Symmetric subcategories, tilting modules and derived recollements

TL;DR

This paper explores the relationship between good tilting modules, symmetric subcategories, and derived recollements, establishing a framework that connects endomorphism rings and module categories through derived category decompositions.

Contribution

It introduces the concept of n-symmetric subcategories associated with tilting modules and constructs derived recollements linking their derived categories.

Findings

Existence of n-symmetric subcategories for good tilting modules.

Derived category of the endomorphism ring forms a recollement with these subcategories.

The kernel of the tensor functor is equivalent to the derived category of the symmetric subcategory.

Abstract

For any good tilting module $T$ over a ring $A$, there exists an $n$-symmetric subcategory $\mathscr{E}$ of a module category such that the derived category of the endomorphism ring of $T$ is a recollement of the derived categories of $\mathscr{E}$ and $A$ in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of $\mathscr{E}$.