Special Reductive Groups Over an Arbitrary Field

Huruguen, Mathieu

doi:10.1007/s00031-016-9378-5

Cited by 9 publications

(7 citation statements)

References 21 publications

(22 reference statements)

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“…We denote by M = Hom ks−gp (T ks , G m,ks ) its character group; this is a free abelian group of finite rank equipped with a continuous action of the absolute Galois group Γ = Γ k . By an unpublished result of Colliot-Thélène (see [Hu16,Thm. 18]; this result is implicitly contained in [CS87]), T is special if and only if the Γ-module M is invertible, i.e., a direct factor of a permutation Γ-module.…”

Section: Very Special Torimentioning

confidence: 99%

On the fundamental groups of commutative algebraic groups

Brion¹

2020

Annales Henri Lebesgue

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Section: Very Special Torimentioning

confidence: 99%

On the fundamental groups of commutative algebraic groups

Brion¹

2020

Annales Henri Lebesgue

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“…General statements and low cohomological dimension. An algebraic group G over k is special if and only if H 1 (K, G K ) = * for every field extension K/k (see [Hur16]).…”

Section: Coflasque Algebraic Groups Over Particular Fieldsmentioning

confidence: 99%

Separable algebras and coflasque resolutions

Ballard¹,

Duncan²,

Lamarche³

et al. 2020

Preprint

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Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories.Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.

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“…We denote by M = Hom k s −gp (T k s , G m,k s ) its character group; this is a free abelian group of finite rank equipped with a continuous action of the absolute Galois group Γ = Γ k . By an unpublished result of Colliot-Thélène (see [8,Thm. 18]; this result is implicitly contained in [2]), T is special if and only if the Γ-module M is invertible, i.e., a direct factor of a permutation Γmodule.…”

Section: Very Special Torimentioning

confidence: 99%

Very special algebraic groups

Brion¹,

Peyre²

2020

Comptes Rendus. Mathématique

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We say that a smooth algebraic group G over a field k is very special if for any field extension K /k, every G K-homogeneous K-variety has a K-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions. Résumé. Nous disons qu'un groupe algébrique lisse G sur un corps k est très spécial si pour toute extension de corps K /k, toute K-variété homogène sous G K a un point K-rationnel. On sait que tout groupe linéaire résoluble scindé est très spécial. Dans cette note, nous obtenons la réciproque et nous discutons ses relations avec la classification birationnelle des actions de groupes algébriques.

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Special Reductive Groups Over an Arbitrary Field

Cited by 9 publications

References 21 publications

On the fundamental groups of commutative algebraic groups

On the fundamental groups of commutative algebraic groups

Separable algebras and coflasque resolutions

Very special algebraic groups

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