Primary Decomposition: Compatibility, Independence and Linear Growth

Abstract

For finitely generated modules $N⊊M$ over a Noetherian ring $R$, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if $\ass (M/N)=\{P_1, P_2, ..., P_s\}$ and $Q_i$ is a $P_i$-primary component of $N⊊M$ for each $i=1,2,...,s$, then $N=Q_1\cap Q_2\cap ...\cap Q_s$; (2) For a given subset $X=\{P_1, P_2, ..., P_r \} \subseteq \ass(M/N)$, $X$ is an open subset of $\ass(M/N)$ if and only if the intersections $Q_1\cap Q_2\cap ...\cap Q_r=Q_1^{\prime }\cap Q_2^{\prime }\cap ...\cap Q_r^{\prime }$ for all possible $P_i$-primary components $Q_i$ and $Q_i^{\prime }$ of $N⊊M$; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals $I_1,I_2,...,I_t$ of $R$, there exists a $k\in \mathbb{N}$ such that for any $n_1,n_2,...,n_t\in \mathbb{N}$ there exists a primary decomposition of $I_1^{n_1}{I}_2^{n_2}...I_t^{n_t}M\subset M$ such that every $P$-primary component $Q$ of that primary decomposition contains $P^{k(n_1+n_2+...+n_t)}M$.

AMS-LaTeX