Exercise 4, § 2

Let A be a Noetherian ring and M a finitely generated A-module. Let p be an element of Ass(M) and let $\mathscr{E}_p$ be the set of submodules Q of M such that $\operatorname{Ass}(M/Q) = \{p\}$ and $\operatorname{Ass}(Q) = \operatorname{Ass}(M) - \{p\}$ (a set which is not empty by § 1, no. 1, Proposition 4). We write $e_p(M) = \inf_{Q \in \mathscr{E}_p} e(M/Q)$ (Exercise 3).

(a) Let n, be an integer $\geq e_p(M)$ and let $\mathscr{F}(n_p)$ be the subset of $\mathscr{E}_p$ consisting of the submodules Q such that $e(M/Q) \leq n_p$. Show that $\mathscr{F}(n_p)$ has a least element $Q(p, n_p)$.

(b) Let $(n_p)_{p \in \operatorname{Ass}(M)}$ be a family of integers such that $n_p \geq e_p(M)$ for all $p \in \operatorname{Ass}(M)$. Show that the submodules $Q(p, n_p)$ corresponding to this family form a reduced primary decomposition of {0} in M said to be canonically determined by the family $(n_p)$. If we take $n_p = e_p(M)$ for all $p \in \operatorname{Ass}(M)$, the primary decomposition consisting of the $Q(p) = Q(p, e_p(M))$ is called the canonical primary decomposition of {0} in M. Let $\{0\} = \bigcap_{p \in \operatorname{Ass}(M)} Q'(p)$ be any reduced primary decomposition of {0} in M. Show that $e(M/Q(p)) \leq e(M/Q'(p))$ for all $p \in \operatorname{Ass}(M)$ and that, if $e(M/Q(p)) = e(M/Q'(p))$ for some p, then $Q(p) \subset Q'(p)$ ("Theorem of Ortiz").

(c) Show that $Q(p, n_p)$ is the saturation of $p^{n_p} M$ with respect to p (use § 1, no. 2, Proposition 6); p is the least element of $\operatorname{Supp}(M/p^n M)$.

(d) Let S be a multiplicative subset of A not meeting a prime ideal $p \in \operatorname{Ass}(M)$ and let Q be a p-primary submodule of M. Show that $e(M/Q) = e(S^{-1}M/S^{-1}Q)$. Show that, if S is a multiplicative subset of A, $\Phi$ the set of $p \in \operatorname{Ass}(M)$ such that $S \cap p = \varnothing$ and N the saturation of {0} in M with respect to S, the $Q(p, n)$ where $p \in \Phi$ form the reduced primary decomposition of N in M canonically determined by the family $(n_p)_{p \in \Phi}$ and the $S^{-1}Q(p, n)$ where $p \in \Phi$ the reduced primary decomposition of (0) in $S^{-1}M$ canonically determined by the family $(n_p)_{p \in \Phi}$. Consider the particular case of canonical primary decompositions.