The gamma-filtration and the Rost invariant

TL;DR

This paper investigates the gamma-filtration on the variety of Borel subgroups of a simple algebraic group, revealing its torsion structure, and connects it with the Rost invariant and essential dimension bounds.

Contribution

It establishes that the torsion part of the second gamma-filtration quotient is cyclic of order given by the Dynkin index, providing explicit generators and bounds.

Findings

The torsion part is cyclic of order equal to the Dynkin index.

An explicit cycle generating this torsion is constructed.

A lower bound for the essential dimension of simple algebraic groups is derived.

Abstract

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.