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A Study in Derived Algebraic Geometry - Volume I: Correspondences and Duality (Paperback)
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A Study in Derived Algebraic Geometry - Volume I: Correspondences and Duality (Paperback)
Series: Mathematical Surveys and Monographs
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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 (?,2)-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
(?,2)-categories needed for the third part.
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