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Degree-zero sheaf cohomology is global sections

Statement

Assume the Axiom of Choice, let $X$ be a topological space, and let $H^{q} (X, -)$ be sheaf cohomology computed from the supplied functorial injective resolution datum $I$ on $A b (X)$ (Sheaf cohomology as right derived global sections). Then for every abelian sheaf $F$ on $X$ there is a canonical isomorphism $H^{0} (X, F) \tilde{\to} Γ (X, F),$ natural in $F$ ; it identifies $H^{0} (X, F)$ with the kernel of $Γ (X, I^{0} (F)) \to Γ (X, I^{1} (F))$ .

Facts & Assumptions

[F1]

$H^{0} (X, F) = R_{I}^{0} Γ (X, F)$ is the zeroth right derived object of $Γ (X, -)$ relative to the supplied datum $I$ , and $H^{0} (X, -) : = R_{I}^{0} Γ (X, -)$ is functorial in $F$ (Sheaf cohomology as right derived global sections).

[F2]

For an additive left exact functor $F$ and a supplied injective resolution datum $I$ on a class $D$ , every $A \in D$ carries a canonical isomorphism $R_{I}^{0} F (A) \to F (A)$ , natural in $A$ (The zero-th right derived functor of a left exact functor recovers the functor).

[F3]

$Γ (X, -)$ is additive and left exact (Global sections of an abelian sheaf).

[F4]

AC implies DC in ZF (AC implies DC implies countable choice).

[F5]

$I$ assigns to every abelian sheaf on $X$ a specific injective resolution, and $A b (X)$ has enough injectives (Enough injective abelian sheaves).

Proof

Given: The Axiom of Choice, a topological space $X$ and an abelian sheaf $F$ on $X$ .

1.1F4given

By [F4] the Axiom of Choice gives the Axiom of Dependent Choice in ZF, which is the hypothesis under which [F2] and the comparison theorems for right derived functors are stated.

1.2F3F5

By [F3] the functor $Γ (X, -)$ is additive and left exact, and by [F5] the supplied datum $I$ assigns a specific injective resolution to every abelian sheaf on $X$ , so $F$ lies in the domain of $I$ . Hence the hypotheses of [F2] are met by $F = Γ (X, -)$ , the datum $I$ and the object $F$ .

2.1F1F2

Applying [F2] gives a canonical isomorphism $H^{0} (X, F) = R_{I}^{0} Γ (X, F) \tilde{\to} Γ (X, F)$ , using the identification $H^{0} = R_{I}^{0} Γ$ of [F1]; the isomorphism is natural in $F$ because [F2] provides a natural isomorphism of functors. [F1, F2, step 1.2]

3.1F1F3∎

Spelling out the derived object, $H^{0} (X, F)$ is the zeroth cohomology of the complex $Γ (X, I^{0} (F)) \to Γ (X, I^{1} (F)) \to \dots$ [F1], that is the kernel of $Γ (X, I^{0} (F)) \to Γ (X, I^{1} (F))$ ; the isomorphism of step 2.1 is the composite of the canonical map $Γ (X, F) \to \ker (Γ (X, I^{0} (F)) \to Γ (X, I^{1} (F)))$ coming from the exactness of $0 \to Γ (X, F) \to Γ (X, I^{0} (F)) \to Γ (X, I^{1} (F))$ with its inverse, so the identification with the kernel is the canonical one. [F1, F3, step 2.1] ∎

Degree-zero sheaf cohomology is global sections

Statement

Facts & Assumptions

Proof

Depends on

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