Editing Derived functor (section)

====[[Sheaf cohomology]]====
If <math>X</math> is a [[topological space]], then the category <math>Sh(X)</math> of all [[sheaf (mathematics)|sheaves]] of [[abelian group]]s on <math>X</math> is an abelian category with enough injectives. The functor <math>\Gamma: Sh(X)\to Ab</math> which assigns to each such sheaf <math>\mathcal{F}</math> the group <math>\Gamma(\mathcal{F}) := \mathcal{F}(X)</math> of global sections is left exact, and the right derived functors are the [[sheaf cohomology]] functors, usually written as <math>H^i(X,\mathcal{F})</math>. Slightly more generally: if <math>(X,\mathcal{O}_X)</math> is a [[ringed space]], then the category of all sheaves of <math>\mathcal{O}_X</math>-modules is an abelian category with enough injectives, and we can again construct sheaf cohomology as the right derived functors of the global section functor.

There are various notions of cohomology which are a special case of this:

* '''[[De Rham cohomology]]''' is the sheaf cohomology of the sheaf of [[Locally constant function|locally constant]] <math>\R</math>-valued functions on a [[manifold]]. The De Rham complex is a resolution of this sheaf not by injective sheaves, but by [[fine sheaf|fine sheaves]].
* '''[[Étale cohomology]]''' is another cohomology theory for sheaves over a scheme. It is the right derived functor of the global sections of abelian sheaves on the [[Étale topology|étale site]].