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?


This is due to Emmanouil as far as I know. See this.


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