# Differentiable Manifolds/De Rham cohomology

**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 .