σ ◦ τ where we have a tree of P1 in the middle. x1 σ, β0

x2 x3 x1 x2

C1 , β1 x3 p

σ ◦ τ ∈ M 0,5

C2 , β2 x4 x5 C3 , β3

x4 τ, β4

x5

Figure 6. Example of a stable map above σ ◦ τ with a tree of P1 in the middle. The tree C1 ◦ C2 ◦ C3 is contracting by p to the node of σ ◦ τ .

Using the universal property of the fiber product we get the morphism (see (3.4.3)) a (5.3.2) α: R M 0,n1 +1 (X, β 0 ) ×X R M 0,n2 +1 (X, β 00 ), → Zβ β 0 +β 00 =β

where the left hand side is defined by the following homotopical fiber product (5.3.3) R M 0,n1 +1 (X, β 0 ) ×X R M 0,n2 +1 (X, β 00 )

/ R M 0,n +1 (X, β 0 ) × R M 0,n +1 (X, β 00 ) 1 2

 X

 / X × X.

e1 ,en2 +1

∆

The heart of the associativity of the quantum products in cohomology (see Theorem 5.3.9 for G0 -theory) is the following statement.

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Theorem 5.3.4 (Theorem 5.2 [37]). – We have the following equality in the Chow ring of the truncation of Zβ . (5.3.5)  α∗ 

X

∆!



  [ M 0,n1 +1 (X, β 0 )]vir ⊗ [ M 0,n2 +1 (X, β 00 )]vir )  = g ! [ M 0,n (X, β)]vir .

β 0 +β 00 =β

Remark 5.3.6. – In [1], Behrend proves that the virtual class satisfies five properties, called orientation (see §7 in [4]), namely: mapping to a point, products, cutting edges, forgetting tails and isogenies. The Formula (5.3.5) is a combination of cutting tails and isogenies. The analogue statement in G0 -theory need a bit more of notations. To have a shorter notation, we denote RXg,n,β := R M g,n (X, β). Let r, n1 , n2 be in N with n1 + n2 = n and let β be in NE(X). Let β = (β0 , . . . , βr ) be a partition of β. Notice that there is only a finite number of partition. We denote by RX0,n1 ,n2 ,β := RX0,n1 +1,β0 ×X RX0,2,β1 ×X · · · ×X RX0,2,βr−1 ×X RX0,n2 +1,βr .

We generalize the situation of (5.3.5) by the following homotopical cartesian diagram   / RX0,n +1,β × Qr−1 RX0,2,β × RX0,n +1,β RX0,n1 ,n2 ,β (5.3.7) 1 0 i 2 r k=1  Xr

 / (X × X)r .

∆r

Gluing all the stable maps and using the universal property of Zβ , we have a morphism a (5.3.8) αr : RX0,n1 ,n2 ,β → Zβ . P β= ri=0 βi

Notice that α1 is the α of (3.4.3). Finally, we can state the analogue of Theorem 5.3.4 in G0 -theory. Theorem 5.3.9 (Proposition 11 in [35]). – We have the following equality in the G0 -group of the truncation of Zβ .     X X vir vir  = g ! O vir (−1)r αr∗  (∆r )! O vir ⊗ O vir X0,n +1β ⊗ O X0,2,β ⊗ · · · ⊗ O X0,2,β X0,n +1,βr X0,n,β . 1

r∈N

Pr

i=0

0

1

r−1

2

βi =β

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Remark 5.3.10. – (1) Comparing Theorem 5.3.4 with Theorem 5.3.9, we see that the formulas are more complicated in G0 -theory. We see that moduli spaces of the kind M 0,2 (X, β) appears in G0 -theory. This corresponds to stable curve with tree of P1 in the middle (see Figure 6). Notice that this is the same reason why the action of the main Theorem 3.1.2 is lax. (2) Also in G0 -theory, there are 5 axioms, called orientation (see Remark 5.3.6), . They are proved by Lee in [35]. for the virtual sheaf O vir M (X,β) g,n

Denote by a Xr,β := RX0,n1 +1,β0 ×X RX0,2,β1 ×X · · · ×X RX0,2,βr−1 ×X RX0,n2 +1,βr . P

βi =β

We deduce a semi-simplicial object in the category of derived stacks where the r + 1-morphisms from Xr+1,β → Xr,β are given by gluing two stable maps together. We have X0,β

X1,β

···

X2,β

Moreover, for any r we have a morphism of gluing all stable maps from Xr,β → Zβ hence a morphism colim X•,β → Zβ . The following theorem was not proved in [42]. We will prove it in the appendix. Theorem 5.3.11. – We have that colim X•,β = Zβ . Remark 5.3.12. – (1) We will see that this theorem implies Theorem 5.3.9. Indeed, in Corollary 5.4.5, we will have the analogue of Theorem 5.3.9 with the sheaves vir,DAG O X0,n,β (see (5.4.2) for the definition of these sheaves). Then in Theorem 5.6.2, h i h i we will prove that the equality O vir,DAG = O vir X0,nβ in G0 -theory. X0,n,β (2) The relation with associativity is more complicated. Indeed, associativity of the quantum product in cohomology or G0 -theory follows from the so-called WDVV equation. But to prove the WDVV equations in cohomology (resp. G0 -theory), one uses the relations of Theorem 5.3.4 (resp. Theorem 5.3.9). 5.4. Virtual object from derived algebraic geometry. – In this section, we explain how derived algebraic geometry will provide a sheaf in G0 ( M g,n (X, β)) that we will compare to the virtual sheaf of Lee. Lemma 5.4.1 (See for example [52] p.192-193). – Let X be a quasi-smooth derived algebraic stack. Denote by t0 (X) its truncation. Denote by ι : t0 (X) ,→ X be the closed embedding. The morphism ι∗ : G0 (t0 (X)) → G0 (X) is an isomorphism. Moreover we have that X (ι∗ )−1 [ F ] = (−1)i [πi ( F )]. i

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Notice that the hypothesis quasi-smooth in the lemma above is done to have that πi ( F ) are coherent module and they vanish for i big enough. Applying this lemma to the situation where X = R M g,n (X, β), we put   vir,DAG (5.4.2) := ι−1 O ∗ [ OR M (X,β) ]. M (X,β) g,n

g,n

where the DAG means Derived Algebraic Geometry. Notice that the sheaf O vir,DAG M (X,β) g,n

depends on the derived structure that we put on the moduli space of stable maps. The following theorem was not stated in [42]. Theorem 5.4.3. – The DAG-virtual sheaf O vir,DAG satisfies the orientation axiom M (X,β) g,n

in G0 -theory. That is (1) Mapping to a point. Let β = 0, we have O

vir,DAG M g,n (X,0)

=

X

(−1)i

i ^ (R1 π∗ O C  TX )∨ ,

i

where C is the universal curve of M g,n and π : C → M g,n . (2) Product. We have O

vir,DAG M g ,n (X,β1 )× M g ,n (X,β2 ) 1 1 2 2

= O vir,DAG  O vir,DAG . M M (X,β ) (X,β ) 1

g1 ,n1

g2 ,n2

2

(3) Cutting edges. With the notation of Diagram (5.3.3), we have O

vir,DAG M g ,n (X,β1 )×X M g ,n (X,β2 ) 1 1 2 2

= ∆! O vir,DAG M (X,β g1 ,n1

1 )× M g2 ,n2 (X,β2 )

.

(4) Forgetting tails. Forgetting the last marked point marked points, we get a morphism π : M g,n+1 (X, β) → M g,n (X, β). We have the following equality. π ∗ O vir,DAG = O vir,DAG . M (X,β) M (X,β) g,n

g,n+1

(5) Isogenies. The are two formulas. The morphism π above induces a morphism ψ : M g,n+1 (X, β) → M g,n+1 × M M g,n (X, β). With notation of Diagram g,n (5.3.1), we have = g ! O vir,DAG . ψ∗ O vir,DAG M (X,β) M (X,β) g,n+1

g,n

The second formula is X X vir,DAG OX (−1)r αr∗ 0,n +1 (X,β0 )×X X0,2 (X,β1 )×X ···×X X0,2 (X,βr−1 )×X X0,n r∈N

Pr

i=0

1

2 +1

(X,βr )

= g ! O vir,DAG X0,n (X,β) ,

βi =β

where g is defined in the key diagram (5.3.1).

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Before proving this theorem, we need a preliminary result. Consider a homotopical cartesian morphisms of schemes X 0 := X ×Y Y 0 ι

' X ×hY Y 0

 / Y0

fe

g

 -X

 / Y.

f

Denote by X ×hY Y 0 the homotopical pullback so that we have the closed immersion ι : X 0 → X ×hY Y 0 . Assume that f is a regular closed immersion. We have a rafined Gysin morphism (see [35, p.4], [19, ex.18.3.16] or Chapter 6 in [20]) which turns to be f ! : G(Y 0 ) → G(X 0 )

(5.4.4)

[ F Y 0 ] 7→ (ι∗ )−1 ◦ fe∗ [ F Y 0 ]. Proof of Theorem 5.4.3. – (1). Strangely this proof is not easy and we postpone to the Appendix B. (2). This follows from the Künneth formula. (3). We have the following diagram. Xg1 ,n1 ,β1 ×X Xg2 ,n2 ,β2

h

k

 Xg1 ,n1 ,β1 ×hX Xg2 ,n2 ,β2

g

 / Xg ,n ,β × Xg ,n ,β 1 1 1 2 2 2

j

RXg1 ,n1 ,β1

i

 ×X RXg2 ,n2 ,β2

f

/ RXg ,n ,β 1 1 1

 × RXg2 ,n2 ,β2 ei ,ej

 X

∆

 / X × X.

We deduce the following equalities ∆! O vir,DAG Xg ,n ,β 1

1

1

×Xg2 ,n2 ,β2

= ∆! (i∗ )−1 O RXg1 ,n1 ,β1 ×RXg2 ,n2 ,β2 = (k∗ )−1 g ∗ (i∗ )−1 O RXg1 ,n1 ,β1 ×RXg2 ,n2 ,β2 by definition of rafined Gysin morphism, = (k∗ )−1 (j∗ )−1 f ∗ O RXg1 ,n1 ,β1 ×RXg2 ,n2 ,β2 by derived base change, = (k∗ )−1 (j∗ )−1 O RXg1 ,n1 ,β1 ×X RXg2 ,n2 ,β2 = O vir,DAG Xg ,n ,β 1

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1

1

×X Xg2 ,n2 ,β2 .

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(4). As π e : R M g,n+1 (X, β) → R M g,n (X, β) is the universal curve (hence, it is flat) and π is the truncation of π e. The derived base change formula implies the equality. (5). We have the following diagram / M g,n × M

ψ

M g,n+1 (X, β) k



/ M g,n+1 × M

ϕ

R M g,n+1 (X, β)



/ M g,n (X, β) .

a

M g,n (X, β)

g,n

i

j

g,n

R M g,n (X, β)

 1 M g,n+1

 / R M g,n (X, β)

b

 / M g,n

c

Notice that as c is flat, the upper right square is also h-cartesian. We have = c! (i∗ )−1 O R M c! O vir,DAG M (X,β) g,n

∗

−1

= a (i∗ )

g,n (X,β)

OR M

g,n (X,β)

b OR M

g,n (X,β)

−1 ∗

= (j∗ )

−1

= (j∗ )

OM

by derived base change

g,n+1 × M g,n R M g,n (X,β)

.

On the other hand, we have ψ∗ O vir,DAG = ψ∗ (k∗ )−1 O R M M (X,β)

g,n+1 (X,β)

= (j∗ )−1 ϕ∗ O R M

g,n+1 (X,β)

g,n

.

The formula follows from the equality below which is a consequence of the proof of Proposition 9 in [35]. ϕ∗ O R M

g,n+1 (X,β)

= OM

g,n × M g,n R M g,n (X,β)

.

To prove the second formula of (5), we use the key Diagram (5.3.1)) with Theorem 5.3.11. Let g1 , g2 , n1 , n2 be integers. Put g = g1 + g2 and n = n1 + n2 and denote M i := M gi ,ni +1 .

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a

t0 (Zβ ) k

   M 1 × M 2 ×h M g,n (X, β)

b

i

j

 Zβ

M1

 / M g,n (X, β)  / R M g,n (X, β)

c

 × M2

g

 / M g,n .

We have g ! O vir,DAG = g ! (i∗ )−1 O R M M (X,β) g,n

−1 ∗

= (k∗ ) =

g,n (X,β)

−1

b (i∗ )

OR M

g,n (X,β)

(k∗−1 )(j∗ )−1 c∗ O R M (X,β) g,n

by derived base change

= (j ◦ k)−1 ∗ O Zβ . We deduce the formula by observing that Zβ is the colimit of X•,β (see Theorem 5.3.11) and that the structure sheaf of a co-limit is the alternating sum of the terms. The last formula of Theorem 5.4.3 and the third one implies the following corollary. Corollary 5.4.5. – We have the following equality in G0 (t0 (Zβ )).     X X vir,DAG vir,DAG vir,DAG vir,DAG (−1)r αr∗  (∆r )! OX0,n +1 (X,β0 ) ⊗ OX0,2 (X,β1 ) ⊗ · · · ⊗ OX0,2 (X,βr−1 ) ⊗ OX0,n +1 (X,βr )  = g! OXvir,DAG . 0,n (X,β) r∈N

Pr

1

2

i=0 βi =β

5.5. Virtual object from perfect obstruction theory. – Here we follow the approach of Behrend-Fantechi [2] to construct virtual object. In the following, we denote by M a Deligne-Mumford stack. The reader can think of M being M 0,n (X, β) as an example. Definition 5.5.1. – Let M be a Deligne-Mumford stack. An element E • in the derived category D( M ) in degree (−1, 0) is a perfect obstruction theory for M if we have a morphism ϕ : E • → L M that satisfies (1) h0 (ϕ) is an isomorphism, (2) h−1 (ϕ) is surjective. Let E • be a perfect obstruction theory. Following [2], we have the following morphisms.

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The morphism a : C M → h1 /h0 (E•∨ ), where C M is the intrinsic normal cone ∨ and h1 /h0 (E•∨ ) is the quotient stack [E−1 /E0∨ ]. To understand how to construct this morphism, let us simplify the situation. Assume that M is embedded in something smooth, i.e., f : M ,→ Y is a closed embedding with ideal sheaf I . Then the intrinsic normal cone is the quotient stack C M = [C M Y /f ∗ T Y ] L where C M Y := Spec n≥0 I n / I n+1 is the normal cone of f . In this case, the intrinsic normal sheaf is N M = [N M Y /f ∗ T Y ] = h1 /h0 (L∨M ) where N M Y := Spec Sym I / I 2 . As we have a morphism from the normal cone to the normal sheaf C M Y → N M Y , we deduce a morphism from the intrinsic normal cone to the intrinsic normal sheaf, i.e., a morphism CM → NM.

(5.5.2)

Now the morphism of the perfect obstruction theory ϕ : E • → L M induces a morphism from ∨ N M → [E−1 /E0∨ ].

(5.5.3)

The morphism a is the composition of the two morphisms (5.5.2) and (5.5.3). We also have a natural morphism b : M → h1 /h0 (E•∨ ) given by the zero section. From these two morphisms, we can perform the homotopical fiber product (5.5.4)

M

/ CM

×hh1 /h0 (E ∨ ) C M •

r

 / h1 /h0 (E•∨ ).

 M

As the standard fiber product is M , we have that M ×hh1 /h0 (E ∨ ) C M is a derived • enhancement of M with e j : M → M ×hh1 /h0 (E ∨ ) C M the canonical closed em•

bedding. Notice that in the case M = M g,n (X, β), we get a derived enhancement which is different from R M g,n (X, β) (see Remark 5.6.3). We will compare these two structures in § 5.6. Hence we can apply the Lemma 5.4.1 and we denote (5.5.5)

[ O vir,POT ] := e j∗−1 [ O M ×h1 M

∨)C M h /h0 (E•

] ∈ G0 ( M ),

where POT means Perfect Obstruction Theory. The definition of Lee for the virtual sheaf turns to be exactly this one. Indeed, Lee consider the following (not homotopical) la cartesian diagram (5.5.6)

M

∨ C1 ×E−1

/ C1

/ CM

 ∨ / E−1

 / h1 /h0 (E•∨ ).

r

 M

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In [35, p.8], Lee takes as a definition for the virtual sheaf X h1 /h0 vir O M := , (−1)i T ori ( O M , O C1 ) = O M ⊗Lh1 /h0 O C1 = O vir,POT M i

where the last equality follows from Lemma 5.4.1. 5.6. Comparison theorem of the two approachs. – Let M := M 0,n (X, β). In this section, we want to compare O vir,DAG with O vir,POT . The first question is : what is M M the perfect obstruction theory we are choosing ? This is given by the following result. Proposition 5.6.1 ([51]). – Let R M be a derived Deligne-Mumford stack. Denote by M its truncation and its truncation morphism by j : M ,→ R M . Then j ∗ LR M → L M is a perfect obstruction theory. Now the original question makes perfectly sense and we have the following result that says that they are the same sheaves. Theorem 5.6.2 (See Proposition 4.3.2 in [42]). – In G0 ( M ), we have [ O vir,DAG ] = [ O vir,POT ]. M M Remark 5.6.3. – Notice that the two enhancements R M or M ×hh1 /h0 (E ∨ ) C M are • not the same. Indeed, the second one has a retract r : M ×hh1 /h0 (E ∨ ) C M → M • given in the diagram (5.5.4) that is r ◦ e j = id M where e j is the closed immersion from M to M ×h h1 /h0 (E•∨ ) C M . From this we get the following exact triangle of cotangent complexes (5.6.4)

Lej [−1] → e j ∗ L M ×h1

(5.6.5)

r∗ L M → L M ×h1

∨)C M h /h0 (E•

∨)C M h /h0 (E•

→ LM

→ Lr .

