On standard norm varieties

Karpenko, Nikita A.; Merkurjev, Alexander

doi:10.24033/asens.2187

Cited by 26 publications

(31 citation statements)

References 26 publications

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“…The p ‐portion of the Rost invariant for G produces a symbol in the Galois cohomology group

H^{3} (k, μ_{p}^{\otimes 2})

, see for references. Since the Rost invariant has trivial kernel (see ), the symbol is non‐zero and the upper motive of the variety X is a Rost motive R corresponding to the symbol (in the sense of ). It follows by (as well as by ) that the Chow motive of the variety X decomposes in a finite direct sum of shifts of R .…”

Section: Simple Groupsmentioning

confidence: 99%

“…It follows by (as well as by ) that the Chow motive of the variety X decomposes in a finite direct sum of shifts of R . The Chow groups of R , computed in (in characteristic 0), are as follows:

{CH}^{j} R

double-struckZ

for

j = 0

;

p Z

for

j = (p + 1) k

with

k = 1, \dots, p - 1

;

Z / p Z

for

j = (p + 1) k - 2

with

k = 1, \dots, p - 1

; and 0 for the remaining values of j . Let n be the number of summands in the decomposition of the motive of X into a direct sum of shifted copies of R .…”

Section: Simple Groupsmentioning

confidence: 99%

“…It follows by [14] (as well as by [8]) that the Chow motive of the variety decomposes in a finite direct sum of shifts of . The Chow groups of , computed in [7] (in characteristic 0), are as follows: CH is ℤ for = 0; ℤ for = ( + 1) with = 1, … , − 1; ℤ∕ ℤ for = ( + 1) − 2 with = 1, … , − 1; and 0 for the remaining values of .…”

Section: And Simply-connectedmentioning

confidence: 99%

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Chow ring of generic flag varieties

Karpenko

2017

Mathematische Nachrichten

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Let G be a split semisimple algebraic group over a field k and let X be the flag variety (i.e., the variety of Borel subgroups) of G twisted by a generic G‐torsor. We start a systematic study of the conjecture, raised in in form of a question, that the canonical epimorphism of the Chow ring of X onto the associated graded ring of the topological filtration on the Grothendieck ring of X is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. We reduce its proof to the case of k=Q. For simply‐connected or adjoint G, we reduce the proof to the case of simple G. Finally, we provide a list of types of simple groups for which the conjecture holds. Besides of some classical types considered previously (namely, A, C, and the special orthogonal groups of types B and D), the list contains the exceptional types G2, F4, and simply‐connected E6.

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“…The p ‐portion of the Rost invariant for G produces a symbol in the Galois cohomology group

H^{3} (k, μ_{p}^{\otimes 2})

Section: Simple Groupsmentioning

confidence: 99%

“…It follows by (as well as by ) that the Chow motive of the variety X decomposes in a finite direct sum of shifts of R . The Chow groups of R , computed in (in characteristic 0), are as follows:

{CH}^{j} R

double-struckZ

for

j = 0

;

p Z

for

j = (p + 1) k

with

k = 1, \dots, p - 1

;

Z / p Z

for

j = (p + 1) k - 2

with

k = 1, \dots, p - 1

; and 0 for the remaining values of j . Let n be the number of summands in the decomposition of the motive of X into a direct sum of shifted copies of R .…”

Section: Simple Groupsmentioning

confidence: 99%

Section: And Simply-connectedmentioning

confidence: 99%

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Chow ring of generic flag varieties

Karpenko

2017

Mathematische Nachrichten

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“…The group of K 1 -zero-cycles is an invariant of smooth projective varieties [KM,Cor. RC.13], so that we have an isomorphism…”

Section: Reduction To Algebras Of Square Degreementioning

confidence: 99%

The group of K1-zero-cycles on the second generalized Severi–Brauer variety of an algebra of index 4

McFaddin

2017

Journal of Algebra

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Abstract. In this manuscript, it is shown that the group of K 1 -zero-cycles on the second generalized Severi-Brauer variety of an algebra A of index 4 is given by elements of the group K 1 (A) together with a square-root of their reduced norm. Utilizing results of Krashen concerning exceptional isomorphisms, we translate our problem to the computation of cycles on involution varieties. Work of Chernousov and Merkurjev then gives a means of describing such cycles in terms of Clifford and spin groups and corresponding R-equivalence classes. We complete our computation by giving an explicit description of these algebraic groups.

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“…In some places, the new proofs are "better organized" and this makes them look simpler and less technical, but substantially the proofs are the same. Techniques of the present paper have been further developed in more recent [3] and [5,Appendix SC].…”

mentioning

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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On standard norm varieties

Cited by 26 publications

References 26 publications

Chow ring of generic flag varieties

Chow ring of generic flag varieties

The group of K1-zero-cycles on the second generalized Severi–Brauer variety of an algebra of index 4

Variations on a theme of rationality of cycles

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