Hilbert coefficients and sequentially Cohen–Macaulay modules

The purpose of this paper is to present a characterization of sequentially Cohen–Macaulay modules in terms of their Hilbert coefficients with respect to distinguished parameter ideals. The formulas involve arithmetic degrees. Among the corollaries of the main result we obtain a short proof of the Vasconcelos Vanishing Conjecture for modules and an upper bound for the first Hilbert coefficient.