Applying e j ∗ to the second line, we get LM → e j ∗ L M ×h1

∨)C M h /h0 (E•

→e j ∗ Lr .

This means that (5.6.4) has a splitting that is (5.6.6)

e j ∗ L M ×h1

∨) h /h0 (E•

CM

= Lej [−1] ⊕ L M .

Comparing to the cotangent complex of R M that has no reason to split, we get a priori two different derived enhancement of M .

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Notice that in the work of Fantechi-Göttsche [18, Lemma 3.5], they prove that for a scheme X with a perfect obstruction theory E • := [E −1 → E 0 ], we have (5.6.7)

τX ( O vir,POT ) = Td(T X vir ) ∩ [X vir,POT ], X

where T X vir ∈ G0 (X) is the class of [E0 ] − [E1 ] where [E0 → E1 ] is the dual complex of E • and τX : G0 (X) → A∗ (X)Q . Notice that we expect that the Formula (5.6.7) with Theorem 5.6.2 should imply that [ M g,n (X, β)]vir,POT = τ ( O R M

g,n (X,β)

) Td(TR M

g,n (X,β)

)−1 .

We can not apply the result of Fantechi-Göttsche because these moduli spaces are not schemes. Notice that there is step in this direction in the work [29] of Roy Joshua.

Appendix A Proof of Theorem 5.3.11 Theorem A.1. – The map f : colimDM X•,β → Zβ of [42, (4.2.9)] is an equivalence of derived Deligne-Mumford stacks. Proof. – It follows from the discussion in the proof of [42, Prop. 4.2.1] that f∗

Perf(Zβ )

P P P P P P P P P P P P h '

/ Perf(colimDM X•,β ) ❦ ❦ ❦ ❦ ❦ ❦ ❦ g❦ ❦ ❦ ❦ ❦ u❦ ❦ ❦

lim∆ Perf(X•,β )

commutes with the morphism h being an equivalence after h-descent for perfect complexes [25, 4.12] and the morphism g being fully faithful after the result of gluing along closed immersions [40, 16.2.0.1]. This immediately implies that the map f ∗ is an equivalence of categories because we have g ◦ f ∗ = h and g is conservative as it is fully faithful. As both source and target of f are perfect stacks (the first being a colimit of perfect stacks along closed immersions and second being pullback of perfect stacks), f ∗ induces an equivalence Qcoh(Zβ )

f∗

/ Qcoh(colimDM X•,β )

We conclude that f is an equivalence using Tannakian duality [40, 9.2.0.2 ].

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Appendix B Proof of Theorem 5.4.3.(1) Let X be a derived stack. We will use the linear derived stacks V(E) (See [52, p.200] ) where E is a complex of quasi-coherent sheaf on X. We have a morphism V(E) → X and a zero section s : X → V(E). One should understand that V(E) as a vector bundle where the fibers are E. It is a derived generalization of Spec Sym E for a coherent sheaf E . If E is a two terms complex with cohomology in degree 0 and 1, then we have that t0 (V(E∨ [−1])) = [h1 /h0 (E)] (See §2 in [2] for the definition of the quotient stacks). Let recall some notation of §5.5 and §5.6. Let g, n ∈ N and β ∈ H2 (X, Z). Denote by j the closed immersion M g,n (X, β) → R M g,n (X, β). To simplify the notation, put M = M g,n (X, β) and R M = R M g,n (X, β). From the exact triangle j ∗ LR M → L M → Lj , we deduce that following cartesian diagram / V(L M [−1])

V(Lj [−1])

(B.1)

 / V(j ∗ LR M [−1]).

 M

Recall that j ∗ LR M (X,β) is a two terms complex in degree −1 and 0 but in general g,n it is not the case for Lj and L M (X,β) . Comparing with Behrend-Fantechi, we have g,n t0 (V(L M [−1])) is the intrinsic normal sheaf N M (See §5.5) and we have the following cartesian diagram (B.2)

M

/ CM

×hV(j ∗ LR M [−1]) C M  V(Lj [−1])

 / V(L M [−1])



 / V(j ∗ LR M [−1]).

M

From Gaitsgory (see Proposition 2.3.6 p 18 Chapter IV.5 [22]), we can construct an derived stack Y scaled such that the following diagram has two homotopical fiber products (B.3)

RM O

h

× {0}

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v

V(Lj [−1]) O

σ

j

M

o / Y scaled O

i0

/ M × A1 o

s i1

M

× {1}.

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For any β, this construction is the deformation to the intrinsic normal bundle and not to the intrinsic normal cone. Notice that if β 6= 0 than the complex Lj has cohomology in degree −1 and 0, so that the structure sheaf of V(Lj [−1]) has an element in degree 2 which leads to an infinite sum in G0 ( M g,n (X, β)) which does not make sense. Corollary B.4. – For stable maps of degree 0, we have that O

vir,DAG M g,n (X,0)

=

i X ^ (−1)i (T X  R1 π∗ O C ). i

Remark B.5. – Notice that in the case of β = 0, we have that M g,n (X, β = 0) = M g,n × X which is smooth. Nevertheless, it has a derived enhancement, given by the R Map which has a retract given by the projection and the evaluation. For β 6= 0, this retract does not exist. Proof. – For β = 0, the smoothness of M implies that the intrinsic normal cone is the intrinsic normal sheaf that is we have the C M = V(L M [−1]) in the diagram (B.2). The second thing which is different is that j : M → R M has a retract. This implies that Lj [−1] ' L M [−1] ⊕ j ∗ LR M . We apply the deformation to the normal sheaf of Diagram (B.3). Let prove that (B.6)

O

vir,DAG M g,n (X,0)

:= j∗−1 O R M

g,n (X,0)

= s−1 ∗ ( O V(Lj )[−1] )

Using Diagram (B.3), as we have that v ∗ σ∗ = s∗ i∗1 , we deduce that (s∗ )−1 O V(Lj [−1]) = (s∗ )−1 v ∗ O Y scaled = i∗1 (σ∗ )−1 O Y scaled = i∗0 (σ∗ )−1 O Y scaled . The last equality follows from the A1 -invariance of the G-theory. That is, we have that G0 ( M × A1 ) → G0 ( M ) and i∗0 = (π ∗ )−1 = i∗1 where π is the projection. Applying the same computation as above with the other homotopical fiber product, we get Formula (B.6). This implies that we need to comppute s−1 ∗ O V(Lj [−1]) which is by standard compuP V i i 1 tation i (−1) (T X  R π∗ O C ) where C is the universal curve of M g,n . From the proof, we see that the RHS of the formula is the structure sheaf of V(Lj [−1]). In fact, we think that R M g,n (X, 0) is isomorphic to V(Lj [−1]). This should follow from a general argument that we will detail in the next section for the affine case.

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Appendix C Alternative proof of Corollary B.4 in the affine case Proposition C.1. – Let F = Spec A be an affine quasi-smooth algebraic derived stack. Let F0 = Spec π0 (A) its truncation and denote j : F0 → F its closed immersion. Assume that F0 is smooth and that F admit a retract r : F → F0 . Then F = V(Lj [−1]). This proposition is a way of proving Corollary B.4 in the affine case without using the deformation argument of Gaitsgory. We believe that we can drop the affine assumption in the previous proposition. Notice that we can drop the existence of the retract in the hypothesis because when F 0 is smooth, there always exists a retract (see the Remark C.6). Lemma C.2. – With the previous hypothesis, we have π0 (Lj ) = π1 (Lj ) = 0 π2 (Lj ) = π1 (j ∗ LF ) = π2 (Lπ0 (A)/τ≤1 A ) = π1 (A) Lj [−1] ' π1 (A)[1]. Proof. – We have the triangle j ∗ LF → LF0 → Lj . Applying the hypothesis, we get (1) As F is quasi-smooth, we have that π2 (j ∗ LF ) = 0. (2) As F0 is smooth, we have that π2 (LF0 ) = π1 (LF0 ) = 0. (3) As j ∗ LF → LF0 is a perfect obstruction theory, we deduce π0 (j ∗ LF ) ' π0 (LF0 ) and π1 (j ∗ LF ) → π1 (LF0 ) is onto. Applying the three properties above to the associated long exact sequence, we get (1) As F is quasi-smooth, we have that π2 (j ∗ LF ) = 0. (2) As F0 is smooth, we have that π2 (LF0 ) = π1 (LF0 ) = 0. (3) As j ∗ LF → LF0 is a perfect obstruction theory, we deduce π0 (j ∗ LF ) ' π0 (LF0 ) and π1 (j ∗ LF ) → π1 (LF0 ) is onto. 0

0

π2 (Lj )

π1 (j ∗ LF )

0

0

π0 (j ∗ LF )

π0 (LF0 )

0

We conclude that (1) π2 (Lj ) = π1 (j ∗ LF ). (2) Lj is 2-connective.

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To prove the second equality of the lemma, we use the Postnikov tower that is we consider the closed immersion j1 : F0 → F1 and j2 : F1 → F where F1 is Spec τ≤1 A. We deduce the exact triangle j1∗ Lj2 → Lj → Lj1 . As we have j and j1 are 1-connected and j2 is 2-connected, we deduce from connective estimates that Lj and Lj1 are 2-connective and Lj2 is 3-connective (See Corollary 5.5 in [44]). We deduce from the long exact sequence that π2 (Lj ) = π2 (Lj1 ). How we apply Lemma 2.2.2.8 in [54] that implies that π2 (Lj1 ) = π1 (A). As we have that πk (Lj ) = 0 for all k 6= 2 and π2 (Lj ) = π1 (A), we deduce that Lj [−1] ' π1 (A)[1]. Proof of Proposition C.1. – To prove the proposition, we will show that (C.3)

B := Symπ0 (A) (π1 (A)[1]) ' A.

First, we will construct a morphism f : B → A. Notice that π1 (A) is a free π0 (A) module by the last statement of Lemma C.2. Then we get an inclusion π1 (A)[1] → A of π0 (A)-modules which induces f : B → A. Moreover f is an equivalence on π0 and π1 that is τ≤1 B ' τ≤1 A. Then we construct an inverse from A → B using the Postnikov tower. We have ϕ : A → τ≤1 A ' τ≤1 B. As B is the colimit of its Postnikov tower, we will proceed by induction on the Postnikov tower. First, we want to lift the morphism ϕ : A → τ≤1 B to A → τ≤2 B. We use the following cartesian diagram (See Remark 4.3 in [44]) (C.4)

/ τ≤1 B

τ≤2 B

d



τ≤1 B

id,0

 / τ≤1 B ⊕ π2 (B)[3].

Hence, we need to construct a commutative diagram (C.5)

ϕ

A ϕ



τ≤1 B

/ τ≤1 B d

id,0

 / τ≤1 B ⊕ π2 (B)[3].

As LA has a tor-amplitude in [−1, 0], we have that π0 (Map(LA , π2 (B)[3])) = 0, π1 (Map(LA , π2 (B)[3])) = 0.

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Hence we deduce a morphism from ψ : A → A ⊕d◦ϕ π2 (B)[3]. Hence we get the morphism from A → Bτ≤2 . A ψ

& A ⊕d◦ϕ π2 (B)[3]

/, A &

/ Bτ ≤1

Bτ≤2 

8 Bτ≤1

ϕ

v

d◦ϕ

d

/ Bτ ≤1

0

 ⊕ π2 (B)[3] h

ϕ

  A

ϕ,id 0

 / A ⊕ π2 (B)[3].

Hence by induction, we get a morphism from g : A → B. The composition g ◦ f : B → A → B is the identity on π1 (B) and by the universal property of Sym, we Vi deduce that g ◦ f = idB . This implies that πi (B) = π1 (A) → πi (A) is injective. To finish the proof, we will prove that these morphisms are surjective. For this purpose we use another characterization of afffine quasi-smooth derived scheme. Let us fix generators of π0 (A). This choice is determined a surjective map of commutative k-algebras k[x1 , . . . , xn ] → π0 (A). As the polynomial ring is smooth, we proceed by induction on the Postnikov tower of A to construct a morphism from k[x1 , . . . , xn ] → τ≤n A. We use the same idea as above for constructing the morphism A → B. We get a map of cdga’s k[x1 , . . . , xn ] → A which remains a closed immersion. Moreover, one can now choose generators for the kernel I of k[x1 , . . . , xn ] → π0 (A), say, f1 , . . . , fm whose image in I/I 2 form a basis. The fact that k[y1 , . . . , ym ] is smooth allows us to extend the zero composition map k[y1 , . . . , ym ] → k[x1 , . . . , xm ] → π0 (A) to map k[y1 , . . . , ym ] → k[x1 , . . . , xm ] → A together with a null-homotopy. This puts A in a commutative square of cdga’s k[y1 , . . . , ym ]

/ k[x1 , . . . , xn ]

 k

 / A,

which we is a pushout square. Indeed, it suffices to show that the canonical map k ⊗Lk[y1 ,...,ym ] k[x1 , . . . , xn ] → A

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induces an isomorphism between the cotangent complexes. But as Spec(A) is quasismooth, its cotangent complex is perfect in tor-amplitudes −1, 0, meaning that it can be written as Am → An and this identifies with the standard description of the cotangent complex of the derived tensor product k ⊗Lk[y1 ,...,ym ] k[x1 , . . . , xn ]. This implies that surjectivity of the morphisms πi (B) → πi (A). Remark C.6. – As F = Spec A is a derived scheme (not necessarily quasi-smooth) and its truncation is F0 is smooth, we have that F0 → F admits a retract. We proceed by induction on the Postnikov tower of A to construct a lift :A

π0 (A)

id

 / π0 (A).

We use the same kind of diagrams as (C.4) and (C.5) Indeed, as LF0 is concentrated in degree 0, all the groups π0 (Map(LF0 , πn (A)[n + 1])) = π1 (Map(LF0 , πn (A)[n + 1])) = 0 vanish for n ≥ 1 saying that the liftings exist at each level of the Postnikov tower the space of choices of such liftings is connected.

References [1] K. Behrend – “Gromov-Witten invariants in algebraic geometry”, Invent. math. 127 (1997), p. 601–617. [2] K. Behrend & B. Fantechi – “The intrinsic normal cone”, Invent. math. 128 (1997), p. 45–88. [3] K. Behrend & Y. Manin – “Stacks of stable maps and Gromov-Witten invariants”, Duke Math. J. 85 (1996), p. 1–60. [4]

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[5] A. Brini & R. Cavalieri – “Crepant resolutions and open strings II”, Épijournal Géom. Algébrique 2 (2018), Art. 4, 30. [6] A. Brini, R. Cavalieri & D. Ross – “Crepant resolutions and open strings”, J. reine angew. Math. 755 (2019), p. 191–245. [7] J. Bryan & T. Graber – “The crepant resolution conjecture”, in Algebraic geometry— Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., 2009, p. 23–42. [8] H. Chu, R. Haugseng & G. Heuts – “Two models for the homotopy theory of ∞-operads”, J. Topol. 11 (2018), p. 857–873.

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[9] T. Coates, A. Corti, H. Iritani & H.-H. Tseng – “Computing genus-zero twisted Gromov-Witten invariants”, Duke Math. J. 147 (2009), p. 377–438. [10] T. Coates & A. Givental – “Quantum Riemann-Roch, Lefschetz and Serre”, Ann. of Math. 165 (2007), p. 15–53. [11] T. Coates, H. Iritani & Y. Jiang – “The crepant transformation conjecture for toric complete intersections”, Adv. Math. 329 (2018), p. 1002–1087. [12] T. Coates, H. Iritani & H.-H. Tseng – “Wall-crossings in toric Gromov-Witten theory. I. Crepant examples”, Geom. Topol. 13 (2009), p. 2675–2744. [13] K. Costello – “Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products”, Ann. of Math. 164 (2006), p. 561–601. [14] D. A. Cox & S. Katz – Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, Amer. Math. Soc., 1999. [15] P. Deligne & D. Mumford – “The irreducibility of the space of curves of given genus”, Inst. Hautes Études Sci. Publ. Math. 36 (1969), p. 75–109. [16] B. Dubrovin – “Geometry of 2D topological field theories”, in Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math., vol. 1620, Springer, 1996, p. 120–348. [17] T. Dyckerhoff & M. Kapranov – Higher Segal spaces, Lecture Notes in Math., vol. 2244, Springer, 2019. [18] B. Fantechi & L. Göttsche – “Riemann-Roch theorems and elliptic genus for virtually smooth schemes”, Geom. Topol. 14 (2010), p. 83–115. [19] W. Fulton – Intersection theory, second ed., Ergebn. Math. Grenzg., vol. 2, Springer, 1998. [20] W. Fulton & S. Lang – Riemann-Roch algebra, Grundl. math. Wiss., vol. 277, Springer, 1985. [21] W. Fulton & R. Pandharipande – “Notes on stable maps and quantum cohomology”, in Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., 1997, p. 45–96. [22] D. Gaitsgory – “Notes on geometric Langlands”, http://www.math.harvard.edu/ ~gaitsgde/GL/, 2017. [23] A. B. Givental – “Symplectic geometry of Frobenius structures”, in Frobenius manifolds, Aspects Math., E36, Friedr. Vieweg, Wiesbaden, 2004, p. 91–112. [24] T. Graber & R. Pandharipande – “Localization of virtual classes”, Invent. math. 135 (1999), p. 487–518. [25] D. Halpern-Leistner & A. Preygel – “Mapping stacks and categorical notions of properness”, preprint arXiv:1402.3204. [26] G. Heuts, V. Hinich & I. Moerdijk – “On the equivalence between Lurie’s model and the dendroidal model for infinity-operads”, Adv. Math. 302 (2016), p. 869–1043. [27] H. Iritani – “An integral structure in quantum cohomology and mirror symmetry for toric orbifolds”, Adv. Math. 222 (2009), p. 1016–1079. , “Ruan’s conjecture and integral structures in quantum cohomology”, in New [28] developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008), Adv. Stud. Pure Math., vol. 59, Math. Soc. Japan, Tokyo, 2010, p. 111– 166.

