Derived categories of sheaves on singular schemes with an application to reconstruction

TL;DR

This paper establishes an equivalence between certain derived categories of sheaves on singular schemes and functor categories, introduces new concepts like pseudo-adjoints and Rouquier functors, and extends reconstruction results to Gorenstein varieties.

Contribution

It introduces pseudo-adjoints and Rouquier functors, and extends the reconstruction theorem to Gorenstein projective varieties.

Findings

Equivalence of derived categories and functor categories for singular schemes

Introduction of pseudo-adjoints and Rouquier functors

Extension of reconstruction results to Gorenstein varieties

Abstract

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties.