Algebraic cycles on Severi-Brauer Schemes of prime degree over a curve
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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 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, then CHd (X) is ﬁnitely generated for every d in the indicated range.