Sheaves and Cohomology

A sheaf records data that is defined locally and glued consistently, and sheaf cohomology measures the obstruction to passing from local solutions to a global one.

Definition

A sheaf on a space assigns to each open set a set (or group, ring, or module) of sections compatible under restriction and gluing; sheaf cohomology is the sequence of derived functors of taking global sections, quantifying the failure of local sections to glue globally.

Scope

This topic introduces presheaves and sheaves on a topological space or scheme, stalks, sheafification, and morphisms of sheaves, with the central examples of the structure sheaf, ideal sheaves, and coherent and quasi-coherent sheaves. It develops sheaf cohomology via derived functors of the global-sections functor and the computational tool of Čech cohomology, the cohomology of coherent sheaves on projective space, and foundational results such as Serre's finiteness and vanishing theorems and Serre duality.

Core questions

• How do the gluing axioms make a sheaf the right tool for local-to-global data?
• What do coherent and quasi-coherent sheaves capture about geometry over a scheme?
• Why is sheaf cohomology defined as a derived functor, and how does Čech cohomology compute it?
• What do Serre's finiteness, vanishing, and duality theorems tell us about coherent cohomology?

Key concepts

• Presheaves, sheaves, stalks, and sheafification
• Coherent and quasi-coherent sheaves
• Sheaf cohomology as a derived functor
• Čech cohomology and its agreement with derived cohomology
• Serre finiteness, vanishing, and Serre duality

Clinical relevance

Sheaf cohomology is the central computational engine of algebraic geometry, controlling sections of line bundles, deformations, and obstruction theory; the same machinery underlies the étale cohomology used to prove the Weil conjectures and is pervasive in topology and complex geometry.

History

Leray introduced sheaves and their cohomology in the 1940s; Serre's FAC (1955) brought coherent sheaf cohomology into algebraic geometry, and Grothendieck recast cohomology as derived functors in his Tôhoku paper (1957), the framework adopted in modern treatments.

Key figures

• Jean Leray
• Jean-Pierre Serre
• Alexander Grothendieck

Seminal works

• hartshorne1977
• maclane1971

Frequently asked questions

What is the difference between a presheaf and a sheaf?

A presheaf assigns data to open sets with restriction maps; a sheaf additionally requires that local sections agreeing on overlaps glue to a unique global section, which is exactly the locality needed for geometry.

Why does sheaf cohomology matter geometrically?

Its dimensions count global sections, obstructions, and invariants such as the genus; vanishing of higher cohomology is what allows local geometric data — for instance, sections of a line bundle — to be assembled globally.