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I have a finite poset $I$ and an inverse system $A: I^{op}\longrightarrow \mathscr C$ taking values in an abelian category $\mathscr C$.

I suppose that $\mathscr C$ has direct sums. Given that my indexing set $I$ is finite, do I need anything more to conclude that

(1) Derived functors of the inverse limit $lim^I: Fun(I^{op},\mathscr C)\longrightarrow \mathscr C$ exist?

(2) Can the derived functors of the inverse limit be computed from the cohomology of the usual Roos complex, i.e., for $F:I^{op}\longrightarrow \mathscr C$, I have

$$ N_k(F):=\underset{i_0\longrightarrow ...\longrightarrow i_k\in N_k(I)}{\prod}F(i_0)$$

with standard differentials.

This seems to be given in proof of Lemma A.3.2 in Neeman's Triangulated Categories. What is confusing me is that Roos had a "theorem" in 1961 about derived functors of inverse limits and Mittag-Leffler sequences, and Neeman found a counterexample around the time this book was written. In the book (section A.6), Neeman seems to hint at something being suspect about Roos' results, but appears to stop short of saying that there is a mistake.

My worry is whether this mistake of Roos affects statements (1) & (2). I suppose things could get complicated if the category $I$ is infinite. But all I am asking is if the derived functor exists for $I$ finite and can be computed from the Roos complex.

I am grateful for any leads, explanations or references.

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  • $\begingroup$ Have you checked out Amnon's Inventiones paper detailing the counterexample publications.ias.edu/sites/default/files/counterexample.pdf? There's also some discussion at mathoverflow.net/questions/291151/… $\endgroup$
    – David Roberts
    Mar 22, 2022 at 5:27
  • $\begingroup$ Hi, thanks for replying. I have tried to understand. But I am not a specialist, and I am really worried I am misunderstanding something. To me it looks as if Roos' mistake is a different issue. In Lemma A.3.2 in Neeman's book, he gives a proof of why the complex computes the derived functor of the inverse limit. I suppose the proof is correct. But I am worried if Neeman is repeating Roos' error (which also I don't understand). $\endgroup$
    – FDR
    Mar 22, 2022 at 6:07
  • $\begingroup$ ah, ok, sorry. I don't really know the details. I just wanted to make sure there was a link to Neeman's paper on the issue for people who may come by later. $\endgroup$
    – David Roberts
    Mar 22, 2022 at 6:09
  • $\begingroup$ In Neeman's Inventiones paper, he refers to his book as well. Hopefully, if there was an error in his book, he would have said so. But the issue seems technical, and beyond my ability to judge. $\endgroup$
    – FDR
    Mar 22, 2022 at 6:16
  • $\begingroup$ Roos wrote a follow-up paper: ROOS, J. (2006). DERIVED FUNCTORS OF INVERSE LIMITS REVISITED. Journal of the London Mathematical Society, 73(1), 65-83. doi.org/10.1112/S0024610705022416, which may help... $\endgroup$
    – David Roberts
    Mar 22, 2022 at 6:23

1 Answer 1

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In response to Neeman's finding a counterexample to the claimed result in the earlier paper, Roos wrote a follow-up paper re-examining the issue:

In particular he proves

that if $C$ is an abelian category satisfying the Grothendieck axioms AB3 and AB4* and having a set of generators then the first derived functor of projective limit vanishes on so-called Mittag-Leffler sequences in $C$.

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