Sheaf cohomology in topology

Builds on Topological space Abelian group

Definition

Let $X$ be a topological space and let $\mathrm{Sh}(X)$ be the category of sheaves of abelian groups on $X$ . The global sections functor $\Gamma(X, -) \colon \mathrm{Sh}(X) \to \mathrm{Ab}$ , $F \mapsto F(X)$ , is left exact, and $\mathrm{Sh}(X)$ has enough injective objects. For a sheaf $F$ , choose an injective resolution $0 \to F \to I^0 \to I^1 \to I^2 \to \cdots$ in $\mathrm{Sh}(X)$ . The $q$ -th cohomology group of $X$ with coefficients in $F$ is $H^q(X, F) = H^q\big(\Gamma(X, I^0) \to \Gamma(X, I^1) \to \Gamma(X, I^2) \to \cdots\big).$ So $H^q(X, -)$ is the $q$ -th right derived functor of $\Gamma(X, -)$ ; it does not depend on the choice of resolution up to canonical isomorphism.

Remarks

Always $H^0(X, F) = F(X)$ , by left exactness of $\Gamma$ . A short exact sequence of sheaves gives a long exact sequence of cohomology groups.

Example: if $X$ is a single point, a sheaf is just an abelian group $A$ , and $\Gamma$ is the identity functor, which is exact. So $H^0(X, A) = A$ and $H^q(X, A) = 0$ for $q > 0$ .

Non-example: the rule $U \mapsto$ (constant functions $U \to \Z$ ) is a presheaf but in general not a sheaf. Two different constants on two disjoint open sets do not glue to a constant function on their union. The definition needs a sheaf, here its sheafification, the sheaf of locally constant functions.

In practice one uses acyclic resolutions instead of injective ones: flasque, soft or fine resolutions compute the same groups. If $X$ is paracompact, Hausdorff and locally contractible, then $H^q(X, \underline{A})$ for the constant sheaf $\underline{A}$ is isomorphic to singular cohomology $H^q(X; A)$ .

Reading path

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Tag	`VRZDAWNTCR`
URI	`m2://VRZDAWNTCR`
Kind	definition
Writing	W0 stub
MSC	55N30, 14F06
Defines	Sheaf cohomology in topology
Uses	Topological space, Abelian group
Changed	2 November 2026

Sheaf cohomology in topology

Definition

Remarks

Reading path

Further reading

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