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Higher analogs of wild kernel. (Analogues supérieurs du noyau sauvage.) (French) Zbl 0783.11042

Let \(F\) be a number field, \(p\) an odd prime, \(S\) the set of the \(p\)-adic primes of \(F\) and \(o_ F^ S\) the ring of \(S\)-integers of \(F\). It is well-known that for \(i\geq 2\) the \(p\)-adic Chern characters \[ ch_{i,2}: K_{2i-2} (o_ F^ S)\otimes \mathbb{Z}_ p\to H^ 2(o_ F^ S,\mathbb{Z}_ p(i)), \qquad ch_{i,1}: K_{2i-1} (o_ F^ S)\otimes \mathbb{Z}_ p\to H^ 1(o_ F^ S,\mathbb{Z}_ p(i)) \] are surjective and isomorphisms for \(i=2\). Moreover, \(ch_{i,1}\) is split surjective. The author constructs a canonical partial splitting of \(ch_{i,2}\) on the subgroup \[ \text{ Ш}_ S^ 2(\mathbb{Z}_ p (i)):= \ker(H^ 2 (o_ F^ S, \mathbb{Z}_ p(i))\to \oplus_{v\in S} H^ 2(F_ v, \mathbb{Z}_ p(i))). \] This group should be viewed as the \(p\)-primary part of a higher wild kernel, this being true for \(i=2\). The author also shows that \(\text{ Ш}_ S^ 2(\mathbb{Z}_ p(i))\) is annihilated by the higher Stickelberger ideal \(S_{i-1}(F)\) introduced by Coates and Sinnott, in case \(F\) is abelian over \(\mathbb{Q}\).

MSC:

11R70 \(K\)-theory of global fields
11R23 Iwasawa theory
11R34 Galois cohomology
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References:

[1] Banaszak, G., Algebraic K-theory of number fields, rings of integers, and the Stickelberger ideal, Ann. of Math.135 (1992), 325-360. · Zbl 0756.11037
[2] Banaszak, G., Generalization of the Moore exact sequence and the wild Kernel for higher K-groups, preprint (1992). · Zbl 0778.11066
[3] Coates, J., On K2 and some classical conjectures in algebraic number theory, Ann. of Math.95 (1972), 99-116. · Zbl 0245.12005
[4] Coates, J. & Sinnott, W., An analogue of Stickelberger’s theorem for the higher K-groups, Invent. math.24 (1974), 149-161. · Zbl 0282.12006
[5] Dwyer, W. & Friedlander, E., Algebraic and étale K-theory, Trans. Amer. Math. Soc.292, n°1 (1985), 247-280. · Zbl 0581.14012
[6] Harris, B. & Segal, G., Ki of rings of algebraic integers, Ann. of Math.101 (1975), 20-33. · Zbl 0331.18015
[7] Kahn, B., Deux théorèmes de comparaison en cohomologie, applications, prépu blication (1991).
[8] Kurihara, M., Some remarks on conjectures about cyclotomic fields and K-groups of Z, Compos. Math.81 (1992), 223-236. · Zbl 0747.11055
[9] Levine, M., The indecomposable K3 of a field, Ann. Sci. ENS22 (1989), 255-344. · Zbl 0705.19001
[10] Lichtenbaum, S., Values of zeta functions, étale cohomology, and algebraic K-theory, in “Algebraic K-theory II”, , 342 (1973). · Zbl 0284.12005
[11] Merkurjev, A.S. & Suslin, A.A., On the K3 of a field, Math. USSR Izv.36 (1990), 541-565. · Zbl 0725.19003
[12] Schneider, P., Über gewisse Galoiscohomologiegruppen, Math. Zeit.168 (1979), 181-205. · Zbl 0421.12024
[13] Soulé, C., K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. math.55 (1979), 251-295. · Zbl 0437.12008
[14] Tate, J., Relations between K2 and Galois cohomology, Invent. math.36 (1976), 257-274. · Zbl 0359.12011
[15] Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math.131 (1990), 493-540. · Zbl 0719.11071
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