Abstract

The aim of this postscript is to provide a Leitfaden between the articles in this volume and their interlinkages, thereby clearly delineating the results from Galois cohomology and K-theory that are used in proving the main results in [BK90]. One reason to do this is to explicitly spell out the K-theoretic results that are used in [BK90], especially those of Soulé. We shall also indicate a proof of the finiteness of the global Tate–Shafarevich groups as considered in [BK90, 5.13] for M = ℤ(m); and note its relation to the Tate–Shafarevich groups considered by Fontaine and Perrin-Riou [FP94], as well as the Tate–Shafarevich groups that can be defined from the Poitou–Tate sequence. For simplicity, we only consider the base field ℚ, indicating briey how the results generalize to an arbitrary totally real abelian number field. As in the previous articles, p will denote an odd prime. All other notation is as in [Su15]. Nguyen Quang Do's contribution in putting this note together is gratefully acknowledged.

Leitfaden

In the article [Li15], Lichtenbaum used K-theory and étale cohomology along with the Chern class maps to prove Soulé's result on the vanishing of the cohomology groups for all m ≥ 2. Soulé [So79] proved that these groups are torsion and Schneider [Sc75] proved that they are divisible. The Quillen–Lichtenbaum conjecture asserts that the Chern class maps

are isomorphisms for p odd, k = 1, 2 and m ≥ 2. Soulé in [So79, Théoréme 6 (iii)] proved the surjectivity for even positive integers m. Dwyer and Friedlander [DF85] proved it for all integers m with k = 1 or 2 or 2m − k > 1, using étale K-theory. The reader should also see [Ka93]. The vanishing of for all m ≥ 2 is equivalent to the finiteness of for m ≥ 2 and this will play an important role in the proof of the Bloch–Kato conjecture, as explained in the article by Coates [Co15] in this volume.