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Twisted Poincaré duality between Poisson homology and Poisson cohomology

A version of the twisted Poincaré duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure in a canonical way, which is constructed from the modular derivation. In the case that the Poisson structure is unimodular, the twisted Poincaré duality reduces to the Poincaré duality in the usual sense. The main result generalizes the work of Launois-Richard \cite{LR} for the quadratic Poisson structures and Zhu \cite{Zhu} for the linear Poisson structures.