English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $R=\bigoplus_{\underline{n} \in \mathbb{N}^t}R_{\underline{n}}$ be acommutative Noetherian $\mathbb{N}^t$-graded ring, and $L =\bigoplus_{\underline{n}\in\mathbb{N}^t}L_{\underline{n}}$ be a finitelygenerated $\mathbb{N}^t$-graded $R$-module. We prove that there exists apositive 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 zerosubmodule $O_{\underline{n}}$ of $L_{\underline{n}}$ such that for any $P \in{\rm Ass}_{R_0}(L_{\underline{n}})$, the $P$-primary component $Q$ in thatprimary decomposition contains $P^k L_{\underline{n}}$. We also give an examplewhich shows that not all primary decompositions of $O_{\underline{n}}$ in$L_{\underline{n}}$ have this property. As an application of our result, weprove 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 $\underline{n} \in\mathbb{N}^t$.

Existence of special primary decompositions in multigraded modules