Sheaves turn “information defined locally” into a precise mathematical system for asking whether compatible local pieces determine a global object.
The preceding guide, Schemes and Local Algebra, built spaces from affine spectra and localisations. Sheaves explain how functions, modules, differential forms, sections of bundles and other algebraic data live on every open subset and communicate across overlaps.
Cohomology enters when local compatibility is not enough for straightforward global reconstruction. It measures, in a systematic derived way, the failure of exactness of global sections and records obstructions that ordinary pointwise data cannot see.
Level: advanced undergraduate / beginning graduate enrichment. Prerequisites: rings and modules, exact sequences, localisation, affine schemes and basic topology.
Reading route: presheaves → sheaf axioms → stalks → sheafification → O-modules → quasi-coherent sheaves → affine modules → Čech gluing → exact sequences → derived cohomology → line bundles and divisors → practice → solutions.
1. A presheaf assigns data to every open set
Let X be a topological space. A presheaf F assigns an object F(U) to every open set U⊆X and a restriction map
res^U_V:F(U)→F(V)
whenever V⊆U.
Restrictions must satisfy identity and composition: restricting from U to U does nothing, and restricting U→V→W agrees with restricting directly U→W.
The objects may be sets, groups, rings, modules or vector spaces. The sheaf concept is structural rather than tied to one type of mathematical object.
2. A sheaf adds locality and gluing
A presheaf F is a sheaf when compatible local sections glue uniquely.
Suppose U=∪U_i and sections s_i∈F(U_i) agree on every overlap U_i∩U_j.
The sheaf axiom requires a unique s∈F(U) whose restriction to each U_i is s_i.
There are two logical parts:
- Locality: two global sections equal on every member of a cover are equal globally.
- Gluing: compatible local sections actually arise from a global section.
Both are needed. Compatibility without existence is incomplete; existence without uniqueness would make the local data ambiguous.
3. Continuous functions form the model example
For a topological space X, let C(U) be the continuous real-valued functions U→R.
Restriction is ordinary restriction of functions.
If continuous functions on an open cover agree on overlaps, they glue pointwise to a unique continuous function on the union.
Thus U↦C(U) is a sheaf.
Smooth functions on a smooth manifold and holomorphic functions on a complex manifold give analogous sheaves.
4. The structure sheaf is the scheme’s function system
For X=Spec A, the structure sheaf O_X is characterised on distinguished opens by
O_X(D(f))≈A_f.
So sections over D(f) are algebraic functions allowed to use powers of f in denominators.
At a prime p, the stalk is O_{X,p}=A_p.
A scheme is therefore not only a set of primes. Its structure sheaf remembers which functions are regular on each open region and how those functions restrict and glue.
5. Stalks isolate the germ of local behaviour
For a sheaf F and point x∈X, the stalk F_x collects sections defined on some neighbourhood of x, identifying two sections if they agree on a smaller neighbourhood.
An element of F_x is called a germ at x.
Stalks forget how a section behaves far away and preserve only its eventual local behaviour near x.
A morphism of sheaves is an isomorphism exactly when it induces an isomorphism on every stalk.
This is a strong local test: sheaf equality can be checked point by point at the level of germs.
6. Worked stalk example: regular functions on A¹
Let X=Spec k[t] and p=(t−a).
The stalk O_{X,p}=k[t]_(t−a) consists of fractions f/g with g(a)≠0.
Thus 1/(t−b) is a valid germ at a whenever b≠a.
But 1/(t−a) is not a regular germ at a because its denominator belongs to the maximal ideal and cannot be inverted in the local ring.
7. A presheaf can fail the sheaf axiom
Consider the presheaf assigning to each open set U the bounded continuous functions U→R.
On an increasing open cover, each local function may be bounded while the glued global function is unbounded.
So locally compatible bounded sections may fail to produce a global bounded section.
This illustrates the point of the sheaf axiom: “locally valid” and “globally valid” need not coincide for an arbitrary presheaf.
8. Sheafification repairs a presheaf
Every presheaf F has an associated sheaf F⁺, its sheafification.
Sheafification preserves the stalks: F_x≈F⁺_x.
