63.12 Compactly supported cohomology

Let $k$ be a field. Let $\Lambda$ be a ring. Let $X$ be a separated scheme of finite type over $k$ with structure morphism $f : X \to \mathop{\mathrm{Spec}}(k)$. In Section 63.9 we have defined the functor $Rf_! : D^+_{tors}(X_{\acute{e}tale}, \Lambda ) \to D^+_{tors}(\mathop{\mathrm{Spec}}(k), \Lambda )$ and the functor $Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(\mathop{\mathrm{Spec}}(k), \Lambda )$ if $\Lambda$ is a torsion ring. Composing with the global sections functor on $\mathop{\mathrm{Spec}}(k)$ we obtain what we will call the compactly supported cohomology.

Definition 63.12.1. Let $X$ be a separated scheme of finite type over a field $k$. Let $\Lambda$ be a ring. Let $K$ be an object of $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or of $D(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda$ is torsion. The cohomology of $K$ with compact support or the compactly supported cohomology of $K$ is

$R\Gamma _ c(X, K) = R\Gamma (\mathop{\mathrm{Spec}}(k), Rf_!K)$

where $f : X \to \mathop{\mathrm{Spec}}(k)$ is the structure morphism. We will write $H^ i_ c(X, K) = H^ i(R\Gamma _ c(X, K))$.

We will check that this definition doesn't conflict with Definition 63.3.7 by Lemma 63.12.3. The utility of this definition lies in the following result.

Lemma 63.12.2. Let $f : X \to Y$ be a finite type separated morphism of schemes with $Y$ quasi-compact and quasi-separated. Let $K$ be an object of $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or of $D(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda$ is torsion. Then there is a canonical isomorphism

$(Rf_!K)_{\overline{y}} \longrightarrow R\Gamma _ c(X_{\overline{y}}, K|_{X_{\overline{y}}})$

in $D(\Lambda )$ for any geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$.

Proof. Immediate consequence of Lemma 63.9.4 and the definitions. $\square$

Lemma 63.12.3. Let $X$ be a separated scheme of finite type over a field $k$. If $\mathcal{F}$ is a torsion abelian sheaf, then the abelian group $H^0_ c(X, \mathcal{F})$ defined in Definition 63.3.7 agrees with the abelian group $H^0_ c(X, \mathcal{F})$ defined in Definition 63.12.1.

Proof. Choose a compactification $j : X \to \overline{X}$ over $k$. In both cases the group is defined as $H^0(\overline{X}, j_!\mathcal{F})$. This is true for the first version by Lemma 63.3.10 and for the second version by construction. $\square$

Lemma 63.12.4. Let $k$ be an algebraically closed field. Let $X$ be a separated scheme of finite type type over $k$ of dimension $\leq 1$. Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X$ which is torsion. Then $H^ q_ c(X, \mathcal{F})$ is a finite $\Lambda$-module.

Proof. This is a consequence of Étale Cohomology, Theorem 59.84.7. Namely, choose a compactification $j : X \to \overline{X}$. After replacing $\overline{X}$ by the scheme theoretic closure of $X$, we see that we may assume $\dim (\overline{X}) \leq 1$. Then $H^ q_ c(X, \mathcal{F}) = H^ q(\overline{X}, j_!\mathcal{F})$ and the theorem applies. $\square$

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

1. $f$ is étale, or

2. $f$ is flat and quasi-finite, or

3. $f$ is quasi-finite and $Y$ is geometrically unibranch, or

4. $f$ 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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