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

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