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In this paper, we study the graded ring gr*(X) defined by K-theory of a twist flag variety X. In particular, the Kunneth map gr*(R') ⊗ gr*(R') → gr*(R) is studied explicitly for an original Rost motive R' and a generalized Rost motive R. Using this, we give examples Tor(X)² ≠ 0 for the ideal Tor(X) of torsion elements in the Chow ring CH*(X). [ABSTRACT FROM AUTHOR]

In this article we study in detail the category of noncommutative motives of separable algebras Sep ( k ) over a base field k . We start by constructing four different models of the full subcategory of commutative separable algebras CSep ( k ) . Making use of these models, we then explain how the category Sep ( k ) can be described as a “fibered Z -order” over CSep ( k ) . This viewpoint leads to several computations and structural properties of the category Sep ( k ) . For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (= CSA) and sequences of elements in the Brauer group of k . As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant ( e.g. algebraic K -theory, cyclic homology, and topological Hochschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras. [ABSTRACT FROM AUTHOR]

In a recent paper, Colliot-Thélène, Parimala and Suresh conjectured that a local–global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves. [ABSTRACT FROM AUTHOR]

In the present paper we introduce and study the twisted γ-filtration on K0(Gs), where Gs is a split simple linear algebraic group over a field k of characteristic prime to the order of the center of Gs. We apply this filtration to construct nontrivial torsion elements in γ-rings of twisted flag varieties. [ABSTRACT FROM AUTHOR]