**Mathematical Surveys and Monographs**

Volume: 221;
2017;
533 pp;
Hardcover

MSC: Primary 14; 18;

Print ISBN: 978-1-4704-3569-1

Product Code: SURV/221.1

List Price: $124.00

Individual Member Price: $99.20

**Electronic ISBN: 978-1-4704-4085-5
Product Code: SURV/221.1.E**

List Price: $124.00

Individual Member Price: $99.20

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# A Study in Derived Algebraic Geometry: Volume I: Correspondences and Duality

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*Dennis Gaitsgory; Nick Rozenblyum*

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.

#### Readership

Graduate students and researchers interested in new trends in algebraic geometry and representation theory.