Remark 48.32.8 (0G57)—The Stacks project

Remark 48.32.8. Let us generalize the covariance of compactly supported cohomology given in Remark 48.32.7 to étale morphisms. Namely, in Situation 48.16.1 suppose given a commutative diagram

\[ \xymatrix{ U \ar[rr]_ h \ar[rd]_ g & & X \ar[ld]^ f \\ & Y } \]

of $\textit{FTS}_ S$ with $h$ étale. Then there is a canonical morphism

\[ Rg_!(h^*K) \longrightarrow Rf_!K \]

functorial in $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. We define this transformation using the sequence of maps

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L) & = \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) \\ & \to \mathop{\mathrm{Hom}}\nolimits _ U(h^*K, h^*(f^!L)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(h^*K, h^!f^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(h^*K, g^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rg_!(h^*K), L) \end{align*}

functorial in $L$ and $K$. Here we have used Proposition 48.32.2 twice, we have used the equality $h^* = h^!$ of Lemma 48.18.2, and we have used the equality $h^! \circ f^! = g^!$ of Lemma 48.16.3. The functoriality in $L$ shows by Categories, Remark 4.22.7 that we obtain a canonical map $Rg_!(h^*K) \to Rf_!K$ in $\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ which is functorial in $K$ by the functoriality of the arrow above in $K$.

The Stacks project

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