In Chapter 14 we discussed some structure theory for finite difference modules over a complete isometric nonarchimedean difference field. This theory can be applied to the field ℇ, which is the p-adic completion of the bounded Robba ring ℇ±; however, the information it gives is somewhat limited.

For the purposes of studying Frobenius structures on differential modules (see Part V), it would be useful to have a structure theory over ℇ± itself. This is a bit too much to ask for; what we can provide is a structure theory that applies over the Robba ring ℛ, which is somewhat analogous to what we obtain over ℇ. In particular, with an appropriate definition of pure modules, we obtain a slope filtration theorem analogous to Theorem 14.4.15 but valid over ℛ.

Given a difference module over ℇ±, one obtains slope filtrations and Newton polygons over both ℇ and ℛ. For a module over K〚t〛0 these turn out to match the generic and special Newton polygons, and so in particular they need not coincide. However, they do admit a specialization property analogous to Theorem 15.3.2.

Unfortunately, a proof of the slope filtration theorem over ℛ would take us rather far afield, so we do not include one here. Instead, we limit ourselves to a brief overview of the proof and consign further discussion and references to the notes.

Hypothesis 16.0.1. Throughout this chapter, let ϕ be a Frobenius lift on the Robba ring ℛ.