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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