Let $S$ be a smooth curve, $\eta$ the generic point of $S$. $X$ is a smooth proper scheme over $S$.
My question is:
How to define an action of etale fundamental group $\pi_{1}(S)$ on the etale cohomology $H^{j}(X_{\overline \eta})$?
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1 Answer
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This does not directly answer your question, but I hope it helps you forward. The keyword is lisse $\ell$-adic sheaf. Let $f \colon X \to S$ be the structure morphism. Then $R^{j}f_{*}\mathbb{Q}_{\ell}$ is a lisse $\ell$-adic sheaf on $S$. Moreover, the stalk at $\eta$ is isomorphic to $H^{j}(X_{\bar{\eta}})$.
Lisse $\ell$-adic sheaves are the analogues of local systems. There is a dictionary between local systems and representations of the (topological) fundamental group. The analogue extends to a dictionary between lisse $\ell$-adic sheaves and representations of the étale fundamental group.
In the topological picture, it is pretty easy to setup this dictionary. Given a local system $F$ on a space $Y$, a base point $y$, and a class of paths $\gamma \in \pi_{1}(Y,y)$, look at the stalk $F_{y}$. Any element $f \in F_{y}$ can be “followed” along $\gamma$, to give an element $\gamma \cdot f \in F_{y}$. (One can make this precise by pulling $F$ back to the unit interval along $\gamma$, and trivializing it there.
The dictionary in the $\ell$-adic case is more involved (after all the étale fundamental group does not really consist of classes of “paths”). Furthermore, the precise definition of lisse $\ell$-adic sheaves is already quite subtle. I would suggest keeping the analogue in mind, and accepting that the dictionary exists.
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