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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 $X$ .

1.1

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.

F4given

1.2

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 $F$ .

F3F5

2.1

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]

F1F2

3.1

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] ∎

F1F3∎

Depends on

Used by

A global section of the quotient that does not lift, and its nonzero connecting class Counterexample
Cohomological dimension relative to a sheaf class Definition
Two-affine Mayer–Vietoris on the projective line Example
Acyclic directions of the Čech–Godement double complex Lemma
Cofinal Čech vanishing implies derived acyclicity Lemma
Filtered colimits and sheaf cohomology on Noetherian spaces Lemma
Sheaf cohomology classes as derived morphisms Lemma
Variance of sheaf cohomology Lemma
Fixed-cover Čech can miss derived cohomology Remark
A point has no higher sheaf cohomology Theorem
Canonical map from fixed-cover Čech to sheaf cohomology Theorem
Cohomology of a finite disjoint union Theorem
Flasque abelian sheaves are Γ-acyclic Theorem
Mayer–Vietoris sequence for sheaf cohomology Theorem

Dependency tree · two levels

28 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)