Hilbert coefficients and depths of form rings

摘要

We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a modulo. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e(1). As an application, we derive bounds on the higher Hilbert coefficient e(i) in terms of e(0).