In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem globally when it can be solved locally. 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. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp.
Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Many earlier results such as the Riemann–Roch theorem and the Hodge theorem have been generalized or understood better using sheaf cohomology.
Definition The category of sheaves of abelian groups on a topological space X is an abelian category, and so it makes sense to ask when a morphism f: B → C of sheaves is injective (a monomorphism) or surjective (an epimorphism). One answer is that f is injective (respectively surjective) if and only if the associated homomorphism on stalks Bx → Cx is injective (respectively surjective) for every point x in X. It follows that f is injective if and only if the homomorphism B(U) → C(U) of sections over U is injective for every open set U in X. Surjectivity is more subtle, however: the morphism f is surjective if and only if for every open set U in X, every section s of C over U, and every point x in U, there is an open neighborhood V of x in U such that s restricted to V is the image of some section of B over V. (In words: every section of C lifts locally to sections of B.)
As a result, the question arises: given a surjection B → C of sheaves and a section s of C over X, when is s the image of a section of B over X? This is a model for all kinds of local-vs.-global questions in geometry. Sheaf cohomology gives a satisfactory general answer. Namely, let A be the kernel of the surjection B → C, giving a short exact sequence
of sheaves on X. Then there is a long exact sequence of abelian groups, called sheaf cohomology groups:
where H0(X,A) is the group A(X) of global sections of A on X. For example, if the group H1(X,A) is zero, then this exact sequence implies that every global section of C lifts to a global section of B. 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. The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups on X to abelian groups. In more detail, start with the functor E ↦ E(X) from sheaves of abelian groups on X to abelian groups. This is left exact, but in general not right exact. Then the groups Hi(X,E) for integers i are defined as the right derived functors of the functor E ↦ E(X). This makes it automatic that Hi(X,E) is zero for i < 0, and that H0(X,E) is the group E(X) of global sections. The long exact sequence above is also straightforward from this definition.
The definition of derived functors uses that the category of sheaves of abelian groups on any topological space X has enough injectives; that is, for every sheaf E there is an injective sheaf I with an injection E → I. It follows that every sheaf E has an injective resolution:
Then the sheaf cohomology groups Hi(X,E) are the cohomology groups (the kernel of one homomorphism modulo the image of the previous one) of the chain complex of abelian groups:
Standard arguments in homological algebra imply that these cohomology groups are independent of the choice of injective resolution of E.
The definition is rarely used directly to compute sheaf cohomology. It is nonetheless powerful, because it works in great generality (any sheaf of abelian groups on any topological space), and it easily implies the formal properties of sheaf cohomology, such as the long exact sequence above. For specific classes of spaces or sheaves, there are many tools for computing sheaf cohomology, some discussed below.
Functoriality For any continuous map f: X → Y of topological spaces, and any sheaf E of abelian groups on Y, there is a pullback homomorphism
for every integer j, where f*(E) denotes the inverse image sheaf or pullback sheaf. If f is the inclusion of a subspace X of Y, f*(E) is the restriction of E to X, often just called E again, and the pullback of a section s from Y to X is called the restriction s|X.
Pullback homomorphisms are used in the Mayer–Vietoris sequence, an important computational result. Namely, let X be a topological space which is a union of two open subsets U and V, and let E be a sheaf on X. Then there is a long exact sequence of abelian groups:
Sheaf cohomology with constant coefficients For a topological space and an abelian group , the constant sheaf means the sheaf of locally constant functions with values in . The sheaf cohomology groups with constant coefficients are often written simply as , unless this could cause confusion with another version of cohomology such as singular cohomology.
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.
For any spaces X and Y and any abelian group A, two homotopic maps f and g from X to Y induce the same homomorphism on sheaf cohomology:
It follows that two homotopy equivalent spaces have isomorphic sheaf cohomology with constant coefficients.
Let X be a paracompact Hausdorff space which is locally contractible, even in the weak sense that every open neighborhood U of a point x contains an open neighborhood V of x such that the inclusion V → U is homotopic to a constant map. Then the singular cohomology groups of X with coefficients in an abelian group A are isomorphic to sheaf cohomology with constant coefficients, H*(X,AX). For example, this holds for X a topological manifold or a CW complex.
As a result, many of the basic calculations of sheaf cohomology with constant coefficients are the same as calculations of singular cohomology. See the article on cohomology for the cohomology of spheres, projective spaces, tori, and surfaces.
