Resolving Subcategories Closed Under Certain Operations and a Conjecture of Dao and Takahashi

Abstract

Let R be a commutative Noetherian local ring with residue field k. Let $\mathcal{X}$ be a resolving subcategory of finitely generated R-modules. This paper mainly studies when $\mathcal{X}$ contains k or consists of totally reflexive modules. It is proved that $\mathcal{X}$ does so if $\mathcal{X}$ is closed under cosyzygies. A conjecture of Dao and Takahashi is also shown to hold in several cases.