The radius of a subcategory of modules

TL;DR

This paper introduces a new invariant called the radius for subcategories of modules over local rings, linking it to Cohen-Macaulay modules and various homological properties, with implications for module classification.

Contribution

It defines the radius invariant for subcategories of modules and explores its implications for Cohen-Macaulay modules over complete intersection and Cohen-Macaulay rings.

Findings

Finiteness of radius implies subcategory contains only maximal Cohen-Macaulay modules.

Category of maximal Cohen-Macaulay modules has finite radius over Cohen-Macaulay complete local rings.

Radius relates to stable category dimension, Cohen-Macaulay representation type, and Auslander conditions.

Abstract

We introduce a new invariant for subcategories X of finitely generated modules over a local ring R which we call the radius of X. We show that if R is a complete intersection and X is resolving, then finiteness of the radius forces 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.