Existence of special primary decompositions in multigraded modules

TL;DR

This paper proves the existence of special primary decompositions in multigraded modules over Noetherian rings, showing uniform bounds on primary components and applications to local cohomology.

Contribution

It establishes the existence of a uniform exponent for primary decompositions in multigraded modules, a result not previously known, and applies it to local cohomology.

Findings

Existence of a uniform exponent k for primary decompositions.

Not all primary decompositions have this uniform property.

Application to boundedness of local cohomology modules.

Abstract

Let $R=\bigoplus_{\underline{n} \in \mathbb{N}^t}R_{\underline{n}}$ be a commutative Noetherian $\mathbb{N}^t$-graded ring, and $L = \bigoplus_{\underline{n}\in\mathbb{N}^t}L_{\underline{n}}$ 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 $L_{\underline{n}} \neq 0$, 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 $H_I^0(L_{\underline{n}}) = \big(0 :_{L_{\underline{n}}} I^l\big)$ for all ideals $I$ of $R_0$ and for all $\underline{n} \in \mathbb{N}^t$.