Codimension 2 cycles on Severi–Brauer varieties and decomposability

Abstract

In this text we show that one can generalize results showing that $$\mathrm {CH}^2(X)$$ , for various Severi–Brauer varieties X, is sometimes torsion free. In particular we show that for any pair of odd integers (n, m), with m dividing n and sharing the same prime factors, one can find a central simple k-algebra A of index n and exponent m that moreover has $$\mathrm {CH}^2(X)$$ torsion free for $$X=\mathrm {SB}(A)$$ . One can even take $$k={\mathbb {Q}}$$ in this construction.