Sheaf CohomologyEdit

Sheaf cohomology is a central tool in modern mathematics for translating locally defined data into global invariants. At its heart, it studies how sections of a sheaf can be pieced together over a space, and how the obstructions to gluing give rise to a sequence of abelian groups that encode global information. The construction unifies perspectives from algebraic geometry, differential geometry, and topology, and provides a flexible language for proving deep theorems such as Riemann–Roch, Serre duality, and many cohomological phenomena in geometry and analysis.

On a Topological space X, a Sheaf F assigns to each open set U ⊆ X a set (often an abelian group or a module) F(U), together with restriction maps for inclusions of open sets. The global sections functor Γ(X, -) takes a sheaf to its sections over the whole space, Γ(X, F) = F(X). The cohomology groups H^i(X, F) are the right derived functors of this global sections functor, and they measure the failure of local data to glue together into global data. In practical terms, H^0(X, F) recovers global sections, while higher cohomology groups detect more subtle obstructions to gluing and global structure. For a compact formulation, these groups are computed as the cohomology of a suitable complex of abelian groups built from F, such as an injective resolution or a Čech complex.

Foundations

Presheaves and sheaves

A presheaf associates to each open set U a set F(U) with restriction maps to smaller open sets. A sheaf strengthens this data with gluing conditions: if sections over an open cover agree on overlaps, they glue to a unique section over the union. The machinery applies to sheaves valued in abelian groups, modules over a ring, or more general abelian categories. The structure sheaf O_X on a space X, for example, encodes rings of functions on open sets and is fundamental in algebraic geometry and complex geometry. Presheaf and Sheaf are therefore foundational terms in this area, as are Global sections and the related functors used to extract global information.

Derived functors and cohomology

Cohomology arises from the language of derived functors: H^i(X, F) is the i-th derived functor of Γ(X, -). This viewpoint explains many formal properties, such as long exact sequences in cohomology coming from short exact sequences of sheaves. The basic full machinery involves injective resolutions, where F is embedded into an exact complex of injective sheaves, and the cohomology of the global sections of that complex computes H^i(X, F). For readers who prefer a more hands-on computational method, the Čech construction provides a concrete way to compute cohomology using covers of X.

Computational tools

Čech cohomology: Given an open cover {U_i} of X, one builds a Čech complex from sections over finite intersections of the cover; its cohomology often computes H^i(X, F) under suitable conditions. See Čech cohomology for a detailed construction and comparisons with sheaf cohomology.
Resolutions: Injective resolutions or other acyclic resolutions allow the cohomology groups to be computed as the cohomology of a global sections complex. This is the standard approach in many algebraic settings and is central to the derived picture.
Spectral sequences: In many geometric situations, spectral sequences (e.g., the Leray spectral sequence) organize complex computations into manageable stages. They link cohomology on a target space with higher direct image sheaves and COhomology on the source.

Key properties

Global sections and vanishing: H^0(X, F) is the group of global sections F(X). Higher groups capture obstructions to gluing and global construction.
Functoriality and exact sequences: A morphism of spaces or a map of sheaves induces maps on cohomology. Short exact sequences of sheaves yield long exact sequences in cohomology, enabling inductive and comparative techniques.
Acyclic objects: If F is acyclic with respect to Γ(X, -) (for instance, certain flabby or fine sheaves on suitable spaces), then higher cohomology can be computed more readily, often by simpler resolutions or by Čech methods.
Relationship to geometric structures: When F encodes geometric data (such as the structure sheaf O_X, differential forms, or line bundles), H^i(X, F) becomes a bridge between topology and the geometry of X.

Examples

A point: Let X be a single point and F a constant sheaf G. Then H^0(X, F) ≅ G and H^i(X, F) = 0 for i > 0. This reflects the idea that a point carries no interesting global obstructions.
A circle: For X = S^1 and F the constant sheaf Z, one has H^0(S^1, Z) ≅ Z and H^1(S^1, Z) ≅ Z, with higher cohomology vanishing. This is a classical illustration of how topology influences cohomology.
Projective space and line bundles: On projective space P^n over a field k, the cohomology of the structure sheaf O_{P^n} is concentrated in degree 0, with H^0(P^n, O_{P^n}) ≅ k and H^i(P^n, O_{P^n}) = 0 for i > 0. For twists by line bundles O(d), one has a rich pattern of nontrivial cohomology, which is central to the construction of embeddings and to the study of sheaves on projective varieties. See Line bundle and Projective space for related terms and results.

Applications

Algebraic geometry: Sheaf cohomology is a foundational tool in the study of coherent sheaves on varieties. It underpins the formulation of the Riemann–Roch theorem and its refinements, and it feeds into modern developments through the cohomology of line bundles, the construction of moduli spaces, and the use of spectral sequences and derived categories. Key results such as the Serre duality theorem connect cohomology groups of dualizing objects to dual vector spaces over the base field. See Riemann–Roch theorem and Serre duality for these connections.
Complex geometry and differential geometry: On complex manifolds, one relates sheaf cohomology of the sheaf of holomorphic forms to other invariants via the Dolbeault isomorphism and Hodge theory. The de Rham cohomology of smooth manifolds is linked to sheaf cohomology of constant sheaves through de Rham’s theorem. See Dolbeault cohomology and de Rham cohomology.
Topology and index theory: The Leray spectral sequence connects the cohomology of the total space to the cohomology of the base with coefficients in higher direct images, providing a powerful computational framework in fiber bundles and fibration contexts. See Leray spectral sequence.
Étale and arithmetic geometry: In algebraic geometry over arbitrary fields, étale cohomology yields robust invariants that mirror topological cohomology in a setting without classical topology. See Étale cohomology for more.

Historical notes

The study of cohomology from the local-to-global perspective was developed in the mid-20th century, with early contributions from Leray and later a sweeping formalism due to Grothendieck that unified many geometric phenomena. The development of derived functors and homological algebra gave a flexible, general framework that continues to influence modern mathematics and its applications.