The de Rham complex of a manifold is the cochain complex which in degree has the vector space of degree- differential forms on . The coboundary map is the deRham exterior derivative. The cohomology of the de Rham complex is de Rham cohomology. De Rham cohomology has a rather subtle generalization for possibly singular varieties due to Grothendieck.
Under the wedge product, the deRham complex becomes a differential graded algebra. This may be regarded as the Chevalley–Eilenberg algebra of the tangent Lie algebroid of .