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Invariants of degree 3 and torsion in the Chow group of a versal flag

Part of: (Co)homology theory Forms and linear algebraic groups Special varieties

Published online by Cambridge University Press:  16 April 2015

Alexander Merkurjev ,
Alexander Neshitov  and
Kirill Zainoulline
Alexander Merkurjev
Affiliation:
Department of Mathematics, University of California at Los Angeles, CA 90095, USA email merkurev@math.ucla.edu
Alexander Neshitov
Affiliation:
St. Petersburg Department of Steklov Mathematical Institute RAS, 27 Fontanka, 191023 St. Petersburg, Russia email neshitov@yandex.ru Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N 6N5, Canada
Kirill Zainoulline
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N 6N5, Canada email kirill@uottawa.ca
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Abstract

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We prove that the group of normalized cohomological invariants of degree $3$ modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group $G$ is isomorphic to the torsion part of the Chow group of codimension-$2$ cycles of the respective versal $G$-flag. In particular, if $G$ is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of $G$. As an application, we construct nontrivial cohomological invariants for indecomposable central simple algebras.

Keywords

cohomological invariantlinear algebraic grouptorsorGalois cohomologyChow group

MSC classification

Primary: 11E72: Galois cohomology of linear algebraic groups 14M17: Homogeneous spaces and generalizations 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 8 , August 2015 , pp. 1416 - 1432
Copyright
© The Authors 2015 

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