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Jinvariant of linear algebraic groups
"... Let G be a linear algebraic group over a field F and X be a projective homogeneous Gvariety such that G splits over the function field of X. In the present paper we introduce an invariant of G called Jinvariant which characterizes the splitting properties of the Chow motive of X. This generalizes ..."
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Let G be a linear algebraic group over a field F and X be a projective homogeneous Gvariety such that G splits over the function field of X. In the present paper we introduce an invariant of G called Jinvariant which characterizes the splitting properties of the Chow motive of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (SeveriBrauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G2 and F4varieties) as well as provide new ones (exceptional varieties of types E6, E7 and E8). We also discuss applications to torsion indices, canonical dimensions and splitting properties of the group G.
Applications of AtiyahHirzebruch spectral sequence for motivic cobordism
 Department of Mathematics, Faculty of Education, Ibaraki University
"... Abstract. We study applications of AtiyahHirzebruch spectral sequences for motivic cobordisms found by Hopkins and Morel. 1. ..."
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Abstract. We study applications of AtiyahHirzebruch spectral sequences for motivic cobordisms found by Hopkins and Morel. 1.
Canonical pdimensions of algebraic groups and degrees of basic polynomial invariants
 Viktor Petrov PIMS, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada Nikita Semenov Mathematisches Institut der LMU München, Germany Kirill Zainoulline Mathematisches Institut der LMU München, Ther
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ALGEBRAIC COBORDISM OF SIMPLY CONNECTED LIE GROUPS
"... Abstract. Let GC be the algebraic group over C corresponding a simply connected Lie group G. The algebraic cobordism Ω(GC) defined by Levine and Morel is showed isomorphic to MU ∗subalgebra of MU ∗ (G) with some modulous and is computed explicitely. 1. ..."
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Abstract. Let GC be the algebraic group over C corresponding a simply connected Lie group G. The algebraic cobordism Ω(GC) defined by Levine and Morel is showed isomorphic to MU ∗subalgebra of MU ∗ (G) with some modulous and is computed explicitely. 1.
Equivariant pretheories and invariants of torsors
"... Abstract. In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant Ktheory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a R ..."
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Abstract. In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant Ktheory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev’s (equivariant Ktheory) spectral sequence. As an application we generalize the theorem of KarpenkoMerkurjev on Gtorsors and rational cycles; to every Gtorsor E and a Gequivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the Jinvariant of E and in the case of Grothendieck’s K0 – indexes of the respective Tits algebras. 1.
Appendix to “The Chow rings of generalized Grassmannians”
"... In this appendix we tabulate preliminary data required to establish Theorem 112 in the paper “The Chow rings of generalized Grassmannians”. This appendix is set to record the intermediate data required to establish Theorem 112 in [DZ3]. At the first sight, the data may be seen as many. However, th ..."
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In this appendix we tabulate preliminary data required to establish Theorem 112 in the paper “The Chow rings of generalized Grassmannians”. This appendix is set to record the intermediate data required to establish Theorem 112 in [DZ3]. At the first sight, the data may be seen as many. However, they are all produced (and are processed again) by computer. We begin with a brief guide for the content tabulated. Sections §1–§6 are devoted, respectively, to the six cases of (G, H) concerned by Theorem 16 (resp. Theorem 7–12) in [DZ3]. a) In 1.1–6.1, the sets W(H; G) (cf. [DZ3, 2.1]) of left cosets of the Weyl group of G by the Weyl group of H are presented both in terms of the minimized decompositions of its elements, and index system [DZ3,(2.3)] imposed by the decompositions. These are produced using the Decomposition in [DZ1]. b) In 1.2–6.2, the matrices Ak (cf. [DZ3, 4.3, Step 1]), k ≥ 1, required to determine the additive cohomology of G/Hs are listed. They are computed from the L–R coefficients in [DZ1]. c) In 1.3–6.3, the multiplicative rule for the basis elements of H even (G/Hs) are presented (cf. [DZ3, 4.3, Step 2]). They are also obtained by applying the L–R coefficients in [DZ1]. d) In 1.4–6.4, with respect to the ordered monomial basis B(m) (cf. [DZ3, 3.3]) for certain values of m, the structure matrices M(πm) are computed by using the L–R coefficients in [DZ1]. e) Applying the built–in function “Nullspace ” in Mathematica to the matrices M(πm) in 1.4–6.4, one obtains the Nullspace N(πm) listed in 1.5–6.5. These are required to specify the relations ri’s in the proofs of Theorem 1–6 in [DZ3,§6]. We conclude this introduction with the Cartan matrices for the Lie groups of types F4, E6 and E7 [Hu, p.59]. All data in this text are generated from them.
EXAMPLES FOR THE MOD p MOTIVIC COHOMOLOGY OF CLASSIFYING SPACES
"... Abstract. Let BG be the classifying space of a compact Lie group G. Some examples to compute the motivic cohomology H∗, ∗ (BG; Z/p) are given, by comparing with H ∗ (BG; Z/p), CH ∗ (BG) and BP ∗ (BG). 1. ..."
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Abstract. Let BG be the classifying space of a compact Lie group G. Some examples to compute the motivic cohomology H∗, ∗ (BG; Z/p) are given, by comparing with H ∗ (BG; Z/p), CH ∗ (BG) and BP ∗ (BG). 1.
THE JINVARIANT AND THE TITS ALGEBRAS OF A LINEAR ALGEBRAIC GROUP.
"... Abstract. In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group G and the degree one parameters of its Jinvariant. Our main technical tool is the second Chern class map in the RiemannRoch theorem without denominators. As an applicat ..."
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Abstract. In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group G and the degree one parameters of its Jinvariant. Our main technical tool is the second Chern class map in the RiemannRoch theorem without denominators. As an application we recover some known results on the Jinvariant of quadratic forms of small dimension; we describe all possible values of the Jinvariant of an algebra with involution up to degree 8 and give explicit examples; we establish several relations between the Jinvariant of an algebra A with involution and the Jinvariant(of the quadratic form) over the function