Existence of special primary decompositions in multigraded modules

Let $R=bigoplus_{underline{n} in mathbb{N}^t}R_{underline{n}}$ be a commutative Noetherian $mathbb{N}^t$-graded ring, and $L = bigoplus_{underline{n}inmathbb{N}^t}L_{underline{n}}$ 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 $L_{underline{n}} eq 0$, there exists a primary decomposition of the zero submodule $O_{underline{n}}$ of $L_{underline{n}}$ such that for any $P in {rm Ass}_{R_0}(L_{underline{n}})$, 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 $0^{rm th}$ local cohomology $H_I^0(L_{underline{n}}) = big(0 :_{L_{underline{n}}} I^lbig)$ for all ideals $I$ of $R_0$ and for all $underline{n} in mathbb{N}^t$.