In the study of étale cohomology of a complex variety X one proves, that the cohomological dimension equals 2d, where d is the dimension of the variety X: Hi(X, F) = 0 for i > 2d and all torsion sheaves F on X.
In the case of real varieties, the situation is totally different: As soon as there is a real point on X, the cohomology becomes unbounded: Let : X Spec() be the structure map of X, : P X the inclusion of a real point. Then the inclusion * : Hi(, M) Hi(X, M) for a G module M is split by the map *. But Hi(, M) is periodic.
To study this relationship between the real points on X and the étale cohomology, the étale Farrell cohomology is introduced.
This can in general be computed more easily than étale cohomology and coincides with this in high dimensions. Étale Farrell cohomology appears naturally in the context of duality theorems for étale cohomology.