The notion of abelian covers

Question

I have some doubts about what an abelian covering is, and I'll try my best to articulate them.

In Serre's Algebraic groups and class fields Chapter VI.2, he fixed a base field $k$ with algebraic closure $\bar{k}$, and $V$ a normal irreducible algebraic variety over $\bar{k}$ with function field $K = \bar{k}(V)$. For a finite separable extension $L/K$, the normalization $W$ of $V$ in $L$ is the variety whose local rings are obtained by decomposing the integral closure in $L$ of the local rings $\mathcal{O}_P$ of points $P \in V$. The variety $W$ thus defined comes with a projection $\pi: W \rightarrow V$ and we say that the covering $\pi$ is abelian if the field extension $L/K$ is.

Now let $G$ be a finite abelian etale group scheme over $k$. Then the classes of $G$-torsors $Y \rightarrow V$ over $V$ are elements of the etale cohomology group $H^1(V,G)$. So an element $[Y] \in H^1(V,G)$ is such that we have $Y \times G \cong Y \times Y$.

I know that the $G$-torsors $Y$ are called abelian coverings of $V$, but is it in the same sense as the one given by Serre? In other words, is $Y$ the normalization of $V$ in some finite extension $L$ of $\bar{k}(V)$ such that $\mathrm{Aut}(L/K) \cong G$? Also, how do we check if a $G$-torsor is unramified, are there convenient results?

Any references or ideas are much appreciated.

Answer 1

(1) $G$-torsors are always unramified, because they are étale-locally trivial, and unramifiedness may be checked etale-locally.

(2) A $G$-torsor is a $G$-covering in the sense of Serre as long as it is irreducible. We simply take $L$ to be the function field $\overline{k}(Y)$. Since the morphism $Y \to V$ is finite, $Y \times_V \operatorname{Spec} \mathcal O_P \to \operatorname{Spec} \mathcal O_P$ is finite, so every function on $Y \times_V\operatorname{Spec} \mathcal O_P$ is integral over $\mathcal O_P$, and because $V$ is normal and $Y \to V$ is etale, $Y$ is normal, so every element of $L$ integral over $\mathcal O_P$ is a function in $Y \times_V\operatorname{Spec} \mathcal O_P$.

Thus the ring of functions on $Y \times_V\operatorname{Spec} \mathcal O_P$ is the integral closure, which decomposes into local rings corresponding to the fibers, as desired.