On the homotopy invariance of the twisted Lie algebroid cohomology

Twisted Lie algebroid cohomologies, i.e. with values in representations, are shown to be Lie algebroid homotopy-invariant. Several important classes of examples are discussed. As an application, a generalized version of the Poincaré lemma is given in the transitive case. Together with the Mayer-Vietoris theorem, which holds in this more general context as well, this leads to a Künneth formula for the cohomologies of Lie algebroids with values in representations. In particular, this comprehensive paper gives a systematic way to compute explicitly the twisted Lie algebroid cohomologies of some transitive and almost transitive Lie algebroids.