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

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.

Dedicated to Professor Craig Huneke on the occasion of his sixtieth birthday

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
Received: 14 July 2012
Revised: 10 August 2013
Accepted: 14 September 2013
Published: 20 April 2014
Hailong Dao
Department of Mathematics
University of Kansas
405 Snow Hall
1460 Jayhawk Blvd
Lawrence, KS 66045
United States
Ryo Takahashi
Department of Mathematics
University of Nebraska
Lincoln, NE 68588-0130
United States
Graduate School of Mathematics
Nagoya University
Furocho, Chikusaku
Nagoya 464-8602