75 research outputs found
Zero cycles on homogeneous varieties
In this paper we study the group of zero dimensional cycles of
degree 0 modulo rational equivalence on a projective homogeneous algebraic
variety . To do this we translate rational equivalence of 0-cycles on a
projective variety into R-equivalence on symmetric powers of the variety. For
certain homogeneous varieties, we then relate these symmetric powers to moduli
spaces of \'etale subalgebras of central simple algebras which we construct.
This allows us to show for certain classes of homogeneous
varieties, extending previous results of Swan / Karpenko, of Merkurjev, and of
Panin.Comment: Significant revisions made to simplify exposition, also includes
results for symplectic involution varieties. Main arguments now rely on
Hilbert schemes of points and are valid with only mild characteristic
assumptions. 32 page
Period and index, symbol lengths, and generic splittings in Galois cohomology
We use constructions of versal cohomology classes based on a new notion of
"presentable functors," to describe a relationship between the problems of
bounding symbol length in cohomology and of finding the minimal degree of a
splitting field. The constructions involved are then also used to describe
generic splitting varieties for degree 2 cohomology with coefficients in a
commutative algebraic group of multiplicative type.Comment: 17 page
Motives of unitary and orthogonal homogeneous varieties
Grothendieck-Chow motives of quadric hypersurfaces have provided many
insights into the theory of quadratic forms. Subsequently, the landscape of
motives of more general projective homogeneous varieties has begun to emerge.
In particular, there have been many results which relate the motive of a one
homogeneous variety to motives of other simpler or smaller ones. In this paper,
we exhibit a relationship between motives of two homogeneous varieties by
producing a natural rational map between them which becomes a projective bundle
morphism after being resolved. This allows one to use formulas for projective
bundles and blowing up to relate the motives of the two varieties. We believe
that in the future this ideas could be used to discover more relationships
between other types of homogeneous varieties.Comment: 5 page
Corestrictions of algebras and splitting fields
Given a field , an \'etale extension and an Azumaya algebra ,
one knows that there are extensions such that is a split
algebra over . In this paper we bound the degree of a minimal
splitting field of this type from above and show that our bound is sharp in
certain situations, even in the case where is a split extension. This
gives in particular a number of generalizations of the classical fact that when
the tensor product of two quaternion algebras is not a division algebra, the
two quaternion algebras must share a common quadratic splitting field.
In another direction, our constructions combined with results of Karpenko
also show that for any odd prime number , the generic algebra of index
, and exponent cannot be expressed nontrivially as the corestriction
of an algebra over any extension field if .Comment: 13 page
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