Existence of special primary decompositions in multigraded modules

Abstract

Let $R={⨁}_{\underline{n}\in {\mathbb{N}}^t}{R}_{\underline{n}}$ be a commutative Noetherian ${\mathbb{N}}^t$-graded ring, and $L={⨁}_{\underline{n}\in {\mathbb{N}}^t}{L}_{\underline{n}}$ be a finitely generated ${\mathbb{N}}^t$-graded $R$-module. We prove that there exists a positive integer $k$ such that for any $\underline{n}\in {\mathbb{N}}^t$ with $L_{\underline{n}}\ne 0$, there exists a primary decomposition of the zero submodule $O_{\underline{n}}$ of $L_{\underline{n}}$ such that for any $P\in {\mathrm{Ass}}_{R_0}(L_{\underline{n}})$, the $P$-primary component $Q$ in that primary decomposition contains $P^k{L}_{\underline{n}}$. We also give an example which shows that not all primary decompositions of $O_{\underline{n}}$ in $L_{\underline{n}}$ have this property. As an application of our result, we prove that there exists a fixed positive integer $l$ such that the $0^{\mathrm{th}}$ local cohomology $H_I^0(L_{\underline{n}})=\left(0{:}_{L_{\underline{n}}}{I}^l\right)$ for all ideals $I$ of $R_0$ and for all $\underline{n}\in {\mathbb{N}}^t$.

5 pages. Comments are welcome