The radius of a subcategory of modules

Abstract

We introduce a new invariant for subcategories $\mathcal{X}$ of finitely generated modules over a local ring $R$ which we call the radius of $\mathcal{X}$. We show that if $R$ is a complete intersection and $\mathcal{X}$ is resolving, then finiteness of the radius forces $\mathcal{X}$ to contain only maximal Cohen–Macaulay modules. We also show that the category of maximal Cohen–Macaulay modules has finite radius when $R$ is a Cohen–Macaulay complete local ring with perfect coefficient field. We link the radius to many well-studied notions such as the dimension of the stable category of maximal Cohen–Macaulay modules, finite/countable Cohen–Macaulay representation type and the uniform Auslander condition.