Sheaf cohomology

Sheaf cohomology assigns groups to a sheaf on a space, measuring how local data fail to assemble into global sections and higher-dimensional analogues.

Global section functor: The functor that assigns to a sheaf its sections over the entire space. Sheaf cohomology derives this functor to detect failures of global sections to preserve exact sequences.
Sheaf: A structure assigning data to open sets, with restriction maps satisfying locality and gluing conditions. Sheaf cohomology studies the global assembly of the data encoded by a sheaf.
Line bundle: A locally trivial vector bundle of rank one over a topological space or geometric object. First cohomology with values in invertible functions classifies line bundles in common settings.
Étale cohomology: A cohomology theory for schemes defined using the étale topology and sheaves on that site. It extends sheaf-cohomological methods to arithmetic geometry where ordinary topology is inadequate.
Derived functor: A construction that extends a non-exact functor into a sequence of functors recording its failures of exactness. Sheaf cohomology is commonly defined as the right derived functors of global sections.
Presheaf: A contravariant assignment of data to open sets with restriction maps, without necessarily satisfying sheaf gluing axioms. Presheaves are the starting structures from which sheaves and their cohomology arise.
Divisor (algebraic geometry): A formal integer combination of codimension-one subvarieties on an algebraic variety. Divisors and line bundles connect geometric data to cohomology groups such as Picard groups.
Group cohomology: A cohomology theory describing invariants and extensions associated with group actions on modules. It is a related derived-functor theory, and sheaf cohomology can connect to it through quotient spaces.
Injective sheaf: A sheaf that is an injective object in the category of sheaves of modules or abelian groups. Injective resolutions provide one standard way to compute sheaf cohomology.
Exact sequence: A sequence of algebraic objects and maps in which each map's image equals the next map's kernel. Long exact sequences expose how cohomology responds to short exact sequences of sheaves.

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