Derived equivalences of functor categories

TL;DR

This paper extends Rickard's theorem to categories of modules over small categories, providing criteria for derived equivalences and applications to recollements, path rings, and graded rings.

Contribution

It generalizes Rickard's derived equivalence theorem to functor categories and explores applications to recollements and specific classes of rings.

Findings

Established a version of Rickard's theorem for $ ext{Mod} extbf{C}$.

Derived criteria for derived equivalences in functor categories.

Constructed recollements of derived categories in various contexts.

Abstract

Let $\Mod \CS$ denote the category of $\CS$-modules, where $\CS$ is a small category. In the first part of this paper, we provide a version of Rickard's theorem on derived equivalence of rings for $\Mod \CS$. This will have several interesting applications. In the second part, we apply our techniques to get some interesting recollements of derived categories in different levels. We specialize our results to path rings as well as graded rings.