Brauer-Thrall type theorem for maximal Cohen-Macaulay modules
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).
scientific article; zbMATH DE number 3995848