A Poincaré lemma for sigma models of AKSZ type

For a sigma model of AKSZ type, we show that the local BRST cohomology is isomorphic to the cohomology of the target space differential when restricted to coordinate neighborhoods both in the base and in the target. An analogous result is shown to hold for the cohomology in the space of functional multivectors. Applications of these latter cohomology classes in the context of the inverse problem of the calculus of variation for general gauge systems are also discussed.