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 two-volume monograph develops generalization of various topics in algebraic geometry in the context derived algebraic geometry.
Volume 1 presents the theory of ind-coherent sheaves, which 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.
Volume 2 develops deformation theory, Lie theory and the theory of algebroids in the context of derived algebraic geometry. To that end, it introduces the notion of inf-scheme, which is an infinitesimal deformation of a scheme and studies ind-coherent sheaves on inf-schemes. As an application of the general theory, the six-functor formalism for D-modules in derived geometry is obtained.
Reihe
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Maße
Höhe: 254 mm
Breite: 178 mm
ISBN-13
978-1-4704-3568-4 (9781470435684)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Dennis Gaitsgory, Harvard University, Cambridge, MA.
Nick Rozenblyum, University of Chicago, IL.
Contents for Volume I:
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
Contents for Volume II:
Inf-schemes: Introduction
Deformation theory
Ind-schemes and inf-schemes
Ind-coherent sheaves on ind-inf-schemes
An application: Crystals
Formal geometry: Introduction
Formal moduli
Lie algebras and co-commutative co-algebras
Formal groups and Lie algebras
Lie algebroids
Infinitesimal differential geometry
Bibliography
Index of notations
Index.
