2
$\begingroup$

Consider the following situation. Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$ schemes $f:X\rightarrow Y$. Moreover, let $K$ and $L$ be two extension fields of $k$ such that $K\subseteq L$ is a finite Galois extension, and suppose we have an $L$-point of $X$, say $x\in X(L)$ such that its image under $f$ defines a point $y\in Y(K)$. Now consider the absolute Galois Group of $K$, say $G=\text{Gal}(\bar{K},K)$, and it's action on $X(\bar{K})$, sending a point $\bar{x}:X\rightarrow\text{Spec}(\bar{K})$ to $\sigma^{\ast}\circ\bar{x}$. Is it true that for each element $\sigma$ of the Galois group $G$, the image of $\sigma^{\star}\circ\bar{x}$ is again $y$? Thank you for any suggestion!

$\endgroup$
1

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.