Algebraic connective $K$-theory of a Severi-Brauer variety with prescribed reduced behavior

Abstract

We show that Chow groups of low dimension cycles are torsion free for a class of sufficiently generic Severi-Brauer varieties. Using a recent result of Karpenko, this allows us to compute the algebraic connective $K$-theory in low degrees for the same class of varieties. Independently of these results, we show that the associated graded ring for the topological filtration on the Grothendieck ring is torsion free in the same degrees for an arbitrary Severi-Brauer variety.