Conceptually, it keeps the same local germs but adds exactly the local-gluing behaviour required of a sheaf.
The construction satisfies a universal property: every morphism from F to a sheaf factors uniquely through F⁺.
This makes sheafification a canonical repair rather than an arbitrary completion.
9. Sheaves of modules are linear data varying over a scheme
Let (X,O_X) be a ringed space. An O_X-module F assigns to each open U an O_X(U)-module F(U), compatibly with restriction.
Examples include the structure sheaf itself, sheaves of ideals, differential forms and sections of algebraic vector bundles.
Module sheaves allow linear algebra to vary from point to point while remaining tied to the functions on the underlying scheme.
At each point x, F_x is naturally an O_{X,x}-module.
10. Modules over a ring produce quasi-coherent sheaves
Let X=Spec A and M be an A-module.
There is an associated sheaf M~ satisfying
M~(D(f))≈M_f.
Such sheaves are the basic quasi-coherent sheaves on affine schemes.
The construction converts ordinary module localisation into geometric restriction.
For M=A, the associated sheaf is O_X.
11. Quasi-coherent sheaves are locally module sheaves
An O_X-module F is quasi-coherent if every point has an affine neighbourhood U=Spec A on which F|_U≈M~ for some A-module M.
Coherent sheaves add finite-generation conditions when the ambient scheme is suitably Noetherian.
Quasi-coherent sheaves are the natural algebraic analogue of fields of module data. They behave well under many scheme operations and include ideal sheaves, structure sheaves of closed subschemes and vector bundles.
12. Affine schemes are unusually cohomologically simple
For X=Spec A and a quasi-coherent sheaf M~, global sections recover M:
Γ(X,M~)≈M.
More strongly, for quasi-coherent sheaves on affine schemes, higher sheaf cohomology vanishes:
H^i(X,F)=0 for i>0
under the standard scheme-theoretic hypotheses.
This is one reason affine schemes are the computational building blocks of algebraic geometry: local-to-global obstruction disappears for quasi-coherent sheaves on a single affine piece.
13. Global sections are left exact but not generally exact
Given a short exact sequence of sheaves
0→F’→F→F”→0,
taking global sections gives
0→Γ(F’)→Γ(F)→Γ(F”).
Injectivity is preserved, but surjectivity can fail: a global section of F” may have local lifts to F that cannot be glued into one global lift.
Cohomology measures this failure and its higher analogues.
14. Cohomology is derived local-to-global information
The sheaf cohomology groups H^i(X,F) are the right derived functors of global sections.
H^0(X,F)=Γ(X,F), the ordinary global sections.
H^1 often measures first gluing obstructions or extension data.
Higher H^i record deeper failures of global exactness.
The phrase “cohomology measures holes” is useful in topology but too narrow here. Algebraic sheaf cohomology measures failures of local data to globalise and packages extension phenomena, line-bundle sections, deformation information and duality structures.
15. Short exact sequences generate long exact cohomology sequences
From
0→F’→F→F”→0
one obtains the long exact sequence
0→H⁰(F’)→H⁰(F)→H⁰(F”)→H¹(F’)→H¹(F)→H¹(F”)→⋯
The connecting map H⁰(F”)→H¹(F’) sends a global section to its obstruction to lifting globally through F.
If that obstruction vanishes, the section lifts; if not, no global lift exists even if local lifts do.
16. Čech cohomology starts from an open cover
Choose an open cover U={U_i}. A 0-cochain is a collection of local sections s_i∈F(U_i).
A 1-cochain is a collection on pairwise overlaps U_i∩U_j.
The Čech differential records alternating differences on overlaps.
A 0-cocycle is exactly a compatible family of local sections, so the sheaf axiom identifies H⁰ with global sections.
Under suitable covers and hypotheses, Čech cohomology computes sheaf cohomology. In general one must state those hypotheses rather than identify the two automatically.
17. Worked gluing on P¹ uses two affine charts
Cover P¹ by U₀=Spec k[t] and U₁=Spec k[s], with s=t⁻¹ on the overlap.
A regular global function is a pair f(t)∈k[t], g(s)∈k[s] that agree after substituting s=t⁻¹ on k[t,t⁻¹].
