Symmetric operations in algebraic cobordism

Vishik, Alexander

doi:10.1016/j.aim.2006.12.012

Cited by 42 publications

(58 citation statements)

References 12 publications

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“…In particular, using the results of P.Brosnan on S i (see [1]), we get: . In reality, these maps can be lifted to a well defined, socalled, symmetric operations [16]. Since over algebraically closed field all our varieties are cellular, and thus, the Chow groups of them are torsion-free, we will not need such subtleties, but we will keep the notation from [16], and denote our maps Ω m → CH m+i as φ t i−m , and the ( mod 2) version Ω m → Ch m+i as ϕ t i−m (in our situation below there is no need to mod-out the 2-torsion even over the ground field, since all our maps end up in the CH 0 of quadrics, which are torsion-free, anyway).…”

Section: Algebraic Cobordismmentioning

confidence: 99%

Excellent connections in the motives of quadrics

Vishik¹

2011

Ann. Sci. École Norm. Sup.

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In this article we prove the Conjecture claiming that the connections in the motives of excellent quadrics are minimal ones for anisotropic quadrics of given dimension. This imposes severe restrictions on the motive of arbitrary anisotropic quadric. As a corollary we estimate from below the rank of indecomposable direct summand in the motive of a quadric in terms of its dimension. This generalises the well-known Binary Motive Theorem. Moreover, we have the description of Tate-motives involved. This, in turn, gives another proof of Karpenko's Theorem on the value of the first higher Witt index. But also other new relations among higher Witt indices follow.

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Section: Algebraic Cobordismmentioning

confidence: 99%

Excellent connections in the motives of quadrics

Vishik¹

2011

Ann. Sci. École Norm. Sup.

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“…In particular, one has the action of the Landweber-Novikov operations there ( [6]). In [9,11] the author constructed some new cohomological operations on Algebraic Cobordisms, so-called, symmetric operations. In the questions related to 2-torsion these operations behave in a more subtle way than the Landweber-Novikov operations.…”

Section: Symmetric Operationsmentioning

confidence: 99%

Generic points of quadrics and Chow groups

Vishik¹

2007

manuscripta math.

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“…In characteristic 0, all of this has been proved several years ago by Alexander Vishik in [7,9] (exact references are given right before each statement) with a help of the algebraic cobordism theory and especially symmetric operations of [8].…”

Section: Msc: 14c25 11e04mentioning

confidence: 99%

Variations on a theme of rationality of cycles

Karpenko

2013

Open Mathematics

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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 -invariant 2 + 1, for ≥ 3, is extended to arbitrary characteristic = 2. MSC: 14C25, 11E04Keywords:The main results of this note are Theorem 1.1 (the basic result) with its enhancement Theorem 2.1, Proposition 3.1 with its enhancement Proposition 4.1 implying Theorems 3.2 and 3.3 (which go a little bit beyond the basic result), and (a quite special) Proposition 5.3 (going in a special situation even more beyond the basic result). The main application is Theorem 5.1.In characteristic 0, all of this has been proved several years ago by Alexander Vishik in [7,9] (exact references are given right before each statement) with a help of the algebraic cobordism theory and especially symmetric operations of [8].In fact, the original versions of the most results are stronger: they do not require to mod out 2-torsion elements of Chow groups (as our weak versions do). In particular, they do not require the assumption that the group CH(Y ) (notation introduced in the beginning of Section 1) is 2-torsion-free (as do our very weak versions).Note that it has been explained in [7, Remark on p. 370] that the weak versions can be obtained in characteristic 0 with a help of the Landweber-Novikov operations (in the algebraic cobordism theory) replacing the symmetric operations.Although the very weak versions are already sufficient for existing applications, we prove the weak versions as well, see Theorem 2.1 and Proposition 4.1. The proofs here are only a bit more complicated (than in the very weak case) *

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Symmetric operations in algebraic cobordism

Cited by 42 publications

References 12 publications

Excellent connections in the motives of quadrics

Excellent connections in the motives of quadrics

Generic points of quadrics and Chow groups

Variations on a theme of rationality of cycles

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