Mod-p Poincare Duality in p-adic analytic geometry

We show Poincare<acute accent> Duality for Fp-e<acute accent>tale cohomology of a smooth proper rigid-analytic space over a non-archimedean field K of mixed characteristic (0, p). It positively answers the question raised by P. Scholze in his paper "p-adic Hodge theory for rigid-analytic varieties." We prove duality via constructing Faltings' trace map relating Poincare<acute accent> Duality on the generic fiber to (almost) Grothendieck Duality on the mod-p fiber of a formal model. We also formally deduce Poincare<acute accent> Duality for Z/pnZ, Zp, and Qp-coefficients.