How does $ \text{Gal}(K / k) $ act on ideles?

Question

Let $K/k$ be cyclic of degree $N$, Galois group $G$. I want to define some action of $G$ on the group of ideles $J_K$ which commutes with multiplication. A natural way to do this is to take each component $$K^v = K \otimes_k k_v = \bigoplus\limits_{w \mid v} K_w$$ and define an action of $G$ here. Should I define $\sigma(x \otimes 1) = \sigma(x) \otimes 1$? And does that satisfy $\sigma(zz') = \sigma(z) \sigma(z')$? It doesn't seem to.

Answer 1

The issue is splitting behaviour, i.e. we might have $\sigma w' = w$ for different places $w,w'|v$ of $K$, $\sigma \in G$. Then this induces a (component-wise) $k_v$-isomorphism $\sigma\!: K_{w'} \to K_{w}$.

Accordingly, the $G$-action on $K^v$ is defined by $\sigma.(x_w)_w = (\sigma( x_{\sigma^{-1}w}))_w$.