Differentiable Manifolds/De Rham cohomology

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

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

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

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

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

Definition (de Rham cohomology):

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

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

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