On the finiteness of radii of resolving subcategories

Abstract

Let R be a commutative Noetherian ring. Denote by \({\text {mod}}R\) the category of finitely generated R-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of \({\text {mod}}R\) with respect to a fixed semidualizing module. As an application, we give a partial positive answer to a conjecture of Dao and Takahashi: we prove that for a Cohen–Macaulay local ring R, a resolving subcategory of \({\text {mod}}R\) has infinite radius whenever it contains a canonical module and a non-MCM module of finite injective dimension.