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Algebraic geometry Definition

Sheaf cohomology

Builds on Ringed spaceSheaf of modulesSheafAbelian groupDerived functorInjective resolutionSheaf cohomology in topology

Definition

Let (X,π’ͺX)(X, \mathcal{O}_X) be a ringed space and let Ab(X)\mathrm{Ab}(X) be the category of sheaves of abelian groups on XX. The global sections functor Ξ“(X,βˆ’) ⁣:Ab(X)β†’Ab\Gamma(X, -) \colon \mathrm{Ab}(X) \to \mathrm{Ab} is left exact, and Ab(X)\mathrm{Ab}(X) has enough injective objects. For a sheaf β„±\mathcal{F} in Ab(X)\mathrm{Ab}(X), the ii-th cohomology group Hi(X,β„±)H^i(X, \mathcal{F}) is the value at β„±\mathcal{F} of the ii-th right derived functor of Ξ“(X,βˆ’)\Gamma(X, -). Concretely, for an injective resolution 0→ℱ→ℐ0→ℐ1β†’β‹―0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots in Ab(X)\mathrm{Ab}(X), Hi(X,β„±)=Hi(Ξ“(X,ℐ0)β†’Ξ“(X,ℐ1)β†’β‹―).H^i(X, \mathcal{F}) = H^i\big(\Gamma(X, \mathcal{I}^0) \to \Gamma(X, \mathcal{I}^1) \to \cdots\big). For a sheaf of π’ͺX\mathcal{O}_X-modules β„±\mathcal{F}, the groups Hi(X,β„±)H^i(X, \mathcal{F}) are taken of its underlying sheaf of abelian groups; they are modules over Ξ“(X,π’ͺX)\Gamma(X, \mathcal{O}_X).

Remarks

The groups Hi(X,β„±)H^i(X, \mathcal{F}) for an π’ͺX\mathcal{O}_X-module β„±\mathcal{F} agree with the derived functors of Ξ“\Gamma computed in the category of π’ͺX\mathcal{O}_X-modules, since injective π’ͺX\mathcal{O}_X-modules are flasque and flasque sheaves are acyclic. This is the same construction as sheaf cohomology in topology, applied to the topological space of a scheme.

Example: always H0(X,β„±)=β„±(X)H^0(X, \mathcal{F}) = \mathcal{F}(X). For X=β„™knX = \mathbb{P}^n_k and β„±=π’ͺ(d)\mathcal{F} = \mathcal{O}(d), H0H^0 is the space of homogeneous polynomials of degree dd, Hnβ‰…kH^n \cong k when d=βˆ’nβˆ’1d = -n-1, and Hi=0H^i = 0 for 0<i<n0 < i < n.

Non-example: a resolution of β„±\mathcal{F} by sheaves that are not acyclic, for example by arbitrary locally free sheaves, does not in general compute Hi(X,β„±)H^i(X, \mathcal{F}). In practice one uses flasque resolutions or, on separated schemes with quasi-coherent sheaves, the Čech complex of an affine cover.

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To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Abelian group, concept, no evidence levelAbelian groupconceptDerived functor, concept, no evidence levelDerived functorconceptInjective resolution, concept, no evidence levelInjective resolutionconceptRinged space, concept, no evidence levelRinged spaceconceptSheaf, concept, no evidence levelSheafconceptSheaf cohomology in topology, concept, no evidence levelSheaf cohomology in t…conceptSheaf of modules, concept, no evidence levelSheaf of modulesconceptSheaf cohomology, definition, no evidence levelSheaf cohomologydefinition
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  1. Abelian group concept
  2. Derived functor concept
  3. Injective resolution concept
  4. Ringed space concept
  5. Sheaf concept
  6. Sheaf cohomology in topology concept
  7. Sheaf of modules concept
  8. Sheaf cohomology definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Sheaf_cohomology
About this page
TagC7E5EMV89Z
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Kinddefinition
WritingW0 stub
MSC14F06, 55N30, 18G10
DefinesSheaf cohomology
UsesRinged space, Sheaf of modules, Sheaf, Abelian group, Derived functor, Injective resolution, Sheaf cohomology in topology
Changed2 November 2026

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