J-invariant of linear algebraic groups
TLDR
In this paper, the authors introduce le J-invariant de G which caracterises le comportement motivique de X and generalise le J invariant defini par A. Vishik dans le cadre des formes quadratiques.Abstract:
Soit G un groupe algebrique lineaire semi-simple de type interieur sur un corps F et soit X un G-espace homogene projectif tel que le groupe G soit deploye sur le point generique de X. Nous introduisons le J-invariant de G qui caracterise le comportement motivique de X et generalise le J-invariant defini par A. Vishik dans le cadre des formes quadratiques. Nous utilisons cet invariant pour obtenir les decompositions motiviques de tous les G-espaces homogenes projectifs qui sont generiquement deployes, par exemple les varietes de Severi-Brauer, les quadriques de Pfister, la grassmannienne des sous-espaces totalement isotropes maximaux d'une forme quadratique, la variete des sous-groupes de Borel de G. Nous discutons egalement les relations avec les indices de torsion, la dimension canonique et les invariants cohomologiques du groupe G.read more
Citations
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Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties
TL;DR: In this paper, the authors studied the structure of Chow motives with coefficients in a finite field of the generalized Severi-Brauer varieties of the algebra A. They showed that the Chow motives of any variety in XG decomposes into a sum of indecomposable motives, and described the indecompositionable summands which appear in the decompositions.
Journal ArticleDOI
Motivic construction of cohomological invariants
TL;DR: In this paper, a cohomological invariant of degree 5 for groups of type E8 with trivial Rost invariant over any field k of characteristic 0, and putting k=Q answer positively this question of Serre, and a variety which possesses a special correspondence of Rost is a norm variety.
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Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties
TL;DR: In this paper, the authors studied the structure of the Chow motives with coefficients in a finite field of the varieties in a semisimple affine algebraic group of inner type over a field F. They showed that the motives of any variety in C decomposes (in a unique way) into a sum of indecomposable summands which appear in the decompositions.
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Generically split projective homogeneous varieties
Viktor Petrov,Nikita Semenov +1 more
TL;DR: In this paper, the Rost invariant of groups of type E7 and their isotropy was shown to be trivial in the sense that the kernel of Rost invariants for such groups is not trivial.
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Equivariant pretheories and invariants of torsors
Stefan Gille,Kirill Zainoulline +1 more
TL;DR: In this paper, the notion of equivariant pretheory was introduced and studied in the context of rational cycles and G-torsors, and a ring which serves as an invariant of E was introduced.
References
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Frank W. Anderson,Kent R. Fuller +1 more
TL;DR: In this paper, the authors provide a self-contained account of much of the theory of rings and modules, focusing on the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules.
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TL;DR: The classical theory of symmetric bilinear forms and quadratic forms: Bilinear form Quadratic form forms over rational function fields Function fields of quadrics Bilinverse forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture on the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratically forms and cyclic cycles as mentioned in this paper.