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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.

FDR
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  • Have you checked out Amnon's Inventiones paper detailing the counterexample https://publications.ias.edu/sites/default/files/counterexample.pdf? There's also some discussion at https://mathoverflow.net/questions/291151/what-was-the-error-in-the-proof-of-roos-theorem?rq=1 – David Roberts Mar 22 '22 at 05:27
  • 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). – FDR Mar 22 '22 at 06:07
  • 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. – David Roberts Mar 22 '22 at 06:09
  • 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. – FDR Mar 22 '22 at 06:16
  • 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. https://doi.org/10.1112/S0024610705022416, which may help... – David Roberts Mar 22 '22 at 06:23
  • Okay! In his 2006 paper addressing the issue, Roos says clearly that the complex does actually compute the derived functor as long as the category has AB4* (and he cites Neeman's book). So I suppose this resolves the question. Thanks :) – FDR Mar 22 '22 at 06:24
  • OK, cool. I'll add it as an answer – David Roberts Mar 22 '22 at 06:35
  • feel free to accept the answer :-) – David Roberts Mar 22 '22 at 08:28

1 Answers1

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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 $C$.

David Roberts
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