We construct N-complexes of non completely antisymmetric irreducible tensorfields on $\mathbb R^D$ generalizing thereby the usual complex (N=2) ofdifferential forms. These complexes arise naturally in the description ofhigher spin gauge fields. Although, for $N\geq 3$, the generalized cohomologyof these N-complexes is non trivial, we prove a generalization of thePoincar\'e lemma. Several results which appeared in various contexts are shownto be particular cases of this generalized Poincar\'e lemma.
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机译：我们在$ \ mathbb R ^ D $上构造非完全反对称不可约张量场的N个复合物，从而推广了通常的微分形式的复合物（N = 2）。这些配合物在较高自旋规场的描述中自然而然地出现。尽管对于$ N \ geq 3 $，这些N络合物的广义同构是不平凡的，但我们证明了Poincar'e引理的广义。在各种情况下出现的一些结果表明是这种广义庞加莱引理的特殊情况。
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