On the K-theory coniveau epimorphism for products of Severi–Brauer varieties

Abstract

For X a product of Severi–Brauer varieties, we conjecture that if the Chow ring of X is generated by Chern classes, then the canonical epimorphism from the Chow ring of X to the graded ring associated to the coniveau filtration of the Grothendieck ring of X is an isomorphism. We show this conjecture is equivalent to the condition that if G is a split semisimple algebraic group of type AC, B is a Borel subgroup of G and E is a standard generic G-torsor, then the canonical epimorphism from the Chow ring of E∕B to the graded ring associated with the coniveau filtration of the Grothendieck ring of E∕B is an isomorphism. In certain cases we verify this conjecture.