Algebraic cobordisms of a Pfister quadric

Vishik, Alexander; Yagita, Nobuaki

doi:10.1112/jlms/jdm056

Cited by 43 publications

(49 citation statements)

References 11 publications

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“…where {f β } β∈B can be any set of elements such that {pr(f β )} β∈B form a Zbasis of CH * (Q| k ) -see Section 2 of [13]. In particular, we can take the set…”

Section: Proofmentioning

confidence: 99%

Generic points of quadrics and Chow groups

Vishik¹

2007

manuscripta math.

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“…where {f β } β∈B can be any set of elements such that {pr(f β )} β∈B form a Zbasis of CH * (Q| k ) -see Section 2 of [13]. In particular, we can take the set…”

Section: Proofmentioning

confidence: 99%

Generic points of quadrics and Chow groups

Vishik¹

2007

manuscripta math.

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“…So, one needs only to find out which elements over algebraic closure are defined over the base field. But the proof of injectivity from [18,Proposition 4.4] uses the original computation by M. Rost of the Chow groups of a Pfister quadric (in the case of MGL 2 * , * there is an independent computation, using motivic homotopy theory-see [18,Theorem 7.2]). In the current subsection we would like to give another proof of this fact, which is based on symmetric operations, does not use the computations of M. Rost, and, in turn, gives a new way to compute the Chow groups.…”

Section: Algebraic Cobordism Of a Pfister Quadricmentioning

confidence: 99%

“…The Pfister quadric is a rare example of non-cellular variety for which the ring of algebraic cobordism is computed-see [18]. This was possible since the extension of scalars map is injective in this case.…”

Section: Algebraic Cobordism Of a Pfister Quadricmentioning

confidence: 99%

Symmetric operations in algebraic cobordism

Vishik

2007

Advances in Mathematics

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In this article we describe certain new cohomological operations in algebraic cobordisms. These operations give the natural obstructions for the cobordism class to be represented by the embedding. Also, they permit to work with algebraic cobordisms and Chow groups in a more subtle way than the LandweberNovikov operations (related to 2-torsion effects). We describe applications to the computation of the algebraic cobordisms of a Pfister quadrics and to the problem of rationality of cycles.

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“…Remark : If Λ is a finite, commutative and connected ring, complete motivic decompositions in CM (F ; Λ) remain complete when the coefficients are extended to the residue field of Λ by [7,Corollary 2.6], hence the study of motivic decompositions in CM G (F ; Λ), where Λ is any finite connected ring whose residue field is of characteristic p, is reduced to the study motivic decompositions in CM G (F ; F p ).…”

Section: Application To Motivic Decompositions Of Projective Homogenementioning

confidence: 99%

Motivic decompositions of projective homogeneous varieties and change of coefficients

Clercq¹

2010

Comptes Rendus. Mathématique

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We prove that under some assumptions on an algebraic group G, indecomposable direct summands of the motive of a projective G-homogeneous variety with coefficients in Fp remain indecomposable if the ring of coefficients is any field of characteristic p. In particular for any projective G-homogeneous variety X, the decomposition of the motive of X in a direct sum of indecomposable motives with coefficients in any finite field of characteristic p corresponds to the decomposition of the motive of X with coefficients in Fp. We also construct a counterexample to this result in the case where G is arbitrary.To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005). RésuméDécompositions motiviques des variétés projectives homogènes et changement des coefficients. Nous prouvons que sous certaines hypothèses sur un groupe algébrique G, tout facteur direct indécomposable du motif associéà une variété projective G-homogèneà coefficients dans Fp demeure indécomposable si l'anneau des coefficients est un corps de caractéristique p. En particulier pour toute variété projective G-homogène X, la décomposition du motif de X comme somme directe de motifs indécomposablesà coefficients dans tout corps fini de caractéristique p correspondà la décomposition du motif de Xà coefficients dans Fp. Nous exhibons de plus un contre-exempleà ce résultat dans le cas où le groupe G est quelconque.

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Algebraic cobordisms of a Pfister quadric

Cited by 43 publications

References 11 publications

Generic points of quadrics and Chow groups

Generic points of quadrics and Chow groups

Symmetric operations in algebraic cobordism

Motivic decompositions of projective homogeneous varieties and change of coefficients

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