Cohen-Macaulay Rees algebras and degrees of polynomial relations
type: Item
instance of: scholarly article
title: Cohen-Macaulay Rees algebras and degrees of polynomial relations
zbMATH Open document ID: 0819.13003
DOI: 10.1007/BF01446637
author: Bernd Ulrich
author: Aron Simis
author: Wolmer V. Vasconcelos
published in: Mathematische Annalen
publication date: 1995-08-27T00:00:00Z
full work available at URL: https://eudml.org/doc/165299
review text: The Rees algebra of an ideal \(I\) of a Noetherian ring \(R\) is the subalgebra of the ring of polynomials \(R[t]\), \(R[It] = \sum_{i \geq 0} I^ it^ i\). It realizes algebraically the notion of blowing up a variety along a subvariety, so that it is of interest to uncover its arithmetical properties, such as normality or Cohen-Macaulayness. Here we are concerned with how the latter occurs in two distinct ways. In a manner of speaking, \(R[It]\) is made up of two parts, \(R\) itself and the associated graded ring \(\text{gr}_ I (R) = \sum_{i \geq 0} I^ i/I^{i+1}\). One basic set of issues is how the Cohen-Macaulayness of \(R[It]\) depends on that of \(R\) and \(\text{gr}_ I(R)\). On account of an argument of Huneke, the issue can be narrowed down to expressing the Cohen-Macaulayness of \(R[It]\) vis-a-vis \(G = \text{gr}_ I (R)\) and \(a(G)\), the so-called \(a\)-invariant of \(G\). After clarifying several cases where \(G\) is Gorenstein, we give our first main result, a calculation of \(a(G)\), in terms of the analytic spread \(\ell (I)\) and reduction number \(r(I)\) of \(I\), under the hypothesis that \(I\) satisfies a mild condition on the local number of generators in codimension below \(\ell (I)\). Our second main result bypasses the use of \(\text{gr}_ I (R)\), focusing instead on the degrees of the generators of modules of syzygies and the polynomial relations amongst the elements of a generating set of the ideal. We explore how the Cohen-Macaulayness of \(R[It]\) is described in terms of these sets of degrees and other integers that arise from a specification of the ideal through a presentation matrix. Some of the results are explicit formulas for the degrees of polynomial relations in terms of homological properties of the ideal, therefore linking aspects of elimination theory with homology.
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 759275
zbMATH Keywords: \(a\)-invariant
zbMATH Keywords: Rees algebra
zbMATH Keywords: Cohen-Macaulayness
zbMATH Keywords: associated graded ring
zbMATH Keywords: analytic spread
zbMATH Keywords: reduction number
zbMATH Keywords: number of generators
zbMATH Keywords: elimination theory
reviewed by: Aron Simis
MaRDI profile type: Publication
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OpenAlex ID: W2037707696
description: scientific article; zbMATH DE number 759275