Intersection multiplicity, canonical element conjecture and the syzygy problem
type: Item
instance of: scholarly article
title: Intersection multiplicity, canonical element conjecture and the syzygy problem
zbMATH Open document ID: 1181.13013
author: Phillip Griffith
author: S. P. Dutta
published in: Michigan Mathematical Journal
publication date: 2009-09-17T00:00:00Z
review text: This paper consists of independent work by the two authors, but published in a single document. Both authors' work pertains to the direct summand conjecture (DSC) and monomial conjecture (MC), and related homological conjectures, all of which are stated for the reader's convenience. The first author's work focuses on the question (Q) of when \(\chi^A(M,N) > \chi^A_2(M,N)\), where \(A\) is regular local ring and \(M\) and \(N\) are finitely generated \(A\)-modules with length\((M \otimes_A N) < \infty\). The first result states that, under the additional assumption that \(A\) is equicharacteristic or unramified, the answer to (Q) is negative when dim \(M\) + dim \(N <\) dim \(A\) and depth \(M\) + depth \(N <\) dim \(A-1\), and positive when both inequalities are replaced with equalities. Although the answer to (Q) is unknown when one of the modules, say \(N\), has the form \(N=A/I\), where \(I\) is an ideal of \(A\) generated by an \(A\)-sequence of length \(n\) such that length\((M/IM) < \infty\) and dim \(M\) + dim \(A/I =\) dim \(A\), the author shows that, in this case, there exists a minimal set of generators \(x_1, \dots, x_n\) of \(I\) and \(s >> 0\) for which (Q) has a positive answer if \(N = A/(x_1, \dots, x_{n-1}, x_n^s)\). The proof uses superficial elements. Next, the author relates the validity of MC to pairs of ideals \(I\) and \(J\) in \(A\) such that \(I\) is a complete intersection, \(J\) is an almost complete intersection, ht \(I\) + ht \(J\) = dim \(A\), \(I+J\) is primary to the maximal ideal, and \(I+J\) is not a complete intersection ideal in \(A\). One consequence of the results in this section is that for a complete local domain \(A\) with a system of parameters \(x_1, \dots, x_n\), there exist \(y_1, \dots, y_n\) in \((x_1, \dots, x_n)\) such that \((y_1, \dots, y_{n-1}, x_n) = (x_1, \dots, x_n)\) and \((y_1, \dots, y_{n-1}, x_n^s)\) satisfies MC for \(s>>0\). Additionally, the author advances some of his earlier work [Math Z. 216, 379--388…
reviewed by: Sandra Spiroff
Mathematics Subject Classification ID: 13D22
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5604529
zbMATH Keywords: homological conjectures
zbMATH Keywords: intersection multiplicities
zbMATH Keywords: syzygies
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1220879406
OpenAlex ID: W2018649968
Wikidata QID: Q123256461
cites work: Homological Dimension in Local Rings
cites work: Q4274969
cites work: On the Canonical Element Conjecture
cites work: On the canonical element conjecture. II
cites work: Dualizing Complex and the Canonical Element Conjecture
cites work: Dualizing Complex and the Canonical Element Conjecture II
cites work: A Note on the Monomial Conjecture
cites work: Splitting of local cohomology of syzygies of the residue field
cites work: Intersection multiplicity of modules in the positive characteristics
cites work: The syzygy problem
cites work: Q3309969
cites work: A graded syzygy theorem in mixed characteristic.
cites work: The direct summand conjecture in dimension three.
cites work: Contracted ideals from integral extensions of regular rings
cites work: Q4055756
cites work: Canonical elements in local cohomology modules and the direct summand conjecture
cites work: Euler characteristics over unramified regular local rings
cites work: Degree \(p\) extensions of an unramified regular local ring of mixed characteristic \(p\)
cites work: Almost complete intersections are not Gorenstein rings
cites work: On the vanishing of Tor in regular local rings
cites work: Q4166835
cites work: Dimension projective finie et cohomologie locale. Applications à la demonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck
cites work: Q3749157
cites work: Q5823327
cites work: Monomial conjecture and complete intersections
DOI: 10.1307/MMJ/1220879406
description: scientific article; zbMATH DE number 5604529