16 - Frobenius modules over the Robba ring

Summary

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 p-adic completion of the bounded Robba ring; 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 the bounded Robba ring itself. This is a bit too much to ask for; what we can provide is a structure theory that applies over the Robba ring, which is somewhat analogous to what we obtain over the p-adic completion. In particular, with an appropriate definition of pure modules, we obtain a slope filtration theorem over the Robba ring. Given a difference module over the bounded Robba rings, one gets slope filtrations and Newton polygons over both the p-adic completion and the Robba ring; these need not coincide, but they do admit a specialization property.