Principal $G$-bundles in Zariski vs étale topology

Solution 1:

Let $X=Spec ~~\mathbb C [X,X^{-1}]$ and let $Y=Spec ~~\mathbb C [X,X^{-1},Y]/(Y^2-X)$, with $f:Y \rightarrow X$ the natural map. Here $Y$ is a degree 2 covering of $X$ with a $\mathbb Z /2 \mathbb Z$ action taking $y$ to $-y$. This is the punctured affine line wound twice around itself.

In the complex (and hence etale) topology, this action makes $Y \rightarrow X$ into a principal $\mathbb Z /2 \mathbb Z$-bundle, but it's not the case in the Zariski topology. A Zariski open $U$ is the complement of a finite point set of $X$, so the restriction of $Y$ to $U$ is also the complement of a finite point set of $Y$. But if $Y$ were locally trivial then this means that $Y_{|U}=U \bigsqcup U$ with each $U$ open in $Y$. This contradicts the fact that each $U$ must be infinite.