Ratliff-Rush filtration, regularity and depth of higher associated graded modules. II. (Q338229)

This paper is a convincing evidence in favor of a shocking idea: one can study the graded module associated to a finitely generated Cohen-Macaulay module over a Noetherian local ring and to an ideal of definition via a non-finitely generated module over the corresponding Rees-algebra! The context is the following: \(A\) is a Noetherian local ring, \(M\) a finitely generated Cohen-Macaulay module of dimension \(\geq 2\), \(I\) an ideal of definition for \(M\), \(G_I(M)\) the graded module associated to these data, and \(\mathcal R(I)\) the Rees-algebra of \(I\). In a previous paper [J. Pure Appl. Algebra 208, No. 1, 159--176 (2007; Zbl 1106.13003)], the author showed that \(L^I(M):=\sum _{n\geq 0} M/I^{n+1}M\) is a non-finitely generated \(\mathcal R(I)\)-module.NEWLINENEWLINEA first result proven here relates the vanishing of the first \(s\) local cohomology modules of \(L^I(M)\) with respect to the maximal homogeneous ideal of \(\mathcal R(I)\) to the behaviour of the Ratliff-Rush filtration with respect to \(I\) when passing from \(M\) to its image modulo a length \(s\) \(M\)-superficial sequence with respect to \(I\). For \(G_I(M)\) generalised Cohen-Macaulay of zero depth, this behaviour can be detected numerically, being tantamount to the fact that the Stückrad-Vogel invariant attains the minimal possible value given in Theorem 5.4. Under the same hypothesis -- \(G_I(M)\) generalised Cohen-Macaulay module -- the local cohomology of \(G_{I^n}(M)\) for \(n\gg 0\) and that of \(L^I(M)\) are related explicitly in Theorem 7.1. The main results in the other two sections give characterisations for the vanishing of Hilbert coefficients of \(M\) with respect to \(I\) in terms of objects mentioned above.