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Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated).

If (the cofinality of) $I$ is the natural numbers $\mathbb{N}$, then the Mittag-Leffler condition ensures the vanishing of the $\lim^1$-term and also of all higher lim-terms. In general, only the sufficiently high lim-terms $\lim^n X_I$ vanish (how high depends on the cofinality of $I$).

Now in my case, I have the directed set $\mathbb{N}^J$ for an infinite set $J$, where the order is defined as $f \le g$ if and only if $f(j) \le g(j)$ for all $j \in J$. The cofinality of this is larger than $\mathbb{N}$ but do we maybe still have that the Mittag-Leffler condition implies vanishing of all higher lim-terms (including the $\lim^1$)? In my case, the connecting maps of the inverse system are even split surjective (which is a stronger condition than what Mittag-Leffler needs).

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  • $\begingroup$ Under what conditions do sufficiently high $\varprojlim^n$ terms vanish? I believe that if the cofinality of $I$ is $\aleph_n$, then $\varprojlim^{\geq n+2} X = 0$, but I believe that if the cofinality of $I$ is $\geq \aleph_{\omega+1}$, then $\varprojlim^n X$ can be nonvanishing for all $n$. $\endgroup$
    – Tim Campion
    Commented Jun 12, 2023 at 21:42

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It seems that the following comment of Dustin Clausen answers my question negatively: Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?

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