A Study in Derived Algebraic Geometry

Released on by Dennis Gaitsgory
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  • A Study in Derived Algebraic Geometry Book Detail

  • Author : Dennis Gaitsgory
  • Release Date : 2019-12-31
  • Publisher : American Mathematical Society
  • Genre : Mathematics
  • Pages : 533
  • ISBN 13 : 1470452847
  • File Size : 87,87 MB
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A Study in Derived Algebraic Geometry by Dennis Gaitsgory PDF Summary

Book Description: 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 $mathrm{(}infty, 2mathrm{)}$-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 $mathrm{(}infty, 2mathrm{)}$-categories needed for the third part.

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