Almost complete intersections
type: Item
instance of: scholarly article
title: Almost complete intersections
zbMATH Open document ID: 1152.13010
author: Sumi Seo
published in: Journal of Algebra
publication date: 2008-11-12T00:00:00Z
review text: Let \(R\) be a polynomial ring in \(n\) variables over a field \(k\) with standard grading and let \(I\subseteq R\) be a homogeneous ideal of codimension \(h\). Consider the graded minimal free \(R\)-resolution of \(R/I\): \[ 0\to \bigoplus R(-j)^{\beta_{p,j}}\to \dots \to \bigoplus R(-j)^{\beta_{1,j}}\to R\to R/I\to 0. \] The multiplicity conjecture of Herzog-Huneke-Srinivasan asserts that if \(R/I\) is a Cohen-Macaulay ring, then the multiplicity of \(R/I\) satisfies the inequalities \[ \frac{\prod_{i=1}^{h}m_i}{h!}\leq e(R/I)\leq \frac{\prod_{i=1}^{h}M_i}{h!}, \] where \[ m_i=\min\{j\mid b_{i,j}\neq 0\}\quad\text{and}\quad M_i=\max\{j\mid b_{i,j}\neq 0\}. \] Herzog, Huneke, and Srinivasan also conjecture that the inequality on the right holds even if \(R/I\) is not Cohen-Macaulay. One motivation for this conjecture is the theorem of \textit{C. Huneke} and \textit{M. Miller} [Can. J. Math. 37, 1149--1162 (1985; Zbl 0579.13012)] which proves the conjecture when the resolution is pure; that is, when \(M_i=m_i\) for all \(i\). After the present paper was written, but before it appeared, Boij and Söderberg [arXiv:0803.1645] proved the monomial conjecture using results of \textit{D. Eisenbud} and \textit{F.-O. Schreyer} [Betti Numbers of Graded Modules and Cohomology of Vector Bundles, \url{arXiv:0712.1843}] and \textit{D. Eisenbud, G. Fløystad}, and \textit{J. Weyman} [The Existence of Pure Free Resolutions, \url{arXiv:0709.1529}]. The present paper focuses on codimension three almost complete intersections. Such ideals are one link from a codimension three Gorenstein ideal; and therefore, it is possible to use the results of \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Am. J. Math. 99, 447--485 (1977; Zbl 0373.13006)] in order to record their minimal resolutions, and thereby verify the monomial conjecture for these ideals. The present paper also studies almost complete intersections which are one link from a complete intersection, such ideals are sometimes call…
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 5365182
zbMATH Keywords: multiplicity
zbMATH Keywords: complete intersection
zbMATH Keywords: Gorenstein ideal
reviewed by: Andrew Kustin
describes a project that uses: BoijSoederberg
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.04.019
OpenAlex ID: W2023151667
cites work: Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3
cites work: Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case
cites work: Irreducibility and dimension theorems for families of height 3 Gorenstein algebras
cites work: Bound on the Multiplicity of Almost Complete Intersections
cites work: Betti numbers of graded modules and cohomology of vector bundles
cites work: Q3592296
cites work: Betti numbers and degree bounds for some linked zero-schemes
cites work: Bounds for multiplicities
cites work: A Note on the Multiplicity of Cohen-Macaulay Algebras with Pure Resolutions
cites work: The multiplicity conjecture in low codimensions
cites work: Liaison des variétés algébriques. I
cites work: Note on bounds for multiplicities
DOI: 10.1016/J.JALGEBRA.2008.04.019
description: scientific article; zbMATH DE number 5365182