Ascent of finite Cohen-Macaulay type
type: Item
instance of: scholarly article
title: Ascent of finite Cohen-Macaulay type
zbMATH Open document ID: 0963.13020
author: Graham J. Leuschke
author: Roger Wiegand
published in: Journal of Algebra
publication date: 2000-10-23T00:00:00Z
review text: Let \(A\) be a Noetherian local ring. A finitely generated \(A\)-module~\(M\) is said to be maximal Cohen-Macaulay if \(\operatorname {depth} M = \dim A\). We say that \(A\) has finite Cohen-Macaulay (representation) type if there are only finitely many indecomposable maximal Cohen-Macaulay modules up to isomorphism. There are many studies on complete Cohen-Macaulay rings having finite Cohen-Macaulay type. In the present paper, the authors study the relation between the ring~\(A\) and its completion~\(\widehat A\). They show that an excellent Cohen-Macaulay ring~\(A\) has finite Cohen-Macaulay type if and only if \(\widehat A\) has finite Cohen-Macaulay type. The second author already showed the if part. The only if part is new. The key of the proof is the existence of the canonical module. The authors use Hinich's result: An excellent Henselian local ring has a canonical module [\textit{V. Hinich}, Math. Nachr. 163, 289-296 (1993; Zbl 0799.13003)].
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13J10
zbMATH DE Number: 1471836
zbMATH Keywords: maximal Cohen-Macaulay
zbMATH Keywords: finite Cohen-Macaulay representation type
zbMATH Keywords: Henselian rings
zbMATH Keywords: canonical module
zbMATH Keywords: completion
describes a project that uses: classifyCeq.lib
MaRDI profile type: Publication
full work available at URL: jabr.2000.8294
OpenAlex ID: W1986227848
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DOI: 10.1006/JABR.2000.8294
description: scientific article; zbMATH DE number 1471836