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 $$M$$ be a smooth manifold, and let $$U, V \subseteq M$$ be open such that $$M = U \cup V$$. 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 $$k$$-th level is given by
 * $$\Omega^k(M) \longrightarrow \Omega^k(U) \oplus \Omega^k(V) \longrightarrow \Omega^k(U \cap V)$$}}

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