Let $A\_1\leftarrow A\_2\leftarrow A\_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p\_{ij}\colon A\_j\to A\_i$, $j\ge i$. The derived functor of projective limit $\varprojlim\_n^1 A\_n$ is constructed as the cokernel of the map
$$
 \mathrm{id}-\mathit{shift}\colon\prod\nolimits\_{n=1}^\infty A\_n\longrightarrow\prod\nolimits\_{n=1}^\infty A\_n,
$$
where the map $\mathit{shift}\colon\prod\_{n=1}^\infty A\_n\to\prod\_{n=1}^\infty A\_n$ takes a sequence $(a\_n\in A\_n)\_{n=1}^\infty$ to the sequence $(p\_{n,n+1}(a\_{n+1})\in A\_n)\_{n=1}^\infty$. The kernel of the map $\mathrm{id}-\mathit{shift}$ is the projective limit $\varprojlim\_n A\_n$.


The following condition, called the Mittag-Leffler condition, is sufficient for the vanishing of $\varprojlim\_n^1 A\_n$. Suppose that for every fixed $m\ge1$ the nonincreasing sequence of subgroups $p\_{m,n}(A\_n)\subset A\_m$ stabilizes for $n$ large enough. Then $\varprojlim\_n^1 A\_n=0$.


The Mittag-Leffler condition is not necessary for the vanishing of $\varprojlim\_n^1 A\_n$. The simplest counterexample would be $A\_n=x^nk[[x]]\subset k[[x]]$, where $k[[x]]$ denotes the right of formal Taylor power series in one variable $x$ over a field $k$ and the maps $p\_{i,j}\colon x^jk[[x]]\hookrightarrow x^ik[[x]]$ are the identity inclusions. Then $\varprojlim\_n^1 A\_n=\varprojlim\_n A\_n=0$, but the sequence of images of the maps $p\_{m,n}$ never stabilizes as a sequence of subgroups in $A\_m$.


Notice that, replacing the power series with the polynomials in the above example, one obtains the projective system of groups/vector spaces $C\_n=x^nk[x]$ and identity inclusions between them, whose derived projective limit does not vanish. In fact, one has $\varprojlim\_n^1 C\_n=k[[x]]/k[x]$.


I believe I can prove the following result, which is of relevance in connection with the weak proregularity condition in the MGM duality theory. Let $(A\_n)$ be a projective system of abelian groups. Denote by $(B\_n)=\bigoplus\_\omega (A\_n)$ the direct sum of a countably infinite family of copies of the projective system $(A\_n)$. Then the following three conditions are equivalent:


1. the projective system $(A\_n)$ satisfies the Mittag-Leffler condition;
2. the projective system $(B\_n)$ satisfies the Mittag-Leffler condition;
3. $\varprojlim\_n^1 B\_n=0$.


My question is: this sounds like a too elementary result in a too popular area of algebra to remain unknown till the present time. Is there a reference where a proof of the equivalence of these conditions can be found?



The Mittag-Leffler condition as necessary and sufficient
This is due to Emmanouil as far as I know. See [this](http://dx.doi.org/10.1016/0040-9383(94)00056-5).


