Let $L/K$ be an extension of global fields, and $O_K$, $O_L$ be respectively integer rings of $K$, $L$. Let $p$ be a prime ideal of $O_K$.
When is the canonical homomorphism $$(\varprojlim_n O_K/p^n) \otimes_{O_K} O_L \to \varprojlim_n (O_K/p^n \otimes_{O_K} O_L)$$ an isomorphism?
I already know that the above map is an isomorphism if $O_L$ is free $O_K$-module. However, $O_L$ is not always free $O_K$-module. In general, is the map an isomorphism?