Motivic cohomology with Z/2-coefficients

Voevodsky, Vladimir

doi:10.1007/s10240-003-0010-6

Cited by 343 publications

(358 citation statements)

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“…Recall that it is equivalent to the Milnor-Bloch-Kato conjecture relating Milnor's K-theory with Galois cohomology [36], [12]. It seems to be now a theorem (see [44]), thanks to work of Rost and Voevodsky; accepted proofs are certainly that of Merkurjev and Suslin in the special case of weight 2 [25] and that of Voevodsky at the prime 2 (in all weights) [41]. Since this seems important to some people, we shall specify in what weights we need the Beilinson-Lichtenbaum (or Milnor-Bloch-Kato) conjecture for our statements.…”

Section: Introductionmentioning

confidence: 99%

Motives of Azumaya algebras

Kahn¹,

Levine²

2010

J. Inst. Math. Jussieu

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Section: Introductionmentioning

confidence: 99%

Motives of Azumaya algebras

Kahn¹,

Levine²

2010

J. Inst. Math. Jussieu

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“…We will use the fact that multiplication by b K is an isomorphism on K n (F ; Z/m) for all n ≥ 1. This is a consequence of the Milnor-Bloch-Kato conjecture (see [6] [12] [11]), and has been observed by several people.…”

mentioning

confidence: 62%

Bott Periodicity for group rings An Appendix to “Periodicity of Hermitian K-groups”

Weibel¹

2011

J. K-Theory

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Abstract. We show that the groups Kn(RG; Z/m) are Bott-periodic for n ≥ 1 whenever G is a finite group, m is prime to |G|, R is a ring of S-integers in a number field and 1/m ∈ R.For any positive integer m there is a Bott elementwhere the period p = p(m) is: 2(ℓ − 1)ℓ ν−1 if m = ℓ ν and ℓ is an odd prime; max{8, 2 ν−1 } if m = 2 ν ; and p(m i ) if m = m i is the factorization of m into primary factors.In this appendix we consider a finite group G of order prime to m, and consider the Bott periodicity map; Z/m) for rings of integers R in local and global fields.Theorem 0.1. Assume that m is relatively prime to |G|, and that R is a ring of S-integers in a number field F with 1/m ∈ R. Then the Bott periodicity mapsTheorem 0.2. Assume that m is relatively prime to |G|, and that R is the ring of integers in a local field F with 1/m ∈ R. Then the Bott periodicity maps b K : The oldest result of this kind is due to Browder, who proved in [1, 2.6] that the Bott periodicity map b K is an isomorphism for finite fields and n ≥ 0 (when m is prime, which implies periodicity for all m).Almost as old is the following folklore result, which includes finite group rings.Lemma 0.3. If B is a finite ring and 1/m ∈ B, the the Bott periodicity mapProof. If m is the nilradical of B, then B red = B/m is semisimple. As such it is a product of matrix rings over finite fields. Now K * (B; Z/m) ∼ = K * (B red ; Z/m) by [9, 1.4]. By Morita invariance, we are reduced to the Browder's theorem that the Bott periodicity map is an isomorphism for finite fields.Remark 0.4. The finite groups K n (F 2 [C 2 ]; Z/8) were computed by Hesselholt and Madsen in [4]; they are not Bott periodic as their order goes to infinity with n.

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“…The case n = 2 of the Bloch-Kato conjecture is the Milnor conjecture, which was proved by Voevodsky [23], and a proof of the corresponding case of the Lichtenbaum-Quillen conjecture appears in [18]. A proof of the full Bloch-Kato conjecture is outlined, subject to some missing details, in [24].…”

Section: éTale K-theorymentioning

confidence: 97%

The K–theory presheaf of spectra

Jardine¹

2009

Geometry &Amp; Topology Monographs

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Motivic cohomology with Z/2-coefficients

Cited by 343 publications

References 30 publications

Motives of Azumaya algebras

Motives of Azumaya algebras

Bott Periodicity for group rings An Appendix to “Periodicity of Hermitian K-groups”

The K–theory presheaf of spectra

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