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Čech H0 equals global sections

Statement

Let $X$ be a topological space, $F$ a sheaf of abelian groups on $X$ , and $U = (U_{i})_{i \in I}$ an open cover of $X$ indexed by a linearly ordered set, with fixed-cover Čech cohomology ${\overset{ˇ}{H}}^{∙} (U, F)$ (Fixed-cover Čech cohomology). Then restriction of global sections, $Γ (X, F) ⟶ {\overset{ˇ}{H}}^{0} (U, F), s ⟼ (s ∣_{U_{i}})_{i \in I},$ is an isomorphism, and it is natural in $F$ : for a morphism $φ : F \to G$ of abelian sheaves the square $\begin{array}{ccc} Γ (X, F) & ⟶ & {\overset{ˇ}{H}}^{0} (U, F) \\ ↓ Γ (X, φ) & ↓ {\overset{ˇ}{H}}^{0} (U, φ) \\ Γ (X, G) & ⟶ & {\overset{ˇ}{H}}^{0} (U, G) \end{array}$ commutes.

Facts & Assumptions

[F1]

$C^{0} (U, F) = \prod_{i \in I} F (U_{i})$ and $(δ^{0} s)_{i j} = s_{j} ∣_{U_{i} \cap U_{j}} - s_{i} ∣_{U_{i} \cap U_{j}}$ for $i < j$ (Ordered Čech cochain complex of a cover).

[F2]

A sheaf satisfies locality: if $s, t \in F (X)$ have $s ∣_{U_{i}} = t ∣_{U_{i}}$ for all $i$ in a cover, then $s = t$ ; and gluing: compatible sections $s_{i} \in F (U_{i})$ with $s_{i} ∣_{U_{i} \cap U_{j}} = s_{j} ∣_{U_{i} \cap U_{j}}$ have a section restricting to each $s_{i}$ (A sheaf on a topological space).

[F3]

${\overset{ˇ}{H}}^{0} (U, F) = \ker (δ^{0}) / im (δ^{- 1}) = \ker (δ^{0})$ , since $C^{- 1} (U, F) = 0$ (Fixed-cover Čech cohomology).

[F4]

$Γ (X, F) = F (X)$ and $Γ (X, φ) = φ_{X}$ , and a morphism of sheaves commutes with restrictions (Global sections of an abelian sheaf, Sections, restrictions, and global sections of a presheaf).

[F5]

A section over the empty open set is zero, $F (\emptyset) = 0$ (A set-valued sheaf has a unique section over the empty open set); in particular for the empty space and empty cover the product over no indices is the zero group.

Proof

Given: A topological space $X$ , a sheaf of abelian groups $F$ and an ordered open cover $U$ of $X$ .

1.1

Define $Φ : Γ (X, F) \to C^{0} (U, F) = \prod_{i \in I} F (U_{i})$ by $Φ (s) : = (s ∣_{U_{i}})_{i \in I}$ [F1, F4]. For $s \in Γ (X, F)$ and $i < j$ one has $(δ^{0} Φ (s))_{i j} = s ∣_{U_{j}} ∣_{U_{i} \cap U_{j}} - s ∣_{U_{i}} ∣_{U_{i} \cap U_{j}} = s ∣_{U_{i} \cap U_{j}} - s ∣_{U_{i} \cap U_{j}} = 0$ by compatibility of restrictions [F1, F4], so $Φ$ takes values in $\ker (δ^{0})$ ; it is a group homomorphism because restrictions are homomorphisms.

F1F4

2.1

$Φ$ is injective: if $Φ (s) = 0$ then $s ∣_{U_{i}} = 0$ for every $i$ , and since the $U_{i}$ cover $X$ the locality half of the sheaf condition [F2] gives $s = 0$ .

F2step 1.1

2.2

$Φ$ is surjective onto $\ker (δ^{0})$ : let $(s_{i})_{i \in I} \in \ker (δ^{0})$ . For $i < j$ the equation $(δ^{0} s)_{i j} = 0$ reads $s_{j} ∣_{U_{i} \cap U_{j}} = s_{i} ∣_{U_{i} \cap U_{j}}$ [F1], so the family is compatible on all pairwise intersections and the gluing half of the sheaf condition [F2] produces $s \in F (X)$ with $s ∣_{U_{i}} = s_{i}$ for all $i$ ; then $Φ (s) = (s_{i})$ by [step 1.1] and this section is unique by locality. (If some $U_{i}$ is empty its component group is $0$ by [F5], so the corresponding entry is forced to be zero and imposes no condition.) [F1, F2, F5, step 1.1]

F1F2F5

3.1

Hence $Φ$ is an isomorphism of abelian groups from $Γ (X, F)$ onto $\ker (δ^{0})$ , and by [F3] the latter is ${\overset{ˇ}{H}}^{0} (U, F)$ ; so the displayed restriction map is an isomorphism. For naturality, let $φ : F \to G$ be a morphism of abelian sheaves. Since $φ$ commutes with restrictions [F4], for every $s \in Γ (X, F)$ and every $i$ one has $φ_{U_{i}} (s ∣_{U_{i}}) = φ_{X} (s) ∣_{U_{i}}$ , so $Φ_{G} (φ_{X} (s)) = C^{0} (U, φ) (Φ_{F} (s))$ ; as ${\overset{ˇ}{H}}^{0} (U, φ)$ is the map induced on $\ker (δ^{0})$ [F1, F3], the square commutes and the isomorphism is natural in $F$ . [F3, F4, step 2.2] ∎

F3F4∎

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Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Cohomology of Sheaves (standard reference, not scraped)
Jiahui Gao and Shuwu Zhang, Lectures on Algebraic Geometry (standard reference, not scraped)