Slope filtrations for relative Frobenius

TL;DR

This paper provides a new, simplified proof of the slope filtration theorem for Frobenius-semilinear endomorphisms over the Robba ring, extending it to more general coefficient actions relevant to p-adic Galois representations.

Contribution

It introduces a third-generation proof using faithfully flat descent and extends the theorem to arbitrary coefficient actions, broadening its applicability.

Findings

Simplified proof of the slope filtration theorem

Extension to arbitrary coefficient actions

Relevance to (phi, Gamma)-modules and p-adic Galois representations

Abstract

The slope filtration theorem gives a partial analogue of the eigenspace decomposition of a linear transformation, for a Frobenius-semilinear endomorphism of a finite free module over the Robba ring (the ring of germs of rigid analytic functions on an unspecified open annulus of outer radius 1) over a discretely valued field. In this paper, we give a third-generation proof of this theorem, which both introduces some new simplifications (particularly the use of faithfully flat descent, to recover the theorem from a classification theorem of Dieudonne-Manin type) and extends the result to allow an arbitrary action on coefficients (previously the action on coefficients had to itself be a lift of an absolute Frobenius). This extension is relevant to a study of (phi, Gamma)-modules associated to families of p-adic Galois representations, presently being initiated by Berger and Colmez.