Derived functors of inverse limit in abelian categories?

Question

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.

Answer 1

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:

• Jan-Erik Roos, Derived functors of inverse limits revisited, J. London Math. Soc. 73 (1) (2006) 65–83, https://doi.org/10.1112/S0024610705022416

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 $C$.