On the Bumpy Fundamental Group Scheme

Marco Antei ()
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Marco Antei: Université de Nice Sophia-Antipolis, Laboratoire J.A.Dieudonné

A chapter in Analytic and Algebraic Geometry, 2017, pp 1-17 from Springer

Abstract: Abstract In this short paper we first recall the definition and the construction of the fundamental group scheme of a scheme X in the known cases: when it is defined over a field and when it is defined over a Dedekind scheme. It classifies all the finite (or quasi-finite) fpqc torsors over X. When X is defined over a noetherian regular scheme S of any dimension we do not know if such an object can be constructed. This is why we introduce a new category, containing the fpqc torsors, whose objects are torsors for a new topology. We prove that this new category is cofiltered thus generating a fundamental group scheme over S, said bumpy as it may not be flat in general. We prove that it is flat when S is a Dedekind scheme, thus coinciding with the classical one.

Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-5648-2_1

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DOI: 10.1007/978-981-10-5648-2_1

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