Generic points of quadrics and Chow groups

Vishik, Alexander

doi:10.1007/s00229-007-0074-6

Cited by 19 publications

(30 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A stronger version of the following result has been proved in characteristic 0 in [7,Corollary 3.5 (1)]. …”

Section: Basic Result: Very Weak Versionmentioning

confidence: 89%

“…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: 90%

“…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.…”

Section: Msc: 14c25 11e04mentioning

confidence: 99%

“…Proof. According to [7,Statement 3.10], the rationality of ensures that for any element ∈ Ch (Q × Y ) there exists an element ∈ Ch (Q × Y ) such that the image of under the composition…”

Section: Let Be An Element Of Ch (Q × Y ) If the Image Of Under The mentioning

confidence: 99%

“…We refer to [7] for an introduction into the subject. Notation is introduced in the beginning of Section 1.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Variations on a theme of rationality of cycles

Karpenko

2013

Open Mathematics

View full text Add to dashboard Cite

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) *

show abstract

“…A stronger version of the following result has been proved in characteristic 0 in [7,Corollary 3.5 (1)]. …”

Section: Basic Result: Very Weak Versionmentioning

confidence: 89%

Section: Msc: 14c25 11e04mentioning

confidence: 90%

Section: Msc: 14c25 11e04mentioning

confidence: 99%

Section: Let Be An Element Of Ch (Q × Y ) If the Image Of Under The mentioning

confidence: 99%

“…We refer to [7] for an introduction into the subject. Notation is introduced in the beginning of Section 1.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Variations on a theme of rationality of cycles

Karpenko

2013

Open Mathematics

View full text Add to dashboard Cite

show abstract

Fields of u-Invariant 2 r + 1

Vishik

2009

Progress in Mathematics

View full text Add to dashboard Cite

Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics

Scully

2018

Math. Ann.

View full text Add to dashboard Cite

Let p and q be anisotropic quadratic forms over a field F of characteristic = 2, let s be the unique non-negative integer such that 2 s < dim(p) ≤ 2 s+1 , and let k denote the dimension of the anisotropic part of q after scalar extension to the function field F (p) of p. We conjecture that dim(q) must lie within k of a multiple of 2 s+1 . This can be viewed as a direct generalization of Hoffmann's Separation theorem. Among other cases, we prove that the conjecture is true if k < 2 s−1 . When k = 0, this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism W (F ) → W (F (p)) has dimension divisible by 2 s+1 .

show abstract

Generic points of quadrics and Chow groups

Cited by 19 publications

References 10 publications

Variations on a theme of rationality of cycles

Variations on a theme of rationality of cycles

Fields of u-Invariant 2 r + 1

Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics

Contact Info

Product

Resources

About