Cycles on Severi - Brauer Varieties

For a given sequence of integers (n i) 1 i=1 we consider all the central simple algebras A (over all elds) satisfying the condition ind A i = n i and nd among them an algebra having the biggest torsion in the second Chow group CH 2 of the corresponding Severi-Brauer variety (\biggest" means that it can be mapped epimorphically onto each other). We describe this biggest torsion in a way in general and more explicitly in some important special situations. As an application we prove indecom-posability of certain algebras. 0. Introduction We consider nite dimensional over elds central simple algebras. Let A be such an algebra and X = SB(A) the corresponding Severi-Brauer variety 2, x1]. We are interested to describe the torsion in the second Chow group CH 2 (X) of 2-codimensional cycles on X modulo rational equivalence (the question seems more natural if one takes in account that the groups CH 0 (X) and CH 1 (X) never have a torsion). Here are some preliminary observations. The group Tors CH 2 (X) is nite and annihilated by ind A. Further, if A 0 is another algebra Brauer equivalent to A and X 0 = SB(A 0) then 16, lemma (1.12)], 12, cor. 1.3.2]: Tors CH 2 (X) ' Tors CH 2 (X 0) : Finally, if A = O p A p is the decomposition of an algebra A into the tensor product of its primary components and X p = SB(A p) for each prime p then Tors CH 2 (X) ' M p Tors CH 2 (X p) or in other word, p-primary part of the group Tors CH 2 (X) is isomorphic to Tors CH 2 (X p). Summarizing we see that the problem to compute Tors CH 2 (X) for all algebras reduces itself to the case of primary division algebras. Support and hospitality of Sonderforschungsbereich 343 (Bielefeld University) during the last stage is gratefully acknowledged.