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.



The notion of abelian covers
(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.


