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. $\square$

Comments (2)

Comment #7787 by Bogdan on

Is it clear that the isomorphism constructed in the proof coincides with the morphism induced by the cup product?

Comment #8027 by on

First of all, yes. But I think you are saying that we didn't write out the proof of this fact. And you would be right about that. This issue comes up in several guises in discussing variants of Kunneth, for example, see proof of Lemma 33.29.2. We should try to come up with a suitably general discussion that allows us to prove these maps agree in many different settings.

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