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 $Z\subset U$, $ H_{Z}^{r}(U,\mathcal{F})=\oplus_{z\in Z}H_{z}^{r}(U,\mathcal{F}). $ Therefore, if $U$ is smooth of dimension $m$ and $\Lambda$ is the constant sheaf $\mathbb{Z}/n\mathbb{Z}$, then $ H_{c}^{r}(U,\Lambda)=dlim H_{Z}^{r}(U,\Lambda)=\oplus_{z\in U}H_{z}% ^{r}(U,\Lambda)=\oplus_{z\in U}\Lambda(-m)$ if $r=2m$, and it is 0 otherwise These groups are not even finite. We need a different definition...

From Section 16 of my notes: Lectures on etale cohomology.

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 $Z\subset U$, $ H_{Z}^{r}(U,\mathcal{F})=\oplus_{z\in Z}H_{z}^{r}(U,\mathcal{F}). $ Therefore, if $U$ is smooth of dimension $m$ and $\Lambda$ is the constant sheaf $\mathbb{Z}/n\mathbb{Z}$, then $ H_{c}^{r}(U,\Lambda)=dlim H_{Z}^{r}(U,\Lambda)=\oplus_{z\in U}H_{z}% ^{r}(U,\Lambda)=\oplus_{z\in U}\Lambda(-m)$ if $r=2m$, and it is 0 otherwise These groups are not even finite. We need a different definition...

If $j\colon\ U\rightarrow X$ 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.