## Algebra & Number Theory

### The radius of a subcategory of modules

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 8, Number 1 (2014), 141-172.

Dates
Revised: 10 August 2013
Accepted: 14 September 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730137

Digital Object Identifier
doi:10.2140/ant.2014.8.141

Mathematical Reviews number (MathSciNet)
MR3207581

Zentralblatt MATH identifier
1308.13015

#### Citation

Dao, Hailong; Takahashi, Ryo. The radius of a subcategory of modules. Algebra Number Theory 8 (2014), no. 1, 141--172. doi:10.2140/ant.2014.8.141. https://projecteuclid.org/euclid.ant/1513730137

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