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Taming wild extensions: Hopf algebras and local Galois module theory. (English) Zbl 0944.11038

Mathematical Surveys and Monographs. 80. Providence, RI: American Mathematical Society (AMS). viii, 215 p. (2000).
For a commutative ring \(R\) and any finite group \(G\), the group ring \(R[G]\) is an \(R\)-Hopf algebra, but there are many more Hopf algebras over \(R\). In geometric parlance, \(R\)-Hopf algebras \(H\) are the representing algebras of affine algebraic groups \(G=\text{Spec}(H)\) over \(R\). The focus of the book under review is the action of such Hopf algebras on other \(R\)-algebras in the local case, that is, \(R\) is a ring of integers in a \(p\)-adic field. Concomitantly, one also learns a lot about the Hopf algebras themselves, for instance about their classification.
Although the group ring case is very classical (it encompasses the whole theory of Galois extensions of local fields), the subject as such is fairly recent. Its roots are in Noether’s theorem which gives a criterion for the freeness of the \(R[G]\)-module \(S=O_L\), where \(L/K\) is a \(G\)-Galois extension of local fields and \(R=O_K\); to wit, this happens if and only if the extension is (at most) tamely ramified. However, the first time where actions of \(R\)-Hopf algebras other than group rings were considered, seems to be in the work of L. G. Roberts (1973) who classified the principal homogeneous spaces over certain finite algebraic groups by calculating a first cohomology group. A very important paper in the subject was published in 1987 by Childs. There he produced families of wildly ramified local extensions \(L/K\) which become Galois in a suitable sense as soon as the group ring \(O_K[G]\) of the Galois group \(G\) is replaced by a slightly larger, appropriately chosen Hopf algebra \(H\), a so-called Hopf order in \(K[G]\). All this was done for \(G\) of prime order. At about the same time, Pareigis and the reviewer showed that Hopf Galois theory has something to say even on the field level, that is: there is a systematic way of deciding whether a separable extension \(L/K\) is Hopf Galois over a suitable \(K\)-Hopf algebra. Here \(L\) and \(K\) may be arbitrary fields, and \(L\) need not be Galois over \(K\). Already much earlier, Chase had put Hopf algebras to work for purely inseparable field extensions. Since the late eighties, the subject has developed steadily, acquiring a lot of new facets. It is still in a very lively, which also means unfinished, stage. Lindsay Child’s timely monograph is a very well-done report on where things stand.
This review will not try to go linearly through all the chapters. Instead, it is warmly recommended to look at the book itself. In particular, the introduction gives a much more detailed history of the subject than this review attempts to do. The book is very well organized and pretty much self-contained, and therefore a pleasure to read; it is excellent material for seminars or study groups. highlights include:
– (1) Basic theory of Hopf algebras, with a particularly valuable section on short exact sequences.
– (2) Hopf-Galois structure on separable field extensions (initiated by Pareigis and the reviewer), including Childs’ and Byott’s contributions.
– (3) Classification of rank \(p\) Hopf algebras, due to Tate and Oort; integral Galois structures on cyclic extensions of prime degree. This is work of Childs and others. In particular, every rank \(p\) Hopf algebra \(H\) is realizable, that is, there is an \(H\)-Galois extension \(A/R\) such that \(A\) is the valuation ring in a field extension \(L\) of \(K=\text{Quot}(R)\).
– (4) Hopf orders in groups rings of cyclic groups of order \(p^2\), and their action on cyclic extensions of degree \(p^2\). This is work of the reviewer and others. It turns out that there are in general plenty of Hopf orders, but only a fairly small minority among them is realizable in the above sense.
– (5) Formal groups and their role in Galois theory of local extensions (mainly work of Childs and collaborators, motivated by Lubin-Tate theory).
– This short list is far from complete.
All these matters are, in the reviewer’s opinion, of interest to everybody in algebraic number theory, particularly those with a bias towards the local theory.
The subject of the book has close ties to representation theory and to cohomology theory, the former coming up in the proof of Noether’s theorem and certainly in global Galois module theory, developed by the Fröhlich school. Cohomology plays a major role in the extension theory of Hopf orders, and also in Agboola’s recent breakthrough concerning the surjectivity of the Picard map: every primitive class in \(\text{Pic}(H)\) comes from an \(H\)-Galois extension, under a very mild hypothesis on the finite \(R\)-Hopf algebra \(H\). These important matters are (rather briefly) mentioned in the book, and references are provided. An interested reader will want to pursue these tracks for himself.
This monograph, authored by a leading expert and main contributor in the field, is very painstakingly written. The reviewer only found a single typo: \(S\) is the valuation ring of \(L\), not of \(R\), on line 6 from the bottom on p. 107. The book contains an extensive bibliography, with the extra feature that each reference comes with a list of the sections of the book which quote it, which is a really neat idea. The book offers an ideal road of entry into the subject, for everybody with an interest in local Galois module theory or in the theory of Hopf algebras.

MSC:

11S23 Integral representations
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11S15 Ramification and extension theory
11S20 Galois theory
12F10 Separable extensions, Galois theory
14L05 Formal groups, \(p\)-divisible groups
14L15 Group schemes
16T05 Hopf algebras and their applications
14L30 Group actions on varieties or schemes (quotients)
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