Let $A = (\dotsc \twoheadrightarrow A_2 \twoheadrightarrow A_1 \twoheadrightarrow A_0)$ be a (commutative) pro-ring with surjective transition maps. Consider the category $\mathcal{M} := \varprojlim_i \,\mathsf{Mod}(A_i)$: Objects are families of right $A_i$-modules $M=(M_i)$ together with isomorphisms $M_{i+1} \otimes_{A_{i+1}} A_i \cong M_i$. We let $\widehat{M} := \varprojlim_i M_i$. For each $j$, there is a natural epimorphism of $A_j$-modules
$$\alpha_j : \widehat{M} \otimes_{\widehat{A}} A_j \to M_j,~ (m_i)_i \otimes a \mapsto m_j \cdot a.$$
**Question.** Is $\alpha_j$ an isomorphism?

Of course we may assume $j=0$. The answer is yes when $A_i = R/p^i$ for some commutative ring $R$ and some element $p \in R$. The proof for this requires some calculations and doesn't generalize to the case of arbitrary $A_i$.

Geometrically speaking, the question aims at understanding quasi-coherent sheaves on affine ind-schemes. Any literature about this is also appreciated.