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Mathematical Surveys and Monographs Volume 221 A Study in Derived Algebraic Geometry Volume I: Correspondences and Duality Dennis Gaitsgory Nick Rozenblyum American Mathematical Society A Study in Derived Algebraic Geometry Volume I: Correspondences and Duality 10.1090/surv/221.1 Mathematical Surveys and Monographs Volume 221 A Study in Derived Algebraic Geometry Volume I: Correspondences and Duality Dennis Gaitsgory Nick Rozenblyum American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 14-02, 14A20, 14F05, 18G55, 18D05. Library of Congress Cataloging-in-Publication Data Names: Gaitsgory, D. (Dennis) | Rozenblyum, Nick, 1984- Title: A study in derived algebraic geometry/ Dennis Gaitsgory, Nick Rozenblyum. Description: Providence, Rhode Island : American Mathematical Society, 2017 | Series: Mathe- matical surveys and monographs ; volume 221.1 | Includes bibliographical references and index. Contents: volume 1. Correspondences and duality – volume 2. Deformations, Lie theory, and formal geometry. Identifiers: LCCN 2016054809 | Volume I ISBN 9781470435691 (alk. paper) | Volume II ISBN 9781470435707 (alk. paper) Subjects: LCSH:Geometry, Algebraic. | Duality theory (Mathematics) | Lie algebras. | Geometry. | AMS: Algebraic geometry – Research exposition (monographs, survey articles).msc | Algebraic geometry – Foundations – Generalizations (algebraic spaces, stacks).msc | Algebraic geome- try – (Co)homology theory – Sheaves, derived categories of sheaves and related constructions. msc | Algebraic geometry – Families, fibrations – Formal methods; deformations. msc | Alge- braic geometry – (Co)homology theory – Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials. msc | Category theory; homological algebra – Homolog- ical algebra – Homotopical algebra. msc | Category theory; homological algebra – Categories with structure – Double categories, 2-categories, bicategories and generalizations.msc Classification: LCC QA564.G35 2017 | DDC 516.3/5–dc23 LC record available at https://lccn. loc.gov/2016054809 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the authors. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Contents Preface ix Acknowledgements xxvii Introduction xxix Part I. Preliminaries 1 Introduction 3 Why do we need these preliminaries? 3 1. ∞-categories and higher algebra 3 2. Basics of derived algebraic geometry 4 3. Quasi-coherent sheaves 5 Chapter 1. Some higher algebra 7 Introduction 7 1. (∞, 1)-categories 16 2. Basic operations with (∞, 1)-categories 26 3. Monoidal structures 35 4. Duality 45 5. Stable (∞, 1)-categories 50 St,cocmpl 6. The symmetric monoidal structure on 1 -Catcont 56 7. Compactly generated stable categories 64 8. Algebra in stable categories 71 9. Rigid monoidal categories 80 10. DG categories 87 Chapter 2. Basics of derived algebraic geometry 95 Introduction 95 1. Prestacks 99 2. Descent and stacks 110 3. (Derived) schemes 121 4. (Derived) Artin stacks 130 Chapter 3. Quasi-coherent sheaves on prestacks 141 Introduction 141 1. The category of quasi-coherent sheaves 143 2. Direct image for QCoh 151 3. The symmetric monoidal structure 157 v vi CONTENTS Part II. Ind-coherent Sheaves 167 Introduction 169 1. Ind-coherent sheaves vs quasi-coherent sheaves 169 2. How to construct IndCoh? 172 Chapter 4. Ind-coherent sheaves on schemes 177 Introduction 177 1. Ind-coherent sheaves on a scheme 180 2. The direct image functor 183 3. The functor of ‘usual’ inverse image 187 4. Open embeddings 193 5. Proper maps 196 6. Closed embeddings 201 7. Groupoids and descent 207 Chapter 5. Ind-coherent sheaves as a functor out of the category of correspondences 211 Introduction 211 1. Factorizations of morphisms of DG schemes 218 2. IndCoh as a functor from the category of correspondences 222 3. The functor of !-pullback 226 4. Multiplicative structure and duality 231 5. Convolution monoidal categories and algebras 237 Chapter 6. Interaction of QCoh and IndCoh 241 Introduction 241 1. The (∞, 2)-category of pairs 246 2. The functor of IndCoh, equipped with the action of QCoh 249 3. The multiplicative structure 259 4. Duality 263 Part III. Categories of Correspondences 269 Introduction 271 1. Why correspondences? 