Rost motives, affine varieties, and classifying spaces

TL;DR

This paper explores the structure of Rost motives, providing decompositions, studying Chow groups, and relating these motives to classifying spaces, advancing understanding in algebraic geometry and motivic theory.

Contribution

It introduces equivariant and ordinary Rost motives over various bases and connects them with classifying spaces, offering new insights into their structure and relationships.

Findings

Decomposition of equivariant motives of flag varieties

Analysis of torsion in Chow groups of twisted forms

Relation between Rost motives and classifying spaces

Abstract

In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.