The second Hilbert coefficient of modules with almost maximal depth

摘要

Let M = {M-n} be a good q-filtration of a finitely generated R-module M of dimension d, where (R, m) is a local ring and q is an m-primary ideal of R. In the case of depth M >= d-1, we give an upper bound for the second Hilbert coefficient e(2)(M) generalizing the results by Huckaba-Marley, and Rossi-Valla proved that, M is Cohen-Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module gr(M)(M). A lower bound on e(2)(M) is proved generalizing a result by Rees and Narita.