Brauer-Thrall type theorem for maximal Cohen-Macaulay modules
type: Item
instance of: scholarly article
title: Brauer-Thrall type theorem for maximal Cohen-Macaulay modules
zbMATH Open document ID: 0615.13008
author: Yuji Yoshino
published in: Journal of the Mathematical Society of Japan
publication date: 1987-01-01T00:00:00Z
review text: Let k be a perfect valuation field and let R be a local analytic k- algebra which is assumed to be Cohen-Macaulay. In this paper we are interested in the category C(R) consisting of all maximal Cohen-Macaulay modules over R. R is said to be of finite representation type provided that C(R) has only a finite number of isomorphism classes of indecomposable objects. We show that the Auslander-Reiten theory is available for the category C(R). Precisely the following is proved: Theorem (1.1). Let \(\Gamma\) be the AR-graph of C(R) and let \(\Gamma ^ 0\) be a connected component of \(\Gamma\). Assume that R is an isolated singularity and assume that there is an upper bound for the multiplicities (and/or ranks) of modules in \(\Gamma ^ 0\). Then \(\Gamma =\Gamma ^ 0\) and \(\Gamma\) is a finite graph. In particular, R is of finite representation type. - From this theorem we may obtain the following corollary (1.4) (Brauer-Thrall type theorem): Let R be an isolated singularity. If there is a bound for multiplicities of indecomposable objects in C(R), then R is of finite representation type. One of the applications of the theorem is given in {\S} 4. Actually we prove the following: Suppose R is a normal Gorenstein local analytic domain of dimension \(2\) over an algebraically closed field of characteristic \(0\). Assume that R is none of the Klein singularities. Then, for any positive \(integer\quad n\), there exist infinitely many classes of indecomposable reflexive modules of \(rank\quad n\). In the appendix we give a brief introduction to the Auslander-Reiten theory for the category C(R).
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 32B05
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 3995848
zbMATH Keywords: analytic algebra
zbMATH Keywords: Auslander-Reiten quiver
zbMATH Keywords: maximal Cohen-Macaulay modules
zbMATH Keywords: bound for the multiplicities
zbMATH Keywords: finite representation type
zbMATH Keywords: Brauer-Thrall type theorem
zbMATH Keywords: isolated singularity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03940719
OpenAlex ID: W2094193393
DOI: 10.2969/JMSJ/03940719
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description: scientific article; zbMATH DE number 3995848