On Filter-regular Sequences of Multi-graded Modules

Abstract

Let $S$ be an $\mathbf{N}^d$-graded algebra over a noetherian ring and a finitely generated $\mathbf{N}^d$-graded $S$-module $M$. This paper will study the relationship of filter-regular sequences of $M$ to joint reductions and homogeneous parameter systems. As an application, we show that any maximal filter-regular sequence is a joint reduction of $(S_1,\ldots,S_d)$ with respect to $M,$ and any maximal strong-filter-regular sequence is a reduction of $S_+$ with respect to $M$. And we characterize the existence of parts of homogeneous parameter systems for $M$ consisting of elements of total degree 1 via strong-filter-regular sequences.