# Variations on a theme of rationality of cycles

Open Mathematics (2013)

- Volume: 11, Issue: 6, page 1056-1067
- ISSN: 2391-5455

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topNikita Karpenko. "Variations on a theme of rationality of cycles." Open Mathematics 11.6 (2013): 1056-1067. <http://eudml.org/doc/269452>.

@article{NikitaKarpenko2013,

abstract = {We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.},

author = {Nikita Karpenko},

journal = {Open Mathematics},

keywords = {Chow groups; Quadrics; Steenrod operations; u-invariant; quadrics; -invariant},

language = {eng},

number = {6},

pages = {1056-1067},

title = {Variations on a theme of rationality of cycles},

url = {http://eudml.org/doc/269452},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Nikita Karpenko

TI - Variations on a theme of rationality of cycles

JO - Open Mathematics

PY - 2013

VL - 11

IS - 6

SP - 1056

EP - 1067

AB - We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

LA - eng

KW - Chow groups; Quadrics; Steenrod operations; u-invariant; quadrics; -invariant

UR - http://eudml.org/doc/269452

ER -

## References

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- [7] Vishik A., Generic points of quadrics and Chow groups, Manuscripta Math., 2007, 122(3), 365–374 http://dx.doi.org/10.1007/s00229-007-0074-6[WoS][Crossref] Zbl1154.14003
- [8] Vishik A., Symmetric operations in algebraic cobordism, Adv. Math., 2007, 213(2), 489–552 http://dx.doi.org/10.1016/j.aim.2006.12.012[WoS][Crossref] Zbl1129.14034
- [9] Vishik A., Fields of u-invariant 2r + 1, In: Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin, II, Progr. Math., 270, Birkhäuser, Boston, 2009, 661–685 http://dx.doi.org/10.1007/978-0-8176-4747-6_22[Crossref]
- [10] Voevodsky V., Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci., 2003, 98, 1–57 http://dx.doi.org/10.1007/s10240-003-0009-z[Crossref] Zbl1057.14027

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