  We prove the hard Lefschetz property for pseudomanifolds and cycles in any
characteristic with respect to an appropriate Artinian reduction. The proof is
a combination of Adiprasito's biased pairing theory and a generalization of a
formula of Papadakis-Petrotou to arbitary characteristic. In particular, we
prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a
generalization of the g-conjecture due to Stanley. We also provide a simplified
presentation of the characteristic 2 case of the Papadakis-Petrotou formula,
and generalize it to pseudomanifolds and cycles.
