sheaf cohomology

Let X be a topological spaceMathworldPlanetmath. The category of sheaves of abelian groupsMathworldPlanetmath on X has enough injectives. So we can define the sheaf cohomology Hi(X,) of a sheaf to be the right derived functorsMathworldPlanetmath of the global sections functor Γ(X,).

Usually we are interested in the case where X is a scheme, and is a coherent sheaf. In this case, it does not matter if we take the derived functors in the category of sheaves of abelian groups or coherent sheaves.

Sheaf cohomology can be explicitly calculated using Čech cohomologyPlanetmathPlanetmath ( Choose an open cover {Ui} of X. We define


where the productPlanetmathPlanetmath is over i+1 element subsets of {1,,n} and Uj0ji=Uj0Uji. If s(Uj0ji) is thought of as an element of Ci(), then the differential


makes C*() into a chain complexMathworldPlanetmath. The cohomology of this complex is denoted Hˇi(X,) and called the Čech cohomology of with respect to the cover {Ui}. There is a natural map Hi(X,)Hˇi(X,) which is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath for sufficiently fine covers. (A cover is sufficiently fine if Hi(Uj,)=0 for all i>0, for every j and for every sheaf ). In the categoryMathworldPlanetmath of schemes, for example, any cover by open affine schemesMathworldPlanetmath has this property. What this means is that if one can find a finite fine enough cover of X, sheaf cohomology becomes computable by a finite process. In fact in [2], this is how the cohomology of projective spaceMathworldPlanetmath is explicitly calculated.


  • 1 Grothendieck, A. Sur quelques points d’algèbre homologique, Tôhoku Math. J., Second Series, 9 (1957), 119–221.
  • 2 Hartshorne, R. Algebraic GeometryMathworldPlanetmathPlanetmath, Springer-Verlag Graduate Texts in Mathematics 52, 1977
Title sheaf cohomology
Canonical name SheafCohomology
Date of creation 2013-03-22 13:50:59
Last modified on 2013-03-22 13:50:59
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Definition
Classification msc 14F25
Related topic EtaleCohomology
Related topic LeraysTheorem
Related topic AcyclicSheaf
Related topic DeRhamWeilTheorem
Defines sufficiently fine