271 2. The six functor formalism 273 3. Constructing functors 278 4. Extension theorems 280 5. (Symmetric) monoidal structures 282 Chapter 7. The (∞, 2)-category of correspondences 285 Introduction 285 1. The 2-category of correspondences 296 2. The category of correspondences via grids 304 3. The universal property of the category of correspondences 319 4. Enlarging the class of 2-morphisms at no cost 324 5. Functors constructed by factorization 334 CONTENTS vii Chapter 8. Extension theorems for the category of correspondences 351 Introduction 351 1. Functors obtained by bivariant extension 354 2. Limits and colimits of sequences 360 3. The core of the proof 368 4. Proof of Proposition 1.2.5: easy reduction steps 374 5. End of the proof of Proposition 1.2.5 378 6. Functors obtained by horizontal extension 381 Chapter 9. The (symmetric) monoidal structure on the category of correspondences 387 Introduction 387 1. (Symmetric) monoidal structures: recollections 390 2. (Symmetric) monoidal structures and correspondences 394 3. Extension results in the symmetric monoidal context 400 4. Monads and associative algebras in the category of correspondences 406 Appendix. (∞, 2)-categories 419 Introduction 421 1. Why do we need them? 421 2. Setting up the theory of (∞, 2)-categories 423 3. The rest of the Appendix 424 Chapter 10. Basics of 2-Categories 427 Introduction 427 1. Recollections: (∞, 1)-categories via complete Segal spaces 431 2. The notion of (∞, 2)-category 436 3. Lax functors and the Gray product 442 4. (∞, 2)-categories via squares 449 5. Essential image of the functor Sq●,● 455 6. The (∞, 2)-category of (∞, 2)-categories 458 Chapter 11. Straightening and Yoneda for (∞, 2)-categories 463 Introduction 463 1. Straightening for (∞, 2)-categories 465 2. Straightening over intervals 471 3. Locally 2-Cartesian and 2-Cartesian fibrations over Gray products 475 4. Proof of Theorem 1.1.8 481 5. The Yoneda embedding 484 A. The universal right-lax functor 489 B. Localizations on 1-morphisms 494 Chapter 12. Adjunctions in (∞, 2)-categories 497 Introduction 497 1. Adjunctions 499 2. Proof of Theorem 1.2.4 507 3. Adjunction with parameters 515 4. An alternative proof 520 viii CONTENTS Bibliography 525 Index of Notations 527 Index 531 Preface Kto ? Ne kamenwik prmo˘i, Ne krovelwik, ne korabelwik, – Dvuruxnik , s dvo˘ino˘iduxo˘i, noqi drug, dn zastrelwik. O. Mandelxtam. Grifelna oda. Who am I? Not a straightforward mason, Not a roofer, not a shipbuilder, – I am a double agent, with a duplicitous soul, I am a friend of the night, a skirmisher of the day. O. Mandelshtam. The Graphite Ode. 1. What is the object of study in this book? The main unifying theme of the two volumes of this book is the notion of ind- coherent sheaf, or rather, categories of such on various geometric objects. In this section we will try to explain what ind-coherent sheaves are and why we need this notion. 1.1. Who are we? Let us start with a disclosure: this book is not really about algebraic geometry. Or, rather, in writing this book, its authors do not act as real algebraic geome- ters. This is because the latter are ultimately interested in geometric objects that are constrained/enriched by the algebraicity requirement. We, however, use algebraic geometry as a tool: this book is written with a view toward applications to representation theory. It just so happens that algebraic geometry is a very (perhaps, even the most) convenient way to formulate representation-theoretic problems of categorical na- ture. This is not surprising, since, after all, algebraic groups are themselves objects of algebraic geometry. The most basic example of how one embeds representation theory into algebraic geometry is this: take the category Rep(G) of algebraic representations of a linear algebraic group G. Algebraic geometry allows us to define/interpret Rep(G) as the category of quasi-coherent
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