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 $\mathit{AC}$, $B$ 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.