nLab
de Rham complex

The de Rham complex of a manifold is the cochain complex which in degree $n\in \mathbb{N}$ has the vector space ${\Omega }^n(X)$ of degree-$n$ differential forms on $X$. 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 $TX$ of $X$.