On topological filtration for Severi-Brauer varieties II

The topological filtration on K ′ 0 of a Severi-Brauer variety is computed if the quotient of its index and exponent is a squarefree number and for each prime p dividing this quotient the p -primary component of the corresponding division algebra is decomposable. This gives in particular a description of Ch for such varieties. Let D be a central simple algebra over a field F and X = SB(D) the SeveriBrauer variety of D [1]. In [3] the topological filtration on the Grothendieck group K(X) has been computed provided that indD = expD. The topic of this note is the case when the quotient indD/ expD is any squarefree number but with one more additional restriction on D: for each prime p | indD/ expD the p -primary component of the corresponding division algebra should be decomposable (i.e. isomorphic to a tensor product D1 ⊗F D2 with Dj ̸= F for both j). In addition to notations introduced above we fix the following: notations relative to the Grothendieck group as introduced in [3], in particular GK(X) is the factorgroup of the topological filtration of codimension i; GiK(X) is the image of the homomorphism GK(X) → GK(X̄) = Z where X̄ is the variety X over the algebraic closure of F . For a prime p, vp is the p -adic valuation on Q; C k n is the binomial coefficient; ( , ) is the greatest common divisor. I owe to A.S. Merkurjev the idea that the cycle SB(D1)×SB(D2) on the variety SB(D1 ⊗D2) might be an interesting one. Theorem 1 Let D be a central simple algebra with indD = r, expD = e and let X = SB(D). If r/e is a squarefree number and for each prime p | r/e the p -primary component of the similar to D division algebra is decomposable then the map GK(X) → GK(X̄) (0 ≤ i ≤ dimX)