The radius of a subcategory of modules

Dao, Hailong; Takahashi, Ryo

doi:10.2140/ant.2014.8.141

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The radius of a subcategory of modules

Hailong Dao and Ryo Takahashi

Vol. 8 (2014), No. 1, 141–172

DOI: 10.2140/ant.2014.8.141

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.

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

Milestones

Received: 14 July 2012

Revised: 10 August 2013

Accepted: 14 September 2013

Published: 20 April 2014

Authors

Hailong Dao
Department of Mathematics University of Kansas 405 Snow Hall 1460 Jayhawk Blvd Lawrence, KS 66045 United States
http://www.math.ku.edu/~hdao/
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 Japan
http://www.math.nagoya-u.ac.jp/~takahashi/

Vol. 8, No. 1, 2014