The Primitive Comparison Theorem in characteristic p

We prove an analogue of Scholze’s Primitive Comparison Theorem for proper rigid spaces over an algebraically closed non-archimedean field K of characteristic p. This implies a v-topological version of the Primitive Comparison Theorem for proper finite type morphisms \(f{:}\,X\rightarrow Y\) $f:X\to Y$ of analytic adic spaces over \(\mathbb {Z}_p\) ${\mathbb{Z}}_p$ . We deduce new cases of the Proper Base Change Theorem for p-torsion coefficients and the Künneth formula in this setting.