Codimension 2 Cycles on Severi–Brauer Varieties

Karpenko, Nikita A.

doi:10.1023/a:1007705720373

Cited by 56 publications

(83 citation statements)

References 12 publications

(5 reference statements)

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“…4, we show that certain products of (generalized) Severi-Brauer varieties considered as schemes over certain subproducts via the projection can be naturally identified with grassmanians bundles (Corollary 4.4). Similar assertions were already proved in [24,Cor. 6.4] and in [25,Prop.…”

Section: Plan Of Worksupporting

confidence: 86%

Some new examples in the theory of quadratic forms

Izhboldin¹,

Karpenko²

2000

Mathematische Zeitschrift

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We construct a 6-dimensional anisotropic quadratic form φ and a 4-dimensional quadratic form ψ over some field F such that φ becomes isotropic over the function field F (ψ) but every proper subform of φ is still anisotropic over F (ψ). It is an example of non-standard isotropy with respect to some standard conditions of isotropy for 6-dimensional forms over function fields of quadrics, known previously. Besides of that, we produce an 8-dimensional quadratic form φ with trivial determinant such that the index of the Clifford invariant of φ is 4 but φ can not be represented as a sum of two 4-dimensional forms with trivial determinants. Using this, we find a 14-dimensional quadratic form with trivial discriminant and Clifford invariant, which is not similar to a difference of two 3-fold Pfister forms. The proofs are based on computations of the topological filtration on the Grothendieck group of certain projective homogeneous varieties. To do these computations, we develop several methods, covering a wide class of varieties and being, to our mind, of independent interest.

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Section: Plan Of Worksupporting

confidence: 86%

Some new examples in the theory of quadratic forms

Izhboldin¹,

Karpenko²

2000

Mathematische Zeitschrift

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“…More precisely, an explicit central division algebra of degree 8 and exponent 2 which has no quaternion subalgebra is constructed in [2]. Other examples of such algebras were given by Karpenko (see [11] and [12]) by computing torsion in Chow groups. In fact, 8 is the smallest possible degree for such an algebra by a well-known theorem of Albert which asserts that every algebra of exponent 2 and degree 4 is decomposable.…”

Section: Introductionmentioning

confidence: 99%

“…To produce explicit examples, one may take for A any indecomposable algebra, as those constructed in [2] or [12]. Alternately, we give in [3] an example of a decomposable algebra satisfying the hypothesis of Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Decomposable and indecomposable algebras of degree $$8$$ 8 and exponent 2

Barry

2013

Math. Z.

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Abstract. We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let B be a biquaternion algebra over F ( √ a) with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether B has a descent to F . This invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2-cohomological dimension 3 and over a field M(t) where the u-invariant of M is 8 and t is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by A. S. Merkurjev in Appendix.

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“…In this section we adopt the notation of [Kar98]. In particular, given a finite-dimensional central simple algebra A/F let X be the Severi-Brauer variety of A and let P be the projective space X F where F is an algebraic closure of F .…”

Section: Torsion In Chmentioning

confidence: 99%

Indecomposable p-algebras and Galois subfields in generic abelian crossed products

McKinnie

2008

Journal of Algebra

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Let F be a Henselian valued field with char(F ) = p and D a semi-ramified, "not strongly degenerate" p-algebra. We show that all Galois subfields of D are inertial. Using this as a tool we study generic abelian crossed product p-algebras, proving among other things that the noncyclic generic abelian crossed product p-algebras defined by non-degenerate matrices are indecomposable p-algebras. To construct examples of these indecomposable p-algebras with exponent p and large index we study the relationship between degeneracy in matrices defining abelian crossed products and torsion in CH 2 of Severi-Brauer varieties.

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Codimension 2 Cycles on Severi–Brauer Varieties

Cited by 56 publications

References 12 publications

Some new examples in the theory of quadratic forms

Some new examples in the theory of quadratic forms

Decomposable and indecomposable algebras of degree $$8$$ 8 and exponent 2

Indecomposable p-algebras and Galois subfields in generic abelian crossed products

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