View source for Differentiable Manifolds/The Mayer–Vietoris sequences in (compact) de Rham cohomology

{{proposition|short exact sequence giving rise to the Mayer–Vietoris sequence is exact|

{{definition|Mayer–Vietoris sequence in de Rham cohomology|Let <math>M</math> be a smooth manifold, and let <math>U, V \subseteq M</math> be open such that <math>M = U \cup V</math>. Then the '''Mayer–Vietoris sequence in de Rham cohomology''' is the long exact sequence on cohomology associated to the short exact sequence on cochain complexes that on the <math>k</math>-th level is given by
:<math>\Omega^k(M) \longrightarrow \Omega^k(U) \oplus \Omega^k(V) \longrightarrow \Omega^k(U \cap V)</math>}}

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