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Tame-wild dichotomy for Cohen-Macaulay modules. (English) Zbl 0760.16005
Let \(R\) be the complete local ring of a reduced curve singularity over an algebraically closed field \(K\). It is shown that, with respect to Cohen- Macaulay (CM) representations, the rings of infinite type (i.e. having infinitely many indecomposable CM-representations) split into two classes. For the first one, called tame, indecomposable CM-modules of any fixed rank form a finite set of at most 1-parameter families of non- isomorphic indecomposable modules, while for the second one, called wild, there exist families of non-isomorphic indecomposable modules of arbitrarily large dimension.
The theorem is actually proved for not necessarily commutative Cohen- Macaulay \(K\)-algebras of Krull dimension 1. The tame-wild dichotomy was known before for finite dimensional (over \(K\)) algebras and proved by the first author [in Representations and quadratic forms, Kiev 1979, 39-74 (1979; Zbl 0454.16014); English translation in: Transl., II. Ser., Am. Math. Soc. 128, 31-55 (1986)].
The proof of this paper is based on the same method of “matrix problems”, or so called representations of bocses. Because of the restriction to CM-representations the authors had to consider a new situation, namely that of “open subcategories”. This new shape seems to be unavoidable in the case of Cohen-Macaulay modules but it should also be of use to other questions in representation theory.

MSC:
16G50 Cohen-Macaulay modules in associative algebras
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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