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

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

Definition (Mayer–Vietoris sequence in de Rham cohomology):

Let be a smooth manifold, and let be open such that . 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 -th level is given by