On the finiteness of radii of resolving subcategories

TL;DR

This paper studies the conditions under which resolving subcategories of finitely generated modules over a noetherian ring have finite or infinite radii, with implications for a conjecture in Cohen-Macaulay rings.

Contribution

It provides new criteria for the finiteness of radii of resolving subcategories, partially confirming a conjecture by Dao and Takahashi.

Findings

Resolving subcategories containing a canonical module and a non-MCM module of finite injective dimension have infinite radius.

The paper offers a partial positive answer to Dao and Takahashi's conjecture.

Conditions for finiteness of radii are characterized in the context of Cohen-Macaulay local rings.

Abstract

Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of mod R with respect to a fixed semidualizing module. As an application, we give a partial positive answer to a conjecture of Dao and Takahashi: we prove that for a Cohen-Macaulay local ring R, a resolving subcategory of mod R has infinite radius whenever it contains a canonical module and a non-MCM module of finite injective dimension.