The Stacks project

Lemma 58.88.5. 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

\[ Rf_*E \otimes _\mathbf {Z}^\mathbf {L} K = Rf_*(E \otimes _\mathbf {Z}^\mathbf {L} f^{-1}K) \]

in $D^+(Y_{\acute{e}tale})$.

Proof. There is a canonical map from left to right by Cohomology on Sites, Section 21.48. 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

\[ (E \otimes _\mathbf {Z}^\mathbf {L} f^{-1}K)_{\overline{x}} = E_{\overline{x}} \otimes _\mathbf {Z}^\mathbf {L} K_{\overline{y}} \]

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

\[ (E \otimes _\mathbf {Z}^\mathbf {L} f^{-1}K) \otimes _{\mathbf{Z}}^\mathbf {L} \mathbf{Q} = (E \otimes _\mathbf {Z}^\mathbf {L} \mathbf{Q}) \otimes _{\mathbf{Q}}^\mathbf {L} (f^{-1}K \otimes _{\mathbf{Z}}^\mathbf {L} \mathbf{Q}) \]

which is zero in the derived category. In this way we see that Lemma 58.87.13 applies to both sides to see that it suffices to show

\[ R\Gamma (X_{\overline{y}}, E|_{X_{\overline{y}}} \otimes _\mathbf {Z}^\mathbf {L} (X_{\overline{y}} \to \overline{y})^{-1}K_{\overline{y}}) = R\Gamma (X_{\overline{y}}, E|_{X_{\overline{y}}}) \otimes _\mathbf {Z}^\mathbf {L} K_{\overline{y}} \]

This is shown in Lemma 58.88.4. $\square$


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