This entry is to record the reference:
Dennis Gaitsgory, Nick Rozenblyum:
A study in derived algebraic geometry
Mathematical Surveys and Monographs 221
Americal Mathematical Society (2017)
on derived algebraic geometry with an eye towards the categorical geometric Langlands conjecture.
Dennis Gaitsgory et. al.: Geometric representation theory, graduate seminar (2009-10) [web]
Preface
Introduction to Part I: Preliminaries
Chapter I.1: Some higher algebra
An introduction to -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 -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 -categories)
Chapter A.1: Basics of -categories
Defines -categories and introduces some basic constructions
Chapter A.2: Straightening and Yoneda -categories
Constructs the straightening/unstraightening procedures and the Yoneda embedding
Chapter A.3: Adjunctions in -categories
Studies the procedure of passing to the adjoint 1-morphism
Last revised on July 11, 2025 at 12:52:15. See the history of this page for a list of all contributions to it.