Sheaf (mathematics)
Sheaf cohomology

In contexts where the open set U {\displaystyle U} is fixed, and the sheaf is regarded as a variable, the set F (U) {\displaystyle F(U)} is also often denoted Γ (U, F). {\displaystyle \Gamma (U,F).}

As was noted above, this functor does not preserve epimorphisms. Instead, an epimorphism of sheaves F → G {\displaystyle {\mathcal {F}}\to {\mathcal {G}}} is a map with the following property: for any section g ∈ G (U) {\displaystyle g\in {\mathcal {G}}(U)} there is a covering U = { U i } i ∈ I {\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}} where

U = ⋃ i ∈ I U i {\displaystyle U=\bigcup _{i\in I}U_{i}}

of open subsets, such that the restriction g | U i {\displaystyle g|_{U_{i}}} are in the image of F (U i) {\displaystyle {\mathcal {F}}(U_{i})}. However, g {\displaystyle g} itself need not be in the image of F (U) {\displaystyle {\mathcal {F}}(U)}. A concrete example of this phenomenon is the exponential map

between the sheaf of holomorphic functions and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function g {\displaystyle g} (on some open subset in C {\displaystyle \mathbb {C} }, say), admits a complex logarithm locally, i.e., after restricting g {\displaystyle g} to appropriate open subsets. However, g {\displaystyle g} need not have a logarithm globally.

Sheaf cohomology captures this phenomenon. More precisely, for an exact sequence of sheaves of abelian groups

(i.e., an epimorphism F 2 → F 3 {\displaystyle {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}} whose kernel is F 1 {\displaystyle {\mathcal {F}}_{1}}), there is a long exact sequence 0 → Γ (U, F 1) → Γ (U, F 2) → Γ (U, F 3) → H 1 (U, F 1) → H 1 (U, F 2) → H 1 (U, F 3) → H 2 (U, F 1) → … {\displaystyle 0\to \Gamma (U,{\mathcal {F}}_{1})\to \Gamma (U,{\mathcal {F}}_{2})\to \Gamma (U,{\mathcal {F}}_{3})\to H^{1}(U,{\mathcal {F}}_{1})\to H^{1}(U,{\mathcal {F}}_{2})\to H^{1}(U,{\mathcal {F}}_{3})\to H^{2}(U,{\mathcal {F}}_{1})\to \dots } By means of this sequence, the first cohomology group H 1 (U, F 1) {\displaystyle H^{1}(U,{\mathcal {F}}_{1})} is a measure for the non-surjectivity of the map between sections of F 2 {\displaystyle {\mathcal {F}}_{2}} and F 3 {\displaystyle {\mathcal {F}}_{3}}.

There are several different ways of constructing sheaf cohomology. Grothendieck (1957) introduced them by defining sheaf cohomology as the derived functor of Γ {\displaystyle \Gamma }. This method is theoretically satisfactory, but, being based on injective resolutions, of little use in concrete computations. Godement resolutions are another general, but practically inaccessible approach.
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