Differentiable Manifolds/De Rham cohomology

Proposition (the differentiable forms of a differentiable manifold and the Cartan derivative constitute a cochain complex):

Let $M$ be a differentiable manifold of class ${\mathcal {C}}^{k}$. Then the diagram

$0\longrightarrow \Omega ^{0}(M){\overset {d}{\longrightarrow }}\Omega ^{1}(M){\overset {d}{\longrightarrow }}\Omega ^{2}(M){\overset {d}{\longrightarrow }}\cdots$

constitutes a chain complex of modules over ${\mathcal {C}}^{k}(M)$, where $d$ shall denote the Cartan derivative.

Proof: This follows immediately from the fact that applying the Cartan derivative twice always yields zero. $\Box$

Definition (de Rham cohomology):

Let $M$ be a differentiable manifold of class ${\mathcal {C}}^{k}$. The cohomology arising from the chain complex

$0\longrightarrow \Omega ^{0}(M){\overset {d}{\longrightarrow }}\Omega ^{1}(M){\overset {d}{\longrightarrow }}\Omega ^{2}(M){\overset {d}{\longrightarrow }}\cdots$

is called de Rham cohomology. The $k$-th ${\mathcal {C}}^{k}(M)$-module of this cohomology is commonly denoted $H_{\text{dR}}^{k}(M)$.