Annihilation of cohomology and strong generation of module categories

TL;DR

This paper introduces the concept of cohomology annihilators for noetherian rings and explores their connection to strong generators in module categories, extending existing results to broader classes of rings.

Contribution

It establishes a link between cohomology annihilators and strong generators, generalizing prior results to excellent local rings and rings essentially of finite type over a field.

Findings

Link between non-trivial cohomology annihilators and strong generators

Extension of results to excellent local rings

Generalization to rings of finite type over a field

Abstract

The cohomology annihilator of a noetherian ring that is finitely generated as a module over its center is introduced. Results are established linking the existence of non-trivial cohomology annihilators and the existence of strong generators for the category of finitely generated modules. Exploiting this link, results of Popescu and Roczen, and Wang concerning cohomology annihilators of commutative rings, and also results of Aihara and Takahashi, Keller and Van den Bergh, and Rouquier on strong finite generation of the corresponding bounded derived category, are generalized to cover excellent local rings and also rings essentially of finite type over a field.