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[29] R. Joshua – “Bredon-style homology, cohomology and Riemann-Roch for algebraic stacks”, Adv. Math. 209 (2007), p. 1–68. [30] L. Katzarkov, M. Kontsevich & T. Pantev – “Hodge theoretic aspects of mirror symmetry”, in From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, Amer. Math. Soc., 2008, p. 87–174. [31] F. F. Knudsen – “The projectivity of the moduli space of stable curves. II. The stacks Mg,n ”, Math. Scand. 52 (1983), p. 161–199. [32] M. Kontsevich – “Enumeration of rational curves via torus actions”, in The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser, 1995, p. 335– 368. [33] M. Kontsevich & Y. Manin – “Gromov-Witten classes, quantum cohomology, and enumerative geometry”, Comm. Math. Phys. 164 (1994), p. 525–562. [34]

, “Quantum cohomology of a product”, Invent. math. 124 (1996), p. 313–339.

[35] Y.-P. Lee – “Quantum K-theory. I. Foundations”, Duke Math. J. 121 (2004), p. 389– 424. [36] J. Li – “A degeneration formula of GW-invariants”, J. Differential Geom. 60 (2002), p. 199–293. [37] J. Li & G. Tian – “Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties”, J. Amer. Math. Soc. 11 (1998), p. 119–174. [38] T. Lowrey, Parkerand Schürg – “Grothendieck-Riemann-Roch for derived schemes”, preprint arXiv:1208.6325, 2012. [39] J. Lurie – “Higher algebra”, https://www.math.ias.edu/~lurie/papers/HA.pdf, 2017. [40]

, “Spectral algebraic geometry”, https://www.math.ias.edu/~lurie/papers/ SAG-rootfile.pdf, 2018.

[41] Y. Manin – “Quantum cohomology, motives, derived categories I, II”, preprint https: //indico.math.cnrs.fr/event/2528/, 2013. [42] E. Mann & M. Robalo – “Brane actions, categorifications of Gromov-Witten theory and quantum K-theory”, Geom. Topol. 22 (2018), p. 1759–1836. [43] F. Perroni – “Chen-Ruan cohomology of ADE singularities”, Internat. J. Math. 18 (2007), p. 1009–1059. [44] M. Porta & G. Vezzosi – “Infinitesimal and square-zero extensions of simplicial algebras”, preprint arXiv:1310.3573. [45] M. Porta & T. Y. Yu – “Non-archimedean quantum k-invariants”, preprint arXiv:2001.05515. [46] M. Robalo – “K-theory and the bridge from motives to noncommutative motives”, Adv. Math. 269 (2015), p. 399–550. [47] Y. Ruan – “Topological sigma model and Donaldson-type invariants in Gromov theory”, Duke Math. J. 83 (1996), p. 461–500. [48]

, “The cohomology ring of crepant resolutions of orbifolds”, in Gromov-Witten theory of spin curves and orbifolds, Contemp. Math., vol. 403, Amer. Math. Soc., 2006, p. 117–126.

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[49] Y. Ruan & G. Tian – “A mathematical theory of quantum cohomology”, Math. Res. Lett. 1 (1994), p. 269–278. [50]

, “Higher genus symplectic invariants and sigma models coupled with gravity”, Invent. math. 130 (1997), p. 455–516.

[51] T. Schürg, B. Toën & G. Vezzosi – “Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes”, J. reine angew. Math. 702 (2015), p. 1–40. [52] B. Toën – “Derived algebraic geometry”, EMS Surv. Math. Sci. 1 (2014), p. 153–240. [53]

, “Operations on derived moduli spaces of branes”, preprint arXiv:1307.0405.

[54] B. Toën & G. Vezzosi – “Homotopical algebraic geometry. II. Geometric stacks and applications”, Mem. Amer. Math. Soc. 193 (2008), p. 224. [55] H.-H. Tseng – “Orbifold quantum Riemann-Roch, Lefschetz and Serre”, Geom. Topol. 14 (2010), p. 1–81.

Etienne Mann, Université d’Angers, Département de mathématiques, Bâtiment I, Faculté des Sciences, 2 Boulevard Lavoisier, F-49045 Angers cedex 01, France E-mail : [email protected] Marco Robalo, Sorbonne Université. Université Pierre et Marie Curie, Institut Mathématiques de Jussieu Paris Rive Gauche, CNRS, Case 247, 4, place Jussieu, 75252 Paris Cedex 05, France E-mail : [email protected]

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LECTURES ON SHIFTED POISSON GEOMETRY by Pavel Safronov

Abstract. – These are expanded notes from lectures given at the États de la Recherche workshop on “Derived algebraic geometry and interactions”. These notes serve as an introduction to the emerging theory of Poisson structures on derived stacks. Résumé (Conférences sur la géométrie de Poisson décalée). – Nous introduisons la théorie de structures de Poisson décalées (supérieures) sur les champs algébriques dérivés. Nous discutons de plusieurs résultats fondamentaux, tels que les intersections coisotropes, l’additivité de Poisson, et une comparaison avec la théorie des structures symplectiques décalées. Enfin, nous fournissons plusieurs exemples de structures de Poisson décalées liées aux structures de Poisson-Lie et aux espaces de modules de fibrés.

Introduction Higher Poisson structures. – Poisson and symplectic structures go back to the works of Lagrange and Poisson on classical mechanics. Classical mechanical systems describe individual particles and as such are examples of 1-dimensional classical field theories. A precursor to the theory of higher Poisson structures is the work [2] of Batalin and Vilkovisky on field theories with symmetries. Their work shows how to endow the (derived) critical locus of an action functional with a (−1)-shifted symplectic structure such that an introduction of a symmetry corresponds to a (−1)-shifted symplectic reduction. The first explicit occurrence of higher symplectic geometry in relation to classical field theories is the work [1] describing how to obtain action functionals of n-dimensional topological field theories from NQ manifolds equipped with a symplectic structure of degree (n − 1) (a smooth analog of the notion of an (n−1)-shifted symplectic stack). We can informally summarize the AKSZ construction as follows: 2010 Mathematics Subject Classification. – 53D17; 17B63, 18D50, 18G55. Key words and phrases. – Poisson algebras, operads, derived stacks.

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— All classical topological field theories of dimension n arise from (n − 1)-shifted symplectic stacks. For instance, the case of classical mechanics corresponds to n = 1 in which case we have 0-shifted (i.e., ordinary) symplectic manifolds. The work of Batalin and Vilkovisky corresponds to n = 0 in which case we have (−1)-shifted symplectic stacks. In a different direction, the theory of multiplicative Poisson structures (PoissonLie groups and Poisson groupoids) gave rise to the notions of Courant algebroids and Dirac structures, see [25]. In turn, Courant algebroids and Dirac structures were later interpreted in terms of higher symplectic geometry by Roytenberg [39] and Ševera [51]. Deformation quantization. – Another motivation to study higher Poisson structures is the theory of deformation quantization. In quantum mechanics observables form an associative algebra which is supposed to be a deformation (with deformation parameter ~, the Planck’s constant) of the Poisson algebra of functions on the classical phase space. Once we generalize commutative algebras to commutative dg algebras, it is natural to consider Poisson algebras with the bracket of a nonzero cohomological degree. These are known as Pn -algebras so that a P1 -algebra is the same as a dg Poisson algebra. Let us now consider an n-dimensional topological field theory. As we have mentioned, such a field theory on the classical level is determined by an (n − 1)-shifted symplectic stack whose algebra of functions has a Pn -structure. The operator product expansion endows the cohomology of local quantum observables with a commutative product. It turns out that on the chain level the product is not coherently commutative but instead forms what is known as an En -algebra. We refer to [12] for an explanation of related ideas. For instance, an E∞ -algebra is a commutative dg algebra and an E1 -algebra is an associative algebra. Similar to the previous case, it is expected that the En -algebras of local quantum observables are deformation quantizations of the Pn -algebras of functions on the clasical phase space described by the (n − 1)-shifted symplectic stack. An example of higher deformation quantization is given by the theory of quantum groups which can be considered as deformation quantizations of the 2-shifted symplectic stack BG, the classifying stack of a reductive group G. The theory of higher deformation quantization is still in its infancy, so we will not discuss it in the main body of the text. Let us just mention several recent articles that deal with this topic: [49], [9, Section 3.5], [31, Section 5], [37], [36] and [38]. Implications for classical geometry. – So far we have explained why one is interested in generalizations of Poisson geometry to the derived setting. However, the derived perspective can also be useful for questions involving only classical geometric objects. In Donaldson-Thomas theory one considers moduli spaces M st (X) of stable sheaves on a Calabi-Yau threefold X. The Donaldson-Thomas invariants can be computed in terms of a symmetric obstruction theory on M st (X), see e.g., [4] and [3]. The moduli spaces M st (X) are open substacks in the classical truncation of the

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derived moduli stack Perf(X) of perfect complexes on X. In turn, the symmetric obstruction theory on M st (X) is a shadow of the (−1)-shifted symplectic structure on Perf(X), see [33, Section 3.2]. Even though the symmetric obstruction theory is enough to define ordinary Donaldson-Thomas invariants, for their categorification one needs to consider the full (−1)-shifted symplectic structure on Perf(X), see [5]. Moreover, the existence of the (−1)-shifted symplectic structure on Perf(X) implies that M st (X) can be covered by charts given by a critical locus of a function. Another fruitful application is the construction of Poisson structures on several moduli spaces M . Classically it proceeds as follows. First, one restricts to an open subspace M sm ⊂ M consisting of smooth and non-stacky points. Second, it is usually easy to define a bivector on M sm . The final step involves a nontrivial computation to show that the bivector satisfies the Jacobi identity. Suppose one can realize M as a classical truncation of a derived mapping stack. Then it inherits shifted symplectic and shifted Lagrangian structures by the AKSZ construction. Using a relation between shifted symplectic and shifted Poisson structures one can endow M with a 0-shifted Poisson structure which then gives an ordinary Poisson structure on the smooth locus M sm . We refer to [33, Section 3.1] which treats the Goldman Poisson structure [19] on the character variety and the symplectic structure on the moduli of bundles on a K3 surface [32] and to Section 3.3 for the case of the Feigin-Odesskii Poisson structure [15]. Outline. – The paper consists of three lectures. In the first lecture (Section 1) we study shifted Poisson structures on commutative dg algebras in terms of Pn -algebras. We sketch a proof of the Poisson additivity Theorem 1.13 which is useful to define coisotropic structures and the forgetful map from n-shifted Poisson structures to (n − 1)-shifted Poisson structures. Besides the abstract definition of the space of shifted Poisson structures, we also present a way to compute this space in terms of Maurer-Cartan elements in the Lie algebra of polyvector fields. We also explain how one can transfer definitions of Poisson structures in the affine setting to any derived stack using the technique of formal localization. In the second lecture (Section 2) we define shifted coisotropic structures which are relative versions of shifted Poisson structures. We show (Proposition 2.7) that a derived intersection L1 ×X L2 of two n-shifted coisotropic maps L1 , L2 → X in an n-shifted Poisson stack X carries an (n − 1)-shifted Poisson structure. After a brief reminder on shifted symplectic and shifted Lagrangian structures (Section 2.4) we outline a proof of an equivalence between shifted symplectic structures and the class of nondegenerate shifted Poisson structures (Theorem 2.31). In the last lecture (Section 3) we provide examples of shifted symplectic and shifted Poisson structures. First, as an application of the comparison between shifted Poisson and shifted symplectic structures, we explain how one can generalize the notion of symplectic realization of Poisson manifolds to shifted Poisson structures. We then give several examples of 1-shifted Poisson structures coming from quasi-Poisson groups,

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quasi-Lie bialgebras and classical (dynamical) r-matrices. We end the paper with several examples of shifted Poisson structures related to the fundamental Lagrangian correspondence BH ← BB → BG (11) where G is a simple group, B is a Borel subgroup and H is the Cartan subgroup. Namely, we show that it gives rise to the FeiginOdesskii Poisson structures, symplectic structure on the moduli space of monopoles and the standard dynamical r-matrix on G. In this text we mainly review the papers [9], [30], [31] and [35] on shifted Poisson geometry without many technical details to simplify the exposition. We refer to [34] which treats certain topics omitted in this review. Conventions — Throughout the notes we will freely use the language of ∞-categories. We refer to [27] and [28] as foundational texts which use quasi-categories as models for ∞-categories and to [20] as an introduction to the theory. This language is indispensable in dealing with higher homotopical structures and descent questions. — We will work in the framework of derived algebraic geometry, see [50] for details and [7], [48] and [47] for an introduction. We work over a base field k of characteristic zero. Acknowledgements. – The author would like to thank A. Brochier, D. Calaque, G. Ginot, V. Melani, T. Pantev, B. Pym, N. Rozenblyum and B. Toën for many conversations about shifted Poisson structures. In particular, the author thanks B. Pym for reading a draft of the notes and making several useful suggestions. While writing this overview, the author was supported by the NCCR SwissMAP grant of the Swiss National Science Foundation.

1. Shifted Poisson algebras In the first lecture we review what Poisson and Pn -algebras are, explain how to relate Pn -algebras for different n using Poisson additivity and discuss the definition of an n-shifted Poisson structure on a derived Artin stack. 1.1. Classical definitions. – Let X be a smooth scheme over a field k and O X the structure sheaf. Definition 1.1. – A Poisson structure on X is the data of a Lie bracket {−, −} : O X ⊗k O X → O X satisfying the Leibniz rule {f, gh} = {f, g}h + {f, h}g for any f, g, h ∈ O X . If X is affine, a Poisson structure on X in the above sense can be equivalently stated in terms of a Lie bracket on the algebra of global functions O (X).

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Example 1.2. – Suppose Y is a smooth affine scheme and consider X = T∗ Y . Then O (X) ∼ = Γ(Y, Sym(TY )). It is generated as a commutative algebra by O (Y ) and Γ(Y, TY ). By the Leibniz rule the Poisson bracket is uniquely determined on generators and we let {f, g} = 0,

f, g ∈ O (Y ),

{v, f } = v.f,

f ∈ O (Y ), v ∈ Γ(Y, TY ),

{v, w} = [v, w],

v, w ∈ Γ(Y, TY ).

Here is a slight reformulation of the above definition. A Poisson structure on X gives rise to an antisymmetric biderivation on O X , i.e., an element π ∈ Γ(X, ∧2 TX ). More generally, define Pol(X, 0) = Γ(X, Sym(TX [−1])), the graded commutative algebra of polyvector fields. We call the Sym grading the weight grading, e.g., vector fields have weight 1 and bivectors have weight 2. The algebra Pol(X, 0) has a Lie bracket of cohomological degree −1 called the Schouten bracket which is given by formulas similar to those of Example 1.2. Proposition 1.3. – A Poisson structure on X is equivalent to the data of an element π ∈ Pol(X, 0) of weight 2 satisfying [π, π] = 0. One has yet another equivalent definition of a Poisson structure given as follows. Definition 1.4. – A symplectic Lie algebroid on X is the data of a Lie algebroid L ∼ over X equipped with an isomorphism α : L − → T∗X satisfying (1) (2)

ιa(l) α(l) = 0, α([l1 , l2 ]) = La(l1 ) α(l2 ) − ιa(l2 ) (ddR α(l1 )).

Proposition 1.5. – A Poisson structure on X is equivalent to the data of a symplectic Lie algebroid L on X. Proof. – Suppose X has a Poisson structure π. Let L = T∗X . Define the anchor a : L → TX to be l 7→ ιl (π). Given two differential forms l1 , l2 ∈ L we define the bracket (the so-called Koszul bracket) to be [l1 , l2 ] = La(l1 ) l2 − La(l2 ) l1 − ddR π(l1 , l2 ). We leave it to the reader to check that L is a symplectic Lie algebroid. Conversely, given a symplectic Lie algebroid L , the composite a T∗X ∼ → TX = L−

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defines an element π ∈ Γ(X, TX ⊗ TX ). The Equation (1) is equivalent to the antisymmetry of π and Equation (2) is equivalent to the Jacobi identity for π. To summarize, we have the following three definitions of a Poisson structure on a smooth scheme X: (1) As an enhancement of the structure sheaf of X to a sheaf of Poisson algebras. The derived version is given by Definition 1.53. (2) As a bivector satisfying the Jacobi identity. The derived version is given by Proposition 1.57. (3) As a symplectic Lie algebroid. We refer to Conjecture 3.2 for its derived version. 1.2. Shifted Poisson algebras. – We are now going to explain what higher Poisson structures are. We will discuss only the affine case deferring to Section 1.5 for the case of derived stacks. One can generalize the notion of a Poisson algebra to the dg setting by considering a commutative dg algebra equipped with a bracket satisfying the Leibniz rule and graded antisymmetry. Since we have a grading, it is also natural to consider a generalization where we allow the degree of the bracket to be arbitrary. Definition 1.6. – A Pn -algebra is a commutative dg algebra A over k equipped with a bracket {−, −} : A ⊗k A → A[1 − n], which gives a Lie bracket on A[n − 1] satisfying {f, gh} = {f, g}h + (−1)|g||h| {f, h}g,

∀f, g, h ∈ A.

