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.
[1] Brosnan P., Steenrod operations in Chow theory, Trans. Amer. Math. Soc., 2003, 355(5), 1869–1903 http://dx.doi.org/10.1090/S0002-9947-03-03224-010.1090/S0002-9947-03-03224-0Search in Google Scholar
[2] Elman R., Karpenko N., Merkurjev A., The Algebraic and Geometric Theory of Quadratic Forms, Amer. Math. Soc. Colloq. Publ., 56, American Mathematical Society, Providence, 2008 10.1090/coll/056Search in Google Scholar
[3] Fino R., Around rationality of cycles, Cent. Eur. J. Math., 11(6), 1068–1077 10.2478/s11533-013-0218-8Search in Google Scholar
[4] Izhboldin O.T., Fields of u-invariant 9, Ann. of Math., 2001, 154(3), 529–587 http://dx.doi.org/10.2307/306214110.2307/3062141Search in Google Scholar
[5] Karpenko N.A., Merkurjev A.S., On standard norm varieties, Ann. Sci. Éc. Norm. Supér. (in press), preprint available at http://arxiv.org/abs/1201.1257 Search in Google Scholar
[6] Vishik A., On the Chow groups of quadratic Grassmannians, Doc. Math., 2005, 10, 111–130 Search in Google Scholar
[7] Vishik A., Generic points of quadrics and Chow groups, Manuscripta Math., 2007, 122(3), 365–374 http://dx.doi.org/10.1007/s00229-007-0074-610.1007/s00229-007-0074-6Search in Google Scholar
[8] Vishik A., Symmetric operations in algebraic cobordism, Adv. Math., 2007, 213(2), 489–552 http://dx.doi.org/10.1016/j.aim.2006.12.01210.1016/j.aim.2006.12.012Search in Google Scholar
[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_2210.1007/978-0-8176-4747-6_22Search in Google Scholar
[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-z10.1007/s10240-003-0009-zSearch in Google Scholar
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