The only such pairs are constants.
Hence Γ(P¹,O)=k.
This demonstrates how projective geometry can have far fewer global regular functions than affine geometry even though each chart separately has many.
18. Line bundles are locally free rank-one sheaves
An algebraic line bundle on X can be represented by a locally free O_X-module of rank one.
Locally it looks like O_X, but on overlaps the local trivializations differ by invertible functions.
These transition functions satisfy cocycle relations.
Isomorphism classes of line bundles form the Picard group Pic(X) under tensor product.
On P¹, the standard line bundles are O(d), indexed by integers d.
19. Sections of O(d) on P¹ are homogeneous polynomials
For d≥0, global sections of O(d) on P¹ correspond to homogeneous polynomials of degree d in two variables.
A basis is X^d, X^{d−1}Y, …, Y^d.
Therefore
dim_k H⁰(P¹,O(d))=d+1 for d≥0.
For d<0 there are no nonzero global sections.
This turns a geometric line-bundle question into a finite-dimensional polynomial count.
20. H¹ of O(d) on P¹ detects the negative range
For P¹ over a field,
H¹(P¹,O(d))=0 for d≥−1,
while for d≤−2,
dim_k H¹(P¹,O(d))=−d−1.
This can be computed using the standard two-chart Čech cover.
The complementary dimensions of H⁰ and H¹ are a first concrete appearance of duality and Riemann–Roch behaviour.
21. Divisors and line bundles encode zeros and poles globally
On a smooth curve, a divisor is a formal integer combination of closed points.
A rational function f determines a principal divisor div(f), recording orders of zeros and poles.
A divisor D determines a line bundle O(D), whose sections are rational functions constrained by D’s allowed poles.
Thus local valuation data at points become a global sheaf.
This bridge is central to algebraic curves, arithmetic geometry and the Riemann–Roch theorem.
22. Riemann–Roch balances sections, obstructions and geometry
For a smooth projective curve C of genus g and divisor D, Riemann–Roch can be written
ℓ(D)−ℓ(K−D)=deg D+1−g,
where ℓ(D)=dim H⁰(C,O(D)) and K is a canonical divisor.
Equivalently, using Serre duality, the H¹ term balances the shortage of global sections.
This guide does not prove Riemann–Roch; it uses it as a model of what cohomological bookkeeping achieves: local divisor data, global sections and topology-like invariants are constrained by one exact numerical relation.
23. Exact sequences compute unknown sheaf data from known pieces
Let Z⊂X be a closed subscheme with ideal sheaf I_Z. There is a short exact sequence
0→I_Z→O_X→O_Z→0.
After tensoring by a line bundle L,
0→I_Z⊗L→L→L|_Z→0.
The long exact cohomology sequence compares global sections on X with their restrictions to Z and records whether prescribed values on Z can be extended globally.
This pattern appears in interpolation, embedding theorems and deformation theory.
24. Cohomology distinguishes affine from projective behaviour
Affine schemes have vanishing higher cohomology for quasi-coherent sheaves, while projective schemes can have nontrivial higher cohomology.
That difference is not a defect. It is global information created by the way affine charts are glued.
A local computation on each chart can be completely simple while the overlap data still produce a nontrivial H¹ class.
Cohomology is therefore a controlled record of information that emerges only after local pieces are assembled globally.
25. Sheaf cohomology is not the same as singular cohomology
The topology guide Homology and Cohomology introduced chain-based topological invariants.
Sheaf cohomology is a more general local-to-global construction. Depending on the sheaf and space, it can recover or relate to familiar topological cohomology, but the input data are different.
For example, coherent sheaf cohomology on a projective algebraic variety measures sections and algebraic obstructions, while singular cohomology studies topological cycles.
Deep comparison theorems connect these worlds over C, but they should not be identified without specifying the theory.
26. Serre duality pairs complementary cohomology
For a smooth projective variety X of dimension n over a field, Serre duality relates H^i(X,F) to a dual of H^{n−i}(X,F^∨⊗ω_X) under standard coherence hypotheses.
On a smooth projective curve, H¹ of a line bundle is dual to H⁰ of a complementary bundle involving the canonical sheaf.
