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(The following is crossposted from Math.SE, where the question did not receive any answers.)

I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, Lemme 1):

Lemma. Let $B$ be a ring, $(I, \leq)$ be a directed poset, and $(M_i, f_{ji}:M_j\rightarrow M_i)_{j\geq i},$ $(N_i, g_{ji}:N_j\rightarrow N_i)_{j\geq i}$ two inverse systems of $B$-modules. Let $(h_i: M_i\rightarrow N_i)_i$ be a morphism of the inverse systems, and assume that all the maps $h_i$ are surjective with an Artinian kernel. Then the limit map $$\varprojlim_i h_i: \varprojlim_i M_i\rightarrow \varprojlim_i N_i$$ is surjective.

I am looking either for a proof (I haven't manage to come up with one so far), or for a reference to a proof - Gabriel refers to "Bourbaki, Topologie, I Appendice, $3^{\text{e}}$ éd.", which is a reference I cannot find anywhere.

I do not want to assume that e.g. $I$ is countable - the reason is that I need this lemma, similarly as Gabriel in his thesis, to establish some properties of pseudo-compact modules over a pseudo-compact ring (namely that quotient of a pseudo-compact module by a closed submodule is pseudo-compact, and exactness of inverse limits). For this reason, as far as I can tell, I cannot use countable index sets (i.e. countable bases of neighbourhoods) in general.

Thank you in advance for any help.

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    $\begingroup$ @JochenWengenroth I apologize for the late reaction, I dived into the references you provided but I struggle with French so it takes me longer. Thank you for your help, it seems to be exactly what I was looking for. $\endgroup$
    – Pavel Čoupek
    Commented Jun 5, 2018 at 9:54

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Your question can be interpreted as the vanishing of the first derived projective limit functor ${\lim\limits_{\leftarrow}}^{(1)} \mathcal M$ for the projective spectrum of the kernels. If these kernels are Artenian even all derivatives vanish. This is shown in Corollary 7.2 of C.U. Jensen's Les Foncteurs Dérivés de lim et leurs Applications en Théorie des Modules, Springer Lecture Notes 254 (1972). He also refers to his article On the vanishing of ${\lim\limits_{\leftarrow}}^{(i)}$, J. Algebra 15 (1970), 151-166 (but I did not have a look at it).

It seems to me that Jensen does not claim for originality as he writes "Il est bien connu...". However, he does not give an explicit reference. A main step in his proof is a result in Bourbaki's Topologie Générale.

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  • $\begingroup$ Am I missing something? The result in Bourbaki's book requires a numerable subset, however, I can't see why this is true in his arguments. $\endgroup$
    – user12580
    Commented May 24, 2019 at 6:50
  • $\begingroup$ In the result quoted from Jensen's Lecture Notes countability is not required. $\endgroup$
    – Jochen Wengenroth
    Commented May 24, 2019 at 7:37
  • $\begingroup$ That's the point, my version says that the index set has a numerable subset. Can you show me the quoted result directly? Mine is: Theorem 1 (Mittag-Leffler)- Let (X_{\alpha,f_{\alpha,\beta}}) be a projective system of uniform, complete and separated spaces, indexed by a filtered set $I$ which admits a numerable cofinal subset; we suppose in addition that for any $\alpha\in I$, $X_{\alpha}$ possesses a numerable fundamental system of entourages. $\endgroup$
    – user12580
    Commented May 24, 2019 at 9:19
  • $\begingroup$ This is Bourbaki, in fact originally due to Arens, about spectra of complete metric spaces. This definitely needs some countability for the index set. Jensen's theorem has a certain form of compactness in its assumptions. That's why (I believe) he does not need countability. $\endgroup$
    – Jochen Wengenroth
    Commented May 24, 2019 at 11:52
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    $\begingroup$ I mean the reference is still not correct. I find a probably right reference which uses set theory and no countability is needed. The reference is Bourbaki Theorie des Ensembles, page 162, Theorem 1. Anyway, I'll check the details. $\endgroup$
    – user12580
    Commented May 24, 2019 at 12:03

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