Cohomology of Lie Conformal Algebroids

We study Lie conformal algebroids (LCAd) and their representations using the language of lambda-brackets and Lie conformal algebras. We describe several general constructions, such as the LCAd of conformal derivations of a differential algebra A, the gauge LCAd associated to a differential algebra A, its module M, and an LCAd E, the current LCAd of a Lie algebroid F, and the LCAd structure of the space of Kähler differentials over a Poisson vertex algebra (PVA) . We develop the cohomology theories of LCAd and we relate them to the corresponding cohomology theories of PVA. In particular, we find an isomorphism between the cohomology of a PVA with coefficients in a module M and the corresponding cohomology of the LCAd with coefficients in the same module.