ag.algebraic geometry - Why does the naive definition of compactly supported étale cohomology give the wrong answer?

It is important in etale cohomology, as it is topology, to define cohomology
groups with compact support --- we saw this already in the case of curves in
Section 14. They should be dual to the ordinary cohomology groups.

The traditional definition (Greenberg 1967, p162) is that, for a manifold
$U$,
$
H_{c}^{r}(U,mathbb{Z})=dlim_{Z}H_{Z}^{r}(U,mathbb{Z})
$
where $Z$ runs over the compact subsets of $U$. More generally (Iversen 1986,
III.1) when $mathcal{F}$ is a sheaf on a locally compact topological space
$U$, define
$
Gamma_{c}(U,mathcal{F})=dlim_{Z}Gamma_{Z}(U,mathcal{F})
$
where $Z$ again runs over the compact subsets of $U$, and let $H_{c}%
^{r}(U,-)=R^{r}Gamma_{c}(U,-)$.

For an algebraic variety $U$ and a sheaf $mathcal{F}$ on $U_{mathrm{et}}$,
this suggests defining
$
Gamma_{c}(U,mathcal{F})=dlim_{Z}Gamma_{Z}(U,mathcal{F}),
$
where $Z$ runs over the complete subvarieties $Z$ of $U$, and setting
$H_{c}^{r}(U,-)=R^{r}Gamma_{c}(U,-)$. However, this definition leads to
anomolous groups. For example, if $U$ is an affine variety over an
algebraically closed field, then the only complete subvarieties of $U$ are the
finite subvarieties, and for a finite subvariety $Zsubset
U$,
$
H_{Z}^{r}(U,mathcal{F})=oplus_{zin Z}H_{z}^{r}(U,mathcal{F}).
$
Therefore, if $U$ is smooth of dimension $m$ and $Lambda$ is the constant
sheaf $mathbb{Z}/nmathbb{Z}$, then
$
H_{c}^{r}(U,Lambda)=dlim H_{Z}^{r}(U,Lambda)=oplus_{zin U}H_{z}%
^{r}(U,Lambda)=oplus_{zin U}Lambda(-m)$ if $r=2m$, and it is 0 otherwise
These groups are not even finite. We need a different definition...

If $jcolon Urightarrow X$ is a homeomorphism of the topological space $U$
onto an open subset of a locally compact space $X$, then
$
H_{c}^{r}(U,mathcal{F})=H^{r}(X,j_{!}mathcal{F})
$
(Iversen 1986, p184).
We make this our definition.

From Section 18 of my notes: Lectures on etale cohomology.