The second Hilbert coefficient of modules with almost maximal depth

Abstract

Let [Formula: see text] be a good [Formula: see text]-filtration of a finitely generated [Formula: see text]-module [Formula: see text] of dimension [Formula: see text], where [Formula: see text] is a local ring and [Formula: see text] is an [Formula: see text]-primary ideal of [Formula: see text]. In the case of depth [Formula: see text], we give an upper bound for the second Hilbert coefficient [Formula: see text] generalizing the results by Huckaba–Marley, and Rossi–Valla proved that [Formula: see text] is Cohen–Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module [Formula: see text]. A lower bound on [Formula: see text] is proved generalizing a result by Rees and Narita.