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Indecomposable Cohen-Macaulay modules and their multiplicities. (English) Zbl 0725.13009

It is shown that the first Brauer-Thrall type conjecture is affirmative for a reduced excellent henselian Cohen-Macaulay local ring \((A,m,k)\) such that (i) A is an isolated singularity; (ii) \(A/pA\) is an isolated singularity \((p=char(k))\); (iii) \([k:k^ p]\) is finite if \(p>0\). Let A be a henselian Cohen-Macaulay local ring and \(n_ A(s)\) the cardinal of isomorphism classes of indecomposable maximal Cohen-Macaulay modules whose multiplicity equals s.
First Brauer-Thrall type conjecture: If \(\sum_{s}n_ A(s) =\infty\), then \(n_ A(s)\neq 0\) for infinitely many s. - For the proof, the author studies the defining ideal \(I_ s(R)\) of the singular locus of an excellent local ring R and gives a sufficient condition for \((R,I_ s(r))\) to be a CM-approximation and a sufficient condition for \(I_ s(R)^ r\) to be a CM-reduction ideal for some r. - Let A be a local ring as above which satisfies the conditions (i), (ii), (iii), and let \(\Gamma\) be the AR-quiver of A and \(\Gamma^ 0\) the connected component of \(\Gamma\). Main theorem: If \(\Gamma^ 0\) is of bounded multiplicity type, then \(\Gamma =\Gamma^ 0\) and \(\Gamma\) is a finite graph.
Y. Yoshino proved the same theorem as above for reduced analytic algebras over a perfect valued field which are isolated singularities [J. Math. Soc. Japan 39, 719-739 (1987; Zbl 0615.13008)]. The proof follows as in Yoshino’s paper.

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13H15 Multiplicity theory and related topics
13J15 Henselian rings
16G30 Representations of orders, lattices, algebras over commutative rings
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