Canonical presentation of pro-modules over pro-rings 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.
 A: 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 $j=0$.
A: The kernel $\widehat{K_j}$ of $\widehat M\to M_j$ consists of sequences of the form $(\cdots\mapsto m_{j+2}\mapsto m_{j+1}\mapsto0\mapsto0\mapsto\cdots\mapsto0)$ whereas the kernel $\widehat{I_j}$ of $\widehat A\to A_j$ consists of sequences of the form $(\cdots\mapsto a_{j+2}\mapsto a_{j+1}\mapsto0\mapsto0\mapsto\cdots\mapsto0)$. Thus $$\widehat{K_j}=\varprojlim_iK_{ji}\textrm{, }\widehat{I_j}=\varprojlim_iI_{ji}$$ with $K_{ji}=\mathrm{Ker}(M_{j+i}\to M_i)$ and $I_{ji}=\mathrm{Ker}(A_{j+i}\to A_i)$.
Now the given isomorphisms allow to identify $M_j$ with $M_{j+1}/(M_{j+1}\mathrm{Ker}(A_{j+1}\to A_j))$; more generally they provide identification of $K_{ji}$ with $M_{j+i}I_{ji}$ for all $i$. It follows that $\widehat{K_j}=\widehat M\widehat{I_j}$. 
This in turn implies $M_j=\widehat M/\widehat M\widehat{I_j}$. On the other hand $\widehat M\otimes_{\widehat A}A_j$ is also $\widehat M/\widehat M\widehat{I_j}$ since $A_j=\widehat A/\widehat{I_j}$.
