- ホーム
- > 洋書
- > 英文書
- > Science / Mathematics
Full Description
Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in various parts of mathematics, most prominently in representation theory. This volume develops the theory of ind-coherent sheaves in the context of derived algebraic geometry. Ind-coherent sheaves are a "renormalization" of quasi-coherent sheaves and provide a natural setting for Grothendieck-Serre duality as well as geometric incarnations of numerous categories of interest in representation theory.
This volume consists of three parts and an appendix. The first part is a survey of homotopical algebra in the setting of $\infty$-categories and the basics of derived algebraic geometry. The second part builds the theory of ind-coherent sheaves as a functor out of the category of correspondences and studies the relationship between ind-coherent and quasi-coherent sheaves. The third part sets up the general machinery of the $\mathrm{(}\infty, 2\mathrm{)}$-category of correspondences needed for the second part. The category of correspondences, via the theory developed in the third part, provides a general framework for Grothendieck's six-functor formalism. The appendix provides the necessary background on $\mathrm{(}\infty, 2\mathrm{)}$-categories needed for the third part.
Contents
Preliminaries: Introduction
Some higher algebra
Basics of derived algebraic geometry
Quasi-coherent sheaves on prestacks
Ind-coherent sheaves: Introduction
Ind-coherent sheaves on schemes
Ind-coherent sheaves as a functor out of the category of correspondences
Interaction of Qcoh and IndCoh
Categories of correspondences: Introduction
The $(\infty,2)$-category of correspondences
Extension theorems for the category of correspondences
The (symmetric) monoidal structure on the category of correspondences
$(\infty,2)$-categories: Introduction
Basics of 2-categories
Straightening and Yoneda for $(\infty,2)$-categories
Adjunctions in $(\infty,2)$-categories
Bibliography
Index of notations
Index.