For arbitrary topological spaces, singular cohomology and sheaf cohomology (with constant coefficients) can be different. This happens even for H0. The singular cohomology H0(X,Z) is the group of all functions from the set of path components of X to the integers Z, whereas sheaf cohomology H0(X,ZX) is the group of locally constant functions from X to Z. These are different, for example, when X is the Cantor set. Indeed, the sheaf cohomology H0(X,ZX) is a countable abelian group in that case, whereas the singular cohomology H0(X,Z) is the group of all functions from X to Z, which has cardinality
For a paracompact Hausdorff space X and any sheaf E of abelian groups on X, the cohomology groups Hj(X,E) are zero for j greater than the covering dimension of X. (This does not hold in the same generality for singular cohomology: for example, there is a compact subset of Euclidean space R3 that has nonzero singular cohomology in infinitely many degrees.) The covering dimension agrees with the usual notion of dimension for a topological manifold or a CW complex.
Flabby and soft sheaves A sheaf E of abelian groups on a topological space X is called acyclic if Hj(X,E) = 0 for all j > 0. By the long exact sequence of sheaf cohomology, the cohomology of any sheaf can be computed from any acyclic resolution of E (rather than an injective resolution). Injective sheaves are acyclic, but for computations it is useful to have other examples of acyclic sheaves.
A sheaf E on X is called flabby (French: flasque) if every section of E on an open subset of X extends to a section of E on all of X. Flabby sheaves are acyclic. Godement defined sheaf cohomology via a canonical flabby resolution of any sheaf; since flabby sheaves are acyclic, Godement's definition agrees with the definition of sheaf cohomology above.
A sheaf E on a paracompact Hausdorff space X is called soft if every section of the restriction of E to a closed subset of X extends to a section of E on all of X. Every soft sheaf is acyclic.
Some examples of soft sheaves are the sheaf of real-valued continuous functions on any paracompact Hausdorff space, or the sheaf of smooth (C∞) functions on any smooth manifold. More generally, any sheaf of modules over a soft sheaf of commutative rings is soft; for example, the sheaf of smooth sections of a vector bundle over a smooth manifold is soft.
For example, these results form part of the proof of de Rham's theorem. For a smooth manifold X, the Poincaré lemma says that the de Rham complex is a resolution of the constant sheaf RX:
where ΩXj is the sheaf of smooth j-forms and the map ΩXj → ΩXj+1 is the exterior derivative d. By the results above, the sheaves ΩXj are soft and therefore acyclic. It follows that the sheaf cohomology of X with real coefficients is isomorphic to the de Rham cohomology of X, defined as the cohomology of the complex of real vector spaces:
The other part of de Rham's theorem is to identify sheaf cohomology and singular cohomology of X with real coefficients; that holds in greater generality, as discussed above.
Čech cohomology Čech cohomology is an approximation to sheaf cohomology that is often useful for computations. Namely, let be an open cover of a topological space X, and let E be a sheaf of abelian groups on X. Write the open sets in the cover as Ui for elements i of a set I, and fix an ordering of I. Then Čech cohomology is defined as the cohomology of an explicit complex of abelian groups with jth group
There is a natural homomorphism . Thus Čech cohomology is an approximation to sheaf cohomology using only the sections of E on finite intersections of the open sets Ui.
If every finite intersection V of the open sets in has no higher cohomology with coefficients in E, meaning that Hj(V,E) = 0 for all j > 0, then the homomorphism from Čech cohomology to sheaf cohomology is an isomorphism.
Another approach to relating Čech cohomology to sheaf cohomology is as follows. The Čech cohomology groups are defined as the direct limit of over all open covers of X (where open covers are ordered by refinement). There is a homomorphism from Čech cohomology to sheaf cohomology, which is an isomorphism for j ≤ 1. For arbitrary topological spaces, Čech cohomology can differ from sheaf cohomology in higher degrees. Conveniently, however, Čech cohomology is isomorphic to sheaf cohomology for any sheaf on a paracompact Hausdorff space.
The isomorphism implies a description of H1(X,E) for any sheaf E of abelian groups on a topological space X: this group classifies the E-torsors (also called principal E-bundles) over X, up to isomorphism. (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, on a ringed space (X,OX), it follows that the Picard group of invertible sheaves on X is isomorphic to the sheaf cohomology group H1(X,OX*), where OX* is the sheaf of units in OX.
Relative cohomology