A study in derived algebraic geometry Volume I ...

A study in derived algebraic geometry Volume I: Correspondences and duality

Dennis Gaitsgory Nick Rozenblyum

2010 Mathematics Subject Classification. Primary

Contents

Preface

Acknowledgements

Introduction

Part I. Preliminaries

Introduction Why do we need these preliminaries? 1. -categories and higher algebra 2. Basics of derived algebraic geometry 3. Quasi-coherent sheaves

Chapter 1. Some higher algebra Introduction 1. (, 1)-categories 2. Basic operations with (, 1)-categories 3. Monoidal structures 4. Duality 5. Stable (, 1)-categories 6. The symmetric monoidal structure on 1 -CatScotn,ctocmpl 7. Compactly generated stable categories 8. Algebra in stable categories 9. Rigid monoidal categories 10. DG categories

Chapter 2. Basics of derived algebraic geometry Introduction 1. Prestacks 2. Descent and stacks 3. (Derived) schemes 4. (Derived) Artin stacks

Chapter 3. Quasi-coherent sheaves on prestacks Introduction 1. The category of quasi-coherent sheaves 2. Direct image for QCoh 3. The symmetric monoidal structure

Part II. Ind-coherent sheaves

vii

xi

xxvii

xxix

1

3 3 3 4 5

7 7 16 26 34 45 50 56 64 70 79 86

95 95 99 110 121 129

141 141 143 151 157

167

viii

CONTENTS

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

195

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

228

4. Multiplicative structure and duality

233

5. Convolution monoidal categories and algebras

239

Chapter 6. Interaction of QCoh and IndCoh

243

Introduction

243

1. The (, 2)-category of pairs

248

2. The functor of IndCoh, equipped with the action of QCoh

251

3. The multiplicative structure

261

4. Duality

265

Part III. Categories of correspondences

271

Introduction

273

1. Why correspondences?

273

2. The six functor formalism

275

3. Constructing functors

280

4. Extension theorems

282

5. (Symmetric) monoidal structures

283

Chapter 7. The (, 2)-category of correspondences

287

Introduction

287

1. The 2-category of correspondences

298

2. The category of correspondences via grids

305

3. The universal property of the category of correspondences

320

4. Enlarging the class of 2-morphisms at no cost

326

5. Functors constructed by factorization

336

Chapter 8. Extension theorems for the category of correspondences

351

Introduction

351

1. Functors obtained by bivariant extension

354

CONTENTS

ix

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

377

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

469

3. Locally 2-Cartesian and 2-Cartesian fibrations over Gray products 473

4. Proof of Theorem 1.1.8

480

5. The Yoneda embedding

483

A. The universal right-lax functor

487

B. Localizations on 1-morphisms

493

Chapter 12. Adjunctions in (, 2)-categories

495

Introduction

495

1. Adjunctions

497

2. Proof of Theorem 1.2.4

504

3. Adjunction with parameters

512

4. An alternative proof

517

Bibliography

523

Index of Notations

525

x

CONTENTS

Index

529

Preface

Kto ? Ne kamenwik pr moi, Ne krovel wik, ne korabel wik, ? Dvuruxnik , s dvoinoi duxoi,

noqi drug, dn zastrel wik.

O. Mandel xtam. Grifel na 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 indcoherent 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 geometers. 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 nature. 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 sheaves on the classifying stack BG.

The advantage of this point of view is that many natural constructions associated with the category of representations are already contained in the package of `quasi-coherent sheaves on stacks'. For example, the functors of restriction and

xi

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download