I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the topological space of $X$, and acting as $f \mapsto f^q$ on the structure sheaf. As a particular case, if $X$ is affine (say $Spec(\mathbb{F}_q[x,y])$), then "$F$ is the standard Frobenius map: $(x,y) \mapsto (x^q,y^q)$." Why is that ? Doesn't the action on the structure sheaf implies that every element of $\mathbb{F}_q[x,y]$ is raised to the $q$-th power ?
Secondly, it is mentioned that the geometric and arithmetic Frobenius act in the same way on $X(\overline{\mathbb{F}_q})$. Can anyone explain this in some detail ?
Thanks !