Codimension 2 Cycles on Quadratic Weil Transfer of Biquaternionic Severi-brauer Variety

Let F be a field, B a biquaternion F -algebra, L/F an étale quadratic extension, X the Weil transfer with respect to L/F of the Severi-Brauer variety of BL. We show that the Chow group of codimension 2 cycle classes on X is torsion-free. Our Chow groups are those with integral coefficients. The motives used in the proof are the Grothendieck Chow motives (still with the integral coefficients) as in [1, §64]. Theorem 1. Let F be a field, B a biquaternion F -algebra, L/F an étale quadratic extension, X the Weil transfer with respect to L/F of the Severi-Brauer variety of BL. Then the Chow group CH(X) is torsion-free. Proof. If L is not a field, then L ' F × F and X ' SB(B) × SB(B). Since X is a rank 3 projective bundle over SB(B), the Chow motive of X is then isomorphic to the sum M ⊕M(1) ⊕M(2) ⊕M(3), where M is the motive of the variety SB(B). Therefore the total Chow group of X is a direct sum of four (shifted) copies of the total Chow group of SB(B). Since the total Chow group of SB(B) is torsion-free [5, Corollary 7], the total Chow group of X is also torsion-free; in particular, so is the group CH(X). Now let us assume that L is a field and BL is not a division algebra. In this case, by [8, Proposition 16.2], there exists a quaternion F -algebra Q such that the algebra BL is isomorphic to the algebra of (2 × 2)-matrices over QL. Let now M be the motive (over L) of the Severi-Brauer variety of QL. Then the motive of SB(BL) is isomorphic to M ⊕M(2) [4, Theorem 1.3.1]. Therefore, by [3, Lemma 2.1], the motive (over F ) of X is isomorphic to the sum RL/FM ⊕ (RL/FM)(4)⊕ corL/F (M ⊗ σM)(2), where σ is the nontrivial automorphism of L/F , RL/F is the motivic Weil transfer functor of [6], and corL/F is the motivic corestriction functor introduced in [7]. The motive RL/FM is the motive of the variety RL/F SB(QL) (abusing notation, we write RL/F also for the Weil transfer on varieties). This variety is a smooth projective quadric surface; its total Chow group is torsion-free. Concerning the remaining summand corL/F (M ⊗ σM)(2), we first note that the corestriction functor preserves the Chow group. The motive M ⊗σM is the motive of the variety SB(QL)×SB(QL) which is a rank 1 projective bundle over SB(QL). Since the total Chow group of SB(QL) is torsion-free, the total Chow group of corL/F (M ⊗ σM) is torsion-free and it follows that the total Chow group of X is torsion-free in this case also. To finish the proof of Theorem 1, we consider the remaining case where BL is a skew field (and L is a field). We consider the Grothendieck group K(X) of classes of coherent OX-modules together with its topological filtration, and we are going to show that the