Let $L/K$ be an abelian extension of function fields and let $\vartheta_{L/K}:\mathcal{D} \to \text{Gal}(L/K)$ be the Artin map from the divisors of $K$ to the galois group.
What can be said about the kernel of the artin map when restricted to the S-divisors, $\vartheta_{L/K} : \mathcal{D}_S \to \text{Gal}(L/K)$, where $S$ is a finite set of primes of $K$?
My instincts tell me that the kernel of the map below should just be the principle S-Divisors. $$\mathcal{D}_S \to \mathbb{A}_S \to \text{Gal}(L/K) \to \text{Gal}(L/K) / \vartheta_{L/K}(\mathcal{O}_S^*) $$ where $\mathcal{O}_S^*$ are the integeral S-ideles, the first arrow is a split on the canonical projection, the second arrow the artin map, and the final arrow just taking quotients. I am at a lost on how to show this, though.
In the number field case (we replace "divisor" with "fractional ideals" and don't talk about $S$, so that the above map looks like $J_K \to \text{Gal}(K^{ab}/K) / \vartheta(\mathcal{O}_K^*) $), I've read that the kernel is the narrow principle ideals, though I don't quite understand that proof, either.