In classical electrodynamics on a simply connected space, equation $dF=0$
is equivalent to the existence of an electrodynamic potential (standard
Poincaré lemma). From a mathematical point of view, de Rham complex
(algebra of total antisymmetric tensor fields with operator $d$
fulfilling $d^2=0$) is the foundation of classical electrodynamics. In
other tensor field theories we work with a field T of more complicated
symmetry and one of the field equations is $DT=0$, where $D$ is some first
order differential operator. In this talk I will define an
$N$-complex (algebra of tensor fields of given Young symmetry with
operator $D$ fulfilling $D^N=0$) and a generalized Poincaré lemma, which is
the reason for the existence of a field potential in every integer-spin
field theory.