Let's start with two inverse systems $\{A_p\}_{p \in \mathbb{Z}}$, and $\{B_p \}_{p \in \mathbb{Z}}$ of $C$ modules. Give each $A_p$, $B_p$, and $C$ discrete topology. Consider inverse limit topology on $A := \lim_p A_p$, and $B := \lim_p B_p$. When do we have $\lim_p (A_p \otimes_C B_p) \approx \lim_p A_p \hat{\otimes} \lim_p B_p$? The example I have in mind is the following: For $k$ any field $lim_{m,n} (\frac{k[x]}{x^n} \otimes \frac{k[x]}{x^m}) \approx \lim_n \frac{k[x]}{x^n} \hat{\otimes} \lim_m \frac{k[x]}{x^m}$. That is why we get $k[\![ x]\!] \hat{\otimes} k[\![ x]\!] \approx k [\![ x,y]\!]$. Is there any Mittag-Leffler like condition (for example all the concerning maps in the inverse systems to be epi) for this to work? Any reference on this would be very helpful.
