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

Abstract

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 $C{H}^d(X{)}_{\mathrm{tors}}$ contains a subgroup isomorphic to $C{H}_0(X/C)$ for every $d$ in the range $2\le d\le p$. We deduce that, if $k$ is a number field, then $C{H}^d(X)$ is finitely generated for every $d$ in the indicated range.

6 pages