Tight double linkage of Gorenstein algebras
A Gorenstein ideal K in a local ring R is in the class \({\mathcal H}\) if there is a sequence of linked ideals \(I_ 0\sim I_ 1\sim...\sim I_{2n}=K\) with \(I_ 0\) a complete intersection and \(I_{2i}\) Gorenstein for all i. If K is in \({\mathcal H}\), then K is linked to a complete intersection by a sequence of ''tight double links.'' This means there is a sequence \(I_ 0=J_ 0\sim J_ 1\sim...\sim J_{2m}=K\), such that for each p, \(J_{2p+1}\) is an almost complete intersection of the form (\({\mathfrak b},y,w)\) and its immediate neighbors are linked to it by the ''similar'' regular sequences (\({\mathfrak b},y)\) and (\({\mathfrak b},w)\), respectively. In this sense \(J_{2p}\) and \(J_{2p+2}\) are ''tightly linked'' Gorenstein ideals. Let k be an algebraically closed field and \(A=kX_ 1,...,X_ q/K\) with K in \({\mathcal H}\). The k-algebra A is rigid if and only if it is linked to a regular complete intersection by a sequence of semi-generic tight double links. It follows that if A is rigid, then it is regular in codimension six.
scientific article; zbMATH DE number 3906614
scientific article