Field extensions and completions at possibly infinite places

Question

In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-finite. If $v$ is a finite valuation of $K$, $w_i$ the extensions of $v$ on $L$. Then if we denote $\widehat{K_v}$ and $\widehat{L_i}$ the corresponding completions, roughly :

1. The fields $\widehat{L}_i$ are of degree $e_i f_i$
2. Ramification and inertia in $B/A$ corresponds to local ramification defined through uniformization.
3. We have that $L \otimes \widehat{K_v} \longrightarrow \prod \widehat{L_i}$ is an isomorphism.

I'm mostly interested in the last proposition. Does this hold true if $L/K$ is any Galois extension, and instead $v$ is a (possibly infinite) place ? Serre's proof works by proving the image of $\varphi$ is dense using an approximation lemma (that holds for finite amount of absolute values of any field), so it is surjective (theory of TVSs). To generalize the proof, it would then boil down to proving the RHS is of dimension $n$. I believe it would follow from the fact a place gives an embedding of $L$ into $\overline{K_v}$, and $\operatorname{Gal}(L/K)$ acts on those embeddings, but I can't figure the details out.

Answer 1

Yes, Property 3 holds true for any place. See Theorem 4 in Ch. III-4 of Weil: Basic number theory. See also the 4 corollaries of this theorem.