Lemma 59.97.2. Let $k$ be a separably closed field. Let $X$ be a proper scheme over $k$. Let $Y$ be a quasi-compact and quasi-separated scheme over $k$.

If $E \in D^+(X_{\acute{e}tale})$ has torsion cohomology sheaves and $K \in D^+(Y_{\acute{e}tale})$, then

\[ R\Gamma (X \times _{\mathop{\mathrm{Spec}}(k)} Y, \text{pr}_1^{-1}E \otimes _\mathbf {Z}^\mathbf {L} \text{pr}_2^{-1}K ) = R\Gamma (X, E) \otimes _\mathbf {Z}^\mathbf {L} R\Gamma (Y, K) \]If $n \geq 1$ is an integer, $Y$ is of finite type over $k$, $E \in D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$, and $K \in D(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$, then

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

## Comments (2)

Comment #7787 by Bogdan on

Comment #8027 by Stacks Project on