A generalization of the dimension and radius of a subcategory of modules and its applications

Abstract

Let $R$ be a commutative noetherian local ring and denote by $\mathrm{mod}R$ the category of finitely generated $R$-modules. In this paper, we give some evaluations of the singular locus of $R$ and annihilators of Tor and Ext from a viewpoint of the finiteness of dimensions/radii of full subcategories of $\mathrm{mod}R$. As an application, we recover a theorem of Dey and Takahashi when $R$ is Cohen–Macaulay. Moreover, we obtain the divergence of the dimensions of specific full subcategories of $\mathrm{mod}R$ in the non-Cohen–Macaulay case.