Exact forms are closed, so the vector spaces of *k*-forms along with the exterior derivative are a cochain complex. The vector spaces of closed forms modulo exact forms are called the **de Rham cohomology groups**. The wedge product endows the direct sum of these groups with a ring structure.

**De Rham's theorem**, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold *M*, these groups are isomorphic as real vector spaces with the singular cohomology groups *H ^{p}*(

The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.