Does the isomorphism between the completion of hom and the hom over completion lift to an isomorphism of pro objects?

Question

Let $(A, \mathfrak{m}, k)$ be a Noetherian local ring (complete if you want). Recall that for an $A$-algebra $B$ and $A$-modules $M, N$ there is a natural map $\operatorname{Hom}_A(M, N) \otimes_A B \to \operatorname{Hom}_A(M, N \otimes_A B) (\cong \operatorname{Hom}_B(M \otimes_A B, N \otimes_A B))$, induced by the bilinear map $(\varphi, b) \mapsto (m \mapsto \varphi(m) \otimes b)$, and that this map is an isomorphism when $M$ is finitely generated and $B$ is flat. In particular when $B$ is the completion of $A$ and $M, N$ are finitely generated this says $$\widehat{\operatorname{Hom}_A(M, N)} \cong \operatorname{Hom}_A(M, \hat{N}) \cong \operatorname{Hom}_{\hat{A}}(\hat{M}, \hat{N}).$$ Notice additionally that \begin{align*} \widehat{\operatorname{Hom}_A(M, N)} &\cong \lim\limits_n \operatorname{Hom}_A(M, N) \otimes_A A/\mathfrak{m}^n \\ \operatorname{Hom}_A(M, \hat{N}) &\cong \operatorname{Hom}_A(M, \lim\limits_n N \otimes_A A/\mathfrak{m}^n) \cong \lim\limits_n \operatorname{Hom}_A(M, N \otimes_A A/\mathfrak{m}^n) \end{align*} and because the map is natural we can deduce that the isomorphism $\widehat{\operatorname{Hom}_A(M, N)} \to \operatorname{Hom}_A(M, \hat{N})$ is the limit of a morphism of towers $\{ \operatorname{Hom}_A(M, N) \otimes_A A/\mathfrak{m}^n \to \operatorname{Hom}_A(M, N \otimes_A A/\mathfrak{m}^n) \}_n$.

Is this morphism of towers a pro-isomorphism? If not, is it at least the case that $${\lim_n}^1 \operatorname{Hom}_A(M, N \otimes_A A/\mathfrak{m}^n) = 0?$$

Answer 1

Yes. The key fact is that the functor $F : A\mathsf{-Mod} \to \operatorname{Pro}(A\mathsf{-Mod})$ defined by $F(M) = \{M \otimes_A A/\mathfrak{m}^n\}_{n\geq 1}$ is exact on finite $A$-modules, which is tag 0911 in the stacks project. We can then run the argument in https://math.stackexchange.com/a/81457/408287.

Edit: Also we can get the vanishing of $\lim^1$ by observing $\lim_n^1 M_n = 0$ for any tower of Artinian modules $\{M_n\}_n$, because such a tower satisfies the Mittag-Leffler condition.