Skip Garibaldi and Kirill Zainoulline: The gamma-filtration and the Rost invariant

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Submission: 2010 Jul 21, revised: 2010 Dec 28

Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is a cyclic group of order the Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle that generates this cyclic group; we provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; we relate the generating cycle with the Rost invariant and the torsion of the respective generalized Rost motives; we use this cycle to obtain a uniform lower bound for the essential dimension of (almost) all simple linear algebraic groups.

2010 Mathematics Subject Classification: 20G15, 14C25, 14L30

Keywords and Phrases: linear algebraic group, Rost invariant, algebraic cycle

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