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Algebraic K-groups as Galois modules. (English) Zbl 1011.11074
Progress in Mathematics (Boston, Mass.). 206. Basel: Birkhäuser. x, 309 p. (2002).
The subject of this book (as aptly described by its title) is the study of the structure of K-groups K$$_i(R)$$ as modules over the group ring $${\mathbb Z}[G]$$, where $$R$$ is a commutative ring on which the finite group $$G$$ acts. Actually one assumes more: $$L= \text{Quot}(R)$$ is a $$G$$-Galois extension of another field $$K$$. Mostly, $$R$$ is the ring of integers in a local or global field $$L$$. It is usually difficult to find a priori results about any individual such K-group, and most results involve two “adjacent” K-groups K$$_{2n-1}(R)$$ and K$$_{2n-2}(R)$$. A prototypical example ($$n=1$$) is the analytic class number formula which expresses $$h_LR_L/w_L$$ by arithmetic data; here $$h_L$$, the class number, is the order of the torsion part of K$$_0(O_L)$$, and $$R_L$$ (the regulator) and $$w_L$$ (the number of roots of unity in $$L$$) are quantities attached to $$O_L^*={}$$K$$_1(O_L)$$.
There are now quite a few invariants attached to a set of data $$L/K, G, n$$. For $$n=1$$ we have the now classical Chinburg invariants $$\Omega(L/K,i)$$ with $$i=1,2,3$$, which will be written $$\Omega_0(L/K,i)$$ to leave room for notation of new invariants; for $$n\geq 2$$ there are the constructions $$\Omega_{n-1}(L/K,3)$$ of T. Chinburg, M. Kolster, G. Pappas, and V. P. Snaith [Fields Inst. Commun. 16, 1-29 (1997; Zbl 0886.11063)], along with an alternative construction of Snaith for $$n=2$$. There are constructions of Burns and Flach as well; actually these have a much wider scope, and we only look at the case $$\Omega(L/K,{\mathbb Q}(-n))$$. Then there are the “lifted” versions: on the one hand we have the construction $$T\Omega(L/K,{\mathbb Q}(-n))$$ of Burns (again a special case of a more general construction based upon methods of Bloch and Kato, and of Fontaine and Perrin-Riou), and on the other hand the lifted root number invariants of Gruenberg, Ritter, and Weiss. By results of Burns, Flach, Chinburg, Kolster, and Snaith, these invariants essentially agree whenever this is to be expected: the alternative construction of Snaith for $$n=2$$ does yield $$\Omega_1(L/K,3)$$; the CKPS invariants give the Burns-Flach invariants for commutative $$G$$ on applying the obvious involution on $$K_0(G)$$ induced by $$g\mapsto g^{-1}$$; and the lifted invariants of Burns and Flach for $$n=1$$ do agree with the lifted invariants of Ritter and Weiss. This collection of invariants and their interrelations is a bit overwhelming, but we chose to put the relevant facts together here, for ease of reference.
The book under review provides background, explicit examples, and results, mostly taken from those developments of the theory where the author was actively involved. Chapter 1 contains generalities on $$L$$-functions and a nice example of the invariant $$\Omega_1(M/{\mathbb Q},3)$$ for quaternionic extensions $$M/{\mathbb Q}$$: it is conjectured (as in general) that this comes from the root number class, and this is proved in some cases. Chapter 2 is more technical: it contains useful material in homological algebra and recaps the Fröhlich Hom description. Chapter 3, notwithstanding its heading, also contains the construction of another invariant $$\Omega_n(L/K,2)$$ for global fields $$L/K$$; this proceeds via certain classes associated to local fields, similarly as in Kim’s formula for the classical Chinburg invariant $$\Omega(L/K,2)$$. This chapter, and the next which assumes $$L$$ to be of positive characteristic, provide a lot of calculations related to this invariant. The reviewer cannot judge whether these calculations are conclusive and sufficient to prove an analog of Chinburg’s conjecture.
Chapter 5 recounts the construction of the CKPS invariants and sketches the proof that for $$n=2$$ Snaith’s alternative construction yields the same. For more details we refer to T. Chinburg et al. [K-Theory 14, 319-369 (1998; Zbl 0943.11051)] and T. Chinburg, M. Kolster and the author [Can. J. Math. 52, 47-91 (2000; Zbl 0964.11049)]. It also discusses lifted invariants and very briefly mentions the result of Burns and the reviewer on the vanishing of these invariants in the absolutely Abelian case, which after all entails the vanishing of the unlifted invariants as a corollary.
Chapter 6 puts the focus back on $$L$$-functions and discusses the existence of the so-called Wiles units. Roughly, these are supposed to be units $$\alpha_{n,L/K}$$ of the group ring $${\mathbb Z}_p[[\text{Gal}(L/K)]]$$ (with $$n=2,3,\ldots$$) whose image (in the noncommutative case: whose determinant) under every even character $$\chi$$ of $$L/K$$ is the quotient of the Iwasawa power series associated to $$\chi$$ and its polynomial part, evaluated at $$u^n-1$$. (Here $$u$$ describes the action of a fixed generator $$\gamma$$ of $$\text{Gal}(K_\infty/K)$$ on roots of unity.) Note that this polynomial part is, by the Main Conjecture, the characteristic polynomial of $$\gamma-1$$ acting on the $$\chi$$-part of a standard Iwasawa module. By results of Burns and the reviewer, these Wiles units do exist in case $$L/K$$ is abelian, provided the $$\mu$$-invariant vanishes. Actually there exists a suitable power series which interpolates them all. Section 6.4 (which sketches an approach to the general Wiles unit problem) contains interesting ideas, but it seems rather speculative at this time.
In Chapter 7, the author sketches how one might derive the Coates-Sinnott conjecture from the Wiles unit conjecture. A more precise result to this extent was obtained by Burns and the reviewer: provided $$\mu=0$$ and $$L/K$$ is abelian, a sharpened version of the Coates-Sinnott conjecture is actually true. The author then presents some weaker results which show that the Coates-Sinnott conjecture is true up to radical.
The overview given by the author in this book, the examples, and the technical background material (part of which is difficult to find anywhere else) are quite useful in the opinion of the reviewer. As eloquently indicated in the author’s preface, this is a “story so far”. One might add that it is also to a considerable extent a personal story, and various recent developments are not covered. Nevertheless it provides motivation and preparation for a reader who wishes to consult the recent literature in this rapidly moving field, and the writing is clear and precise throughout.

MSC:
 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11R34 Galois cohomology 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
Keywords:
algebraic K-theory; Galois modules