Strong generation for module categories

TL;DR

This paper characterizes when commutative Noetherian rings have strong generators in their module categories, providing criteria, explicit examples, and bounds on Rouquier dimension, advancing understanding in algebraic module theory.

Contribution

It establishes a criterion for strong generation in module categories of Noetherian rings and identifies explicit generators and bounds for Rouquier dimension.

Findings

Noetherian quasi-excellent rings of finite Krull dimension satisfy the criterion.

Explicit strong generators are identified for rings of prime characteristic.

Upper bounds on Rouquier dimension are established in terms of classical invariants.

Abstract

This article investigates strong generation within the module category of a commutative Noetherian ring. We establish a criterion for such rings to possess strong generators within their module category, addressing a question raised by Iyengar and Takahashi. As a consequence, this not only demonstrates that any Noetherian quasi-excellent ring of finite Krull dimension satisfies this criterion, but applies to rings outside this class. Additionally, we identify explicit strong generators within the module category for rings of prime characteristic, and establish upper bounds on Rouquier dimension in terms of classical numerical invariants for modules.