Onp-adic monodromy

Abstract

Every scheme X carries a sheaf r of generalized Witt vectors. In w we study the cohomology groups H"(X, "WCx) under the reasonable hypothesis (2.4). Implicitly this is a study of the formal groups of Artin and Mazur [1], but for our purpose there is no need to make these formal groups more explicit. In w we consider after base changing X/A to a p-adic situation X | R/R the cohomology of the sheaf of p-typical Witt vectors H " ( X | R, ~r174 The crucial, and fairly restrictive, hypothesis in this section requires that the Frobenius operator Fp acts bijectively on the fibers H"(X~, ~(5'x ) at the geometric points s of Spec(R/pR). We show that after further base changing to R at, the p-adic completion of an infinite &ale extension of the ring R, the ~/r H"(X | R at, "r174 ) has a basis _~ consisting of elements which are fixed by Fp. Let A:= k e r ( F p 1). Then A is a free 2gp-module with basis ~. Projection of Witt vectors onto their first coordinate induces an injection of A into H"(X, (gx) | R at and in fact