Mod $p$ Poincaré duality for $p$-adic period domains

In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a $p$-adic field that satisfy Poincaré duality for étale cohomology with mod $p$-coefficients : the varieties satisfying “primitive comparison with compact support”. We show that almost proper varieties, as well as p-adic (weakly admissible) period domains in the sense of Rappoport-Zink belong to this class. In particular, we recover Poincaré duality for almost proper varieties as first established by Li-Reinecke-Zavyalov, and we compute the étale cohomology with ${\mathbb{F}}_p$-coefficients of p-adic period domains, generalizing a computation of Colmez-Dospinescu-Niziol for Drinfeld’s symmetric spaces. The arguments used in this paper rely crucially on Mann’s six functors formalism for solid ${\mathcal{O}}^{+,a}\mathrm{/}\pi$ coefficients.