The Stacks project

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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GJY. Beware of the difference between the letter 'O' and the digit '0'.