Exercise 21, § 3

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$ and $M$ and $N$ two finitely generated $A$-modules.

(a) Suppose that $M$ is Hausdorff with the $m$-adic topology. Show that in $\operatorname{Hom}_A(M, N)$ with the $m$-adic topology the set of injective homomorphisms is open (use no. 1, Proposition 2 and the Artin-Rees Lemma).

(b) Suppose that $A$ is a complete Zariski ring and $m$ a defining ideal of

A. For every integer $i$, let $A_i = A/m^{i+1}$, $M_i = M/m^{i+1}M$, $N_i = N/m^{i+1}N$; show that the topological $A$-module $\operatorname{Hom}_A(M, N)$ is isomorphic to

Deduce that in $\operatorname{Hom}_A(M, N)$ the set of surjective homomorphisms is open.