Groupes et algébres de lie

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

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Reflection Groups And Coxeter Groups

Quadratic and hermitian forms over rings

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.

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J-invariant of linear algebraic groups

We prove Knebusch’s Norm Principle for finite extensions of semi-local regular rings containing a field of characteristic 0. As an application we prove the Grothendieck-Serre conjecture on principal homogeneous spaces for the case of spinor groups of regular quadratic forms over a field of characteristic 0.

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On Knebusch’s Norm Principle for quadratic forms over semi-local rings

For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute A(X), where X is an isotropic projective homogeneous variety and A means algebraic K-theory, motivic cohomology or algebraic cobordism MGL.

Oriented Cohomology and Motivic Decompositions of Relative Cellular Spaces

In paper [Vi07] A. Vishik using the techniques of symmetric operations in algebraic cobordism proved that changing the base field by the function field of a smooth projective quadric doesn’t change the property of being rational for cycles of small codimension. This fact which he calls Main Tool Lemma plays the crucial role in his construction of fields with u-invariant 2 + 1. In the present paper we prove M.T.L. for the class of varieties introduced by M. Rost in the context of the proof of Bloch-Kato conjecture. Namely, for varieties which possess a special correspondence (see [Ro06, Definition 5.1]). As in the Vishik’s proof the main technical tools are algebraic cobordism of M. Levine and F. Morel [LM07], generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. Therefore, we always assume that our base field k has characteristic 0. We use the following notation. By k we denote the algebraic closure of k and by Xk the respective base change X ×k k of a variety X. By K we denote the function field of X. Given a prime p by Ch ∗ (X) we denote the Chow ring of X modulo p-torsion taken with Z/p-coefficients, i.e., Ch ∗ (X) = CH ∗ (X)/p, where CH ∗ (X) = CH(X)/(p-tors). We say that a cycle from Ch ∗ (Xk) is defined over k if it belongs to the image of the restriction map Ch ∗ (X) → Ch ∗ (Xk). The following notion will be central in this paper

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Special correspondences and Chow traces of Landweber-Novikov operations

Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K_0(X)$ in terms of generators and relations in the case $G=G^{sc}/\mu_2$ is of Dynkin type ${\rm A}$ or ${\rm C}$ (here $G^{sc}$ is the simply-connected cover of $G$); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.

Kirill Zainoulline

Papers

On Knebusch’s Norm Principle for quadratic forms over semi-local rings

Oriented Cohomology and Motivic Decompositions of Relative Cellular Spaces

Special correspondences and Chow traces of Landweber-Novikov operations

The K-theory of versal flags and cohomological invariants of degree 3

Generic injectivity for étale cohomology and pretheories