Postscript

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 Soule. 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 briefly 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 [Sul5]. Nguyen Quang Do's contribution in putting this note together is gratefully acknowledged.