Compactly supported étale cohomology

Definition

Let $X$ be a separated scheme of finite type over a field $k$ , let $f:X\to\operatorname{Spec}k$ , and let $\mathcal F$ be a sheaf on the étale site or a complex with suitable torsion or $\ell$ -adic coefficients. Its compactly supported étale cohomology is

H_c^i(X_{\bar k},\mathcal F) =H^i\!\left(\operatorname{Spec}\bar k, Rf_!\mathcal F\right).

The exceptional direct image $Rf_!$ retains sections whose support is proper over the base.

Compactification description

If $j:X\hookrightarrow\overline X$ is an open immersion into a proper scheme, then

R\Gamma_c(X_{\bar k},\mathcal F) \simeq R\Gamma(\overline X_{\bar k},j_!\mathcal F).

The result is independent of the chosen compactification. If $X$ itself is proper, compactly supported and ordinary étale cohomology agree.

Arithmetic role

For a variety over a finite field, Frobenius acts on $H_c^i$ . The Grothendieck–Lefschetz trace formula expresses point counts as an alternating sum of traces on these groups. In the Langlands program, compactly supported cohomology of shtuka and local-shtuka spaces carries commuting Hecke and Galois actions.

References

Alexander Grothendieck, SGA 4, Exposé XVII, Lecture Notes in Mathematics 305, Springer, 1973.
The Stacks Project Authors, “More Étale Cohomology,” §63.12, “Compactly supported cohomology.” Stacks Project.