Sheaves and Sheaf Cohomology — Mathematics Knowledge System

Overview

A sheaf on a topological space X assigns data (sets, groups, rings) to open sets in a coherent way: restriction maps are compatible and local data glues to global sections uniquely. Sheaf cohomology measures the obstruction to extending local data globally. It unifies and generalises de Rham, Cech, and singular cohomology, and is the primary tool in algebraic geometry (coherent sheaves), complex analysis (holomorphic functions), and modern topology. The derived functor definition via injective resolutions gives a functorial, base-change-compatible theory.

Intuition

A sheaf is like a 'database' that stores local information consistently: each open set U has associated data F(U), and the data on smaller sets is obtained by restriction. The global sections H^0(X, F) are the globally consistent data. Higher cohomology H^k(X, F) for k >= 1 detects obstructions: H^1 measures how local data fails to glue globally (a 'twist'); H^2 measures obstructions to trivialising H^1 data, and so on. The classic example: the sheaf of holomorphic functions on a Riemann surface -- H^1 is related to the number of holes.

Formal Definition

Definition

A presheaf F of abelian groups on X assigns to each open U an abelian group F(U) and to each inclusion V subset U a restriction map rho_{UV}: F(U) -> F(V), functorially. F is a sheaf if it satisfies the gluing axiom: for any open cover {U_i} of U, sections s_i in F(U_i) agreeing on overlaps (s_i|_{U_i cap U_j} = s_j|_{U_i cap U_j}) glue uniquely to s in F(U) with s|_{U_i} = s_i. Sheaf cohomology H^*(X, F) is the derived functor of the global sections functor Gamma(X, -).

H^0(X, \mathcal{F}) = \Gamma(X, \mathcal{F}) = \mathcal{F}(X)

Global sections

0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots \text{ (injective resolution)}

Injective resolution

H^k(X, \mathcal{F}) = H^k(\Gamma(X, \mathcal{I}^\bullet))

Sheaf cohomology via injective resolution

\check{H}^k(\mathcal{U}, \mathcal{F}) = H^k(C^\bullet(\mathcal{U}, \mathcal{F}))

Cech cohomology

Notation

Notation	Meaning
$\mathcal{F}(U)$	Sections of sheaf F over open set U
$\Gamma(X, \mathcal{F})$	Global sections of F
$H^k(X, \mathcal{F})$	k-th sheaf cohomology group
$\mathcal{F}_x$	Stalk of sheaf F at point x

Theorems

Theorem 1: Long Exact Sequence in Sheaf Cohomology

A short exact sequence 0 -> F' -> F -> F'' -> 0 of sheaves gives a long exact sequence: 0 -> H^0(X,F') -> H^0(X,F) -> H^0(X,F'') -> H^1(X,F') -> H^1(X,F) -> ... This is the fundamental tool for computing sheaf cohomology by induction.

Theorem 2: Leray's Theorem

If U is a covering of X such that H^k(U_{i_0...i_p}, F) = 0 for all k >= 1 and all intersections, then the Cech cohomology with respect to U agrees with sheaf cohomology: H^k_Cech(U, F) is isomorphic to H^k(X, F) for all k.

Theorem 3: Serre Duality

For a smooth projective variety X of dimension n over a field with dualising sheaf omega_X, there is a perfect pairing H^k(X, F) tensor H^{n-k}(X, F^* tensor omega_X) -> k (the base field), generalising Poincare duality to sheaf cohomology.

Worked Examples

1
The exponential sequence on S^1: 0 -> Z -> C -> C* -> 0 (where Z is the constant sheaf, C is the sheaf of continuous functions, C* is the sheaf of nowhere-zero continuous functions).
2
C (continuous functions) is fine (admits partitions of unity), so H^k(S^1, C) = 0 for k >= 1.
$H^k(S^1, \underline{\mathbb{C}}) = 0,\quad k \ge 1$
3
Long exact sequence: H^0(C*) -> H^1(Z) -> H^1(C) = 0. So H^1(S^1, Z) = coker(H^0(C*) -> H^1(Z)). Separately H^1(S^1, Z) = Z by topology.
$H^1(S^1, \mathbb{Z}) \cong \mathbb{Z}$

✓ Answer

H^1(S^1, Z) = Z, consistent with the fundamental group and singular cohomology of the circle.

Practice Problems

Hardfree response

Explain the gluing axiom for sheaves and give an example of a presheaf that is NOT a sheaf.

Hardproof writing

Show that if X is a contractible topological space and F is a constant sheaf A, then H^k(X, A) = 0 for all k >= 1.

Common Mistakes

Common Mistake

Thinking Cech cohomology always equals sheaf cohomology.

Cech cohomology agrees with sheaf cohomology for Leray covers (where all intersections are acyclic) and for paracompact spaces with constant coefficients. In general they may differ.

Common Mistake

Confusing sheaves with presheaves.

Every sheaf is a presheaf, but presheaves need not satisfy the gluing axiom. The sheafification functor converts any presheaf to the 'closest' sheaf, but the cohomology of a presheaf and its sheafification can differ.

Quiz

The stalk F_x of a sheaf F at a point x is:

The group F(X) of global sectionsThe direct limit of F(U) over all open U containing xThe kernel of the restriction map to {x}The fiber of F over the basepoint

H^1(X, F) measures:

The global sections of FObstructions to patching local sections to global onesThe number of connected components of XThe dimension of the fiber at each point

Historical Background

Sheaf theory was introduced by Jean Leray during WWII (1945) while a prisoner of war, as a tool for computing cohomology of fibrations. Cartan's seminar (1948-51) developed the modern formalism. Grothendieck's 1957 Tohoku paper reformulated sheaf cohomology as a derived functor, introduced injective resolutions, and laid the foundations of homological algebra in abelian categories. Serre's GAGA theorem (1956) used sheaf cohomology to bridge algebraic geometry and complex analysis.

1945
Leray introduces sheaves and sheaf cohomology
Jean Leray
1950
Cartan seminars develop sheaf theory systematically
Henri Cartan
1957
Grothendieck's Tohoku paper: sheaf cohomology as derived functors
Alexander Grothendieck
1963
Grothendieck introduces etale cohomology via sheaves on the etale site
Alexander Grothendieck

Summary

A sheaf assigns consistent data to open sets; sheaf cohomology measures global obstructions.
H^0(X, F) = global sections; H^k for k >= 1 are derived functors detecting higher obstructions.
The long exact sequence in cohomology is the primary computational tool.
Cech cohomology equals sheaf cohomology for good covers (Leray's theorem).
Sheaf cohomology unifies de Rham, singular, and Dolbeault cohomology theories.

References

BookHartshorne, R. Algebraic Geometry. Springer, 1977. Chapter III.
BookGodement, R. Topologie Algebrique et Theorie des Faisceaux. Hermann, 1958.
WebsiteWikipedia -- Sheaf cohomology