# What was the error in the proof of Roos' theorem?

Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse systems of epimorphisms are exact [2] [3]. The result stood for 41 years before Neeman published a counterexample in 2002. Influenced by Gabber, Roos published a corrected statement a few years later: in an abelian category with arbitrary coproducts (AB3) and exact countable products (AB4$^\ast_\omega$) which contains a generator, limits of inverse systems of epimorphisms are exact [4].

Question: In neither Neeman's paper nor in Roos' 2006 paper can I find a discussion of where in his 1961 paper Roos actually makes a mistake. Can anybody shed any light on this?

Notes:

[1] I can't find a copy of Roos' paper online. The citation is

Roos, Jan-Erik (1961). "Sur les foncteurs dérivés de lim. Applications". C. R. Acad. Sci. Paris. 252: 3702–3704. MR 0132091.

[2] By an "inverse system" $X^\bullet$, I mean a diagram $\dots \to X^n \to X^{n-1} \to \dots$ indexed by $\mathbb N$ (or $\mathbb Z$ -- it doesn't make a difference). By an "inverse system of epimorphisms" $X^\bullet$, I mean an inverse system where each map $X^n \twoheadrightarrow X^{n-1}$ is an epimorphism. In an abelian category $\mathcal C$ with countable products (AB3$^\ast_\omega$), there is a limit functor from inverse systems in $\mathcal C$, to $\mathcal C$, which is left exact. If the countable products in $\mathcal C$ are exact (AB4$^\ast_\omega$), then there is a natural two-term complex $\prod_n X^n \to \prod_n X^n$ which may be used to derive the limit functor (the resulting $\delta$-functor is effaceable via the map $X^\bullet \to \prod_{n \leq \bullet} X^n$ where the latter complex uses the projections for structure maps); because this complex has two terms, the derived functors $\varprojlim^d X^\bullet$ vanish for $d > 1$. When I say that $X^\bullet$ is exact, I mean that $\varprojlim^1 X^\bullet = 0$. I think I understand how to argue that if $X^\bullet$ is an epimorphic system in an AB4$^\ast_\omega$ category and $\varprojlim^1 X^\bullet = 0$, then the map $\varprojlim X^\bullet \twoheadrightarrow X$ is an epimorphism for each $n$ (actually it's the reverse implication that holds); when Roos' theorem holds, this is a formulation of the result which doesn't mention $\varprojlim^1$.

[3] There is a stronger statement about Mittag-Leffler sequences; let me stick with the statement about inverse systems of epimorphisms for simplicity.

[4] In the comments here Leonid Positselski points out that Roos' proof actually establishes that this holds more generally in any AB4$^\ast_\omega$ category with enough projective effacements (I hadn't heard of this concept until recently, but it's right there in Tohoku -- actually, enough local projective effacements suffices). In this form, the statement is actually quite straightforward to prove.

• Here is the paper, on page 1109: ia800701.us.archive.org/23/items/… (pdf is quite large, about 100 MB) – Jānis Lazovskis Jan 21 '18 at 0:12
• The official channel for Comptes Rendus is the Bibliothèque nationale de France. This article is available on BnF's website here. – R. van Dobben de Bruyn Jan 21 '18 at 2:53
• In Grothendieck's Tohoku paper, where I believe the condition (AB3) to (AB6) and their dual conditions were introduced, (AB3) is the existence of arbitrary coproducts, not just countable ones. Similarly (AB4*) is about arbitrary products. Has the definition changed since, or is it just what is needed in this question, or is just a mistake? – Joël Jan 21 '18 at 3:09
• @Joël Ah, sorry, it's just what's needed for the question. – Tim Campion Jan 21 '18 at 3:13