A Study in Derived Algebraic Geometry

Volume I: Correspondences and Duality

Dennis Gaitsgory, Nick Rozenblyum (Autoren)

Buch | Hardcover

553 Seiten

2017
American Mathematical Society (Verlag)
978-1-4704-3569-1 (ISBN)

Titel ist leider vergriffen;
keine Neuauflage

Artikel merken

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.

Dennis Gaitsgory, Harvard University, Cambridge, MA. Nick Rozenblyum, University of Chicago, IL.

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.

Erscheinungsdatum	20.03.2017
Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-3569-1 / 1470435691
ISBN-13	978-1-4704-3569-1 / 9781470435691
Zustand	Neuware