On Topological Filtration for Triquaternion Algebra

The topological filtration on the Grothendieck group of the Severi-Brauer variety corresponding to a triquaternion algebra is computed. In particular, it is shown that the second Chow group of the variety is torsionfree. Let F be a field, A a central simple F -algebra of degree 8 and exponent 2, X = SB(A) the Severi-Brauer variety of A [2]. Consider the second Chow group CH(X), i.e. the group of 2-codimensional algebraic cycles on X modulo rational equivalence [3, 10]. In [6] it is shown that the group CH(X) can contain a non-trivial torsion. In this note we study the case when A decomposes (in a tensor product of two smaller algebras) or (what is equivalent [1]) when A is a product of three quaternion algebras Q1⊗F Q2⊗F Q3. We compute (almost completely) the topological filtration on the Grothendieck group K(X) = K ′ 0(X) [3, 10] and as a consequence show that the second Chow group CH(X) is torsionfree. It deserves to be mentioned that an analogues situation occurs in the case of odd prime exponent too. The group CH(X ′) where X ′ = SB(A′) for an algebra A′ of an odd prime exponent p and degree p can have a non-trivial torsion [7]. But it is known to be torsionfree in the case when A′ decomposes [5].