Generalization of a going-down theorem in the category of Chow-Grothendieck motives due to N. Karpenko

Grothendieck Chow-motives of Severi-Brauer varieties

For any central simple algebra, the Grothendieck Chow-motive of the corresponding Severi-Brauer variety is decomposed in a direct sum where each summand is a twisted motive of the Severi-Brauer variety corresponding to the underlying division algebra. It leads to decompositions in other theories (for instance, of K-cohomologies) because of the universal property of the Chow-motives. In the seco...

Motives and Chow Groups of Quadrics with Application to the U-invariant (after Oleg Izhboldin)

Weil Transfer of Algebraic Cycles

Let L/F be a finite separable field extension of degree n, X a smooth quasi-projective L-scheme, and R(X) the Weil transfer of X with respect to L/F . The map Z 7→ R(Z) of the set of simple cycles Z ⊂ X extends in a natural way to a map Z(X) → Z(R(X)) on the whole group of algebraic cycles Z(X). This map factors through the rational equivalence of cycles and induces this way a map of the Chow g...

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...

Schur-finiteness in Λ-rings

We introduce the notion of a Schur-finite element in a λ-ring. Since the beginning of algebraic K-theory in [G57], the splitting principle has proven invaluable for working with λ-operations. Unfortunately, this principle does not seem to hold in some recent applications, such as the K-theory of motives. The main goal of this paper is to introduce the subring of Schur-finite elements of any λ-r...