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

1.1F1F2given

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.

2.1F3step 1.1∎

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.

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

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, second edition