Quasi-complete intersection ideals of height \(2\)
type: Item
instance of: scholarly article
title: Quasi-complete intersection ideals of height \(2\)
zbMATH Open document ID: 0930.13010
DOI: 10.1016/S0022-4049(97)00085-6
published in: Journal of Pure and Applied Algebra
publication date: 2000-02-07T00:00:00Z
review text: Let \(R = k[x_0,\dots,x_n]\), \(n \geq 3\), and let \({\mathfrak m} = (x_0,\dots,x_n)R\). A homogeneous ideal \(I \subset R\) of height \(m\) is a quasi-complete intersection (q.c.i.) on \(m+1\) elements if \(I\) is unmixed and there exist \(m+1\) homogeneous elements \(F_0,\dots,F_m\) in \(I\) such that \((F_0,\dots,F_m) R = I \cap {\mathfrak q}\) for some \({\mathfrak m}\)-primary ideal \(\mathfrak q\). This paper is concerned with the problem of describing minimal generating sets for such an ideal, assuming that it is prime and its height is 2. One of the main results is the following. Assume \({\mathfrak p}\) is a homogeneous prime ideal of height 2 with at least 3 minimal generators, and assume that \(\mathfrak p\) is a q.c.i.\ on \(F_0,F_1,F_2\). If \(\mathfrak B\) is an arbitrary minimal homogeneous generating set of \(\mathfrak p\), then there are elements \(B_0,B_1,B_2\) in \(\mathfrak B\) such that \( {\mathfrak B}/\{B_0,B_1,B_2\} \cup \{F_0,F_1,F_2\} \) is a minimal generating set of \(\mathfrak p\). Another result, slightly more technical, gives a way of obtaining, from a fixed homogeneous minimal generating set for a q.c.i.\ prime ideal \(\mathfrak p\), a set of three polynomials on which \(\mathfrak p\) is a q.c.i. The authors apply these results to monomial curves, by characterizing those monomial curves with 4 minimal generators which are in fact quasi-complete intersections.
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14M07
zbMATH DE Number: 1308161
zbMATH Keywords: quasi-complete intersection
zbMATH Keywords: monomial curve
zbMATH Keywords: codimension 2
zbMATH Keywords: minimal generating set
zbMATH Keywords: homogeneous prime ideal of height 2
author: H. Bresinsky
author: Peter Schenzel
author: J. Stückrad
reviewed by: Juan C. Migliore
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00085-6
OpenAlex ID: W2087492623
description: scientific article; zbMATH DE number 1308161