Our next goal is to study all ways of endowing a commutative dg algebra A with a Pn -bracket. We can consider the set of such lifts which we temporarily denote g by Pois(A, n−1). It is clear that an isomorphism of commutative dg algebras A1 ∼ = A2 g 2 , n − 1). g 1 , n − 1) ∼ induces an isomorphism of sets Pois(A = Pois(A In derived algebraic geometry we are also interested in the behavior of properties under quasi-isomorphisms A1 → A2 and in this case there is no obvious reg 1 , n − 1) and Pois(A g 2 , n − 1). Nevertheless, by the lation between the sets Pois(A homotopy transfer theorem [26, Section 10.3] given a Pn -algebra structure on A1 we can transfer it to a homotopy Pn -algebra structure on A2 which is uniquely deg fined modulo ∞-quasi-isomorphisms. Thus, we should replace Pois(A, n−1) by the set of ∞-quasi-isomorphism classes of homotopy Pn -structures on A which are compatible with the given commutative algebra structure on A. Instead of passing to equivalence classes we will construct an ∞-groupoid Pois(A, n − 1) whose objects are homotopy Pn -structures on A and (higher) morphisms are given by (higher) homotopies. Such an ∞-groupoid can be constructed quite explicitly by hand; instead, we will define it in a more concise way using the ∞-category of Pn -algebras. We will now switch to the language of dg operads for which the reader is referred to [26]. The above definition of a Pn -algebra gives a quadratic dg operad Pn generated

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by two binary operations: multiplication of degree 0 and bracket of degree 1 − n, so that a Pn -algebra is an algebra over the dg operad Pn . Notation 1.7. – Let O be a dg operad such as the associative operad Ass or the commutative operad Comm. — We denote by Alg O the category of O -algebras in complexes. We also introduce the notations Alg = AlgAss , CAlg = AlgComm for the categories of associative and commutative dg algebras. — Let W ⊂ Alg O be the wide subcategory consisting of quasi-isomorphisms, i.e., those morphisms of O -algebras A1 → A2 which induce isomorphisms H• (A1 ) → H• (A2 ). Definition 1.8. – The ∞-category of O -algebras is the localization Alg

O

= Alg O [W −1 ].

Similarly, we have Alg

= Alg Ass ,

CAlg

= Alg Comm .

We have a functor AlgPn → CAlg given by forgetting the bracket which after localization induces a functor of ∞-categories Alg P

n

→ CAlg .

Given an ∞-category C we denote by C ∼ the underlying ∞-groupoid of objects where we throw away non-invertible morphisms. Definition 1.9. – Let A be a commutative dg algebra. The ∞-groupoid Pois(A, n) of n-shifted Poisson structures on A is the fiber of the forgetful map ∼

Alg P

n+1

−→ CAlg ∼

at A ∈ CAlg ∼ . In other words, an object of Pois(A, n) is given by a pair of a Pn+1 -algebra A˜ and ∼ a quasi-isomorphism of commutative dg algebras A˜ − → A. Remark 1.10. – We will use the terms “∞-groupoid” and “space” interchangeably as both will refer to objects of equivalent ∞-categories. In the model of ∞-categories as simplicial categories, we can take the relevant ∞-category S to be that of Kan complexes. Remark 1.11. – The same definition of the ∞-groupoid of n-shifted Poisson structures can be given for a commutative algebra in a general enough symmetric monoidal dg category M (see [9, Section 1.1] for precise assumptions). This will be used to extend this notion to the non-affine setting in Section 1.5.

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1.3. Additivity. – A natural question is whether Pn -algebras for different n are related in some way. To explain the relation, let us recall the following result. Let En the topological operad of little n-disks. That is, the space of m-ary operations En (m) is the space of rectilinear embeddings of m n-dimensional disks into a fixed n-dimensional disk. For instance, in the case n = 1 E1 (m) is the space of embeddings of m intervals [0, 1] into [0, 1]. Such a space retracts to Sm , the symmetric group, and hence the operad E1 is equivalent to the associative operad. Thus, an E1 -algebra in complexes is simply an associative dg algebra. In the opposite limit the space E∞ (m) is weakly contractible, so E∞ is equivalent to the commutative operad and an E∞ -algebra in complexes is a commutative dg algebra. Given any symmetric monoidal ∞-category C we can consider the ∞-category Alg E ( C ) of En -algebras in C . The following is [28, Theorem 5.1.2.2]. n Theorem 1.12 (Dunn-Lurie). – One has an equivalence of symmetric monoidal ∞-categories Alg E ( C) ∼ = Alg En ( Alg Em ( C )). n+m Let R be a commutative dg algebra over k and denote by Mod R the ∞-category of dg R-modules defined as Mod R

= ModR [W −1 ].

We can also consider the ∞-category of Pn -algebras in Mod R by defining Alg P

n

( Mod R ) = AlgPn (ModR )[W −1 ],

where ModR is the dg category of R-modules. The following is [42, Theorem 2.22] and was previously proved independently by Rozenblyum (unpublished). Theorem 1.13. – One has an equivalence of symmetric monoidal ∞-categories Alg P

n+m

( Mod R ) ∼ = Alg En ( Alg Pm ( Mod R )).

Remark 1.14. – The statement in [42] was proved in the case R = k, but the proof can be generalized to any commutative dg algebra over k. Let us sketch the construction of the equivalence in Theorem 1.13. The reader unfamiliar with the theory of operads can safely skip to the corollaries of the construction (Proposition 1.20 and Proposition 1.22). Given a Lie algebra g and an element x ∈ g we denote by adx : g → g the adjoint action [x, −]. Definition 1.15. – An n-shifted Lie bialgebra is a dg Lie algebra g equipped with a degree −n Lie cobracket δ : g ⊗ g → g[−n] satisfying the relation δ([x, y]) = (adx ⊗ id + id ⊗ adx )δ(y) − (−1)n+|x||y| (ady ⊗ id + id ⊗ ady )δ(x) for any x, y ∈ g.

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Definition 1.16. – A dg Lie coalgebra g is conilpotent if for every x ∈ g there is an N so that the N -fold application of δ to x gives zero. Denote by BiAlgLien the category of n-shifted Lie bialgebras conilpotent as Lie coalgebras. Then one has a bar-cobar adjunction / Alg Ω : BiAlgLien−1 o Pn+1 : B constructed as follows. Suppose g is an (n − 1)-shifted Lie bialgebra. Then A = Sym(g[−n]) is endowed with the Chevalley-Eilenberg differential coming from the Lie coalgebra structure on g. Moreover, the Lie bracket on g induces a Pn+1 -structure on A. Given an R-module V we denote by coLie(V ) the cofree conilpotent Lie coalgebra cogenerated by V . If A is a Pn+1 -algebra we can endow coLie(A[1])[n−1] with the Harrison differential and an (n − 1)-shifted Lie bialgebra structure, see [42, Section 2.4] for details. We call the functor Ω the cobar construction and B the bar construction. We also have another bar-cobar adjunction / Alg o CoAlg Pn+1

Pn+1

between Pn+1 -coalgebras and Pn+1 -algebras constructed in a similar way. Both categories BiAlgLien−1 and CoAlgPn+1 have a class of weak equivalences which are those morphisms which become quasi-isomorphisms after applying Ω. Lemma 1.17. – The bar-cobar adjunctions Ω : BiAlgLien−1 o and CoAlgPn+1 o

/ Alg Pn+1 : B / Alg Pn+1

induce equivalences on the underlying ∞-categories. Recall that given a Lie algebra g its universal enveloping algebra U(g) is a cocommutative bialgebra. Moreover, by the Cartier-Milnor-Moore theorem we have an equivalence of categories U

AlgLie − → Alg(CoAlgComm ), where Alg(−) is the category of associative algebras and Alg(CoAlgComm ) is the category of conilpotent cocommutative bialgebras. Similarly, if g is an (n − 1)-shifted Lie bialgebra, U(g) inherits a natural Pn -cobracket which induces an equivalence of categories U

BiAlgLien−1 − → Alg(CoAlgPn ). The composite B

U

AlgPn+1 − → BiAlgLien−1 − → Alg(CoAlgPn )

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after passing to underlying ∞-categories induces the required functor (3)

Alg P

n+1

−→ Alg ( Alg Pn ).

Note that the category AlgPn+1 has an involution given by flipping the sign of the bracket. Under the bar complex B it goes to the same involution of BiAlgLien−1 . Applying the universal enveloping algebra U it goes to the involution of Alg(CoAlgPn ) given by passing to the opposite algebra. Remark 1.18. – The ∞-category Alg ( Alg Pn ) has two involutions: passing to the opposite associative algebra and passing to the opposite Pn -algebra. The antipode of U(g) gives a natural isomorphism of the two involutions of Alg ( Alg Pn ). Proposition 1.19. – The Poisson additivity equivalence (3) intertwines the involution of Alg Pn+1 given by flipping the sign of the bracket and the involution of Alg ( Alg Pn ) given by passing to the opposite algebra. Let us present some corollaries of this construction of the additivity functor. Observe that the additivity functor gives rise to a forgetful functor ∼

→ Alg ( Alg Pn ) −→ Alg Pn , forgetnn+1 : Alg Pn+1 − which has the following properties. Proposition 1.20. – Let g be an (n−1)-shifted Lie bialgebra and A = Ωg the associated Pn+1 -algebra. Denote by A0 ∈ Alg Pn the same commutative dg algebra with the trivial bracket. Then an equivalence of Pn -algebras forgetnn+1 (A) ∼ = A0 is the same as an equivalence of Pn -coalgebras U(g) ∼ = Sym(g). Observe that U(g) is the algebra of distributions on the formal Poisson-Lie group G integrating g and Sym(g) is the algebra of distributions on the completion b g0 . Thus, the forgetful functor forgetnn+1 is closely related to the theory of formal linearizations of Poisson-Lie groups; we refer to [43, Proposition 2.17] for more details. Even though the forgetul functor forgetn−1 : Alg Pn → Alg Pn−1 is nontrivial, the n

iterated forgetful functors forgetnn−m : Alg Pn → Alg Pn−m for m ≥ 2 are essentially trivial, i.e., the corresponding Pn−m -algebra is automatically commutative. Before we prove this claim, let us recall that Tamarkin [46] has constructed a formality isomorphism C• (E2 ) ∼ = P2 of dg Hopf operads from the data of a λ-Drinfeld associator which acts as λ ∈ k × on the Lie bracket in artiy 2. We will need to know that it is also compatible with the Lie structures. Lemma 1.21. – Two maps Lie{1} → P2 in the ∞-category of dg operads are homotopic iff the underlying maps of complexes Lie{1}(2) → P2 (2) are homotopic.

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Proof. – Given a dg operad O , the collection of arity spaces Y S gO = O (n) n n≥2

has a natural structure of a pro-nilpotent dg Lie algebra, see [26, Section 5.4.3]. Moreover, the mapping space Map(Lie{1}, P2 ) in the ∞-category of operads is equivalent to the Maurer-Cartan space Map(Lie{1}, P2 ) ∼ = MC(gP2 {−2} ) by [30, Proposition 2.2]. By definition P2 (n) is concentrated in cohomological degrees [1−n, 0] and hence P2 {−2}(n) is concentrated in cohomological degrees [n−1, 2(n−1)]. Therefore, the Lie algebra gP2 {−2} is concentrated in cohomological degrees ≥ 1. The degree 1 part of gP2 {−2} is one-dimensional and is spanned by the Lie bracket in arity 2. Therefore, any 1-Drinfeld associator gives a commutative diagram of dg operads / < P2

∼

C• (E2 ) d

Lie{1}. Proposition 1.22. – The choice of a 1-Drinfeld associator gives a homotopy commutativity data in the diagram Alg P

forgetn−1 n+1 n+1

forget

/ Alg Pn−1 : triv

$ CAlg .

Proof. – Let g be an (n − 1)-shifted Lie bialgebra and A = Ωg the corresponding Pn+1 -algebra. The functor Alg Pn+1 → Alg E2 ( Alg Pn−1 ) is given by A 7→ UE2 (g), where UE2 (−) is the E2 enveloping algebra. ∼ P2 Given a Drinfeld associator we obtain an equivalence of Hopf operads C• (E2 ) = and hence an equivalence of universal enveloping functors UE2 (g) ∼ U (g). The latter = P2 object can be identified with UP2 (g) ∼ = Sym(g[−1]) as Pn−1 -coalgebras. But Sym(g[−1]) ∈ CoAlgPn−1 corresponds under Koszul duality to the Pn−1 -algebra A with the trivial bracket.

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1.4. Polyvectors. – Now we are going to give a way to compute the ∞-groupoid Pois(A, n) of n-shifted Poisson structures on A. Let us recall from [28, Section 5.3] the notion of a centralizer in a symmetric monoidal ∞-category C . Definition 1.23. – Let f : A → B a morphism in C . The centralizer of f is the universal object Z(f ) ∈ C equipped with the following data: (1) A morphism 1 → Z(f ), (2) A morphism Z(f ) ⊗ A → B, (3) A commutativity data in the diagram Z(f ) ⊗ A ; A

f

$

/ B.

Definition 1.24. – For an object A ∈ C its center Z(A) is the centralizer Z(id : A → A). Note that a centralizer may not exist in general; we will say that C admits centralizers if any morphism f in C has a centralizer Z(f ) ∈ C . Example 1.25. – Suppose C is a closed symmetric monoidal ∞-category with the internal Hom functor Hom C (−, −). Then it admits centralizers given by Z(f ) = Hom C (A, B) so that the map 1 → Z(f ) is adjoint to f : A → B. Example 1.26. – Consider the symmetric monoidal ∞-category Alg ( Ch ) of dg algebras. The center of a dg algebra A ∈ Alg ( Ch ) is an object Z(A) ∈ Alg ( Alg ( Ch )) which by the Dunn-Lurie additivity Theorem 1.12 is the same as an E2 -algebra. One can show that this E2 -algebra is equivalent to the Hochschild complex CH• (A, A) (see e.g., [18, Section 7.2]). When A is concentrated in degree 0, the zeroth cohomology of the Hochschild complex CH• (A, A) is isomorphic to the ordinary center of A as an associative algebra. Thus, we see that Z(A) is a derived generalization of this notion. Example 1.27. – Let X = Spec A be a smooth affine Poisson scheme. Then Z(A) is quasi-isomorphic to the complex Γ(X, Sym(TX [−1])) equipped with the differential [πX , −], i.e., the Poisson cohomology of X. Its zeroth cohomology coincides with the space of Casimir functions of A, i.e., elements f ∈ A such that {f, g} = 0 for every g ∈ A. f

g

Suppose C admits centralizers and consider a sequence of morphisms A − →B− →C in C . From the universal property we obtain an action morphism Z(f ) ⊗ Z(g) → Z(g ◦ f ). In particular, we can upgrade Z(A) = Z(id) to an object of Alg ( C ). We have the following result given by [28, Theorem 5.3.1.14]:

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Theorem 1.28. – Suppose O is a coherent ∞-operad and C is a symmetric monoidal ∞-category whose tensor product preserves colimits. Then Alg O ( C ) admits centralizers. At the moment the above statement is not known for dg operads (see, however, the recent preprint [10] which discusses the appropriate notion of a linear ∞-operad). Nevertheless, we have the following statement whose proof will appear in a future paper. Theorem 1.29. – Let R be a commutative dg algebra over k. Then the symmetric monoidal ∞-category of R-linear Pn -algebras Alg Pn ( Mod R ) admits centralizers. We are now going to construct the algebra of n-shifted polyvector fields. Definition 1.30. L – A graded dg Lie algebra is a dg Lie algebra g equipped with a weight grading g = n gn such that the Lie bracket has weight −1. Similarly, one can define a graded Pn -algebra to be a Pn -algebra equipped with an extra weight grading such that the Lie bracket has weight −1. Thus, both Lie and Pn lift to operads in graded complexes. We denote by Alg gr the ∞-category of graded Pn Pn -algebras. We denote by trivPn : CAlg −→ Alg gr Pn the functor which sends a commutative dg algebra A to the graded Pn -algebra concentrated in weight 0 with the zero bracket. Definition 1.31. – Let A ∈ CAlg be a commutative dg algebra. The algebra of n-shifted polyvector fields on A is Pol(A, n) = Z(trivPn+1 (A)) ∈ Alg ( Alg gr ). Pn+1 In particular, using Theorem 1.13 we see that Pol(A, n) ∈ Alg gr . Pn+2 Given a commutative dg algebra A ∈ CAlg , its cotangent complex LA ∈ Mod A is defined by the universal property for M ∈ Mod A (4)

HomA (LA , M ) ∼ = Map CAlg

/A

(A, A ⊕ M ),

where CAlg /A is the ∞-category of commutative dg algebras with a map to A and A ⊕ M is a square-zero extension. Note that the right-hand side simply represents derivations from A to M . The following statement then gives a control over Pol(A, n) as a graded commutative dg algebra. Proposition 1.32. – Let A be a commutative dg algebra. Then we have an equivalence of graded commutative dg algebras Pol(A, n) ∼ = HomA (SymA (LA [n + 1]), A).

