Local BRST cohomology for AKSZ field theories: a global approach I

We study the Lagrangian antifield BRST formalism, formulated in terms of exterior horizontal forms on the infinite order jet space of graded fields for topological field theories associated to $Q$-bundles. In the case of a trivial Q-bundle with a flat fiber and arbitrary base, we prove that the BRST cohomology are isomorphic to the cohomology of the target space differential "twisted" by the de Rham cohomology of the base manifold. This generalizes the local result of G. Barnich and M. Grigoriev, computed for a flat base manifold.