Generators of graded modules associated with linear filter-regular sequences

Abstract

A finitely generated graded module E over a polynomial ring R has a filtration by its finitely generated graded submodules over polynomial subrings satisfying the conditions related with a certain kind of direct sum decomposition, if and only if the variables of R form a filter-regular sequence with respect to E when arranged reversely. We obtain this fact as a variation of the existence theorem for a system of Weierstrass polynomials or a standard basis with respect to generic coordinates, without using the ordering on the initial terms. We also show that the properties of the filtration mentioned above are reflected naturally upon the free complex constructed from a free resolution of E by a method described in the present author's previous work.