Mod-p Poincaré Duality in p-adic Analytic Geometry

Abstract

We show Poincaré Duality for ${\mathbf{F}}_p$-étale 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 [Sch13a]. We prove duality via constructing Faltings' trace map relating Poincaré Duality on the generic fiber to (almost) Grothendieck Duality on the mod-$p$ fiber of a formal model. We also formally deduce Poincaré Duality for $\mathbf{Z}/p^n\mathbf{Z}$, ${\mathbf{Z}}_p$, and ${\mathbf{Q}}_p$-coefficients.

Major revision. Comments are very welcome!