Contravariantly finite resolving subcategories over commutative rings

TL;DR

This paper classifies contravariantly finite resolving subcategories over henselian Gorenstein local rings, revealing only three such subcategories, and connects these findings to the Gorenstein property of the ring.

Contribution

It provides a complete classification of contravariantly finite resolving subcategories over a specific class of commutative rings and links this to the Gorenstein property.

Findings

Only three contravariantly finite resolving subcategories exist over henselian Gorenstein local rings.

The classification method recovers a known theorem relating totally reflexive modules and Gorenstein rings.

The work enhances understanding of the structure of module categories over commutative rings.

Abstract

Contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras. In this paper we study contravariantly finite resolving subcategories over commutative rings. The main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian Gorenstein local ring; in fact there exist only three ones. Our method to obtain this classification also recovers as a by-product the theorem of Christensen, Piepmeyer, Striuli and Takahashi concerning the relationship between the contravariant finiteness of the full subcategory of totally reflexive modules and the Gorenstein property of the base ring.