I'm reading the book "Weil conjectures, perverse sheaves and l-adic fourier transform"
I can't understand the following lemma: 
where $X_0$ is a smooth curve over $\kappa=\mathbb{F}_q$, $X=X_0\times_\kappa k$, $k=\bar{\kappa}$,
$U_0\subset X_0$ open subset, $S_0=X_0-U_0$ finite points. We may assume that $X$ is affine.
$F_0$ is a smooth sheaf on $U_0$, $G_0=j_*F_0$
I have no idea of what the ramification group of $X$ in $s\in S$ is.
I know the ramification theory for extension of valuation rings, and its globalization for Dedekind rings. I don't know whether it's related to this question.
I have the following intuition, but I don't know how to express it rigorously:
As a smooth sheaf, every stalk of $F$ (hence every stalk of $G$ in $U$) are isomorphic to $V$
For topological case, we may consider elements of fundamental group as loops.
For a loop $\alpha:s\to s$ whose interior lies in $U$, $\alpha$ induced an automorphism of $V$
In the stalk $G_s$, $\alpha$ should acts trivially. Hence $G_s$ should be something like the fixed point of $V$ under the actions of $\alpha$