An almost p-standard system of parameters and approximately Cohen–Macaulay modules

We characterize the approximate Cohen–Macaulayness of amodule in terms of the length function and the Hilbert coefficient of the modulewith respect to an almost p-standard system of parameters (a strict subclass ofd-sequences). As applications, we characterize the approximate Cohen–Macaulayproperty of Stanley–Reisner rings, localizations, idealizations, and power seriesrings. Furthermore, for power series rings, we construct almost p-standard systemsof parameters of them. From this result, we give a class of Cohen–MacaulayRees algebras and give precise formulas computing all Hilbert coefficients of theformal power series ring with respect to an almost p-standard system of parameters.