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By I.G. Macdonald

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6 . . . . . . . . . . . . . . . . . . . . -P. ): LNM 1835, pp. 25–101, 2004. 8 . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . 15 . . . . . . . . . . . 52 54 55 56 57 60 6 Some Applications . . . . . . . . . . . . . . . .

Then N∨ ∼ = N (r)[2r], where r = m − a(N ) − b(N ). Proof. It is clear that proving the statement for N is equivalent to proving it for N ∨ . So, we can assume that either a(N ) = b(N ) = m/2, or b(N ) ≥ m/2. In the former case, N |k = Z(m/2)[m] ⊕ Z(m/2)[m] = N ∨ |k ∼ N ∨ by (since it should consists of at least two Tate motives), and so N = RNT. So, we can assume that b(N ) ≥ m/2. 14, there exists 1 ≤ t < h(Q) such that iW (q|Ft ) ≤ a(N ), m − b(N ) < iW (q|Ft+1 ). 13, for r = m−a(N )−b(N ), N (r)[2r] is isomorphic to a direct summand of M (Q), and a(N (r)[2r]) = m − b(N ) = a(N ∨ ).

Izv. Akad. Nauk SSSR 54 (1990) Trad. anglaise : Math. USSR Izv. 36, 541–565 (1991) 32. , Voevodsky, V. : An exact sequence for K∗M /2 with applications to quadratic forms. Pr´epublication, 2000 33. Peyre, E. : Corps de fonctions de vari´et´es homog`enes et cohomologie galoisienne. C. R. Acad. Sci. paris 321, 136–164 (1995) 34. Quillen D. : Higher algebraic K-theory, I. Lect. Notes in Math. 341, 85–147, Springer, 1973 35. Rost, M. : Injectivity of K2(D) → K2 (F ) for quaternion algebras. html Cohomologie non ramifi´ee des quadriques 23 36.

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Algebraic geometry : introduction to schemes by I.G. Macdonald


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