Hilbert–Samuel functions of well bifiltered modules

Abstract

In an earlier paper, we studied the Hilbert quasi-polynomial functions of finitely generated bigraded modules in the general framework when the base ring is bigraded and generated by finitely many homogeneous elements of arbitrary degrees. In this paper, we introduce the concept of [Formula: see text]-good bifiltration [Formula: see text] on a finitely generated [Formula: see text]-module [Formula: see text], where [Formula: see text] and [Formula: see text] are specified noetherian filtrations on the noetherian ring [Formula: see text]. The bigraded modules associated with such bifiltrations are shown to be finitely generated under reasonable hypotheses. Their Hilbert functions are studied. The Hilbert–Samuel function of [Formula: see text] with respect to the [Formula: see text]-good bifiltration [Formula: see text] of [Formula: see text] is one of them. It is proved, among others, that this function is a quasi-polynomial function in two variables and that if [Formula: see text] is a noetherian local ring and if the filtrations [Formula: see text] and [Formula: see text] are primary filtrations, then its degree equals the Krull dimension of [Formula: see text].