Bounds for one-dimensional rings of finite Cohen-Macaulay type
type: Item
instance of: scholarly article
title: Bounds for one-dimensional rings of finite Cohen-Macaulay type
zbMATH Open document ID: 0813.13013
DOI: 10.1016/0022-4049(94)90093-0
author: Roger Wiegand
author: Sylvia Wiegand
published in: Journal of Pure and Applied Algebra
publication date: 1995-05-30T00:00:00Z
review text: Let \(R\) be a one-dimensional reduced noetherian ring such that the integral closure \(R^ -\) of \(R\) is a finitely generated \(R\)-module. The rank of an MCM \(R\)-module (i.e., a torsion-free \(R\)-module) \(M\) is the sequence \(\{\text{rank} (M_ p) | p\) is a minimal prime ideal of \(R\}\). The ring \(R\) is said to be of finite CM type if \(R\) has only finitely many indecomposable MCM modules up to isomorphism and is of bounded CM type if there exists a bound on the ranks of MCM \(R\)-modules. Using the notion of modules over ``Artin pairs'', considerations of the case of local rings, and some glueing techniques, the authors show the following: Theorem. (1) If \(R\) is of bounded CM type and any residue field of \(R^ -\) is not purely inseparable of degree 2 over \(R\), then every MCM \(R\)- module of constant rank has a direct summand of modules of constant rank 1, 2, 3, 4, 5, 6, 8, 9, or 12. (2) Let \(k\) be a field admitting a separable non-Galois extension of degree 3. Then there exists a one-dimensional integral domain of finite CM type which is essentially of finite type over \(k\), has exactly two maximal ideals having residue field \(k\), and has indecomposable MCM modules of rank 1, 2, 3, 4, 5, 6, 8, 9, and 12.
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 584669
zbMATH Keywords: maximal Cohen-Macaulay module
zbMATH Keywords: ring of finite CM type
zbMATH Keywords: one-dimensional local ring
zbMATH Keywords: Artin pairs
reviewed by: Akira Ooishi
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90093-0
OpenAlex ID: W2076054293
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description: scientific article; zbMATH DE number 584669