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A Study in Derived Algebraic Geometry

Volume II: Deformations, Lie Theory and Formal Geometry

By: Dennis Gaitsgory,Nick Rozenblyum

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A Study in Derived Algebraic Geometry

Volume II: Deformations, Lie Theory and Formal Geometry

By: Dennis Gaitsgory,Nick Rozenblyum

Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in other parts of mathematics, most prominently in representation theory. This volume 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 such. As an application of the general theory, the six-functor formalism for D-modules in derived geometry is obtained. This volume consists of two parts. The first part introduces the notion of ind-scheme and extends the theory of ind-coherent sheaves to inf-schemes, obtaining the theory of D-modules as an application. The second part establishes the equivalence between formal Lie group(oids) and Lie algebr(oids) in the category of ind-coherent sheaves. This equivalence gives a vast generalization of the equivalence between Lie algebras and formal moduli problems. This theory is applied to study natural filtrations in formal derived geometry generalizing the Hodge filtration.

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Page Count

436

Publisher

American Mathematical Society

Published Date

ISBN 10

1470452855

ISBN 13

9781470452858

Stratified Noncommutative Geometry

By: David Ayala,Aaron Mazel-Gee,Nick Rozenblyum

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Stratified Noncommutative Geometry

By: David Ayala,Aaron Mazel-Gee,Nick Rozenblyum

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Page Count

272

Publisher

American Mathematical Society

Published Date

ISBN 10

1470469626

ISBN 13

9781470469627

A Study in Derived Algebraic Geometry

By: Dennis Gaitsgory,Nick Rozenblyum

Write a review Read reviews Take a Quiz Solve Book Puzzle

A Study in Derived Algebraic Geometry

By: 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 -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 -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 -categories needed for the third part.

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Page Count

577

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470435691

ISBN 13

9781470435691

Stratified Noncommutative Geometry

By: David Ayala,Aaron Mazel-Gee,Nick Rozenblyum

Write a review Read reviews Take a Quiz Solve Book Puzzle

Stratified Noncommutative Geometry

By: David Ayala,Aaron Mazel-Gee,Nick Rozenblyum

View the abstract.

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Page Count

272

Publisher

American Mathematical Society

Published Date

ISBN 10

1470469626

ISBN 13

9781470469627