A simple proof of Dieudonné-Manin classification theorem
type: Item
instance of: scholarly article
title: A simple proof of Dieudonné-Manin classification theorem
zbMATH Open document ID: 1273.14046
author: Yi Wen Ding
author: Yi Ouyang
publication date: 2012-11-07T00:00:00Z
review text: This article offers another proof of Dieudonné-Manin classification of \(\phi\)-modules, i.e. every \(\phi\)-module is a direct sum of pure \(\phi\)-modules of various slope. The authors define a left continuous decreasing filtration on \(\phi\)-modules which allows the authors to capture subquotients of various slopes. More precisely, if \(D\) is a \(\phi\)-module then \(D\) has a filtration \(D^{\mu}\) such that \(D^{\mu}/D^{>\mu}\) is a pure \(\phi\)-module of slope \(\mu\). Moreover, this filtration behaves well with respect to exact sequences. Using the jumps in the above filtration, the authors obtain a finite length increasing filtration on \(D\) such that the sub-quotients are pure of decreasing slope \(\mu\). The authors then proceed to show that this is the unique filtration on \(D\) such that slopes of the sub-quotients are decreasing. In other words, it is the Harder-Narashimhan filtration of \(D\). Regarding \(D\) as \(\phi^{-1}\)-module and using the Harder-Narashimhan filtration on this new module, the authors show that these sub-quotients are in fact direct summand of \(D\).
reviewed by: Manish Kumar
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 13F35
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 6102776
zbMATH Keywords: \(\phi \)-module
zbMATH Keywords: strict ring
zbMATH Keywords: Dieudonné-Manin theorem
zbMATH Keywords: Harder-Narasimhan filtration
MaRDI profile type: Publication
full work available at URL: s10114-012-0490-8
OpenAlex ID: W2051017459
cites work: THE THEORY OF COMMUTATIVE FORMAL GROUPS OVER FIELDS OF FINITE CHARACTERISTIC
cites work: Lectures on \(p\)-divisible groups
cites work: Notes on Crystalline Cohomology. (MN-21)
DOI: 10.1007/S10114-012-0490-8
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6102776