By I. Fesenko, M. Kurihara
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Additional info for Invitation to Higher Local Feilds
At first sight this question looks trivial because the group DKn (F ) consists of all divisible elements of Kn (F ) . However, the following theorem shows that the group DKn (F ) is not necessarily a divisible group! Theorem 8 (Izhboldin, [ I3 ]). For every n > 2 and prime p there is a field F such that char (F ) 6= p , p 2 F and (1) The group DKn (F ) is not divisible, and the group Dp K2 (F ) is not p -divisible, Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 28 O.
Math. IHES 63(1986), 107–152. H. Bass, H. and J. Tate, The Milnor ring of a global field, In Algebraic K -theory II, Lect. Notes in Math. 342, Springer-Verlag, Berlin, 1973, 349–446. I. Fesenko, Class field theory of multidimensional local fields of characteristic 0, with the residue field of positive characteristic, Algebra i Analiz (1991); English translation in St. Petersburg Math. J. 3(1992), 649–678. Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields Part I.
2. Tate’s argument. To prove Bloch–Kato’s theorem we may assume that Indeed, consider the cohomological long exact sequence n = 1. p q ! H q;1 (K Z=p(q)) ;! H q (K Z=pn;1(q)) ;! H (K Z=pn(q)) ! : : : which comes from the Bockstein sequence 0 ;! Z=pn;1 ; ! Z=pn ;;;! Z=p ;! 0: p mod p We may assume p 2 K , so H q;1 (K Z=p(q )) ' hq;1 (K ) and the following diagram is commutative (cf. [ T, x2 ]): g kq;1 (K ) ;f;;;p! yhq; 1 K Kq (K )=pn;1 ? hq K ? y p ;;;; ! Kq (K )=pn ? hq K ? y mod p ;;;; ! kq (K ) ?
Invitation to Higher Local Feilds by I. Fesenko, M. Kurihara