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Softcover ISBN:  9781470452841 
Product Code:  SURV/221.1.S 
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Book DetailsMathematical Surveys and MonographsVolume: 221; 2017; 533 ppMSC: Primary 14; 18
Derived algebraic geometry is a farreaching 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 indcoherent sheaves in the context of derived algebraic geometry. Indcoherent sheaves are a “renormalization” of quasicoherent sheaves and provide a natural setting for GrothendieckSerre 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 indcoherent sheaves as a functor out of the category of correspondences and studies the relationship between indcoherent and quasicoherent 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 sixfunctor formalism. The appendix provides the necessary background on \(\mathrm{(}\infty, 2\mathrm{)}\)categories needed for the third part.
ReadershipGraduate students and researchers interested in new trends in algebraic geometry and representation theory.
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Table of Contents

Preliminaries

Introduction

Some higher algebra

Basics of derived algebraic geometry

Quasicoherent sheaves on prestacks

Indcoherent sheaves

Introduction

Indcoherent sheaves on schemes

Indcoherent 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 2categories

Straightening and Yoneda for $(\infty ,2)$categories

Adjunctions in $(\infty ,2)$categories


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Derived algebraic geometry is a farreaching 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 indcoherent sheaves in the context of derived algebraic geometry. Indcoherent sheaves are a “renormalization” of quasicoherent sheaves and provide a natural setting for GrothendieckSerre 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 indcoherent sheaves as a functor out of the category of correspondences and studies the relationship between indcoherent and quasicoherent 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 sixfunctor formalism. The appendix provides the necessary background on \(\mathrm{(}\infty, 2\mathrm{)}\)categories needed for the third part.
Graduate students and researchers interested in new trends in algebraic geometry and representation theory.

Preliminaries

Introduction

Some higher algebra

Basics of derived algebraic geometry

Quasicoherent sheaves on prestacks

Indcoherent sheaves

Introduction

Indcoherent sheaves on schemes

Indcoherent 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 2categories

Straightening and Yoneda for $(\infty ,2)$categories

Adjunctions in $(\infty ,2)$categories