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One can think of the Lie bracket on Pol(A, n) as a generalization of the Schouten bracket of polyvector fields. A weight 2 element of Pol(A, n) is a bivector on A and by Proposition 1.3 one can describe Pn+1 -brackets on A in terms of bivectors π satisfying [π, π] = 0. Now let k(2)[−1] ∈ Alg gr be the trivial graded dg Lie algebra concentrated in Lie weight 2 and cohomological degree 1. Then a bivector π satisfying [π, π] = 0 is the same as a morphism of graded Lie algebras k(2)[−1] → Pol(A, n)[n + 1]. In fact, this gives the whole ∞-groupoid of n-shifted Poisson structures as shown in [29]. Theorem 1.33. – One has an equivalence of ∞-groupoids Pois(A, n) ∼ = Map gr (k(2)[−1], Pol(A, n)[n + 1]). Alg Lie

To get a more concrete understanding of the previous statement, let us recall the notion of a Maurer–Cartan space from [21] and [17]. Let Ωn = Ω(∆n ) be the commutative dg algebra of polynomial differential forms on the n-simplex. Explicitly, it is given by X X Ωn = k[t0 , . . . , tn , dt0 , . . . , dtn ]/( ti = 1, dti = 0), P i.e., the de Rham algebra of k[t0 , . . . , tn ]/( ti = 1), the commutative dg algebra of polynomial functions on the n-simplex. The simplices ∆• form a cosimplicial simplicial set, so Ω• form a simplicial commutative dg algebra. Therefore, if g is a dg Lie algebra, g ⊗ Ω• is a simplicial dg Lie algebra. Definition 1.34. – Let g be a nilpotent dg Lie algebra. — The set of Maurer-Cartan elements of g is the set MC(g) of elements x ∈ g of degree 1 satisfying 1 dx + [x, x] = 0. 2 — The Maurer-Cartan space of g is the simplicial set MC• (g) = MC(g ⊗ Ω• ). The term “Maurer-Cartan space” indicates that we have an ∞-groupoid, i.e., the simplicial set MC(g) is a Kan complex. This is a theorem of Hinich and Getzler, see [17, Proposition 4.7]. Example 1.35. – Suppose g is a nilpotent Lie algebra concentrated in degree 0. Then MC(g) is weakly equivalent to the nerve of the groupoid ∗/ exp(g), where exp(g) is the unipotent group with Lie algebra g. Thus, MC(−) can be thought of as an integration functor. Definition 1.36. – Let g be a dg Lie algebra. It is pronilpotent if it admits a cofiltration g0  g1  · · · , where g = limn gn and gn is nilpotent.

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Definition 1.37. – Let g be a pronilpotent dg Lie algebra. The Maurer-Cartan space of g is the simplicial set MC(g) = lim MC(gn ). n

Again, for a pronilpotent dg Lie algebra, MC(g) is a Kan complex by [13, Proposition 4.1] and [53, Proposition 2.6]. Given a graded dg Lie algebra g, its completion g≥2 in weights at least 2 is pronilpotent: it is obtained as the inverse limit of the system g≥2 /g≥3  g≥2 /g≥4  · · · of nilpotent dg Lie algebras. Note that a morphism of graded Lie algebras k(2)[−1] → g is the same as an element x2 ∈ g of weight 2 and cohomological degree 1 which satisfies [x2 , x2 ] = 0 up to coherent higher homotopies. The following statement proved in [30, Proposition 1.19] shows that the higher homotopies can be encoded by finding x = x2 + x3 + · · · where xn ∈ g has weight n such that x satisfies the Maurer-Cartan equation. Proposition 1.38. – Let g be a graded dg Lie algebra. Then we have a natural equivalence of ∞-groupoids ∼ MC(g≥2 ). Map gr (k(2)[−1], g) = Alg Lie

1.5. Globalization. – So far we have defined the ∞-groupoid of n-shifted Poisson structures Pois(A, n) on a commutative dg algebra A. We are now going to sketch how this definition extends to the general setting of derived Artin stacks. Such an extension will be provided using the technology of formal localization developed in [9]. We refer the reader to [34] for a more detailed account of formal localization, while here we will just indicate some general ideas. Let Ch ≤0 be the ∞-category of chain complexes of k-vector spaces concentrated in non-positive cohomological degrees (i.e., connective chain complexes). The ∞-category of connective commutative dg algebras is denoted by CAlg ( Ch ≤0 ). There is a notion of étale maps and covers in CAlg ( Ch ≤0 ). Definition 1.39. – A derived prestack is a functor of ∞-categories X : CAlg ( Ch ≤0 ) → S where S is the ∞-category of spaces. — A derived stack is a derived prestack X which satisfies étale descent: for any étale cover A → B in CAlg ( Ch ≤0 ) the natural morphism // // X(B ⊗A B) X(A) −→ Tot( X(B) / ··· ) is an equivalence. The following definition is an informal rephrasing of [50, Definition 1.3.1]: Definition 1.40. – A derived stack X is (−1)-geometric if it is equivalent to an affine derived scheme, i.e., it is a representable functor CAlg ( Ch ≤0 ) → S .

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— A derived stack X is n-geometric ` if the diagonal X → X × X is (n − 1)-geometric and it admits a smooth cover Ui → X where Ui are affine. — A derived stack X is a derived Artin stack (sometimes just called a geometric stack) if it is n-geometric for some n. Alternatively, one can say that X is n-geometric if there is a smooth groupoid o Y0 oo Y1 oo ··· , where each Yi is (n − 1)-geometric and X is equivalent to colim Y• , the quotient of Y0 by this groupoid. Let us also recall the notion of a derived stack locally of finite presentation: Definition 1.41. – A derived stack X is locally of finite presentation if for any filtered system of connective commutative dg algebras Ai ∈ CAlg ( Ch ≤0 ) the natural morphism colim X(Ai ) −→ X(colim Ai ) is an equivalence. Recall that given a commutative dg algebra A ∈ CAlg we have defined the cotangent complex LA ∈ Mod A by the universal property (4). For a derived prestack X we can define a similar universal property. Let f : S → X be a morphism where S is an affine derived scheme and M ∈ QCoh(S) a quasi-coherent sheaf on S. Let S[ M ] = Spec( O (S) ⊕ M ) be the square-zero extension. Then we define LX,f ∈ QCoh(S) by the universal property (5)

MapS/ (S[ M ], X) ∼ = MapQCoh(S) (LX,f , M ),

where MapS/ is the ∞-category of derived prestacks under S. Note that the universal property uniquely determines LX,f if it exists. For a morphism of derived affine schemes g : T → S we have a natural pullback map g ∗ LX,f → LX,f ◦g . Definition 1.42. – Let X be a derived prestack. It admits a cotangent complex if the following two properties are satisfied: (1) For any f : S → X where S is an affine derived scheme there exists LX,f ∈ QCoh(S) satisfying (5). (2) For any morphism of derived affine schemes g : T → S the natural map g ∗ LX,f → LX,f ◦g is a quasi-isomorphism. If X admits a cotangent complex, the assignment (f : S → X) 7→ LX,f defines a quasi-coherent sheaf LX ∈ QCoh(X) so that LX,f = f ∗ LX .

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Proposition 1.43. – Suppose X is a derived Artin stack. Then it admits a cotangent complex LX ∈ QCoh(X). If we assume moreover that X is locally of finite presentation, then LX is perfect. Proof. – The first statement is given by [50, Corollary 2.2.3.3]. To prove the second statement, let f : S → X where S is an affine derived scheme and consider a filtered diagram M • : I → QCoh(S). By the universal property of the cotangent complex, MapS/ (S[ M i ], X) ∼ = MapQCoh(S) (f ∗ LX , M i ). Therefore, we get a diagram of spaces colim MapS/ (S[ M i ], X)

/ MapS/ (S[colim M i ], X)

∼



∼

colim MapQCoh(S) (f ∗ LX , M i )

 / MapQCoh(S) (f ∗ LX , colim M i ).

So, if X is locally of finite presentation, the top horizontal morphism is an equivalence and hence the bottom horizontal morphism is an equivalence, i.e., f ∗ LX ∈ QCoh(S) is compact and hence dualizable. By [28, Corollary 4.6.1.11] this implies that LX ∈ QCoh(X) itself is dualizable, i.e., it is perfect. Example 1.44. – Suppose G is an affine algebraic group. Its classifying stack BG is defined to be the colimit of the groupoid pt ⇔ G · · · in the ∞-category of derived stacks. Note that by construction it is a derived Artin stack. One has an equivalence of symmetric monoidal categories QCoh(BG) ∼ = Rep G, where Rep G is the ∞-category of complexes of G-representations. Under this equivalence the cotangent complex LBG ∈ QCoh(BG) corresponds to g∗ [−1] ∈ Rep G, the coadjoint representation in cohomological degree 1. Definition 1.45. – Suppose X is a derived stack which admits a perfect cotangent complex. Its tangent complex is TX = L∨ X = Hom(LX , O X ) ∈ QCoh(X). Given an affine derived scheme S, we let H0 (S) = Spec H0 ( O (S)). If S is an affine (non-derived) scheme, then we denote by Sred the reduced scheme. Definition 1.46. – Let X be a derived prestack. Its de Rham prestack XdR is given by XdR (S) = X(H0 (S)red ).

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The construction X 7→ XdR preserves limits, so if X is a derived stack, then so is XdR . Note, however, that XdR is almost never a derived Artin stack. Nevertheless, it admits a cotangent complex which is in fact trivial since by definition XdR does not have infinitesimal deformations. Proposition 1.47. – Let X be a derived prestack. Then LXdR = 0. Example 1.48. – Suppose X is a smooth scheme. In particular, it is formally smooth, i.e., p : X → XdR is an epimorphism. Therefore,   o o XdR ∼ X ×XdR X oo ··· . = colim X o We can identify X ×XdR X ∼ × X ∆ , the formal completion of X × X along the = X\ diagonal. Therefore, XdR is a quotient of X by its infinitesimal groupoid. As the above example shows, XdR is a kind of a formal derived stack. Such objects were studied in [16] and [9, Section 2.1]. Consider the projection p : X → XdR . Even though p is not affine, it is a family of formal affine stacks (see [9, Definition 2.1.5]). It was shown in [9] that one can enhance p∗ O X to a prestack of graded mixed commutative dg algebras so that many geometric structures on X can be recovered from it. We can summarize it in the following slogan: — A formal affine stack is determined by its graded mixed commutative dg algebra of global functions. L Definition 1.49. – A graded mixed complex is a graded complex V = n V (n) equipped with a square-zero endomorphism  of weight 1 and cohomological degree 1. Denote by Ch gr, the ∞-category of graded mixed complexes. Definition 1.50. – A graded mixed commutative dg algebra is a commutative algebra in Ch gr, . Definition 1.51. – Suppose A is a graded mixed commutative dg algebra. — Its realization is |A| =

Y

A(n)

n≥0

equipped with the differential dA + . — Its Tate realization is |A|t = colim

m→−∞

equipped with the differential dA + .

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Y n≥m

A(n)

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One can also encode Tate realization in terms of the usual realization as follows. There is a certain ind-object in graded mixed complexes k(∞) which is defined in [9, Section 1.5]. Given any other graded mixed complex V , we denote V (∞) = V ⊗ k(∞). Then we can identify |A|t ∼ = |A(∞)|. Let us stress that the introduction of the twist by k(∞) will merely allow us to omit Tate realizations in what follows. Given a formal affine stack X, [9] introduce a graded mixed commutative dg algebra D(X) such that |D(X)| ∼ = O (X). Definition 1.52. – Let X be a derived stack. — The crystalline structure sheaf DXdR is a prestack of graded mixed commutative dg algebras on XdR defined as (Spec A → XdR ) 7→ D(Spec A). — The sheaf of principal parts BX is a prestack of graded mixed commutative dg algebras on XdR defined as (Spec A → XdR ) 7→ D(X ×XdR Spec A). Note that by construction we have a morphism DXdR → BX . Moreover, we have equivalences of prestacks of commutative dg algebras on XdR | BX | ∼ = p∗ O X ,

|DXdR | ∼ = O XdR .

If X is a derived Artin stack locally of finite presentation, many geometric structures on X can be reconstructed from DXdR → BX : the dg category of perfect complexes Perf(X), cotangent complex LX etc. We refer to [34] and [9, Section 2] for details. We are now ready to define shifted Poisson structures on a derived Artin stack. Definition 1.53. – Let X be a derived Artin stack locally of finite presentation. The ∞-groupoid Pois(X, n) of n-shifted Poisson structures on X is Pois(X, n) = Pois( BX (∞)/DXdR (∞), n). In other words, an n-shifted Poisson structure on X is defined to be a lift of BX (∞) from a DXdR (∞)-linear commutative algebra to a DXdR (∞)-linear Pn+1 -algebra. Remark 1.54. – In the definition of Pois(X, n) we consider BX (∞) as a commutative algebra in the dg category of prestacks of graded mixed DXdR (∞)-modules on XdR . See also Remark 1.11.

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Remark 1.55. – One may also consider a “non-Tate,” version of n-shifted Poisson structures on X defined as Pois( BX /DXdR , n). Even though the two definitions agree for derived schemes, they are different in general and, for instance, Proposition 1.57 fails for X = BG as the non-Tate version does not “see” all polyvectors. In a way, Definition 1.53 asserts that an n-shifted Poisson structure on X is the same as a fiberwise n-shifted Poisson structure on X → XdR . Such a definition is sensible since the natural inclusion of fiberwise n-shifted polyvector fields along X → XdR into all n-shifted polyvector fields on X is an equivalence since TXdR = 0 by Proposition 1.47. Similarly, under the same assumption one can define a graded Pn+2 -algebra Pol(X, n) of n-shifted polyvectors on X using BX . Proposition 1.56. – One has an equivalence of graded commutative dg algebras Pol(X, n) ∼ = Γ(X, Sym(TX [−n − 1])). Proposition 1.57. – One has an equivalence of spaces Pois(X, n) ∼ = Map gr (k(2)[−1], Pol(X, n)[n + 1]). Alg Lie

Example 1.58. – Suppose X is a (possibly singular) scheme over k. Its cotangent complex LX is connective, so the weight p part of Pol(X, n) is concentrated in cohomological degrees [p(n + 1), ∞). Therefore, by Proposition 1.57 the spaces Pois(X, n) are contractible for n > 0, i.e., every n-shifted Poisson structure for n > 0 is canonically zero. The same computation shows that Pois(X, 0) is identified with the set of ordinary Poisson structures on X. Remark 1.59. – The book [16] presents a slightly different formalism for treating formal stacks in terms of inf-schemes. Given a derived Artin stack X, the morphism p : X → XdR is proper, so p∗ is left adjoint to the symmetric monoidal functor p! : IndCoh(XdR ) → IndCoh(X), where IndCoh(−) is the ∞-category of ind-coherent sheaves. In particular, p∗ acquires an oplax symmetric monoidal structure. Therefore, we get a cocommutative coalgebra p∗ ωX ∈ IndCoh(XdR ). One may define an n-shifted Poisson structure on X as a lift of the cocommutative structure on p∗ ωX to a Pn+1 -coalgebra structure. It would be interesting to compare such a definition to Definition 1.53. Such a definition might be useful to deal with derived stacks not locally of finite presentation. Recall that in Section 1.3 we have defined a forgetful functor forgetn−m+1 : Alg Pn+1 → Alg Pn−m+1 . n+1 Since it is compatible with the forgetful functors to CAlg , it induces a forgetful map (6)

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Pois(X, n) −→ Pois(X, n − m)

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for any derived stack X. By Proposition 1.22, the forgetful map (6) is trivial for m ≥ 2, i.e., it sends any n-shifted Poisson structure on X to the zero (n − m)-shifted Poisson structure on X. We refer to Conjecture 2.43 and Proposition 3.12 for examples of the forgetful map Pois(X, n) → Pois(X, n − 1). 2. Coisotropic and Lagrangian structures In the second lecture we explain how to define n-shifted coisotropic structures in terms of relative Poisson algebras and, after a brief reminder on shifted sympletic geometry, explain the relationship between shifted Poisson and shifted symplectic structures. 2.1. Relative Poisson algebras. – Recall the equivalence of ∞-categories from Theorem 1.13: ∼ Alg P = Alg ( Alg Pn ). n+1 Given a symmetric monoidal ∞-category C we denote by LMod ( C ) the ∞-category of pairs (A, M ) of an associative algebra A in C and an A-module M . By definition (see [28, Definition 5.3.1.6]), if C admits centralizers, we can identify LMod ( C )∼ with the space of pairs of an associative algebra A ∈ Alg ( C ), an object M ∈ C and a morphism of associative algebras A → Z(M ). We will now give a similar characterization of the whole ∞-category LMod ( Alg Pn ). Definition 2.1. – Let B ∈ AlgPn be a Pn -algebra. Its strict center is Zstr (B) = HomModB (SymB (Ω1B [n]), B) with the differential twisted by [πB , −]. The strict center is equipped with a Pn+1 -algebra structure where the bracket is a generalization of the Schouten bracket of polyvector fields. The term “strict center” is explained by the following proposition which will be proved in a future paper. Proposition 2.2. – Assume B ∈ AlgPn is cofibrant as a commutative dg algebra. Then we have an equivalence of objects of Alg Pn+1 ∼ = Alg ( Alg Pn ): Zstr (B) ∼ = Z(B), where Z(B) is the center of B in the sense of Definition 1.24. We are now ready to define relative Poisson algebras introduced in [40] under the name “coisotropic morphism”. Definition 2.3. – A P[n+1,n] -algebra is a triple (A, B, F ), where — A is a Pn+1 -algebra, — B is a Pn -algebra, — F : A → Zstr (B) is a morphism of Pn+1 -algebras.

