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# Mathematical Research Letters

## Volume 15 (2008)

### Number 1

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

Pages: 51 – 56

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n1.a5

#### Author

#### 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 $CH^{d}(X)_{{\rm{tors}}}$ contains a subgroup isomorphic to $CH_{0}(X/C)$ for every $d$ in the range $2\leq d\leq p$. We deduce that, if $k$ is a number field, the full Chow ring $CH^{*}\be(X)$ is a finitely generated abelian group.