Exercise 24, § 4

(a) Let $A$ be a ring and $M, N$ two $A$-modules such that $M$ is finitely generated. Show that $\operatorname{Supp}(\operatorname{Hom}_A(M, N))$ is contained in the intersection $\operatorname{Supp}(M) \cap \operatorname{Supp}(N)$.

(b) Give an example where $M$ and $N$ are both finitely presented and $\operatorname{Supp}(\operatorname{Hom}_A(M, N))$ is strictly contained in $\operatorname{Supp}(M) \cap \operatorname{Supp}(N)$ (take $A = \mathbf{Z}_{(p)}$ and $N = A$).

(c) Let $M$ be the $\mathbf{Z}$-module the direct sum of the $\mathbf{Z}/p^k\mathbf{Z}$ ($p$ prime, $k \in \mathbf{N}$). Show that $\operatorname{Supp}(\operatorname{Hom}_{\mathbf{Z}}(M, M))$ is not contained in $\operatorname{Supp}(M)$.