If S is a scheme of characteristic p, we define an F-zip over S to be
a vector bundle with two filtrations plus a collection of semi-linear
isomorphisms between the graded pieces of the filtrations. For every smooth
proper morphism XβS satisfying certain conditions the de Rham bundles
HdRnβ(X/S) have a natural structure of an F-zip. We give a
complete classification of F-zips over an algebraically closed field by
studying a semi-linear variant of a variety that appears in recent work of
Lusztig. For every F-zip over S our methods give a scheme-theoretic
stratification of S. If the F-zip is associated to an abelian scheme over
S the underlying topological stratification is the Ekedahl-Oort
stratification. We conclude the paper with a discussion of several examples
such as good reductions of Shimura varieties of PEL type and K3-surfaces.Comment: 35 pages, minor changes in exposition, major changes to introductio