A simple proof of Dieudonné-Manin classification theorem
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 \(D\).
scientific article; zbMATH DE number 6102776