Modules in resolving subcategories which are free on the punctured spectrum

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.