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

$\Omega ^{k}(M)\longrightarrow \Omega ^{k}(U)\oplus \Omega ^{k}(V)\longrightarrow \Omega ^{k}(U\cap V)$