Suppose W is the Witt ring of a perfect field k of characteristic p, and X is a smooth k-scheme. The Frobenius automorphism of W is a PD morphism, covered by the absolute Frobenius endomorphism F_{X}of X, and it follows that F_{X}acts on the crystalline cohomology of X relative to W. In this chapter we shall study this action, in particular, its relationship to the Hodge filtration on crystalline cohomology (as determined from the ideal J_{X/S}). The main global applications are Mazur's theorem (8.26), which says that (with suitable hypotheses on X) the action of Frobenius determines the