p-adic Cohomology

The purpose of this paper is to survey some recent results in the theory of “p-adic cohomology”, by which we will mean several different (but related) things: the de Rham or p-adic étale cohomology of varieties over p-adic fields, or the rigid cohomology of varieties over fields of characteristic p > 0. Our goal is to update Illusie’s beautiful 1994 survey [I] by reporting on some of the many interesting results that postdate it. In particular, we concentrate more on the present and near-present than the past; [I] provides a much better historical background than we can aspire to, and is a more advisable starting point for newcomers. Before beginning, it is worth pointing out one (but not the only) crucial reason why so much progress has been made since the appearance of [I]. In the mid-1990s, it suddenly became possible to circumvent the resolution of singularities problem in positive characteristic thanks to de Jong’s alterations theorems [dJ1], which provide forms of “weak resolution” and “weak semistable reduction”. (The weakness is the introduction of an unwanted but often relatively harmless finite extension of the function field: for instance, an alteration is a morphism which is proper, dominant, and generically finite rather than generically an isomorphism.) The importance of de Jong’s results cannot be overstated; they underlie almost every geometric argument cited in this paper! See [Brt6] for more context regarding alterations.