## 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.
$\square$

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