nLab A study in derived algebraic geometry

This entry is to record the reference:

• D. Gaitsgory, N. Rozenblyum:

  
  

  A study in derived algebraic geometry

  
  

  Mathematical Surveys and Monographs 221

  Americal Mathematical Society (2017)

  publisher page, author’s website

on derived algebraic geometry with an eye towards the categorical geometric Langlands conjecture.

See also

• Dennis Gaitsgory, Notes on geometric Langlands [web]

• Dennis Gaitsgory et. al.: Geometric representation theory, graduate seminar (2009-10) [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