Cyclic Galois extensions of commutative rings
Let \(R\) be a commutative ring, \(G\) a finite abelian group. The group \(\mathrm{Gal}(R,G)\) of isomorphism classes of Galois extensions of \(R\) with group \(G\), as developed by Auslander, Goldman, Chase, Harrison and Rosenberg in the 1960s, has important connections with algebraic number theory: if \(R\) is the ring of integers of a number field \(K\), then by class field theory \(\mathrm{Gal}(R,G)\) is closely related to the class group of \(K\); more recently, work of Kersten, Michalichek and the author has found important and subtle connections with Leopoldt's and Vandiver's conjectures [c.f. \textit{I. Kersten} and \textit{J. Michalichek} [J. Number Theory 32, 371--386 (1989; Zbl 0709.11058)]. The main purpose of these notes is to describe \(\mathrm{Gal}(R,G)\) in a noncohomological way. Since \(\mathrm{Gal}(R,G)\) respects products in the second variable, the problem immediately reduces to \(G\) cyclic of prime power order \(p^n\), which we assume henceforth. Chapter 0 gives a useful description of the basic theory and a description of \(\mathrm{Gal}(R,G)\) and the subgroup \(\mathrm{NB}(R,G)\) of Galois extensions with normal basis, when \(R\) is connected and contains \(p^{-1}\) and a primitive \(p^n\)-th root of unity \(\zeta\). This generalized Kummer theory goes back to the reviewer [Ill. J. Math. 15, 273--280 (1971; Zbl 0211.37102)] and \textit{A. Z. Borevich} [J. Sov. Math. 11, 514--534 (1979); transl. from Zap. Nauchn. Semin. LOMI Steklova 57, 8--30 (1976; Zbl 0379.13003)]. Chapter I then describes \(\mathrm{Gal}(R,G)\) when \(R\) contains \(p^{-1}\) but not \(\zeta\), by letting \(S=R[\zeta]\) and ``descending'' the Kummer theory over \(S\) to \(R\). This chapter is based on the author's paper [Trans. Am. Math. Soc. 326, 307--343 (1991; Zbl 0743.11060)], and has useful applications to number theory, as is shown in chapter IV. Chapter II describes an alternate approach to obtaining \(\mathrm{Gal}(R,G)\) from \(\mathrm{Gal}(S,G)\) using corestriction, which leads to a proof that the map from \(\mathrm{NB}(R,G)\) to \(\mathrm{NB}(R/N,G)\) is surjective if \(N\subseteq\mathrm{Rad}(R)\). In Chapter III the author specializes to number fields. He obtains the order of \(\mathrm{NB}(R,G)\) (as always, \(G\) is cyclic of order \(p^n)\) if \(p\) is odd and either \(R\) is a finite extension of \(\mathbb Q_p\) or \(R={\mathfrak O}_K(p^{-1})\), where \(K\) is a finite extension of \(\mathbb Q\) with ring of integers \({\mathfrak O}_K\). In the latter case, \(\mathrm{NB}(R,G)=O(1)p^{n(s+1)}\) where \(s\) is the number of pairs of nonreal complex embeddings of \(K\). Chapter IV then presents the author's description of \(\mathrm{Gal}(R,\mathbb Z_ p)\) when \(R={\mathfrak O}_K(p^{-1})\), and its connection with Leopoldt's conjecture in cyclotomic fields. The results are adapted from the author's paper cited above, except for some additional information relating the theory with extensions obtained by adjoining torsion points of abelian varieties of CM type. Chapter V studies \(\mathbb Z^p\)-extensions of the function field \(K\) of a variety defined over a number field. Among the results obtained is that all such extensions arise from \(\mathbb Z_p\)-extensions of the largest number field contained in \(K\). The results in this chapter are previously unpublished. The author views the approach in this chapter as an alternative approach to the geometric class field theory of \textit{N. Katz} and \textit{S. Lang} [Enseign. Math., II. Sér. 27, 285--314 (1981; Zbl 0495.14011)]. The final chapter gives an exposition of the author's paper in [Manuscr. Math. 64, 261--290 (1989; Zbl 0705.13004)] which reformulates and generalizes \textit{H. Hasse}'s description [J. Reine Angew. Math. 176, 174--183 (1936; Zbl 0016.05204)] using the ``Artin-Hasse exponential'', of those \(b\) in \(K^*/(K^*)^q\), \(q=p^n\), \(K\) a local field containing \(\mathbb Q_p[\zeta]\), so that the Kummer extension \(K[\root q \of {b}]\) is unramified over \(K\): this is equivalent to describe \(\mathrm{Gal}(R,G)\) where \(R\) is the valuation ring of \(K\). The chapter concludes with the (previously unpublished) construction of a generic \(G\)-Galois extension. The author's work on \(\mathrm{Gal}(R,G)\) which is presented in this monograph is fundamental to the subject. These notes will become a standard reference in the area.
scientific article; zbMATH DE number 462628
scientific article