Exercise 20, § 3

Let $A$ be a commutative ring and $m$ an ideal of $A$; let $A$ be given the $m$-adic topology. Let $B$ be a commutative topological $A$-algebra ($\S$ 2, Exercise 28); suppose that $B$ is a Zariski ring; let $n$ be a defining ideal of $B$. Show that, if $M$ is a finitely generated $A$-module with the $m$-adic topology, then on the $B$-module $B \otimes_A M$ the tensor product of the topology on $B$ and the $m$-adic topology on $M$ is the $n$-adic topology; then the completion tensor product $(B \otimes_A M)^{\wedge}$ is isomorphic to $\hat{B} \otimes_A M$.