Differentiable Manifolds/De Rham cohomology

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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 .