nLab A study in derived algebraic geometry

This entry is about the book (in progress)

• D. Gaitsgory, N. Rozenblyum, A study in derived algebraic geometry, web.

It develops some aspects of the theory of derived algebraic geometry with the categorical geometric Langlands conjecture in mind.

See also

• Dennis Gaitsgory, Notes on geometric Langlands, web.

• D. Gaitsgory et. al., Geometric representation theory, graduate seminar, Fall 2009–Spring 2010, web.

Contents

• Preface

• Introduction to Part I: Preliminaries

• Chapter I.1: Some higher algebra

An introduction to $\infty$-categories and review of Lurie’s books

• Chapter I.2: Basics of derived algebraic geometry

Introduces (derived) prestacks, stacks, schemes and Artin stacks.

• Chapter I.3: Quasi-coherent sheaves on prestacks

Introduces and studies the basic properties of the category of quasi-coherent sheaves on a prestack.

• Introduction to Part II: Ind-coherent sheaves

• Chapter II.1: Ind-coherent sheaves on schemes, pdf

Introduces and studies elementary properties of the stable (infinity,1)-category of ind-coherent sheaves on derived schemes.

• Chapter II.2: The !-pullback and base change, pdf.

Discusses how the system of stable (infinity,1)-categories of ind-coherent sheaves satisfies the formalism of six operations.

• Chapter II.3: Relation between QCoh and IndCoh, pdf.

Discusses the relationship between the stable (infinity,1)-categories of quasi-coherent sheaves and ind-coherent sheaves.

• Introduction to Part III: Inf-schemes

• Chapter III.1: Deformation theory, pdf.

Sets up derived deformation theory.

• Chapter III.2: (Ind)-inf-schemes, pdf.

Introduces inf-schemes?, which are algebro-geometric objects that include derived schemes and de Rham prestacks of schemes

• Chapter III.3: Ind-coherent sheaves on (ind)-inf-schemes, pdf.

Extends the formalism of ind-coherent sheaves to inf-schemes? and discusses functoriality.

• Chapter III.4: An application: crystals, pdf.

Studies the six operations for D-modules.

• Introduction to Part IV: Formal geometry

• Chapter IV.1: Lie algebras and co-commutative co-algebras, pdf.

Sets up the theory of Lie algebras from the point of view of Koszul-Quillen duality?.

• Chapter IV.2: Formal moduli, pdf.

Reinterprets Jacob Lurie‘s theory of formal moduli problems using the language of inf-schemes?.

• Chapter IV.3: Formal groups and Lie algebras, pdf.

Explains how to pass between Lie algebras and formal groups within the framework of derived algebraic geometry.

• Chapter IV.4: Lie algebroids, pdf.

Introduces Lie algebroids and studies various aspects of infinitesimal geometry?.

• Chapter IV.5: Infinitesimal differential geometry

Studies various aspects of infinitesimal geometry, such as the n-th infinitesimal neighborhood.

• Introduction to Part V: Categories of correspondences

• Chapter V.1: The $(\infty,2)$-category of correspondences

Introduces the formalism of correspondences.

• Chapter V.2: Extension theorems for category of correspondences

Extends the theory of IndCoh from schemes to inf-schemes.

• Chapter V.3: The (symmetric) monoidal structure on the category of correspondences

Shows how the formalism of correspondences encodes Serre duality

• Introduction to Part A (appendix on $(\infty,2)$-categories)

• Chapter A.1: Basics of $(\infty,2)$-categories

Defines $(\infty,2)$-categories and introduces some basic constructions

• Chapter A.2: Straightening and Yoneda $(\infty,2)$-categories

Constructs the straightening/unstraightening procedures and the Yoneda embedding

• Chapter A.3: Adjunctions in $(\infty,2)$-categories

Studies the procedure of passing to the adjoint 1-morphism

• References