Formula omitted]-isocrystals and homotopy types

Abstract

We study a positive characteristic analogue of the nonabelian Hodge structure constructed by Katzarkov, Pantev, and Toen on the homotopy type of a complex algebraic variety. Given a proper smooth scheme X over a perfect field of characteristic p and a Tannakian category C of isocrystals on X , we construct an object X C in a suitable homotopy category of simplicial presheaves whose category of local systems is equivalent to C in a manner compatible with cohomology. We then study F -isocrystal structure on these simplicial presheaves. As applications of the theory, we prove a p -adic analogue of a result of Hain on relative Malcev completions, a generalization to the level of homotopy types of a theorem of Katz relating p -adic étale local systems and F -isocrystals, as well as a p -adic version of the formality theorem in homotopy theory. We have also included a new proof based on reduction modulo p of the formality theorem for complex algebraic varieties.