TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Short exact mayer vietoris sequence of de rham complexes

Statement

Under countable choice, $0 \to Ω^{∙} (M) \overset{r}{\to} Ω^{∙} (U) \oplus Ω^{∙} (V) \overset{s}{\to} Ω^{∙} (U \cap V) \to 0$ is short exact as a sequence of real cochain complexes.

Facts & Assumptions

Given: An open cover $M = U \cup V$ and countable choice.

[F1]

The de rham mayer vietoris sequence is exact at the first two terms: The sequence $0 \to Ω^{k} (M) \overset{r}{\to} Ω^{k} (U) \oplus Ω^{k} (V) \overset{s}{\to} Ω^{k} (U \cap V)$ is exact at the first two nonzero terms.

[F2]

The de rham mayer vietoris difference map is surjective: Assume countable choice. The difference map $s : Ω^{k} (U) \oplus Ω^{k} (V) \to Ω^{k} (U \cap V)$ is surjective in every degree.

[F3]

Short exact sequence of complexes: A short exact sequence of complexes is a sequence of chain maps $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ that is exact in each degree as a sequence in the ambient abelian category.

Proof

technique · direct

1.1

In every degree, the first lemma gives injectivity of $r$ and $\ker s = im r$ , and the second gives surjectivity of $s$ . Hence each degree is a short exact sequence of real vector spaces.

F1F2given

2.1

The maps $r, s$ commute with the differentials because they are the restriction cochain maps of those lemmas. Reindexing by $C_{n} = C^{- n}$ turns this into a sequence of chain maps exact in every degree, precisely the definition of a short exact sequence of complexes. Thus it is the claimed cochain version, including zero terms.

F3step 1.1∎

Source locator

Lee, Theorem 17.20, pp.449–450, and its full proof pp.462–463. This page reverses Lee’s difference convention consistently: $s (α, β) = β - α$ .

Depends on

Used by

Explicit de rham mayer vietoris connecting class Proposition
Mayer vietoris sequence in de rham cohomology Theorem

Dependency tree · two levels

9 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)