Lemma 59.92.5. Let $f : X \to Y$ be a proper morphism of schemes. Let $E \in D^+(X_{\acute{e}tale})$ have torsion cohomology sheaves. Let $K \in D^+(Y_{\acute{e}tale})$. Then
in $D^+(Y_{\acute{e}tale})$.
Proof. There is a canonical map from left to right by Cohomology on Sites, Section 21.50. We will check the equality on stalks. Recall that computing derived tensor products commutes with pullbacks. See Cohomology on Sites, Lemma 21.18.4. Thus we have
where $\overline{y}$ is the image of $\overline{x}$ in $Y$. Since $\mathbf{Z}$ has global dimension $1$ we see that this complex has vanishing cohomology in degree $< - 1 + a + b$ if $H^ i(E) = 0$ for $i \geq a$ and $H^ j(K) = 0$ for $j \geq b$. Moreover, since $H^ i(E)$ is a torsion abelian sheaf for each $i$, the same is true for the cohomology sheaves of the complex $E \otimes _\mathbf {Z}^\mathbf {L} K$. Namely, we have
which is zero in the derived category. In this way we see that Lemma 59.91.13 applies to both sides to see that it suffices to show
This is shown in Lemma 59.92.4. $\square$
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