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The above definition gives rise to a two-colored dg operad P[n+1,n] of relative Poisson algebras. The following statement is proved in [42, Theorem 3.8]. Proposition 2.4. – One has an equivalence of ∞-categories ∼ Alg = LMod ( Alg ). Pn

P[n+1,n]

One can introduce a forgetful functor (see [30, Section 3.5]) U : Alg P

(7)

[n+1,n]

−→ Alg Pnu

n+1

from relative Poisson algebras to non-unital Pn+1 -algebras. Given (A, B) Alg P , this construction has the following properties:

∈

[n+1,n]

(1) One can identify as non-unital commutative dg algebras U(A, B) ∼ = fib(A → B). (2) One has a fiber sequence of Lie algebras B[n − 1] −→ U(A, B)[n] −→ A[n], i.e., we have a pullback square B[n − 1]

/ U(A, B)

 0

 / A[n].

The natural projection Z(B) → B is a morphism of commutative dg algebras, so we also get a forgetful functor AlgP[n+1,n] −→ Arr(CAlg), where Arr( C ) = Fun(∆1 , C ) is the ∞-category of arrows in C whose objects are morphisms in C . After localization it induces a functor of ∞-categories Alg P

[n+1,n]

−→ Arr( CAlg ).

Definition 2.5. – Let f : A → B be a morphism of commutative dg algebras. The space Cois(f, n) of n-shifted coisotropic structures on f is the fiber of ∼

Alg P

[n+1,n]

−→ Arr( CAlg )∼

at f ∈ Arr( CAlg ). By construction we have forgetful maps Cois(f, n) w Pois(B, n − 1)

PANORAMAS & SYNTHÈSES 55

& Pois(A, n).

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We have a natural involution opp : Pois(B, n − 1) −→ Pois(B, n − 1) induced by the automorphism of the operad Pn given by flipping the sign of the Poisson bracket. We denote by B the opposite Pn -algebra. The next statement shows that a derived intersection of n-shifted coisotropic morphisms carries an (n − 1)-shifted Poisson structure. Proposition 2.6. – Suppose A → B1 and A → B2 are two morphisms of commutative dg algebras. We have a morphism of spaces Cois(A → B1 , n) ×Pois(A,n) Cois(A → B2 , n) −→ Pois(B1 ⊗A B2 , n − 1). Moreover, given two n-shifted coisotropic morphisms A → B1 and A → B2 , the projection B1 ⊗B2 → B1 ⊗A B2 is compatible with the (n−1)-shifted Poisson structures on both sides. Proof. – By Proposition 1.19, we have a diagram of ∞-categories LMod ( Alg P

n

)

 Alg P

∼

/ RMod ( Alg ) Pn  / Alg . Pn

opp n

The relative tensor product [28, Section 4.4] gives rise to horizontal functors RMod ( Alg P

n

) × Alg ( Alg P

n

) LMod ( Alg Pn )

 RMod ( CAlg ) × Alg ( CAlg ) LMod ( CAlg )

∼

∼

/ BiMod ( Alg ) Pn

/ Alg Pn

 / BiMod ( CAlg )

 / CAlg .

Using [28, Proposition 2.4.3.9] we can identify LMod ( CAlg )

∼ = Arr( CAlg ),

RMod ( CAlg )

∼ = Arr( CAlg )

and the claim follows by passing to fibers of the vertical functors. Given a morphism f : L → X of derived Artin stacks locally of finite presentation we can define the space Cois(f, n) using the formalism of formal localization as in Section 1.5. Moreover, the previous statement admits a generalization to derived Artin stacks, see [31, Theorem 3.6].

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Proposition 2.7. – Suppose L1 , L2 → X are two morphisms of derived Artin stacks locally of finite presentation. Then we have a diagram of spaces Cois(L1 → X, n) ×Pois(X,n) Cois(L2 → X, n)

/ Pois(L1 ×X L2 , n − 1)

 Pois(L1 , n − 1) × Pois(L2 , n − 1)

 / Pois(L1 , n − 1) × Pois(L2 , n − 1).

opp×id

This is known as a coisotropic intersection theorem. 2.2. Poisson morphisms Definition 2.8. – Let f : A1 → A2 be a morphism of commutative dg algebras. The space Pois(f, n) of n-shifted Poisson structures on f is the fiber of Arr( Alg Pn+1 )∼ −→ Arr( CAlg )∼ at f ∈ Arr( CAlg ). One can similarly define the space Pois(f, n) for a morphism of derived Artin stacks locally of finite presentation. Recall that a morphism of smooth Poisson schemes f : X1 → X2 is Poisson iff its graph g : X1 → X1 × X2 is coisotropic, where X1 is X1 equipped with the opposite Poisson structure. We have a similar statement in the derived context as well, see [31, Theorem 2.8]. Proposition 2.9. – Let f : X1 → X2 be a morphism of derived Artin stacks locally of finite presentation and let g : X1 → X1 × X2 be its graph. One has a fiber square of spaces / Pois(X1 , n) × Pois(X2 , n) Pois(f, n)  Cois(g, n)

id⊕opp

 / Pois(X1 × X2 , n).

2.3. Relative polyvectors. – In this section we generalize results of Section 1.4 to the relative setting. Let f : A → B be a morphism of commutative dg algebras. The center Z(trivPn B) is a graded Pn+1 -algebra and we get a natural morphism of graded Pn+1 -algebras trivPn+1 (A) −→ Z(trivPn (B)). Also recall that we denote by Pol(B/A, n − 1) the graded Pn+1 -algebra of polyvectors of B ∈ CAlg ( Mod A ). Proposition 2.10. – One has an equivalence of graded Pn+1 -algebras Pol(B/A, n − 1) ∼ = Z(trivPn+1 (A) → Z(trivPn (B))).

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From the previous statement and the universal property of centralizers we obtain that (Pol(A, n), Pol(B/A, n − 1)) ∈ LMod ( Alg gr ). Pn+1 Definition 2.11. – Let f : A → B be a morphism of commutative dg algebras. The algebra of relative n-shifted polyvector fields is Pol(f, n) = U(Pol(A, n), Pol(B/A, n − 1)) ∈ Alg gr . Pnu n+1

Given this definition, we have the following explicit way to compute the space of n-shifted coisotropic structures. Proposition 2.12. – Let f : A → B be a morphism of commutative dg algebras. One has an equivalence of spaces Cois(f, n) ∼ (k(2)[−1], Pol(f, n)[n + 1]). = Map Alg gr Lie As in Section 1.5, we can generalize relative polyvectors to morphisms of derived Artin stacks. Proposition 2.13. – Let f : L → X be a morphism of derived Artin stacks locally of finite presentation. The image of the pair (Pol(X, n), Pol(L/X, n−1)) under the functor gr Alg P → Arr( CAlg gr ) is equivalent to the composite [n+2,n+1]

Γ(X, Sym(TX [−n − 1])) −→ Γ(L, Sym(f ∗ TX [−n − 1])) −→ Γ(L, Sym(TL/X [−n])). Proposition 2.14. – Let f : L → X be as before. Then one has an equivalence of spaces Cois(f, n) ∼ (k(2)[−1], Pol(f, n)[n + 1]). = Map Alg gr Lie Example 2.15. – Let f : L → X be an embedding of a closed subscheme. As in Example 1.58, from we see that the weight p part of Pol(f, n) is concentrated in cohomological degrees [p(n + 1), ∞). Therefore, by Proposition 2.14 the spaces Cois(f, n) are contractible for n > 0. Moreover, for n = 0 we recover the set of Poisson structures on X for which L is coisotropic in the usual sense, i.e., for which the bivector πX ∈ Γ(X, ∧2 TX ) vanishes under the restriction Γ(X, ∧2 TX ) −→ Γ(L, ∧2 NL/X ). Example 2.16. – If X is a smooth Poisson scheme, the identity inclusion X → X is coisotropic. In the derived setting one can show that there is a unique coisotropic structure. Let X be an n-shifted Poisson stack and consider the identity map id : X → X. By the property (2) of the forgetful functor U : Alg P → Alg Pnu given in [n+2,n+1] n+2 Section 2.1 we have a fiber sequence of Lie algebras Pol(X/X, n − 1)[n] −→ Pol(id, n)[n + 1] −→ Pol(X, n)[n + 1].

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∼ Γ(X, Sym(TX/X [−n])) By Proposition 1.56 we can identify Pol(X/X, n − 1) = as graded commutative dg algebras and since TX/X = 0, the graded complex Pol(X/X, n − 1) is concentrated in weight 0. In particular, the morphism Pol(id, n)≥2 → Pol(X, n)≥2 is an equivalence and hence the forgetful map Cois(id, n) −→ Pois(X, n) is an equivalence of spaces. In other words, there is a unique n-shifted coisotropic structure on the identity map compatible with the given n-shifted Poisson structure on X. ∼

Using the previous example we get a diagram Pois(X, n) ← − Cois(id, n) → Pois(X, n − 1) and hence a forgetful map Pois(X, n) → Pois(X, n − 1). Proposition 2.17. – The forgetful map Pois(X, n) → Pois(X, n − 1) defined above is equivalent to the one given by (6). 2.4. Shifted symplectic structures. – In this section we briefly remind some facts about shifted symplectic structures. The reader is referred to [33] and [7] for a more complete treatment. Definition 2.18. – Let A be a commutative dg algebra. Its de Rham complex is the graded mixed commutative dg algebra DR(A) = SymA (LA [−1]) with weights given by the Sym grading and the mixed structure given by the de Rham differential. Definition 2.19. – Let X be a derived stack. Its de Rham complex is the graded mixed commutative dg algebra DR(X) = lim DR(A). Spec A→X

Definition 2.20. – Let X be a derived stack. — The space of two-forms of degree n on X is A

2

(X, n) = Map Ch gr (k(2)[−1], DR(X)[n + 1]).

— The space of closed two-forms of degree n on X is A

2,cl

(X, n) = Map Ch gr, (k(2)[−1], DR(X)[n + 1]).

From the definition we get a morphism A

2,cl

(X, n) −→ A 2 (X, n)

which extracts the underlying two-form. Note that by pulling back differential forms to affines we obtain a natural morphism of graded commutative dg algebras (8)

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Γ(X, Sym(LX [−1])) −→ DR(X).

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Unfortunately, for a general derived stack X we do not know if this morphism is an equivalence. However, we have the following statement, see [33, Proposition 1.14]. Proposition 2.21. – Suppose X is a derived Artin stack. Then the morphism of graded commutative dg algebras (8) is an equivalence.

A

In particular, under the previous assumptions we see that a two-form ω ∈ 2 (X, n) is the same as a morphism of complexes k → Γ(X, Sym2 (LX [−1]))[2 + n].

If we assume LX is perfect, by adjunction this morphism gives rise to a morphism ω ] : TX → LX [n]. Definition 2.22. – Let X be a derived Artin stack locally of finite presentation. A two-form ω ∈ A 2 (X, n) is nondegenerate if the morphism ω ] : TX → LX [n] is an equivalence. Definition 2.23. – Let X be a derived Artin stack locally of finite presentation. The space of n-shifted symplectic structures on X is the subspace Symp(X, n) ⊂ 2,cl A (X, n) of closed two-forms whose underlying two-form is nondegenerate. Example 2.24. – Suppose Y is a derived Artin stack locally of finite presentation. Then for any n the n-shifted cotangent stack T∗ [n]Y ∼ = SpecY Sym(TY [−n]) has an exact n-shifted symplectic structure by [8]. If f : L → X is a morphism of derived stacks, we get an induced morphism of graded mixed commutative dg algebras f ∗ : DR(X) −→ DR(L) given by pulling back differential forms. An n-shifted isotropic structure on f is then given by a closed two-form ω on X of degree n and the nullhomotopy of f ∗ ω. Definition 2.25. – Let f : L → X be a morphism of derived stacks. The space of n-shifted isotropic structures on f is Isot(X, n) = Map Ch gr, (k(2)[−1], fib(f ∗ : DR(X) → DR(L))[n + 1]). Suppose f : L → X is a morphism of derived Artin stacks locally of finite presentation. It is easy to see that given an n-shifted isotropic structure on f we obtain a null-homotopy of the composite ω]

TL −→ f ∗ TX −→ f ∗ LX [n] −→ LL [n].

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Definition 2.26. – Let f : L → X be a morphism of derived Artin stacks locally of finite presentation. The space of n-shifted Lagrangian structures on f is the subspace Lagr(f, n) ⊂ Isot(X, n) of those n-shifted isotropic structures whose underlying closed two-form on X is nondegenerate and such that the sequence in QCoh(L) TL −→ f ∗ TX −→ LL [n] is a fiber sequence. Equivalently, an n-shifted isotropic structure λ induces a morphism λ] : TL/X → LL [n − 1] and it is Lagrangian if λ] is an equivalence. Example 2.27. – Suppose Z → Y is a morphism of derived Artin stacks locally of finite presentation. Then we have an n-shifted conormal stack N∗ [n](Z/Y ) = SpecZ Sym(TZ/Y [1 − n]). We have a natural morphism SpecZ Sym(TZ/Y [1 − n]) → SpecZ Sym(f ∗ TY [−n]) → SpecY Sym(TY [−n]) = T∗ [n]Y and it is shown in [8] that N∗ [n](Z/Y ) → T∗ [n](Y ) carries an exact n-shifted Lagrangian structure. Note that by construction we have a forgetful map Lagr(f, n) −→ Symp(X, n). Fix two maps fi : Li → X for i = 1, 2. Suppose X carries a closed two-form ω of degree n and we have two null-homotopies fi∗ ω ∼ 0. Then the intersection L1 ×X L2 carries two null-homotopies of the pullback g ∗ ω where g : L1 ×X L2 → X. The difference between the two null-homotopies is a closed two-form of degree (n − 1) on L1 ×X L2 . This procedure in fact preserves nondegeneracy as shown in [33, Section 2.2], i.e., given two n-shifted Lagrangians L1 , L2 → X, their intersection L1 ×X L2 has an (n − 1)-shifted symplectic structure. Proposition 2.28. – Suppose L1 , L2 → X are two morphisms of derived Artin stacks locally of finite presentation. Then we have a map of spaces Isot(L1 → X, n) × A 2,cl (X,n) Isot(L2 → X, n) −→ A 2,cl (L1 ×X L2 , n − 1). This map preserves nondegeneracy, i.e., it induces a map of spaces Lagr(L1 → X, n) ×Symp(X,n) Lagr(L2 → X, n) −→ Symp(L1 ×X L2 , n − 1). This is known as the Lagrangian intersection theorem.

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215

Example 2.29. – Suppose X is a smooth symplectic scheme and L ⊂ X is a smooth Lagrangian subscheme. Using the Koszul complex we can identify L ×X L ∼ = N[−1](L/X). Since L is Lagrangian, N(L/X) ∼ = T∗ L and hence L ×X L ∼ = T∗ [−1]L. The derived scheme L×X L is a Lagrangian intersection and so by Proposition 2.28 it inherits a (−1)-shifted symplectic structure. One can show that the equivalence L ×X L ∼ = T∗ [−1]L is compatible with the (−1)-shifted symplectic structures on both sides. 2.5. Shifted Poisson and shifted symplectic structures. – If X is a smooth scheme, a symplectic structure on X is the same as a nondegenerate Poisson structure, i.e., a Poisson structure which induces an isomorphism T∗X → TX . This is clear on the level of linear algebra and the only nontrivial computation is that closedness of the twoform ω is equivalent to the Jacobi identity for the bivector π = ω −1 . In the derived context the same result is far from obvious since we need to show that closedness of the two-form up to coherent homotopy is equivalent to the Jacobi identity for the bivector which again holds up to coherent homotopies. Nevertheless, we still have an equivalence between nondegenerate n-shifted Poisson structures and n-shifted symplectic structures. Definition 2.30. – Let X be a derived Artin stack locally of finite presentation. — Suppose π is an n-shifted Poisson structures on X. It is nondegenerate if the morphism π ] : LX −→ TX [−n] induced by the underlying bivector is an equivalence. — Denote by Poisnd (X, n) ⊂ Pois(X, n) the subspace of nondegenerate n-shifted Poisson structures. The following is proved in [9, Theorem 3.2.4] and [35, Theorem 3.13]. Theorem 2.31. – One has an equivalence of spaces Poisnd (X, n) ∼ = Symp(X, n). The proof given in [9] first proceeds by a reduction to a local statement and then using a strictification provided by a Darboux lemma one shows the equivalence. We will sketch an alternative proof given in [35] in the affine case. The theorem in both papers is proved by going through an intermediate space of compatible pairs. Let π be an n-shifted Poisson structure on A, a commutative dg algebra. Then [π, −] defines a mixed structure on the graded Pn+2 -algebra Pol(A, n). Denote by Polπ (A, n) the underlying graded mixed commutative dg algebra. By the

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universal property of the de Rham complex, one has a unique morphism of graded mixed commutative dg algebras DR(A) −→ Polπ (A, n) given by the identity A → A in weight 0 and which sends f ddR g 7→ f [π, g] in weight 1. Splitting the n-shifted Poisson structure π = π2 + π3 + · · · according to the weight, we have a natural morphism of graded mixed complexes σ : k(2)[−1] −→ Polπ (A, n)[n + 1] given by 1 7→

X

(m − 1)πm .

m≥2

Definition 2.32. – Let ω ∈ A 2,cl (A, n) be a closed two-form of degree n on A defining a morphism ω : k(2)[−1] −→ DR(X)[n + 1] of graded mixed complexes. Let π be an n-shifted Poisson structure on A. The compatibility between ω and π is a commutativity data in the diagram of graded mixed complexes / Polπ (X, n) 8

σ

k(2)[−1] ω

% DR(X).

