While reading Katz and Mazur's book "Arithmetic Moduli of Elliptic Curves" (available here), I came across the following statement (p.65).
Let $S$ be an arbitrary scheme and let $E$ be a proper smooth curve over $S$ with geometrically connected fiber. Let $f$ denote the morphism $E\rightarrow S$. Then, we have $$f_{\star}(\mathcal{O}_E)=\mathcal{O_S}$$ Hence, if $\mathcal{L}$ is some invertible sheaf defined on an open subset $U\subset S$, we have the equality of invertible sheaves on $U$ $$f_{\star}f^{\star}(\mathcal{L})=\mathcal{L}$$
The second point is a consequence of the first by the projection formula.
For the first point, I think one may reduce to the case where $S$ is the spectrum of an algebraically closed field $k$. Then, to prove the isomorphism of sheaves over $k$, we only have to prove the isomorphism between the rings of global sections. It is then a known result that any global section of a connected reduced proper $k$-scheme with $k=\overline{k}$ is a constant, which is exactly what we want (see for instance Vakil's note, 10.3.7, page 289, here).
However, I am really unsure about the justification of the reduction to this case. In order to prove that the natural morphism $\mathcal{O}_S \rightarrow f_{\star}\mathcal{O}_E$ is an isomorphism, I would like to show that this map induces isomorphism between sections on open subsets or (equivalently) between stalks. How does one reduces to just looking at fibers and, even more specifically, geometric fibers?
I thank you very much for your help.