A Study in Derived Algebraic Geometry: Volume 1

Dennis Gaitsgory · Nick Rozenblyum

01/2017 · American Mathematical Soc.

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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  -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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Dennis Gaitsgory: Harvard University, Cambridge, MA,
Nick Rozenblyum: University of Chicago, Chicago, IL