We denote by Comp(A, n) the space of such compatible pairs. We have natural forgetful maps Comp(A, n)

A

2,cl

w (A, n)

' Pois(A, n).

Lemma 2.33. – Let (ω, π) ∈ Comp(A, n) be a compatible pair on A. Then the composite π]

ω]

π]

LX −→ TX [−n] −→ LX −→ TX [−n] is homotopic to π ] .

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Let Compnd (A, n) ⊂ Comp(A, n) be the subspace of compatible pairs (ω, π) where π is nondegenerate. In particular, by the previous lemma we get that ω is nondegenerate as well, so we obtain projections Compnd (A, n) ' Poisnd (A, n).

w Symp(A, n)

Lemma 2.34. – The projection Compnd (A, n) → Poisnd (A, n) is an equivalence. Proof. – If π is nondegenerate, the induced morphism of graded mixed commutative dg algebras DR(A) −→ Polπ (A, n) is an equivalence and hence there is a unique extension of π to a compatible pair. Therefore, we just need to show that Compnd (A, n) → Symp(A, n) is an equivalence. The strategy used in [35] is to employ the natural filtrations on both spaces and prove the statement by obstruction theory. That is, let DR≥(m+1) (A) be the truncation of DR(A) in weights at least m + 1. Let Symp≤m (A, n) ⊂ Map Ch gr, (k(2)[−1], DR(A)/DR≥(m+1) (A)) be the subspace of nondegenerate forms. Since   DR(A) = lim DR(A)/DR≥(m+1) (A) , m

we have Symp(A, n) = lim Symp≤m (A, n). m

One can similarly define Poisnd,≤m (A, n) and Compnd,≤m (A, n). Then we can prove a stronger statement: Lemma 2.35. – The projection Compnd,≤m (A, n) → Symp≤m (A, n) is an equivalence. This statement is proved by induction as follows. First, for m = 2 the statement reduces to a problem in linear algebra: given a nondegenerate two-form ω, by Lemma 2.33 we have to show there is a unique non-degenerate bivector π such that π] ∼ = π] ◦ ω] ◦ π] . By nondegeneracy of π ] this is equivalent to π ] ∼ = (ω ] )−1 . The inductive statement requires one to construct obstruction spaces.

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Definition 2.36. – Suppose p

O oo

(9)



s2 s1

X

is a diagram of simplicial sets together with homotopies p ◦ s1 ∼ = idX and p ◦ s2 ∼ = idX . Its vanishing locus is the limit of Xo

p

O oo

s2 s1

X

In other words, the vanishing locus of the diagram (9) consists of a point x ∈ X and a homotopy s1 (x) ∼ s2 (x) in p−1 (x). It is called a vanishing locus since the section s2 in examples will be a “zero section” in some sense. Studying the problem of lifting a Poisson structure π ∈ Pois≤m (A, n) to Pois≤(m+1) (A, n) one naturally encounters obstructions. Indeed, suppose π = π2 + · · · + πm ∈ Pois≤m (A, n). Its lift π 0 = π2 + · · · + πm + πm+1 ∈ Pois≤(m+1) (A, n) satisfies the Maurer–Cartan equation 1 dπ 0 + [π 0 , π 0 ] = 0 2 in the truncation Pol≤(m+1) (A, n). Since π satisfies the Maurer-Cartan equation up to weight m, we need to check the Maurer-Cartan equation only in weight (m + 1) which gives 1 X dπm+1 + [πi , πj ] = 0. 2 i+j=m+2 Thus, the fiber of Pois≤(m+1) (A, n) → Pois≤m (A, n) at π consists of nullhomotopies of 1 X obs(π) = [πi , πj ] ∈ Polm+1 (A, n). 2 i+j=m+2

Denote by Obs(Pois, m + 1) → Pois≤m (A, n) the trivial local system over Pois≤m (A, n) with fiber Polm+1 (A, n). Then we have two sections: the zero section and obs(−). Thus, we obtain an obstruction diagram as in (9). One similarly constructs obstruction diagrams ' 0 Obs(Symp, m + 1) oo Symp≤m (A, n) obs

and Obs(Comp, m + 1) oo

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( Comp≤m (A, n).

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These obstruction diagrams restrict to obstruction diagrams for nondegenerate pairs. Then one proves the following key lemmas. Lemma 2.37. – The vanishing locus of

( Comp≤m (A, n)

Obs(Comp, m + 1) oo is equivalent to

Comp≤(m+1) (A, n) −→ Comp≤m (A, n) and similarly for Symp and Pois. Lemma 2.38. – The square Obs(Comp, m + 1)

/ Compnd,≤m (A, n)

 Obs(Symp, m + 1)

 / Symp≤m (A, n)

is Cartesian. This proves the inductive step and hence Theorem 2.31. One also has a relative analog of Theorem 2.31. Given an n-shifted coisotropic structure (π, γ) on f : L → X, where π is an n-shifted Poisson structure on X, one has an induced morphism γ ] : LL/X −→ TL [−n]. Definition 2.39. – Let f : L → X be a morphism of derived Artin stacks locally of finite presentation. — Suppose (π, γ) is an n-shifted coisotropic structure on f . It is nondegenerate if the n-shifted Poisson structure π on X is nondegenerate and γ ] : LL/X → TL [−n] is an equivalence. — Coisnd (f, n) ⊂ Cois(f, n) is the subspace of nondegenerate n-shifted coisotropic structures. The following result was proved in [38] for n = 0 and [31, Theorem 4.22] for any n. Theorem 2.40. – Suppose f : L → X is a morphism of derived Artin stacks locally of finite presentation. Then one has an equivalence of spaces Coisnd (f, n) ∼ = Lagr(f, n) fitting into a diagram Coisnd (f, n)  Poisnd (X, n)

∼

/ Lagr(f, n)

∼

 / Symp(X, n).

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Under this equivalence the diagram LL/X [−1] ∼ γ]

 TL [−n]

/ f ∗ LX ∼ π]



/ f ∗ TX [−n]

/ LL ∼ γ]

 / TL/X [1 − n]

induced by a nondegenerate n-shifted coisotropic structure (π, γ) is inverse to the diagram / f ∗ TX [−n] / TL/X [1 − n] TL [−n] ∼ λ]



LL/X [−1]

∼ ω]

 / f ∗ LX

∼ λ]

 / LL

induced by the corresponding n-shifted Lagrangian structure (ω, λ). Corollary 2.41. – Under the previous assumptions we have a forgetful map Lagr(f, n) ∼ = Coisnd (f, n) −→ Pois(L, n − 1). Given a Lagrangian structure (ω, λ) on f , the induced (n−1)-shifted Poisson structure on L has bivector given by the composite (λ] )−1

LL −→ LL/X −−−−→ TL [1 − n]. Example 2.42. – Suppose f : L → X is a smooth Lagrangian subscheme of a 0-shifted symplectic scheme X. We have an exact sequence 0 −→ TL −→ f ∗ TX −→ NL/X −→ 0 of vector bundles on L which gives rise to an extension class H1 (L, N∗L/X ⊗ TL ). Since L is Lagrangian, we have an isomorphism N∗L/X ∼ = TL . Therefore, we obtain an element πL ∈ H1 (L, TL ⊗ TL ), which can be shown to be symmetric. We see that πL , the bivector underlying the (−1)-shifted Poisson structure on L, measures the obstruction for the normal bundle to L being split. Let g be a complex equipped with a nondegenerate pairing of degree n. Then d ∗ [−1]) becomes a Pn+3 -algebra. A cyclic L∞ structure on g can be encoded in Sym(g d ∗ [−1])[n + 3] which is a Hamiltonian for the vector field defining a potential h ∈ Sym(g the L∞ structure, see e.g., [6, Section 3]. Suppose (X, ω) is a smooth symplectic scheme. Kapranov [24] has defined a cyclic L∞ structure on TX [−1] with a pairing of degree −2 given by ω whose potential is ≥3 d (TX )) of cohomological degree 1. an element of Γ(X, Sym

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Conjecture 2.43. – Suppose (X, ω) is a smooth symplectic scheme. The (−1)-shifted Poisson structure on X obtained as the image of ω under the forgetful map Pois(X, 0) → Pois(X, −1) gives the potential for the cyclic L∞ structure on TX [−1]. For instance, we claim that the bivector underlying the (−1)-shited Poisson structure on X is trivial. It can be seen as follows. The diagonal X → X × X is Lagrangian and the image of ω under Pois(X, 0) → Pois(X, −1) coincides with the image of the Lagrangian structure on the diagonal under Lagr(X → X × X, 0) → Pois(X, −1). The normal bundle to the diagonal embedding is split, so by Example 2.42 the bivector underlying the (−1)-shifted Poisson structure is trivial.

3. Examples In the last lecture we give examples of shifted Poisson and shifted symplectic structures. 3.1. Symplectic realizations. – Recall from Corollary 2.41 that given a morphism f : L → X of derived Artin stacks locally of finite presentation one has a forgetful map (10)

Lagr(f, n) −→ Pois(L, n − 1).

Definition 3.1. – Let L be a derived Artin stack equipped with an (n − 1)-shifted Poisson structure. A symplectic realization of L is the data of an n-shifted Lagrangian f : L → X which lifts the (n − 1)-shifted Poisson structure on L along (10). The following statement is currently being investigated by Costello-Rozenblyum and Calaque-Vezzosi. Conjecture 3.2. – For a shifted Poisson stack L there is a unique symplectic realization L → X for which L → X is a nil-isomorphism, i.e., an isomorphism on reduced stacks. Remark 3.3. – In fact, generalizing Proposition 1.5 one can define the space Pois(X, n) in terms of such formal symplectic realizations. Such a definition allows one to reduce questions about shifted Poisson structures to questions about shifted symplectic structures which are significantly easier to work with. For instance, the coisotropic intersection theorem and the Poisson version of the AKSZ construction ([33, Theorem 2.5]) follow almost immediately from the corresponding statements for shifted symplectic structures. To motivate this notion, let us recall the following definition from [11].

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Definition 3.4. – Let G ⇒ X be a smooth algebraic groupoid. It is a symplectic groupoid if one is given a multiplicative symplectic structure ω on G for which the unit section X → G is Lagrangian. Lemma 3.5. – Suppose G ⇒ X is a symplectic groupoid. Then its Lie algebroid L is a symplectic Lie algebroid. In particular, given a symplectic groupoid G over X one has an induced Poisson structure on X by Proposition 1.5. Proposition 3.6. – Let G ⇒ X be a smooth algebraic groupoid over a smooth scheme X. Then the space Lagr(X → [X/ G ], 1) is equivalent to the set of symplectic structures ω on G endowing it with the structure of a symplectic groupoid. A symplectic groupoid G ⇒ X lifting a given Poisson structure on X gives an example of a symplectic realization of X in the sense of [52]. Thus, the forgetful map Lagr(X → [X/ G ], 1) −→ Pois(X, 0) gives a different interpretation of the underlying Poisson structure on X. Let us also explain some geometric consequences one can extract from a symplectic realization. Suppose that L → X is a 1-shifted Lagrangian such that L is a smooth Poisson manifold. The Lagrangian structure gives an isomorphism ∼

→ T∗L . λ] : TL/X − By Corollary 2.41 the underlying Poisson structure is given by the composite (λ] )−1

T∗L −−−−→ TL/X → TL . The image of the Poisson bivector under T∗L → TL gives a foliation of L by the symplectic leaves of the Poisson structure which we see is the same as the tangent foliation TL/X → TL . More generally, given a symplectic realization L → X of an n-shifted Poisson structure on L we can think of it as follows: — X is the moduli space of symplectic leaves of the n-shifted Poisson structure on L. — The fibers of L → X are the symplectic leaves. Example 3.7. – Suppose g is a finite-dimensional Lie algebra of an algebraic group G and consider the Poisson scheme g∗ equipped with the Kirillov-Kostant-Souriau Poisson structure. It is well-known that the symplectic leaves of g∗ are given by coadjoint orbits. In fact, it admits a symplectic realization g∗ −→ [g∗ /G]. This 1-shifted Lagrangian can be constructed as follows. We can identify [g∗ /G] ∼ = T∗ [1](BG) which endows it with a 1-shifted symplectic structure. The above Lagrangian can be identified with the 1-shifted conormal bundle of the basepoint pt → BG which carries a 1-shifted Lagrangian structure as in Example 2.27.

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The corresponding symplectic groupoid can be extracted as g∗ ×[g∗ /G] g∗ ∼ = T∗ G ⇒ g∗ , which coincides with the standard symplectic groupoid integrating the Poisson structure on g∗ . 3.2. Classifying stacks. – Given a group G and a polyvector p ∈ ∧k (g) at the unit of G, we can extend it using left or right translations to elements we denote by pL , pR ∈ Vk (TG ). Definition 3.8. – Let G be an affine algebraic group. A quasi-Poisson structure on G is the data of a bivector π ∈ ∧2 TG , a trivector φ ∈ ∧3 (g) satisfying π(g1 g2 ) = Lg1 ,∗ π(g2 ) + Rg2 ,∗ π(g1 ), 1 [π, π] + φL − φR = 0, 2 [π, φL ] = 0. Given a quasi-Poisson structure (π, φ) on G and a bivector λ ∈ ∧2 (g) we can construct a new quasi-Poisson structure on G via a procedure called twisting. We have the following interpretation of quasi-Poisson structures in terms of shifted Poisson structures (see [43, Proposition 2.5] and [43, Theorem 2.8] for details). Proposition 3.9. – Let G be an affine algebraic group. (1) The space Pois(BG, 2) is equivalent to the set Sym2 (g)G . (2) The space Pois(BG, 1) is equivalent to the groupoid of quasi-Poisson structures on G with morphisms given by twists. Let us briefly explain how one computes the spaces Pois(BG, n). Recall from Example 1.44 that under the identification QCoh(BG) ∼ = Rep G we have LBG ∼ = g∗ [−1]. Moreover, the functor of global sections Γ(BG, −) corresponds to taking the group cochains C• (G, −) valued in a representation. The algebra of polyvectors is then quasi-isomorphic to Pol(BG, n) ∼ = C• (G, Sym(g[−n])). By Proposition 1.57 we can compute n-shifted Poisson structures on BG in terms of Maurer-Cartan elements of weight ≥ 2 in Pol(BG, n)[n+1]. Observe that the weight p part of Pol(BG, n) is concentrated in cohomological degrees [pn, ∞) for n ≥ 0. We get the following cases: — For n > 2 all Maurer-Cartan elements are trivial for degree reasons since in this case pn − n − 1 > 1 for p ≥ 2. — For n = 2 a Maurer-Cartan element has to be concentrated in weight 2, i.e., it is given by an element c ∈ C0 (G, Sym2 (g)). The Maurer-Cartan equation forces it to be a cocycle, i.e., it has to be G-invariant.

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— Finally, for n = 1 a Maurer-Cartan element is given by a pair of elements π ∈ C1 (G, ∧2 (g)) and φ ∈ C0 (G, ∧3 (g)). The Maurer-Cartan equation boils down to the quasi-Poisson relations in Definition 3.8. Definition 3.10. – Let G be an affine algebraic group. A Poisson-Lie structure on G is the same as a multiplicative Poisson structure π on G. Equivalently, it is a quasiPoisson structure (π, φ = 0). The Lie algebra g of a Poisson-Lie group G inherits a Lie bialgebra structure. Let b the corresponding formal group. Then a Poisson-Lie structure on G b is us denote by G the same as a Lie bialgebra structure on g. The following statement is shown in [43, Corollary 2.10]. Proposition 3.11. – Let G be an affine algebraic group. (1) The space Pois(pt → BG, 1) is equivalent to the set of Poisson-Lie structures on G. b 1) is equivalent to the set of Lie bialgebra structures (2) The space Pois(pt → BG, on g. These statements essentially follow from the description of the space of 1-shifted Poisson structures on BG given by Proposition 3.9. Indeed, recall that for a Poisson manifold X the inclusion of a point pt → X is Poisson iff the Poisson structure vanishes at the point. Thus, Pois(pt → BG, 1) consists of 1-shifted Poisson structures on BG which vanish at the basepoint. The restriction to the basepoint on the level of polyvector fields is given by Pol(BG, 1) ∼ = C• (G, Sym(g[−1])) −→ C0 (G, Sym(g[−1])), so all such 1-shifted Poisson structures have φ = 0. There is also an interesting interpretation of maps (6) given by forgetting the shift. Given an element c ∈ Sym2 (g)G ⊂ g ⊗ g we denote by c12 ∈ (Ug)⊗3 the element c ⊗ 1 ∈ g ⊗ g ⊗ Ug and similarly for c23 . Then one can see that [c12 , c23 ] ∈ g⊗3 . More k concretely, pick a basis {ei } of g with structure constants fij so that X k [ei , ej ] = fij ek . k

Let c =

P

i,j

ij

c ei ⊗ ej . Then [c12 , c23 ] =

X

a cij ckl fjk ei ⊗ ea ⊗ el .

i,j,k,l,a

The following statement combines [43, Proposition 2.15] and [43, Proposition 2.17]. Proposition 3.12. – Let G be an affine algebraic group.

