The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Proposition 54.54.2. Let $f : X \to Y$ be a finite morphism of schemes.

  1. 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 $\textit{Ab}(X_{\acute{e}tale})$.

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

Proof. Let $X_{\overline{y}}^{sh}$ denote the fiber product $X \times _ Y \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}^{sh})$. By Theorem 54.52.1 the stalk of $R^ qf_*\mathcal{F}$ at $\overline{y}$ is computed by $H_{\acute{e}tale}^ q(X_{\overline{y}}^{sh}, \mathcal{F})$. Since $f$ is finite, $X_{\bar y}^{sh}$ is finite over $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}^{sh})$, thus $X_{\bar y}^{sh} = \mathop{\mathrm{Spec}}(A)$ for some ring $A$ finite over $\mathcal{O}_{Y, \bar y}^{sh}$. Since the latter is strictly henselian, Lemma 54.32.5 implies that $A$ is a finite product of henselian local rings $A = A_1 \times \ldots \times A_ r$. Since the residue field of $\mathcal{O}_{Y, \overline{y}}^{sh}$ is separably closed the same is true for each $A_ i$. Hence $A_ i$ is strictly henselian. This implies that $X_{\overline{y}}^{sh} = \coprod _{i = 1}^ r \mathop{\mathrm{Spec}}(A_ i)$. The vanishing of Lemma 54.54.1 implies that $(R^ qf_*\mathcal{F})_{\overline{y}} = 0$ for $q > 0$ which implies (2) by Theorem 54.29.10. Part (1) follows from the corresponding statement of Lemma 54.54.1. $\square$


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