The Stacks project

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 $k$.

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

    \[ R\Gamma (X \times _{\mathop{\mathrm{Spec}}(k)} Y, \text{pr}_1^{-1}E \otimes _\mathbf {Z}^\mathbf {L} \text{pr}_2^{-1}K ) = R\Gamma (X, E) \otimes _\mathbf {Z}^\mathbf {L} R\Gamma (Y, K) \]
  2. 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

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

Proof. Proof of (1). By Lemma 59.92.5 we have

\[ R\text{pr}_{2, *}( \text{pr}_1^{-1}E \otimes _\mathbf {Z}^\mathbf {L} \text{pr}_2^{-1}K) = R\text{pr}_{2, *}(\text{pr}_1^{-1}E) \otimes _\mathbf {Z}^\mathbf {L} K \]

By proper base change (in the form of Lemma 59.91.12) this is equal to the object

\[ \underline{R\Gamma (X, E)} \otimes _\mathbf {Z}^\mathbf {L} K \]

of $D(Y_{\acute{e}tale})$. Taking $R\Gamma (Y, -)$ on this object reproduces the left hand side of the equality in (1) by the Leray spectral sequence for $\text{pr}_2$. Thus we conclude by Lemma 59.92.4.

Proof of (2). This is exactly the same as the proof of (1) except that we use Lemmas 59.96.6, 59.92.3, and 59.96.5 as well as $\text{cd}(Y) < \infty $ by Lemma 59.96.2. $\square$

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