Hyperbolicity of unitary involutions

Karpenko, Nikita A.

doi:10.1007/s11425-012-4364-4

Cited by 14 publications

(23 citation statements)

References 21 publications

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“…By [12, Theorem A.2] (see also [9]), we may replace F by the function field of the F -variety R K/F SB(D). Since now the F -algebra D ′ splits over the quadratic extension K/F , it is equivalent to a quaternion F -algebra Q.…”

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confidence: 99%

“…There exists a finite field extension L/F of odd degree such that the algebra D ⊗ F L is split (cf. [1, 3.3.1] or [9,Page 938]). Let h be a hermitian form such that the involution σ L on the split algebra D ⊗ F L is adjoint to h. Let q be the quadratic form over L (of dimension 2m) given by h. Let q ′ be a nondegenerate subform of q of dimension 2m − 1.…”

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confidence: 99%

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Unitary Grassmannians of Division Algebras

Karpenko

2015

Transformation Groups

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Abstract. We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [13], where a similar issue for orthogonal and symplectic involutions has been treated, the present paper completes the study of grassmannians of isotropic ideals of division algebras.Let F be a field, K/F a separable quadratic field extension, n an integer ≥ 1, and D a central division K-algebra of degree 2 n endowed with a K/F -unitary involution σ. For definitions as well as for basic facts about involutions on central simple algebras, we refer to [16].For any integer i, we write X i for the F -variety of isotropic (with respect to) For any i, the variety X i is smooth and projective. It is nonempty if and only if 0 ≤ i ≤ 2 n−1 (X 0 is simply Spec F ) in which case it is geometrically connected and has dimensionFor any i, the variety X i is a closed subvariety of the Weil transfer R K/F SB i D, where SB i D is the ith generalized Severi-Brauer variety of D -the K-variety of all right ideals in D of reduced dimension i. We recall that according to [10] (see [15] for a more recent and simple proof), for any r = 0, 1, . . . , n − 1, the variety R K/F SB 2 r D is 2-incompressible. This means, roughly speaking, that any self-correspondenceof odd multiplicity is dominant. In particular, any rational self-mapThe following theorem is the main result of this note. It extends to the unitary setting the results on orthogonal and symplectic involutions obtained in [13]. Theorem 1. For any r = 0, 1, . . . , n − 1, the variety X 2 r is 2-incompressible.

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confidence: 99%

Section: ])mentioning

confidence: 99%

Unitary Grassmannians of Division Algebras

Karpenko

2015

Transformation Groups

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“…The implication (1) ⇒ (2) is known as the Pfister Factor Conjecture, and was proven in [3]. The implication (1) ⇒ (3), and the equivalence (2) ⇔ (3), follows from the Pfister Factor Conjecture and the non-hyperbolic splitting result of [8]. The converse implication, (2) or (3) ⇒ (1), is still open in general.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal Pfister involutions in characteristic two

Dolphin

2014

Journal of Pure and Applied Algebra

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Abstract. We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions in characteristic 2 that completely determines the isotropy behaviour of the involution. We also give an example of a non-totally decomposable algebra with orthogonal involution that becomes totally decomposable over every splitting field of the algebra.

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“…In particular, whether totally decomposable involutions are exactly those involutions on algebras of two-power degree that are either anisotropic or hyperbolic after extending scalars. That totally decomposable orthogonal involutions over a field of characteristic different from two are always either anisotropic or hyperbolic can be shown (see [6, (3.2)]) using the nonhyperbolic splitting results of Karpenko, [10], and that any totally decomposable orthogonal involution on a split algebra is adjoint to a Pfister form. The later result is known as the Pfister Factor Conjecture, and was shown in [3].…”

Section: Introductionmentioning

confidence: 99%

Totally decomposable symplectic and unitary involutions

Dolphin

2016

manuscripta math.

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Abstract. We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.

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Hyperbolicity of unitary involutions

Cited by 14 publications

References 21 publications

Unitary Grassmannians of Division Algebras

Unitary Grassmannians of Division Algebras

Orthogonal Pfister involutions in characteristic two

Totally decomposable symplectic and unitary involutions

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