Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 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 The p-adic local monodromy theorem
- Part VII Global theory
- Appendix A Picard–Fuchs modules
- Appendix B Rigid cohomology
- Appendix C p-adic Hodge theory
- References
- Index of notation
- Subject index
15 - Frobenius modules
from Part IV - Difference Algebra and Frobenius Modules
Published online by Cambridge University Press: 06 August 2022
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 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 The p-adic local monodromy theorem
- Part VII Global theory
- Appendix A Picard–Fuchs modules
- Appendix B Rigid cohomology
- Appendix C p-adic Hodge theory
- References
- Index of notation
- Subject index
Summary
Having introduced the general formalism of difference algebra, and made a more careful study over a complete nonarchimedean field, we specialize to the sort of power series rings over which we studied differential algebra. Most of the rings are ones we have seen before, but we encounter a couple of new variations, notably the Robba ring. This ring consists of power series convergent on some annulus of outer radius 1, but with unspecified inner radius which may vary with the choice of the series. This may seem to be a strange construction at first, but it is rather natural from the point of view of difference algebra: the endomorphisms we will consider (Frobenius lifts) do not preserve the region of convergence of an individual series, but do act on the Robba ring as a whole. This chapter serves mostly to set definitions and notation for what follows. That said, one nontrivial result here is the behavior of the Newton polygon under specialization.
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- p-adic Differential Equations , pp. 281 - 293Publisher: Cambridge University PressPrint publication year: 2022