Local properties of licci ideals
type: Item
instance of: scholarly article
title: Local properties of licci ideals
zbMATH Open document ID: 0769.13006
DOI: 10.1007/BF02571423
author: Craig Huneke
author: Bernd Ulrich
published in: Mathematische Zeitschrift
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174436
review text: Let \(R\) be a Gorenstein local ring and \(I\), \(J\) be two ideals. One says that \(I\) and \(J\) are linked if there is a regular sequence \(\alpha\) contained in \(I\cap J\) such that \(I=(\alpha):J\) and \(J=(\alpha):I\); one says that \(I\) and \(J\) are in the same (even) linkage class if there is a finite (even) chain of links from one to the other; in particular, the licci ideals are those that are in the linkage class of a complete intersection. In previous papers the authors have shown how the property \((R_ k)\) is preserved under linkage for perfect ideals (up to deformation) if \(k\leq 3\), and how the bound on local number of generators \(\mu(I_ P)\leq\max\{\text{codim}I,\dim R_ P-k\}\) is preserved (again, up to deformation, for fixed \(k\leq 3)\) for Cohen-Macaulay ideals under even linkage [cf. \textit{B. Ulrich}, J. Pure Appl. Algebra 39, 165-175 (1986; Zbl 0575.13013) and the authors, Ann. Math., II. Ser. 126, 277-334 (1987; Zbl 0638.13003) and J. Reine Angew. Math. 390, 1-20 (1988; Zbl 0732.13004)]. These results are best possible (e.g., a licci ideal is smoothable in codimension 3, but if non-Gorenstein not in codimension 4; and a Gorenstein licci ideal is smoothable in codimension 6, but if non- complete intersection not in codimension 7), but by combining an estimate on the local number of generators with local hypersurface conditions on a Cohen-Macaulay ideal \(I\) of codimension \(g\), the authors can give local structural descriptions for ideals in the even linkage class of \(I\), up to deformation. For example, if for some fixed \(k\leq 6\), \(I_ P\) is a complete intersection if \(\dim R_ P\leq g+k+2\) and \(\mu(I_ P)\leq\dim R_ P-k\) if \(\dim R_ P\geq g+k+3\), then every \(R\)-ideal \(J\) in the even linkage class of \(I\) admits a deformation to some \(\tilde R\)-ideal \(\tilde J\) such that if \(\dim\tilde R_{\tilde Q}\leq g+k+1\) then \(\tilde J_{\tilde Q}\) is either a perfect \((g-2)\)-fold hypersurface section of deviation at most…
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D10
Mathematics Subject Classification ID: 14M06
zbMATH DE Number: 126403
zbMATH Keywords: deformation
zbMATH Keywords: Gorenstein local ring
zbMATH Keywords: linkage class
zbMATH Keywords: licci ideals
zbMATH Keywords: complete intersection
zbMATH Keywords: smoothable in codimension 3
zbMATH Keywords: embedding codimension
zbMATH Keywords: completion
MaRDI profile type: Publication
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OpenAlex ID: W2080573477
description: scientific article; zbMATH DE number 126403