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(1) The map Pois(BG, 2) → Pois(BG, 1) sends c ∈ Sym2 (g)G to a quasi-Poisson structure with (π = 0, φ) where 1 φ = − [c12 , c23 ] ∈ ∧3 (g)G . 6 (2) The image of a Lie bialgebra structure under b 1) → Pois(pt → BG, b 0) Pois(pt → BG, is trivial iff the formal Poisson-Lie group G∗ integrating g∗ is formally linearizable, i.e., it is Poisson isomorphic to the completion of g∗ at the origin. Now suppose H ⊂ G is an abelian subgroup equipped with a nondegenerate pairing. Given a function r : H → g ⊗ g, its differential ddR r defines a function H → h ⊗ g ⊗ g. We denote by Alt(ddR r) the function H → ∧3 (g) given by its complete antisymmetrization. Definition 3.13. – Let G be an affine algebraic group and H ⊂ G a closed subgroup. A quasi-triangular classical dynamical r-matrix with base H is an H-equivariant function r : H → g ⊗ g satisfying (1) [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] + Alt(ddR r) = 0; (2) The symmetric part of r is a constant element of Sym2 (g)G . Note that the quotient [H/H] has a canonical 1-shifted symplectic structure. Indeed, it can be written as a Lagrangian self-intersection of the diagonal BH ×BH×BH BH, where BH carries a 2-shifted symplectic structure, to which we can apply Proposition 2.28. Alternatively, we can identify [H/H] ∼ = Map(S 1 , BH) B

and use the AKSZ construction given by [33, Theorem 2.5] to transgress the 2-shifted symplectic structure on BH to a 1-shifted symplectic structure on [H/H]. Finally, since H is abelian, we can identify [H/H] ∼ = H × BH and write in coordinates a 1-shifted symplectic structure which pairs H and BH. Using the 1-shifted symplectic on [H/H] we can turn it into a 1-shifted Poisson structure on [H/H] using Theorem 2.31. Proposition 3.14. – The set of quasi-triangular classical dynamical r-matrices with base H is equivalent to the space of pairs: (1) A 1-shifted Poisson map [H/H] −→ BH −→ BG compatible with the given 1-shifted Poisson structure on [H/H]. (2) A lift of the underlying 1-shifted Poisson structure on BG to a 2-shifted Poisson structure.

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3.3. Moduli of bundles. – Let G be a split simple group, let B ⊂ G be the Borel subgroup and H ⊂ B the Cartan subgroup. Choose a nondegenerate pairing in Sym2 (g∗ )G ∼ = Sym2 (h∗ )W which defines 2-shifted symplectic structures on BG and BH. The following is shown in [41, Lemma 3.4]. Proposition 3.15. – The correspondence BB

(11)

BH

|

"

BG

has a unique 2-shifted Lagrangian structure. Let E be an elliptic curve and denote by Bun(−) (E) = Map(E, B(−)) the moduli stack of principal bundles on E. Applying [33, Theorem 2.5] we get a 1-shifted Lagrangian correspondence BunB (E) x BunH (E)

& BunG (E).

In particular, we get a 0-shifted Poisson structure on BunB (E) known as the FeiginOdesskii Poisson structure introduced in [15], see [22] and [43, Example 4.11] for more details. Example 3.16. – Let G = PGL2 , so H = Gm . Fix a line bunde L ∈ BunH (E) and let BunB (E; L ) be the fiber of BunB (E) → BunH (E) at L . Then by the above BunB (E; L ) → BunG (E) has a 1-shifted Lagrangian structure. The stack BunB (E; L ) parametrizes rank 2 vector bundles F on E which fit into an exact sequence 0 −→ L −→ F −→ O −→ 0. Such extensions are parametrized by −1 H1 (E; L ) ∼ = H0 (E; L )∗ .

Suppose deg( L ) = −4. Then BunB (E; L ) ∼ = [A4 /Gm ]. It has an open substack given by P3 . Therefore, we get a Poisson structure on P3 which is known as the Sklyanin Poisson structure (introduced in [44]).

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The AKSZ construction given by [33, Theorem 2.5] shows that the functor Map(E, −) sends n-shifted symplectic stacks to (n − 1)-shifted symplectic stacks. Spaide [45] has given a version of this construction which shows that the moduli space of framed maps Mapf r (P1 , −) sends n-shifted symplectic stacks to (n − 1)-shited symplectic stacks. Here Mapf r (P1 , −) consists of maps from P1 which have a specified value at the basepoint of P1 . In particular, we get a 1-shifted Lagrangian correspondence Mapf r (P1 , BB) v Mapf r (P1 , BH)

( Mapf r (P1 , BG).

The stack Mapf r (P1 , BH) is a discrete set, so restricting to a fixed component λ ∈ Mapf r (P1 , BH), we get a 1-shifted Lagrangian Mapf r,λ (P1 , BB) −→ Mapf r (P1 , BG). One can check that the inclusion of the trivial G-bundle pt −→ Mapf r (P1 , BG) is also a 1-shifted Lagrangian, so Mapf r,λ (P1 , G/B) ∼ = Mapf r,λ (P1 , BB) ×Mapf r (P1 ,BG) pt has a 0-shifted symplectic structure. The scheme Mapf r,λ (P1 , G/B) by a result of Jarvis [23] (following some work by Donaldson) can be identified with the moduli space of G-monopoles on R3 with maximal symmetry breaking at infinity. 1 Finally, let us construct a classical dynamical r-matrix. Applying Map(SB , −) to the 2-shifted Lagrangian correspondence (11) by [33, Theorem 2.5] we get a 1-shifted Lagrangian correspondence [B/B] z [H/H]

$ [G/G].

The projection [B/B] → [H/H] is an isomorphism over the locus [H reg /H] ⊂ [H/H] where the Weyl group acts freely. For instance, in the case of G = SLn , H reg ⊂ H consists of diagonal matrices with distinct eigenvalues. Therefore, we get a 1-shifted Lagrangian correspondence [H reg /H] x [H reg /H]

% [G/G],

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which is a graph of a map [H reg /H] → [G/G] which is hence a symplectomorphism. The composite [H reg /H] −→ [G/G] −→ BG therefore has a 1-shifted Poisson structure. So, by Proposition 3.14 we obtain a dynamical r-matrix. One can check that in this way we get the basic trigonometric dynamical r-matrix, see [14, Proposition 3.3]. References [1] M. Alexandrov, A. Schwarz, O. Zaboronsky & M. Kontsevich – “The geometry of the master equation and topological quantum field theory”, Internat. J. Modern Phys. A 12 (1997), p. 1405–1429. [2] I. A. Batalin & G. A. Vilkovisky – “Quantization of gauge theories with linearly dependent generators”, Phys. Rev. D 28 (1983), p. 2567–2582. [3] K. Behrend – “Donaldson-Thomas type invariants via microlocal geometry”, Ann. of Math. 170 (2009), p. 1307–1338. [4] K. Behrend & B. Fantechi – “Symmetric obstruction theories and Hilbert schemes of points on threefolds”, Algebra Number Theory 2 (2008), p. 313–345. [5] O. Ben-Bassat, C. Brav, V. Bussi & D. Joyce – “A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications”, Geom. Topol. 19 (2015), p. 1287–1359. [6] C. Braun & A. Lazarev – “Unimodular homotopy algebras and Chern-Simons theory”, J. Pure Appl. Algebra 219 (2015), p. 5158–5194. [7] D. Calaque – “Three lectures on derived symplectic geometry and topological field theories”, Indag. Math. (N.S.) 25 (2014), p. 926–947. [8]

, “Shifted cotangent stacks are shifted symplectic”, Ann. Fac. Sci. Toulouse Math. 28 (2019), p. 67–90.

[9] D. Calaque, T. Pantev, B. Toën, M. Vaquié & G. Vezzosi – “Shifted Poisson Structures and Deformation Quantization”, J. Topol. 10 (2017), p. 483–584. [10] H. Chu & R. Haugseng – “Enriched ∞-operads”, Adv. Math. 361 (2020), 106913. [11] A. Coste, P. Dazord & A. Weinstein – Groupoïdes symplectiques, Publ. Dép. Math. Nouvelle Sér. A, vol. 87, Univ. Claude-Bernard, Lyon, 1987. [12] K. Costello & O. Gwilliam – Factorization algebras in quantum field theory. Vol. 1, New Mathematical Monographs, vol. 31, Cambridge Univ. Press, 2017. [13] V. A. Dolgushev & C. L. Rogers – “On an enhancement of the category of shifted L∞ -algebras”, Appl. Categ. Structures 25 (2017), p. 489–503. [14] P. Etingof & O. Schiffmann – “Lectures on the dynamical Yang-Baxter equations”, in Quantum groups and Lie theory (Durham, 1999), London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, 2001, p. 89–129. [15] B. L. Feigin & A. V. Odesskii – “Vector bundles on an elliptic curve and Sklyanin algebras”, in Topics in quantum groups and finite-type invariants, Amer. Math. Soc. Transl. Ser. 2, vol. 185, Amer. Math. Soc., 1998, p. 65–84.

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Pavel Safronov, School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh UK, EH9 3FD • E-mail : [email protected]

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2019 54. F. ANDREATTA, R. BRASCA, O. BRINON, X. CARUSO, B. CHIARELLOTTO, G. FREIXAS I MONTPLET, S. HATTORI, N. MAZZARI, S. PANOZZO, M. SEVESO, G. YAMASHITA – An excursion into p-adic Hodge theory : from foundations to recent trends 2018 53. F. BENAYCH-GEORGES, C. BORDENAVE, M. CAPITAINE, C. DONATI-MARTIN, A. KNOWLES (edited by F. BENAYCH-GEORGES, D. CHAFAÏ, S. PÉCHÉ, B. DE TILIÈRE) – Advanced Topics in Random Matrices 2017 52. P. HABEGGER, G. RÉMOND, T. SCANLON, E. ULLMO, A. YAFAEV – Autour de la conjecture de Zilber-Pink 51. F. MANGOLTE, J.-P. ROLIN, K. KURDYKA, S. BASU, V. POWERS (edited by K. BEKKA, G. FICHOU, J.-P. MONNIER, R. QUAREZ) – Géométrie algébrique réelle 2016 50. A. NOVOTNÝ, R. DANCHIN, M. PEREPELITSA, edited by D. BRESCH – Topics on Compressible Navier-Stokes Equations 49. T. SAITO, L. CLOZEL, J. WILDESHAUS – Autour des motifs. École d’été franco-asiatique de géométrie algébrique et de théorie de nombres 48. T. T. Q. LÊ, C. LESCOP, R. LIPSHITZ, P. TURNER – Lectures on quantum topology in dimension three 2015 47. M. DEMAZURE, B. EDIXHOVEN, P. GILLE, W. VAN DER KALLEN, T.-Y. LEE, S. P. LEHALLEUR, M. ROMAGNY, J. TONG, J.-K. YU – B. EDIXHOVEN, P. GILLE, G. PRASAD, P. POLO, eds. – Autour des schémas en groupes, École d’été « Schémas en groupes », Volume III 46. B. CALMÈS, P.-H. CHAUDOUARD, B. CONRAD, C. DEMARCHE, J. FASEL – B. EDIXHOVEN, P. GILLE, G. PRASAD, P. POLO, eds. – Autour des schémas en groupes, École d’été « Schémas en groupes », Volume II 45. B. DE TILIÈRE, P. FERRARI – C. BOUTILLIER, N. ENRIQUEZ, eds. – Dimer Models and Random Tilings 44. L. BORCEA, H. KANG, H. LIU, G. UHLMANN – H. AMMARI, J. GARNIER, eds. – Inverse Problems and Imaging 2014 42–43. S. BROCHARD, B. CONRAD, J. OESTERLÉ – Autour des schémas en groupes, École d’été « Schémas en groupes », Volume I 2013 41. M. LEVINE, J. WILDESHAUS, B. KAHN – Asian-French summer school on algebraic geometry and number theory 39–40. P. DEGOND, V. GRANDGIRARD, Y. SARAZIN, S. C. JARDIN, C. VILLANI – N. CROUSEILLES, H. GUILLARD, B. NKONGA, E. SONNENDRÜCKER, eds. – Numerical models for fusion

2012 38. V. BANICA, L. VEGA, C. BARDOS, D. LANNES, J. EGGERS, M. A. FONTELOS, A. MELLET, Y. POMEAU, M. LE BERRE – C. JOSSERAND, L. SAINT-RAYMOND, eds. – Singularities in mechanics : formation, propagation and microscopic description 37. D. CHAFAÏ, O. GUÉDON, G. LECUÉ, A. PAJOR – Interactions between compressed sensing random matrices and high dimensional geometry 36. K. BELABAS, H. W. LENSTRA JR., P. GAUDRY, M. STOLL, M. WATKINS, W. MCCALLUM, B. POONEN, F. BEUKERS, S. SIKSEK – Explicit methods in number theory rational points and Diophantine equations 2011 34-35. M. BRUNELLA, S. DUMITRESCU, P. EYSSIDIEUX, A. GLUTSYUK, L. MEERSSEMAN, M. NICOLAU. S. DUMITRESCU, ed. – Complex manifolds, foliations and uniformization 33. V. P. KOSTOV – Topics on hyperbolic polynomials in one variable 2010 32. J. BARRAL, J. BERESTYCKI, J. BERTOIN, A. H. FAN, B. HAAS, S. JAFFARD, G. MIERMONT, J. PEYRIÈRE – Quelques interactions entre analyse, probabilités et fractals 31. L. BONAVERO, B. HASSETT, J. M. STARR, O. WITTENBERG – Variétés rationnellement connexes : aspects géométriques et arithmétiques 30. S. CANTAT, A. CHAMBERT-LOIR, V. GUEDJ – Quelques aspects des systèmes dynamiques polynomiaux 2009 29. M. KIM, R. SUJATHA, L. LAFFORGUE, A. GENESTIER, NGÔ B. C. – École d’été francoasiatique de géométrie algébrique et de théorie des nombres 28. C. BERTHON, C. BUET, J.-F. COULOMBEL, B. DESPRÈS, J. DUBOIS, T. GOUDON, J. E. MOREL, R. TURPAULT – Mathematical models and numerical methods for radiative transfer 2008 27. P.-L. CURIEN, H. HERBELIN, J.-L. KRIVINE, P.-A. MELLIÈS – Interactive Models of Computation and Program Behaviour 26. Z. DJADLI, C. GUILLARMOU, M. HERZLICH – Opérateurs géométriques, invariants conformes et variétés asymptotiquement hyperboliques 25. M. DISERTORI, W. KIRSCH, A. KLEIN, F. KLOPP, V. RIVASSEAU – Random Schrödinger operators 2007 24. M. COSTE, T. FUKUI, K. KURDYKA, C. McCRORY, A. PARUSIŃSKI, L. PAUNESCU – Arc spaces and additive invariants in Real Algebraic and Analytic Geometry 23. R. CERF – On Cramér’s Theory in infinite dimensions 2006 ˜ NGO 22. S. VU C – Systèmes intégrables semi-classiques : du local au global . 21. S. CROVISIER, J. FRANKS, J.-M. GAMBAUDO, P. LE CALVEZ – Dynamique des difféomorphismes conservatifs des surfaces : un point de vue topologique 2005 20. A. CATTANEO, B. KELLER,C. TOROSSIAN, A. BRUGUIÈRES – Déformation, Quantification, Théorie de Lie 19. B. MALGRANGE – Systèmes différentiels involutifs

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In the series Panoramas et Synthèses are published texts from 100 to 150 pages, in French or in English, which give an account of the present state of some mathematical area. The authors aim at explaining the main problems, while giving some flavor of the proofs and an overview of the recent developments of their subject. The texts, which are intended to be read by non-specialists, should be accessible to graduate students.

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Le présent volume est un compte rendu d’une des sessions des « États de la recherche » sur la géométrie algébrique dérivée qui s’est tenue à Toulouse en juin 2017. Il contient les contributions de David BenZvi et David Nadler, Damien Calaque et Julien Grivaux, Etienne Mann et Marco Robalo, Tony Pantev et Gabriele Vezzosi, et de Pavel Safronov, associées à leurs minicours respectifs. Elles couvrent une grande variété de sujets du domaine, depuis les fondements de la géométrie algébrique dérivée et de la théorie des déformation, jusqu’à leurs applications à la géométrie énumérative, la théorie géométrique des représentations et à la catégorification dans le contexte de la géométrie algébrique. The present volume covers the content of one of the session of “États de la recherche” on derived algebraic geometry which has been held in Toulouse in June 2017. It contains the contributions of David BenZvi and David Nadler, Damien Calaque and Julien Grivaux, Etienne Mann and Marco Robalo, Tony Pantev and Gabriele Vezzosi, and Pavel Safronov, taken from their original lectures. These cover a wide variety of subjects, from foundations of derived algebraic geometry and derived deformation theory, to its applications to enumerative geometry, geometric representation theory and categorification in algebraic geometry.

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