Proposition 57.54.2. Let $f : X \to Y$ be a finite morphism of schemes.

For any geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ we have

\[ (f_*\mathcal{F})_{\overline{y}} = \prod \nolimits _{\overline{x} : \mathop{\mathrm{Spec}}(k) \to X,\ f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}. \]for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and

\[ (f_*\mathcal{F})_{\overline{y}} = \bigoplus \nolimits _{\overline{x} : \mathop{\mathrm{Spec}}(k) \to X,\ f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}. \]for $\mathcal{F}$ in $\textit{Ab}(X_{\acute{e}tale})$.

For any $q \geq 1$ we have $R^ q f_*\mathcal{F} = 0$.

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