The Primitive Comparison Theorem in characteristic p

Abstract 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 $$f{:}\,X\rightarrow Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mspace/> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> of analytic adic spaces over $$\mathbb {Z}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> . We deduce new cases of the Proper Base Change Theorem for p -torsion coefficients and the Künneth formula in this setting.