Local properties of licci ideals (Q1204288): Difference between revisions

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 1, or a perfect Gorenstein \((g-3)\)-fold hypersurface section of deviation 2, and if \(\dim\tilde R_{\tilde Q}\geq g+k+2\) then \(\mu(\tilde J_{\tilde Q})\leq\dim\tilde R_{\tilde Q}-k\). If \(I\) and \(J\) are Gorenstein ideals the same result extends to \(k\leq 10\), possible after purely transcendental extension of the residue class field. For rigid licci algebras \(A=k[[x_ 1,\dots,x_ n]]/I\) the situation is even better. In this case the authors had earlier shown that \(A\) is \((R_ 3)\); here they obtain strong bounds on the deviations of localization \(A_ P\), and, if \(\dim A_ P\) is small enough, bounds on the embedding codimension of \(A_ P\) as well as rigidity of its completion.