Special correspondences and Chow traces of Landweber-Novikov operations

Zainoulline, Kirill

doi:10.1515/crelle.2009.023

Cited by 4 publications

(4 citation statements)

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“…is divisible by p for such m. Moreover, the total operation S L−N := i S i L−N is a ring homomorphism (see [Za09], [Vi07], and [LM07]). …”

Section: Examplementioning

confidence: 99%

Motivic construction of cohomological invariants

Semenov

2016

Comment. Math. Helv.

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Abstract. Let G be a group of type E 8 over Q such that G R is a compact Lie group, let K be a field of characteristic 0, and q = −1, −1, −1, −1, −1 a 5-fold Pfister form. J-P. Serre posed in a letter to M. Rost written on June 23, 1999 the following problem: Is it true that G K is split if and only if q K is hyperbolic?In the present article we construct a cohomological invariant of degree 5 for groups of type E 8 with trivial Rost invariant over any field k of characteristic 0, and putting k = Q answer positively this question of Serre. Aside from that, we show that a variety which possesses a special correspondence of Rost is a norm variety. IntroductionLet G be a group of type E 8 over Q such that G R is a compact Lie group. Let now K/Q be a field extension and q = −1, −1, −1, −1, −1 a 5-fold Pfister form. J-P. Serre posed in a letter to M. Rost written on June 23, 1999 the following problem:Is it true that G K is split if and only if q K is hyperbolic?M. Rost replied on July 2, 1999 proving that if q K is hyperbolic, then G K is split (see [GS10] for the proof). One of the goals of the present article is to give a positive answer to Serre's question (see Theorem 8.8).Let us recall first some recent developments in the topics which are relevant for the method of the proof of this result.1.1. Cohomological invariants. The study of cohomological invariants was initiated by J-P. Serre in the 90'ies. Serre conjectured the existence of an invariant in H 3 et (k, Q/Z(2)) of G-torsors, where G is a simply connected simple algebraic group over a field k. This invariant was constructed by M. Rost and is now called the Rost invariant of G (see [KMRT98,).Nowadays there exist numerous constructions and estimations of cohomological invariants for different classes of algebraic objects (see e.g. [GMS03]). Nevertheless, the most constructions of cohomological invariants rely on a specific construction of the object under consideration. Unfortunately, for many groups, like E 8 , there is no classification and no general construction so far.

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“…is divisible by p for such m. Moreover, the total operation S L−N := i S i L−N is a ring homomorphism (see [Za09], [Vi07], and [LM07]). …”

Section: Examplementioning

confidence: 99%

Motivic construction of cohomological invariants

Semenov

2016

Comment. Math. Helv.

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“…) is divisible by p for such m. Moreover, the total operation S L−N := i S i L−N is a ring homomorphism (see [Za09], [Vi07], and [LM07]).…”

Section: Examplementioning

confidence: 99%

Motivic construction of cohomological invariants

Semenov¹

2009

Preprint

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“…Write CH i (Y ) for the Chow group with coefficients in the integers localized at p and CH i (Y ) for the factor group of the Chow group CH i (Y ) modulo p-torsion elements and p CH i (Y ). In [36,Theorem 1.3], K. Zainoulline proved, using the Landweber-Novikov operations in algebraic cobordism theory, that every Y and every norm variety X of s enjoy the following property: if i < (p n − 1)/(p − 1), every class α in CH i (Y F ), where F is an algebraic closure of F , such that α F (X) is F (X)-rational, is F -rational itself, i.e., α belongs to the image of the map CH i (Y ) → CH i (Y F ). This statement is in the spirit of the Main Tool Lemma of A. Vishik [32].…”

Section: Introductionmentioning

confidence: 99%

“…The A-triviality property for X means that the degree map deg : CH 0 (X K ) → Z (p) is an isomorphism (i.e., the kernel A(X K ) of the degree map is trivial) for any field extension K/F such that X has a point over K. (We believe that the A-triviality condition should be also imposed in the statement of [36,Theorem 1.3]. ) Our proof of Theorem 1.1 is "elementary" in the sense that it does not use the algebraic cobordism theory.…”

Section: Introductionmentioning

confidence: 99%

On standard norm varieties

Karpenko

Merkurjev

2013

Ann. Sci. École Norm. Sup.

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Abstract. Let p be a prime integer and F a field of characteristic 0. Let X be the norm variety of a symbol in the Galois cohomology group H n+1 (F, µ ⊗n p ) (for some n ≥ 1), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field F (X) has the following property: for any equidimensional variety Y , the change of field homomorphism CH(Y ) → CH(Y F (X) ) of Chow groups with coefficients in integers localized at p is surjective in codimensions < (dim X)/(p−1). One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in Appendix). Another important ingredient is A-triviality of X, the property saying that the degree homomorphism on CH 0 (X L ) is injective for any field extension L/F with X(L) = ∅. The proof involves the theory of rational correspondences, due to Markus Rost, reviewed in Appendix.

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Special correspondences and Chow traces of Landweber-Novikov operations

Cited by 4 publications

References 5 publications

Motivic construction of cohomological invariants

Motivic construction of cohomological invariants

Motivic construction of cohomological invariants

On standard norm varieties

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