This entry is about the book (in progress)
It develops some aspects of the theory of derived algebraic geometry with the categorical geometric Langlands conjecture in mind.
D. Gaitsgory et. al., Geometric representation theory, graduate seminar, Fall 2009–Spring 2010, web.
Preface
Introduction to Part I: Preliminaries
Chapter I.1: Some higher algebra
An introduction to $\infty$-categories and review of Lurie’s books
Introduces (derived) prestacks, stacks, schemes and Artin stacks.
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.
Discusses how the system of stable (infinity,1)-categories of ind-coherent sheaves satisfies the formalism of six operations.
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.
Introduces inf-schemes?, which are algebro-geometric objects that include derived schemes and de Rham prestacks of schemes
Extends the formalism of ind-coherent sheaves to inf-schemes? and discusses functoriality.
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?.
Reinterprets Jacob Lurie‘s theory of formal moduli problems using the language of inf-schemes?.
Explains how to pass between Lie algebras and formal groups within the framework of derived algebraic geometry.
Introduces Lie algebroids and studies various aspects of infinitesimal 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.
Extends the theory of IndCoh from schemes to inf-schemes.
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
Constructs the straightening/unstraightening procedures and the Yoneda embedding
Studies the procedure of passing to the adjoint 1-morphism
Last revised on January 25, 2016 at 18:27:39. See the history of this page for a list of all contributions to it.