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*}

Lemma 54.89.9. Let $k$ be a separably closed field. Let $X$ and $Y$ be finite type schemes over $k$. Let $n \geq 1$ be an integer invertible in $k$. Then for $E \in D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$ and $K \in D(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$ we have

\[ R\Gamma (X \times _{\mathop{\mathrm{Spec}}(k)} Y, \text{pr}_1^{-1}E \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} \text{pr}_2^{-1}K ) = R\Gamma (X, E) \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} R\Gamma (Y, K) \]

Proof. By Lemma 54.89.8 we have

\[ R\text{pr}_{1, *}( \text{pr}_1^{-1}E \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} \text{pr}_2^{-1}K) = E \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} \underline{R\Gamma (Y, K)} \]

We conclude by Lemma 54.88.5 which we may use because $\text{cd}(X) < \infty $ by Lemma 54.88.2. $\square$


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