COMPLEX GEOMETRY Course notes

Chapter 4
SHEAF COHOMOLOGY
4.1 Sheaves
Definition 4.1.1. Let X be a topological space and A an abelian category. A presheaf F on X is a
collection of objects F(U) of objects in A, for each open subset U ⊆ X, and a collection of morphisms
ρ UV : F(U) −→ F(V )
σ ↦→ σ| V = ρ UV (σ)
for each inclusion of open subsets V ↩→ U such that
ρ UV = ρ V W ◦ ρ UV .
The last equality is known as compatibility.
A presheaf F is called a sheaf if it is saturated, i.e., if it satisfies the following condition: Let s i ∈ F(U i )
be a collection of sections such that
s i | Uij = s j | Uij ,
where U ij = U i ∩ U j , then there exists a unique section s ∈ F(∪U i ) such that s| Ui = s i .
Definition 4.1.2. A morphism of (pre)sheaves is a map ϕ : F −→ G which associates to each open
subset U ⊆ X a morphism
such that for every V ⊆ U open
ϕ U : F(U) −→ G(U)
ρ UV ◦ ϕ U = ϕ V ◦ ρ UV . (compatibility)
Example 4.1.1. Sheaves of sections of vector bundles (C ∞ , C h , C ω for real analytic, O, etc).
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