Exercise 22, § 4

(a) Let $M$ be an $A$-module and $(N_i)$ a finite family of submodules of $M$. Show that $\operatorname{Supp}\left(M / \bigcap_i N_i\right) = \bigcup_i \operatorname{Supp}(M/N_i)$ (note that $M / \bigcap_i N_i$ is isomorphic to a submodule of the direct sum of the $M/N_i$).

(b) Let $p$ be a prime number; the support of the $\mathbf{Z}$-module $Z$ is distinct from the closure of the union of the supports of the $\mathbf{Z}$-modules $\mathbf{Z}/p^k\mathbf{Z}$ ($k \in \mathbf{N}$).

(c) Let $M$ be the direct sum of the $\mathbf{Z}$-modules $\mathbf{Z}/p^k\mathbf{Z}$ ($k \in \mathbf{N}$); show that $\operatorname{Supp}(M)$ is closed but distinct from $V(a)$, where $a$ is the annihilator of $M$.

(d) Let $N$ be the direct sum of the $\mathbf{Z}$-modules $\mathbf{Z}/n\mathbf{Z}$ ($n \geq 1$); show that $\operatorname{Supp}(N)$ is not closed in $\operatorname{Spec}(\mathbf{Z})$.

(e) Deduce from (c) and (d) an example of an $A$-module $M$ such that, if $a$ is its annihilator, $\operatorname{Supp}(M)$ is not closed and its closure is distinct from $V(a)$.

(f) Show that the support of the $\mathbf{Z}$-module $\prod_{k=1}^m (\mathbf{Z}/p^k\mathbf{Z})$ is distinct from the closure of the union of the supports of the factor modules $\mathbf{Z}/p^k\mathbf{Z}$.