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Let $(M_i)_{i\in I}$ and $(N_i)_{i\in I}$ be Mittag-Leffler systems of $R$-modules. I have a map $(h_i)$ of projective systems such that every $h_i$ is surjective. I search for conditions for $\lim \limits_{\leftarrow} M_i \rightarrow \lim \limits_{\leftarrow} N_i$ to be surjective.

I am in a complicated case (from my point of vue) being the following :

  • the systems have no reasons to be countable, nor totally ordered.
  • I tried to follow de IV, §3 of "Categories abéliennes" of P. Gabriel, in SMF. However my ring $R$ is a $R_0$-toplogical algebra, complete and separated but not pseudo-compact (even if $R_0$ is a pseudo compact ring). $R$ itself is not pseudo compact. I tried to raffine the conditions of Gabriel replacing the finite length conditions by artinian conditions. But even this does not work : $R$ has no base of neighborhood $R'$ such that $R/R'$ are artinian as $R_0$-modules (otherwise I would have won).

Cool points :

  • My modules are of finite type.
  • To be precise $R_0=k[[ X_0, \ldots, X_n ]]$ and $R=R_0 [X_i^{-1}]$ for several variables.
  • The modules I look after also have additionnal structures that I may exploit.

Thanks.

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3 Answers 3

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Oh, this is just an announcement for a talk, without slides or links to a paper. You should perhaps ask Andrea Pulita, he is on Math Overflow but one cannot tag people here.

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  • $\begingroup$ I'll try. Thanks ! $\endgroup$
    – Nataniel Marquis
    Commented May 28, 2021 at 14:19
  • $\begingroup$ You can tell him that I suggested you to ask him. We know quite well each other. $\endgroup$
    – Antoine Ducros
    Commented May 28, 2021 at 14:31
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I found the following on the web — I have no idea whether it will be helpful, but try to have a look.

https://www.maths.ox.ac.uk/node/34846

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You are asking for the derived limit functor ${\lim\limits_\leftarrow}^1$ to vanish on the system of the kernels $H_i$ of the morphisms $h_i$. A reference is given in this MO answer Surjectivity of a map on inverse limits

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  • $\begingroup$ As far as I understand in the situation considered above by Nataniel the Artinian assumption is not fulfilled. $\endgroup$
    – Antoine Ducros
    Commented May 28, 2021 at 6:49
  • $\begingroup$ Yes, I thought at first of the theorem by Gabriel, but was not able to fulfuill the artinian assumption ( $k[[X,Y]][(XY)^{-1}]/k[[X,Y]]$ is not artinian over $k[[X,Y]]$). $\endgroup$
    – Nataniel Marquis
    Commented May 28, 2021 at 14:21

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