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.



Canonical presentation of pro-modules over pro-rings
Since this just got bumped ...


I think the answer <https://math.stackexchange.com/a/938076/88262>, to a special case of this question that Martin asked on Math.SE, settles this question by giving a counterexample.


In summary, there is a counterexample where $A\_i=k[x\_1,\dots,x\_i]$ modulo the ideal generated by polynomials of degree two, each $M\_i$ is the direct sum of copies of $A\_i$ indexed by the natural numbers, and $j=0$.


