  We construct $N$-complexes of non completely antisymmetric irreducible tensor
fields on $\mathbb R^D$ which generalize the usual complex $(N=2)$ of
differential forms. Although, for $N\geq 3$, the generalized cohomology of
these $N$-complexes is non trivial, we prove a generalization of the Poincar\'e
lemma. To that end we use a technique reminiscent of the Green ansatz for
parastatistics. Several results which appeared in various contexts are shown to
be particular cases of this generalized Poincar\'e lemma. We furthermore
identify the nontrivial part of the generalized cohomology. Many of the results
presented here were announced in [10].
