Remark 63.12.5 (0GKS): Covariance of compactly supported cohomology

Remark 63.12.5 (Covariance of compactly supported cohomology). Let $k$ be a field. Let $f : X \to Y$ be a morphism of separated schemes of finite type over $k$. If $X$, $Y$, and $f$ satisfies one of the following conditions

$f$ is étale, or
$f$ is flat and quasi-finite, or
$f$ is quasi-finite and $Y$ is geometrically unibranch, or
$f$ is quasi-finite and there exists a weighting $w : X \to \mathbf{Z}$ of $f$

then compactly supported cohomology is covariant with respect to $f$. More precisely, let $\Lambda $ be a ring. Let $K$ be an object of $D^+_{tors}(Y_{\acute{e}tale}, \Lambda )$ or of $D(Y_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion. Under one of the assumptions (1) – (4) there is a canonical map

\[ \text{Tr}_{f, w, K} : f_!f^{-1}K \longrightarrow K \]

See Section 63.5 for the existence of the trace map and Examples 63.5.5 and 63.5.7 for cases (2) and (3). If $p : X \to \mathop{\mathrm{Spec}}(k)$ and $q : Y \to \mathop{\mathrm{Spec}}(k)$ denote the structure morphisms, then we have $Rq_! \circ f_! = Rp_!$ by Lemma 63.9.2 and the fact that $Rf_! = f_!$ for the quasi-finite separated morphism $f$ by Lemma 63.10.3. Hence we can look at the map

\begin{align*} R\Gamma _ c(X, f^{-1}K) & = R\Gamma (\mathop{\mathrm{Spec}}(k), Rp_!f^{-1}K) \\ & = R\Gamma (\mathop{\mathrm{Spec}}(k), Rq_!f_!f^{-1}K) \\ & \xrightarrow {Rq_!\text{Tr}_{f, w, K}} R\Gamma (\mathop{\mathrm{Spec}}(k), Rq_!K) \\ & = R\Gamma _ c(Y, K) \end{align*}

In particular, if $\Lambda $ is a torsion ring, then we obtain an arrow

\[ \text{Tr}_ f : R\Gamma _ c(X, \Lambda ) \longrightarrow R\Gamma _ c(Y, \Lambda ) \]

This map has lots of additional properties, for example it is compatible with taking ground field extensions.

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