of U if and only if U is the union of the open subsets Vα. The motivating example of a Grothendieck topology beyond that case was the étale topology on schemes. Since then, many other Grothendieck topologies have been used in algebraic geometry: the fpqc topology, the Nisnevich topology, and so on. The definition of a sheaf works on any site. So one can talk about a sheaf of sets on a site, a sheaf of abelian groups on a site, and so on. The definition of sheaf cohomology as a derived functor also works on a site. So one has sheaf cohomology groups Hj(X, E) for any object X of a site and any sheaf E of abelian groups. For the étale topology, this gives the notion of étale cohomology, which led to the proof of the Weil conjectures. Crystalline cohomology and many other cohomology theories in algebraic geometry are also defined as sheaf cohomology on an appropriate site. Notes References . English translation. External links The thread "Sheaf cohomology and injective resolutions" on MathOverflow The "Sheaf cohomology" on Stack Exchange Cohomology theories Homological algebra Sheaf theory Topological methods of algebraic geometry