Algebraic cycles on Severi-Brauer schemes of prime degree over a curve

Let k be a perfect field and let p be a prime number different from the characteristic of k. Let C be a smooth, projective and geometrically integral k-curve and let X be a Severi-Brauer C-scheme of relative dimension p - 1. In this paper we show that CHd (X)(tors) contains a subgroup isomorphic to CH0(X/C) for every d in the range 2 <= d <= p. We deduce that, if k is a number field, the full Chow ring CH*(X) is a finitely generated abelian group.