The structure of linkage
type: Item
instance of: scholarly article
title: The structure of linkage
zbMATH Open document ID: 0638.13003
DOI: 10.2307/1971402
author: Craig Huneke
author: Bernd Ulrich
published in: Annals of Mathematics. Second Series
publication date: 1987-01-01T00:00:00Z
review text: Two unmixed ideals I, J in a Cohen-Macaulay local ring R are said to be linked (written \(I\sim J)\) if there is a regular sequence \(x=x_ 1,...,x_ g\) in \(I\cap J\) such that \((x):I=J\) and \((x):J=I\). Two ideals are in the same linkage class if there is a finite chain of links from one to the other; ideals in the linkage class of a complete intersection are called licci. While many properties are invariants of a linkage class (or at least of the even linkage class of an ideal), better results seem to require more control over the intermediate steps of the linking chain. The authors prove a remarkable theorem asserting that given I, there is a universal chain of links \(IS\sim L^ 1(I)\sim...\sim L^ n(I)\) in a suitable extension \(S=R[X]_{{\mathfrak m}_ R[X]}\). Universality means that if \(I\sim I_ 1\sim...\sim I_ n\) is any chain of links in R, then \((S,L^ i(I))\) is essentially a deformation of \((R,I_ i)\) for every i, that is, obtained by deformation, localization, and automorphism. In particular, since I is linked to itself in two steps each \(L^{2k}(I)\) is essentially a deformation of I. The result is an explicit construction based on the idea of generic linkage. If \(I=(f_ 1,...,f_ n)=(f)\) then \(\alpha =fX\) is a regular sequence in \(R'=R[X]\) where X is an \(n\times g\) matrix of indeterminates. Then \(L_ 1(f)=(\alpha):IR'\) is called the first generic link, and \(L^ 1(f)=(\alpha):IS\) is the first universal link. By definition \(L^ i(I)=L^ 1(L^{i-1}(I))\) for \(i>0.\) The utility of the authors' result has two main aspects. First, since it is an explicit construction, they are able to actually compute the non- Gorenstein and non-complete intersection loci of the second universal link (this is the critical step) and to prove that the defining ideals of these loci have prime radicals. This enables them to establish sharp smoothability results. Secondly, there is the philosophy, amply demonstrated here, that if (S,J) is essentially a deformation of (…
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D99
Mathematics Subject Classification ID: 13D10
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 4040042
zbMATH Keywords: Cohen-Macaulay local ring
zbMATH Keywords: linkage class
zbMATH Keywords: licci
zbMATH Keywords: deformation
zbMATH Keywords: complete intersection
zbMATH Keywords: multiplicity
zbMATH Keywords: Cohen-Macaulay type
zbMATH Keywords: embedding codimension
zbMATH Keywords: deviation
Wikidata QID: Q105501297
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971402
OpenAlex ID: W1964853496
description: scientific article; zbMATH DE number 4040042