Abstract
We define and study Harder-Narasimhan filtrations on Breuil-Kisin-Fargues modules and related objects relevant to p-adic Hodge theory.
Highlights
Universal such functor, and a universal realisation category, which he called the category of motives
A top-down approach to Grothendieck’s conjecture aims to cut down the elusive category of motives from the various realisation categories of existing cohomology theories, and this first requires assembling them in some ways
We mostly investigate an hidden but implicit structure of these BKF-modules: they are equipped with some sort of Harder–Narasimhan formalism, adapted from either [Iri16] or [LWE16], which both expanded the original constructions of Fargues [Far19] from p-divisible groups over OC to Breuil–Kisin modules
Summary
Over an algebraically closed complete extension C of Qp, Bhatt, Morrow and Scholze [BMS16] have recently defined a new (integral) p-adic cohomology theory, which specializes to all other known such theories and nicely explains their relations and pathologies. It takes values in the category of Breuil–Kisin–Fargues modules (hereafter named BKF-modules), a variant of Breuil–Kisin modules due to Fargues [Far15]. This new realisation category has various, surprisingly different but equivalent incarnations, see [SW17, 14.1.1], [Sch17, 7.5] or Section 3; beyond its obvious relevance for p-adic motives, it is expected to play a role in the reformulation of the p-adic Langlands program proposed by Fargues [Far16]. We mostly investigate an hidden but implicit structure of these BKF-modules: they are equipped with some sort of Harder–Narasimhan formalism, adapted from either [Iri16] or [LWE16], which both expanded the original constructions of Fargues [Far19] from p-divisible groups over OC to Breuil–Kisin modules
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