On a theorem of Knörrer concerning Cohen-Macaulay modules
type: Item
instance of: scholarly article
title: On a theorem of Knörrer concerning Cohen-Macaulay modules
zbMATH Open document ID: 0967.13008
DOI: 10.1016/S0022-4049(99)00123-1
author: Liam O'Carroll
author: Dorin Popescu
published in: Journal of Pure and Applied Algebra
publication date: 2000-11-21T00:00:00Z
review text: Let \(K\) be a field, \(X = X_1,\ldots,X_n\) and \(Y = Y_1,\ldots,Y_r\) sets of indeterminates. Let \(f \in KX\) be a non-zero formal power series with \(f \in (X)\) and \(s_1,\ldots,s_r\) positive integers with each \(s_i \geq 2\) which are not multiple of the characteristic of \(K\). Put \(R = KX,Y/(f+\sum_{i=1}^rY_i^{s_i})\), \(R_1= KX/(f)\) and \(\widetilde{R} = R/(Y_1^{s_1-1},\ldots,Y_r^{s_r-1})\). The main result of this paper shows that for a maximal Cohen-Macaulay \(R\)-module \(N\) with no free direct summands, we have (i) \(\widetilde N := \widetilde R\otimes_RN\) is a deformation of the maximal Cohen-Macaulay \(R_1\)-module \(N/YN\) to \(\widetilde R\), (ii) \(\Omega_R^r(\widetilde N) \cong (N\oplus \Omega_R^1(N))^{2^{r-1}}\). This generalizes a result of \textit{H. Knörrer} in the case \(r=1\) and \(g = Y_1^2\) [Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. The authors also find resolutions of the higher order syzygy modules which are minimal in certain situation, and express them in terms of matrix factorizations. The approach simplifies and extends that of \textit{J. Herzog} and \textit{D. Popescu} [Math. Ann. 309, No. 4, 677-700 (1997; Zbl 0902.13010)] by using iterated mapping cones.
reviewed by: Trung Ngo Viet
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14B07
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 1517309
zbMATH Keywords: maximal Cohen-Macaulay module
zbMATH Keywords: isolated singularity
zbMATH Keywords: higher order syzygy
zbMATH Keywords: matrix factorization
zbMATH Keywords: mapping cone
zbMATH Keywords: formal power series
zbMATH Keywords: Thom-Sebastiani problems
MaRDI profile type: Publication
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Wikidata QID: Q126407797
full work available at URL: s0022-4049(99)00123-1
OpenAlex ID: W2073153943
description: scientific article; zbMATH DE number 1517309