T. Geisser and M. Levine, The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, preprint (1998). Home/Search Document Details and Download
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Voevodsky's Seattle Lectures: K-Theory and Motivic Cohomology.. - Voevodsky (Correct)
....AND MOTIVIC COHOMOLOGY 15 Theorem 4.10. Suslin Voevodsky [SV1, 5.9] Assume that k admits resolution of singularities. Then BL(q; holds for k if and only H q 1;q L (K; Z ( 0 for every field extension K of k. If k has characteristic p 0, and 6= p, Geisser and Levine proved in [GL] that the analogue of this theorem for BL(q; holds for all q, provided that the groups H pq (K; Z ( are replaced by Bloch s higher Chow groups. Low degree cases 4.11. We know that BL(q; holds for q 1, because we know Z(q) in this range. BL(q; is trivial for q 0 because then ....
T. Geisser and M. Levine, The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, Preprint (1998).
K-Theory And Motivic Cohomology Of Schemes - Levine Self-citation (Levine) (Correct)
....of Thomason s theorem [61] comparing Bott localized algebraic K theory with etale K theory. In Corollary 13.3, we show that the Beilinson Lichtenbaum conjectures for motivic cohomology implies the Quillen Lichtenbaum conjectures for algebraic K theory; we add in the reduction steps of [59] and [22] to reduce the Quillen Lichtenbaum conjectures to the Bloch Kato conjectures. This shows that Voevodsky s verification of the Milnor Conjecture [64] yields the sharp version of the 2 primary part of the Quillen Lichtenbaum conjectures, at least for schemes essentially of finite type over a ....
....(X[1=2] Z(q) has uniquely 2 divisible kernel and cokernel for p 0. Thus, for (1) we may assume that 2 is invertible on X . Since X is a scheme of finite type over Z, the groups H 1 (X; Z=2 (q) are finite for all q ( 11, Th eor eme de finitude] For (1) it follows from [64] 59] and [22], as in the proof of Theorem 14.5 that the etale cycle classes induce the isomorphisms H p (X; Z=2 (q) H p et (X; Z=2 (q) for p q; taking p 0, we get H p (X; Z=2 (q) 0 for p 0. The universal coefficient sequence 0 H p (X; Z=2 (q) 2 H p (X; Z=2 (q) ....
T. Geisser and M. Levine, The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, preprint (1998).
Inverting The Motivic Bott Element - Levine (1998) Self-citation (Levine) (Correct)
....that the cycle class map for i = 1; are also isomorphisms. Remark 6.3. Both Theorem 1.1 and Theorem 6.2 remain true without the hypothosis that k contain p Gamma1. This follows the Milnor conjecture (proved by Voevodsky in [14] which, together with [11] in case char k = 0) or [5] (in general) implies that the cycle class map Z q (X; 2 G et (X; Omega Gamma 2 ) induces an isomorphism in cohomology in degrees q. 7. Some homological algebra For the reader s convenience, we collect a few basic facts and definitions on homotopy limits and colimits. 7.1. ....
Geisser, T. and Levine, M. The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, preprint (1998).
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