Cohen-Macaulayness in graded algebras
type: Item
instance of: scholarly article
title: Cohen-Macaulayness in graded algebras
zbMATH Open document ID: 0844.13006
author: Joseph Lipman
published in: Mathematical Research Letters
publication date: 1996-08-27T00:00:00Z
review text: Cohen-Macaulayness of Rees algebras and associated graded rings of ideals have been actively investigated in recent years. We illustrate here how some methods introduced by \textit{J. B. Sancho de Salas} [in: Géométrie algébrique et applications, La Rabida 1984, Part I, Trav. Cours 22, 201-209 (1987; Zbl 0625.14025)] yield additional insights in this area. Theorem 4.1 relates the Cohen-Macaulay property of a Rees algebra to the vanishing of cohomology of its Proj. As a special case, we get that in a normal local ring \(R\), there exists an ideal \(I\) whose blow-up desingularizes \(R\) and whose Rees algebra \(R[It]\) is Cohen-Macaulay (CM) if and only if \(R\) has a rational singularity -- at least in characteristic zero, and more generally if desingularization theorems such as have been announced by Spivakovsky hold up. [In contrast, theorem 4.3, due to J. B. Sancho de Salas (loc. cit.) implies that for any CM local ring on a variety over \(\mathbb{C}\), there exists a desingularizing \(I\) whose associated graded ring is CM.] Theorem 5 gives an affirmative answer to a question of Huneke: It states that if \(R\) is pseudo-rational and if \(I\) is a non-zero \(R\)-ideal whose associated graded ring is CM, then \(R[It]\) is CM, too.
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 817681
zbMATH Keywords: Cohen-Macaulayness of Rees algebras
zbMATH Keywords: vanishing of cohomology
zbMATH Keywords: desingularization
MaRDI profile type: Publication
full work available at URL: mrl.1994.v1.n2.a2
OpenAlex ID: W2058906530
DOI: 10.4310/MRL.1994.V1.N2.A2
description: scientific article; zbMATH DE number 817681