Resolving subcategories closed under certain operations and a conjecture of Dao and Takahashi

TL;DR

This paper investigates conditions under which resolving subcategories of finitely generated modules over local rings contain the residue field or consist of totally reflexive modules, proving a conjecture in specific cases.

Contribution

It establishes that resolving subcategories closed under cosyzygies contain the residue field or totally reflexive modules, and confirms Dao and Takahashi's conjecture in certain scenarios.

Findings

Resolving subcategories closed under cosyzygies contain the residue field or totally reflexive modules.

The conjecture of Dao and Takahashi holds in several specific cases.

Provides new insights into the structure of resolving subcategories in commutative algebra.

Abstract

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