Lemma 59.97.9. 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 59.97.8 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 59.96.5 which we may use because $\text{cd}(X) < \infty$ by Lemma 59.96.2. $\square$

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