Exercise 25, § 4

Let $A$ be a Noetherian ring, $X = \operatorname{Spec}(A)$ its spectrum, $M$ a finitely generated $A$-module and $Y = \operatorname{Supp}(M)$. Let $Y_k$ ($1 \leq k \leq n$) be the distinct connected components of $Y$.

(a) Show that there exists a unique decomposition of $M$ as a direct sum $M = M_1 \oplus M_2 \oplus \ldots \oplus M_n$ such that $\operatorname{Supp}(M_k) = Y_k$ for all $k$.

(b) If $a = \operatorname{Ann}(M)$ and $a_k = \operatorname{Ann}(M_k)$, show that the $a_k$ are relatively prime in pairs and that $\prod_{k=1}^n a_k = a$.

(Using Proposition 17 of no. 4, reduce it to the case where $a = 0$ and $Y = X$ and apply Proposition 15 of no. 3.)