The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper.
Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria.
These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems.
More broadly, the exact sequence makes knowledge of higher cohomology groups a fundamental tool in aiming to understand sections of sheaves.
Grothendieck's definition of sheaf cohomology, now standard, uses the language of homological algebra.
For specific classes of spaces or sheaves, there are many tools for computing sheaf cohomology, some discussed below.
For a continuous map f: X → Y and an abelian group A, the pullback sheaf f*(AY) is isomorphic to AX.
As a result, the pullback homomorphism makes sheaf cohomology with constant coefficients into a contravariant functor from topological spaces to abelian groups.
(This statement generalizes to any sheaf of groups G, not necessarily abelian, using the non-abelian cohomology set H1(X,G).)
By definition, an E-torsor over X is a sheaf S of sets together with an action of E on X such that every point in X has an open neighborhood on which S is isomorphic to E, with E acting on itself by translation.
For example, if X is a topological space with subspaces Y and U such that the closure of Y is contained in the interior of U, and E is a sheaf on X, then the restriction is an isomorphism.
[17] (So cohomology with support in a closed subset Y only depends on the behavior of the space X and the sheaf E near Y.)
Also, if X is a paracompact Hausdorff space that is the union of closed subsets A and B, and E is a sheaf on X, then the restriction is an isomorphism.
For a sheaf E of abelian groups on X, one can define cohomology with compact support Hcj(X,E).
Also, for an open subset U of a locally compact space X and a sheaf E on X, there is a pushforward homomorphism known as extension by zero:[21] Both homomorphisms occur in the long exact localization sequence for compactly supported cohomology, for a locally compact space X and a closed subset Y:[22] For any sheaves A and B of abelian groups on a topological space X, there is a bilinear map, the cup product for all i and j.
[23] Here A⊗B denotes the tensor product over Z, but if A and B are sheaves of modules over some sheaf OX of commutative rings, then one can map further from Hi+j(X,A⊗ZB) to Hi+j(X,A⊗OXB).
More generally, for any complex of sheaves E (not necessarily bounded below) on a space X, the cohomology group Hj(X,E) is defined as a group of morphisms in the derived category of sheaves on X: where ZX is the constant sheaf associated to the integers, and E[j] means the complex E shifted j steps to the left.
A central result in topology is the Poincaré duality theorem: for a closed oriented connected topological manifold X of dimension n and a field k, the group Hn(X,k) is isomorphic to k, and the cup product is a perfect pairing for all integers j.
If X is an oriented n-manifold, not necessarily compact or connected, and k is a field, then cohomology is the dual of cohomology with compact support: For any manifold X and field k, there is a sheaf oX on X, the orientation sheaf, which is locally (but perhaps not globally) isomorphic to the constant sheaf k. One version of Poincaré duality for an arbitrary n-manifold X is the isomorphism:[25] More generally, if E is a locally constant sheaf of k-vector spaces on an n-manifold X and the stalks of E have finite dimension, then there is an isomorphism With coefficients in an arbitrary commutative ring rather than a field, Poincaré duality is naturally formulated as an isomorphism from cohomology to Borel–Moore homology.
For any locally compact space X of finite dimension and any field k, there is an object DX in the derived category D(X) of sheaves on X called the dualizing complex (with coefficients in k).
The special case where f is a fibration and E is a constant sheaf plays an important role in homotopy theory under the name of the Serre spectral sequence.
Let X be a compact Hausdorff space, and let R be a principal ideal domain, for example a field or the ring Z of integers.
Let E be a sheaf of R-modules on X, and assume that E has "locally finitely generated cohomology", meaning that for each point x in X, each integer j, and each open neighborhood U of x, there is an open neighborhood V ⊂ U of x such that the image of Hj(U,E) → Hj(V,E) is a finitely generated R-module.
[30] For example, for a compact Hausdorff space X that is locally contractible (in the weak sense discussed above), the sheaf cohomology group Hj(X,Z) is finitely generated for every integer j.
A sheaf E on X that is constructible with respect to the given stratification has locally finitely generated cohomology.
[31] If X is compact, it follows that the cohomology groups Hj(X,E) of X with coefficients in a constructible sheaf are finitely generated.
More generally, suppose that X is compactifiable, meaning that there is a compact stratified space W containing X as an open subset, with W–X a union of connected components of strata.
[32] For example, any complex algebraic variety X, with its classical (Euclidean) topology, is compactifiable in this sense.
A great deal is known about the cohomology groups of a scheme or complex analytic space with coefficients in a coherent sheaf.
A topological space X determines a site in a natural way: the category C has objects the open subsets of X, with morphisms being inclusions, and with a set of morphisms Vα → U being called a covering of U if and only if U is the union of the open subsets Vα.