#### Vol. 8, No. 1, 2014

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The radius of a subcategory of modules

### Hailong Dao and Ryo Takahashi

Vol. 8 (2014), No. 1, 141–172
##### Abstract

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

 Dedicated to Professor Craig Huneke on the occasion of his sixtieth birthday
##### Keywords
radius of subcategory, resolving subcategory, thick subcategory, Cohen–Macaulay module, complete intersection, dimension of triangulated category, Cohen–Macaulay representation type
##### Mathematical Subject Classification 2010
Primary: 13C60
Secondary: 13C14, 16G60, 18E30