# 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:

[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) Jan 21, 2018 at 0:12
• The official channel for Comptes Rendus is the Bibliothèque nationale de France. This article is available on BnF's website here. Jan 21, 2018 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, 2018 at 3:09
• @Joël Ah, sorry, it's just what's needed for the question. Jan 21, 2018 at 3:13
• I just want to know, what is the status of papers related to (cited) his paper? Sep 5, 2020 at 17:51