# Why does the naive definition of compactly supported étale cohomology give the wrong answer?

Illusie's article about étale cohomology available here (in French) mentions that the standard definition of compactly supported cohomology (and higher direct images with compact support) does not give the right answers in the case of étale cohomology.

He says Grothendieck had the idea of defining them as follows instead: $$Rf_! = Rg_* \circ j_!$$ where $f = gj$ is a compactification of $f$: $g$ is proper, and $j$ is an open immersion. It takes a bit of work to see that this is well defined (a theorem of Nagata guarantees the existence of this compactification, and the proper base change theorem shows that the result does not depend on which compactification is chosen).

What is the explanation here for the difference between Grothendieck's definition and the usual one, which amounts to $Rf_! = R(g_* j_!)$ ? What is it that allows us to say that either is the "right" one?
It seems to me that a satisfactory explanation should be more than just computing the groups in a special case and appealing to intuition (indeed, the compactly supported cohomology groups are often strange, at first sight, even in the topological situation).
For example, I could imagine that an explanation might involve derived categories: after all, one of the main uses of $Rf_!$ is Verdier duality, which implies many useful properties such as Poincaré duality. This maybe gives some hints as to which definition is preferred.

• It is instructive to prove that for analytification of separated finite-type schemes over the complex numbers, Grothendieck's "alternative" formulation gives "the same" topological theory (respecting all structures). More amusingly, the analogue of the Artin comparison theorem (for proper supports) was proved by Berkovich over non-arch. fields using analytic spaces in his sense, where on his analytic side he uses derived functor of "sections with proper support". Very nice! (He gets P. duality too.) For Huber's adic spaces, there are compactification issues on the analytic side. – BCnrd Apr 12 '10 at 1:35
• If you would like to have a definition of $f_!$ without using a compactification, you may want to follow the approach of Laszlo-Olsson. As Laumon observed, the dualizing complex is "local", so one may apply the glueing lemma in BBD to define the dualizing complex on a (not even necessarily separated) scheme, say locally of finite type over some nice base, and define $f_!,$ using Poincare duality, to be $Df_*D.$ This applies for non-separated $f$ too. Laszlo-Olsson did this for alg. stacks, most of which are not separated and hence cannot be compactified. – shenghao Apr 1 '11 at 22:08

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.