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.