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

We characterize the approximate Cohen–Macaulayness of a module in terms of the length function and the Hilbert coefficient of the module with respect to an almost p-standard system of parameters (a strict subclass of d-sequences). As applications, we characterize the approximate Cohen–Macaulay property of Stanley–Reisner rings, localizations, idealizations, and power series rings. Furthermore, for power series rings, we construct almost p-standard systems of parameters of them. From this result, we give a class of Cohen–Macaulay Rees algebras and give precise formulas computing all Hilbert coefficients of the formal power series ring with respect to an almost p-standard system of parameters.