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Slope filtrations for relative Frobenius. (English) Zbl 1168.11053
Berger, Laurent (ed.) et al., Représentation \(p\)-adiques de groupes \(p\)-adiques I. Représentations galoisiennes et \((\varphi, \Gamma)\)-modules. Paris: Société Mathématique de France (ISBN 978-2-85629-256-3/pbk). Astérisque 319, 259-301 (2008).
Author’s summary: 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 Dieudonné-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.
For the entire collection see [Zbl 1156.14002].

MSC:
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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