  We develop tensor product finite element cochain complexes of arbitrary
smoothness on Cartesian meshes of arbitrary dimension. The first step is the
construction of a one-dimensional $C^m$-conforming finite element cochain
complex based on a modified Hermite interpolation operator, which is proved to
commute with the exterior derivative by means of a general commutation lemma.
Adhering to a strict tensor product construction we then derive finite element
complexes in higher dimensions.
