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Ordinary sheaf cohomology on Euclidean open sets vanishes above real dimension

Physical scope

A bound for ordinary derived sheaf cohomology on Euclidean open sets; no physical model or empirical claim is involved.

Statement

Assume full AC. Let $m \geq 1$ , let $Ω \subseteq R^{m}$ be a nonempty open subset with its Euclidean metric, and let $G$ be any sheaf of abelian groups on $Ω$ . Then ordinary derived-functor sheaf cohomology satisfies $H^{q} (Ω, G) = 0 for every integer q > m .$ This is an ordinary cohomology bound for arbitrary sheaves; it is not a compact-support cohomology assertion.

Statement references

Agol, §6, Theorem 6.5, p. 29

Demailly, Chapter IX §4.B, Lemma 4.4 proof, p. 421

Facts & Assumptions

Given: Full AC, $m \geq 1$ , a nonempty Euclidean open set $Ω \subseteq R^{m}$ , and a sheaf of abelian groups $G$ on $Ω$ .

[F1]

The Axiom of Choice and Sheaf cohomology as right derived global sections: full AC is assumed, and $H^{q} (Ω, G)$ denotes the right derived global-sections cohomology.

[F2]

Ordered Čech cochain complex of a cover: for an ordered open cover, $C^{p}$ is the product of section groups over increasing $(p + 1)$ -fold intersections; an empty intersection contributes the zero group.

[F3]

Fixed-cover Čech cohomology and Refinement-colimit Čech cohomology: fixed-cover Čech cohomology is cohomology of $C^{∙}$ , and the refinement-limit Čech group is the colimit over all open covers.

[F4]

Locally finite refinements of Euclidean open covers have order at most the dimension plus one: every open cover of $Ω$ has a locally finite open refinement of order at most $m + 1$ .

[F5]

Refinement-limit Čech cohomology computes derived sheaf cohomology on metric spaces: on the metric space $Ω$ , refinement-limit Čech cohomology is naturally isomorphic to derived sheaf cohomology.

Proof

technique · direct

1.1F2F4given

Let $U$ be any open cover of $Ω$ . By [F4] it has a refinement $V$ in which every point lies in at most $m + 1$ members. For $q > m$ , an increasing $(q + 1)$ -tuple has at least $m + 2$ distinct members, so its intersection is empty; therefore every factor of $C^{q} (V, G)$ is the zero group by [F2], and ${\overset{ˇ}{H}}^{q} (V, G) = 0$ .

2.1F3step 1.1

Since every cover has such a refinement, the colimit over all covers in [F3] is zero in degree $q > m$ : every representative becomes zero after refinement.

3.1F1F5step 2.1∎

The comparison isomorphism [F5] identifies this zero colimit with $H^{q} (Ω, G)$ , proving the ordinary sheaf cohomology bound. No compact-support comparison is invoked.

Ordinary sheaf cohomology on Euclidean open sets vanishes above real dimension

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources