Proposition (the differentiable forms of a differentiable manifold and the Cartan derivative constitute a cochain complex):
Let be a differentiable manifold of class . Then the diagram
constitutes a chain complex of modules over , where shall denote the Cartan derivative.
Proof: This follows immediately from the fact that applying the Cartan derivative twice always yields zero.
Definition (de Rham cohomology):
Let be a differentiable manifold of class . The cohomology arising from the chain complex
is called de Rham cohomology. The -th -module of this cohomology is commonly denoted .