Pseudo-abelian varieties

Burt Totaro

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 5, page 693-721
  • ISSN: 0012-9593

Abstract

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Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field k to be a smooth connected k -group in which every smooth connected affine normal k -subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension of a pseudo-abelian variety by a smooth connected affine group, in a unique way. We work out much of the structure of pseudo-abelian varieties. These groups are closely related to unipotent groups in characteristic p and to pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad. Many properties of abelian varieties such as the Mordell-Weil theorem extend to pseudo-abelian varieties. Finally, we conjecture a description of  Ext 2 ( 𝐆 a , 𝐆 m ) over any field by generators and relations, in the spirit of the Milnor conjecture.

How to cite

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Totaro, Burt. "Pseudo-abelian varieties." Annales scientifiques de l'École Normale Supérieure 46.5 (2013): 693-721. <http://eudml.org/doc/272234>.

@article{Totaro2013,
abstract = {Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field $k$ to be a smooth connected $k$-group in which every smooth connected affine normal $k$-subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension of a pseudo-abelian variety by a smooth connected affine group, in a unique way. We work out much of the structure of pseudo-abelian varieties. These groups are closely related to unipotent groups in characteristic $p$ and to pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad. Many properties of abelian varieties such as the Mordell-Weil theorem extend to pseudo-abelian varieties. Finally, we conjecture a description of $\mathrm \{Ext\}^2(\mathbf \{G\}_a,\mathbf \{G\}_m)$ over any field by generators and relations, in the spirit of the Milnor conjecture.},
author = {Totaro, Burt},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {algebraic group; pseudo-reductive group; pseudo-abelian variety; unipotent group; Weil restriction},
language = {eng},
number = {5},
pages = {693-721},
publisher = {Société mathématique de France},
title = {Pseudo-abelian varieties},
url = {http://eudml.org/doc/272234},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Totaro, Burt
TI - Pseudo-abelian varieties
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 5
SP - 693
EP - 721
AB - Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field $k$ to be a smooth connected $k$-group in which every smooth connected affine normal $k$-subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension of a pseudo-abelian variety by a smooth connected affine group, in a unique way. We work out much of the structure of pseudo-abelian varieties. These groups are closely related to unipotent groups in characteristic $p$ and to pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad. Many properties of abelian varieties such as the Mordell-Weil theorem extend to pseudo-abelian varieties. Finally, we conjecture a description of $\mathrm {Ext}^2(\mathbf {G}_a,\mathbf {G}_m)$ over any field by generators and relations, in the spirit of the Milnor conjecture.
LA - eng
KW - algebraic group; pseudo-reductive group; pseudo-abelian variety; unipotent group; Weil restriction
UR - http://eudml.org/doc/272234
ER -

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