Classifying Resolving Subcategories

TL;DR

This paper uses Auslander Buchweitz approximations to classify resolving subcategories containing semidualizing or dualizing modules, linking them to grade consistent functions and maximal Cohen-Macaulay modules.

Contribution

It provides a classification of resolving subcategories with dualizing modules using the theory of Auslander Buchweitz approximations, establishing a bijection with grade consistent functions.

Findings

Resolving subcategories containing maximal Cohen-Macaulay modules correspond to grade consistent functions.

In rings with a dualizing module, these subcategories are precisely the dominant resolving subcategories.

The classification offers a new perspective on the structure of resolving subcategories in such rings.

Abstract

We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving subcategories containing maximal Cohen-Macaulay modules are in bijection with grade consistent functions and thus are the precisely the dominant resolving subcategories.