Book contents
- Frontmatter
- Contents
- Preface
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- 14 Formalism of difference algebra
- 15 Frobenius modules
- 16 Frobenius modules over the Robba ring
- Part V Frobenius Structures
- Part VI Areas of Application
- References
- Notation
- Index
16 - Frobenius modules over the Robba ring
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- 14 Formalism of difference algebra
- 15 Frobenius modules
- 16 Frobenius modules over the Robba ring
- Part V Frobenius Structures
- Part VI Areas of Application
- References
- Notation
- Index
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 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.
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- p-adic Differential Equations , pp. 273 - 288Publisher: Cambridge University PressPrint publication year: 2010