Modules in resolving subcategories which are free on the punctured spectrum

TL;DR

This paper explores modules within resolving subcategories over noetherian local rings by relating them to modules free on the punctured spectrum, introducing a new notion of level, and generalizing existing theorems on nonfree loci.

Contribution

It introduces an analogue of the level in triangulated categories for resolving subcategories and generalizes results on the dimension of nonfree loci for categories with countably many indecomposables.

Findings

Established a relation between modules in resolving subcategories and those free on the punctured spectrum.

Introduced a notion of level in the context of resolving subcategories.

Proved a generalized theorem on the dimension of nonfree loci for categories with countably many indecomposables.

Abstract

Let R be a commutative noetherian local ring, and let X be a resolving subcategory of the category of finitely generated R-modules. In this paper, we study modules in X by relating them to modules in X which are free on the punctured spectrum of R. We do this by investigating nonfree loci and establishing an analogue of the notion of a level in a triangulated category which has been introduced by Avramov, Buchweitz, Iyengar and Miller. As an application, we prove a result on the dimension of the nonfree locus of a resolving subcategory having only countably many nonisomorphic indecomposable modules in it, which is a generalization of a theorem of Huneke and Leuschke.