We construct N-complexes of non-completely antisymmetric irreducible tensor fields on R-D which generalize the usual complex (N = 2) of differential forms. Although, for N greater than or equal to 3, the generalized cohomology of these N-complexes is nontrivial, we prove a generalization of the Poincare 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 Poincare lemma. We furthermore identify the nontrivial part of the generalized cohomology. Many of the results presented here were announced in [10]. [References: 25]
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