Cohomological descent theory for a morphism of stacks and for equivariant derived categories

TL;DR

This paper develops a cohomological descent framework for morphisms of stacks and applies it to show that the derived category of equivariant sheaves under a reductive group action can be reconstructed from the non-equivariant derived category with group action data.

Contribution

It introduces a cohomological descent approach for stacks and demonstrates an equivalence between equivariant derived categories and categories with group actions.

Findings

Derived category of equivariant sheaves is equivalent to the category of objects with group actions.

Cohomological descent theory applies to morphisms of stacks and varieties.

Reconstruction of derived categories via descent theory.

Abstract

In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.