In Part III we focus our attention specifically on p-adic ordinary differential equations (although most of our results apply also to complete nonarchimedean fields of residual characteristic 0). To do this with maximal generality, one would need first to introduce a category of geometric spaces over which to work. This would require a fair bit of discussion of either rigid analytic geometry, in the manner of Tate, or nonarchimedean analytic geometry in the manner of Berkovich, neither of which we want either to assume or introduce. Fortunately, since we only need to consider one-dimensional spaces, we can manage by working completely algebraically and considering differential modules over appropriate rings.

In this chapter, we introduce those rings and collect their basic algebraic properties. This includes the fact that they carry Newton polygons analogous to those for polynomials. Another key fact is that there is a form of the approximation lemma (Lemma 1.3.7) valid over some of these rings.

Notation 8.0.1. Throughout this part, let K be a field of characteristic 0 that is complete for a nontrivial nonarchimedean norm | · |. (The assumption of characteristic 0 is not used in this chapter; it will become crucial when we start discussing differential modules again.) Let p denote the characteristic of the residue field kK. We do not assume p > 0 (as the case p = 0 may be useful for some applications), but when p > 0 we do require the norm to be normalized in such a way that |p| = p−1.