Exercise 6, § 5

Let $A$ be a commutative Noetherian ring, $\mathfrak{g}$ an ideal of $A$ and $M$ an $A$-module. Show that condition (v) of Theorem 1 is equivalent to the following: (v') For every finitely generated $A$-module $N$ and every submodule $N'$ of $N$, the canonical mapping $(M \otimes_A N')^\wedge \to (M \otimes_A N)^\wedge$ (where the two sides are the Hausdorff completions with respect to the $\mathfrak{g}$-adic topologies on $M \otimes_A N'$ and $M \otimes_A N$ respectively) is injective. (To prove that (v) implies (v'), argue as in part (E) of the proof of Theorem 1. To see that (v') implies (v), consider $A$-modules $N$ annihilated by a power of 8.)