Kunneth formula for graded rings associated to -theories of Rost motives

In this paper, we study the graded ring g r ∗ ( X ) gr^*(X) defined by K K -theory of a twist flag variety X X . In particular, the Kunneth map g r ∗ ( R ′ ) ⊗ g r ∗ ( R ′ ) → g r ∗ ( R ) gr^*(R’)\otimes gr^*(R’)\to gr^*(R) is studied explicitly for an original Rost motive R ′ R’ and a generalized Rost motive R R . Using this, we give examples T o r ( X ) 2 ≠ 0 Tor(X)^2\not =0 for the ideal T o r ( X ) Tor(X) of torsion elements in the Chow ring C H ∗ ( X ) CH^*(X) .