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L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle, I


Ayoub, Joseph (2014). L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle, I. Journal für die Reine und Angewandte Mathematik, 2014(693):1-149.

Abstract

This is the first article of a series of two, aiming at constructing and studying motivic Galois groups in the context of triangulated motives. We first develop a general formalism that allows us to associate to a monoidal functor f, satisfying some natural conditions, a Hopf algebra in the target category of f. This formalism is then applied to the Betti realization of Morel-Voevodsky motives over a base field k endowed with a complex embedding σ: k → ℂ. This gives a Hopf algebra Hmot(k, σ) in the derived category of ℚ-vector spaces. Using the comparison theorem between singular and de Rham cohomology, we obtain an explicit description of unitary algebra Hmot(k, sigma;) ℂ showing in particular that the complex Hmot(k, σ) has no homology in strictly negative degrees. We deduce from this a structure of a Hopf algebra on the zeroth homology of Hmot(k, σ) whose spectrum will be called the motivic Galois group.

Abstract

This is the first article of a series of two, aiming at constructing and studying motivic Galois groups in the context of triangulated motives. We first develop a general formalism that allows us to associate to a monoidal functor f, satisfying some natural conditions, a Hopf algebra in the target category of f. This formalism is then applied to the Betti realization of Morel-Voevodsky motives over a base field k endowed with a complex embedding σ: k → ℂ. This gives a Hopf algebra Hmot(k, σ) in the derived category of ℚ-vector spaces. Using the comparison theorem between singular and de Rham cohomology, we obtain an explicit description of unitary algebra Hmot(k, sigma;) ℂ showing in particular that the complex Hmot(k, σ) has no homology in strictly negative degrees. We deduce from this a structure of a Hopf algebra on the zeroth homology of Hmot(k, σ) whose spectrum will be called the motivic Galois group.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:August 2014
Deposited On:27 Jan 2015 16:04
Last Modified:05 Apr 2016 18:45
Publisher:De Gruyter
ISSN:0075-4102
Publisher DOI:https://doi.org/10.1515/crelle-2012-0089

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