This explains why the “missing” sections in one degree reappear as dual cohomology in another.
Duality is one of the signatures that cohomology is not merely an error term: it is structured global information.
27. A dependable sheaf-and-cohomology workflow
Identify the space X and the sheaf F. State whether F is a sheaf of sets, groups, rings or O_X-modules.
Choose an affine or otherwise useful open cover. Compute sections and restrictions on each open and overlap.
Use stalks for local questions and quasi-coherent module descriptions on affine charts.
Use exact sequences to replace a hard sheaf with simpler neighbouring sheaves.
When global sections fail to be exact, identify the connecting map and the relevant cohomology group rather than saying only that “gluing failed”.
When using Čech cohomology, state why the chosen cover computes the intended sheaf cohomology.
28. Common misconceptions
A sheaf is just a function on open sets. It also has restriction maps satisfying functoriality and the locality/gluing axioms.
If data agree pairwise locally, they always globalise in any presheaf. That is exactly what can fail before the sheaf condition is imposed.
H¹ means a literal one-dimensional hole. Not generally. Its meaning depends on the sheaf and the geometric problem.
Čech cohomology always equals sheaf cohomology. It does under suitable hypotheses and covers, not automatically in complete generality.
Projective varieties have no useful functions, so sheaves add nothing. Line bundles and coherent sheaves carry rich section and cohomology data even when global regular functions are only constants.
29. Independent practice: twenty questions
- State the restriction compatibility required of a presheaf.
- State the two sheaf axioms in local language.
- Why do continuous functions form a sheaf?
- What is O_X(D(f)) for X=Spec A?
- What is the stalk O_{X,p}?
- Describe the stalk at (t−a) on A¹.
- What does sheafification preserve?
- For an A-module M, what is M~(D(f))?
- What does quasi-coherent mean locally?
- What are Γ(Spec A,M~)?
- What happens to higher quasi-coherent cohomology on affine schemes?
- Why is Γ left exact but not generally exact?
- What does the connecting map H⁰(F”)→H¹(F’) measure?
- What is a Čech 0-cocycle?
- Compute Γ(P¹,O).
- What is dim H⁰(P¹,O(d)) for d≥0?
- When is H¹(P¹,O(d)) nonzero?
- What group classifies line bundles up to isomorphism?
- State the ideal-sheaf exact sequence for a closed subscheme Z⊂X.
- Explain one difference between singular cohomology and coherent sheaf cohomology.
30. Worked solutions and checks
1. Restriction U→U is identity, and U→V→W composes to the direct restriction U→W.
2. Compatible local sections on a cover glue to one global section, and that global section is unique.
3. Continuous functions agreeing on overlaps define one pointwise function, and continuity is local.
4. A_f.
5. A_p.
6. k[t]_(t−a), fractions whose denominators do not vanish at a.
7. Stalks.
8. M_f.
9. On each affine open Spec A it is isomorphic to M~ for some A-module M.
10. M.
11. H^i=0 for i>0 for quasi-coherent sheaves on affine schemes under the standard hypotheses.
12. Injectivity survives, but a global section of the quotient sheaf may have only local lifts and no compatible global lift.
13. The obstruction to globally lifting a section of F” through F.
14. A family of local sections agreeing on pairwise overlaps.
15. k: all global regular functions on P¹ are constant.
16. d+1.
17. For d≤−2; its dimension is −d−1.
18. Pic(X).
19. 0→I_Z→O_X→O_Z→0.
20. Singular cohomology is built from topological chains; coherent sheaf cohomology derives global sections of algebraic sheaves and measures algebraic local-to-global phenomena.
31. Where this guide hands off
R24.07 now turns the scheme-and-sheaf language toward arithmetic: rational points, integral models, reduction modulo primes, local fields, elliptic curves and the local–global questions that connect geometry to number theory.
R24.08 then makes selected parts computational through Gröbner bases, elimination, Hilbert functions, resultants and enumerative intersection methods.
Sources and further study
Use the Stacks Project for sheaves, modules on sites and schemes, cohomology and derived methods; MIT 18.726 Algebraic Geometry for graduate scheme and sheaf foundations; and the BTT topology guide for the distinct chain-based cohomology route.