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.
