**Mathematical Surveys and Monographs**

Volume: 221;
2017;
436 pp;
Softcover

MSC: Primary 14; 18;

**Print ISBN: 978-1-4704-5285-8
Product Code: SURV/221.2.S**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-4087-9
Product Code: SURV/221.2.E**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

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#### Supplemental Materials

# A Study in Derived Algebraic Geometry: Volume II: Deformations, Lie Theory and Formal Geometry

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*Dennis Gaitsgory; Nick Rozenblyum*

Derived algebraic geometry is a far-reaching
generalization of algebraic geometry. It has found numerous
applications in other parts of mathematics, most prominently in
representation theory. This volume develops deformation theory, Lie
theory and the theory of algebroids in the context of derived
algebraic geometry. To that end, it introduces the notion of
inf-scheme, which is an infinitesimal deformation of a scheme and
studies ind-coherent sheaves on such. As an application of the
general theory, the six-functor formalism for D-modules in derived
geometry is obtained.

This volume consists of two parts. The first part introduces the
notion of ind-scheme and extends the theory of ind-coherent sheaves to
inf-schemes, obtaining the theory of D-modules as an application. The
second part establishes the equivalence between formal Lie group(oids)
and Lie algebr(oids) in the category of ind-coherent sheaves. This
equivalence gives a vast generalization of the equivalence between Lie
algebras and formal moduli problems. This theory is applied to study
natural filtrations in formal derived geometry generalizing the Hodge
filtration.

#### Readership

Graduate students and researchers interested in new trends in algebraic geometry and representation theory.

#### Table of Contents

# Table of Contents

## A Study in Derived Algebraic Geometry: Volume II: Deformations, Lie Theory and Formal Geometry

- Cover Cover11
- Title page iii4
- Copyright iv5
- Contents v6
- Preface ix10
- Acknowledgements xxvii28
- Introduction xxix30
- Inf-schemes 138
- Introduction 340
- Deformation theory 1148
- Ind-schemes and inf-schemes 77114
- Ind-coherent sheaves on ind-inf-schemes 117154
- An application: Crystals 159196
- Formal geometry 191228
- Introduction 193230
- Formal moduli 203240
- Lie algebras and co-commutative co-algebras 223260
- Formal groups and Lie algebras 285322
- Lie algebroids 327364
- Infinitesimal differential geometry 381418
- Bibliography 429466
- Index of notations 431468
- Index 435472
- Back Cover Back Cover1474