# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.