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Exact sequences and Galois module structure. (English) Zbl 0567.12010

From the introduction: ”Let N/K be a finite normal extension of number fields and let \(G=Gal(N/K)\). In Invent. Math. 74, 321-349 (1983; Zbl 0564.12016) a parallel was developed between the theory of the G-structure of the additive group \({\mathfrak O}_ N\) of integers of N and an analogous theory of the G-structure of the group \(U=U_{N,S}\) of S-units of N when S is a sufficiently large finite set of places of N stable under G. In this paper we study the G-structure of objects which incorporate both the structure of \({\mathfrak O}_ N\) and of U.
Let \(N_ v\) be the completion of N at the place v, and let \(N^{\times}_ v\) be the multiplicative group of \(N_ v\). Define \(U_ v\) to be \(N_ v^{\times}\) if v is Archimedean, and to be the group of integral units of \(N^{\times}_ v\) if v is non-Archimedean. The S- idèles \(J=J_{N,S}\) are \(\prod_{v\in S}N^{\times}_ v \times \prod_{v\not\in S}U_ v\). The objects which we consider are exact sequences \(0\to U\to J_ f\to C_ f\to 0\) of finitely generated modules which are constructed from \({\mathfrak O}_ n\), U, J and the exponential maps associated to finite places in S.
By means of the sequences just mentioned we define invariants \(\Omega\) (N/K,i) \((i=1,2,3)\) in the class group Cl(\({\mathbb{Z}}[G])\) of G which are associated to the G-structure of \(C_ f\), \(J_ f\), and U, respectively. These invariants are not dependent on the choice of the sequences or on any of the other arbitrary choices made in the course of their definition, and are thus invariants of N/K. One is also led to certain natural questions about the relationship of the invariants \(\Omega\) (N/K,i) to the class q(\({\mathcal W}'(N/K))\) of Cassou-Noguès and Fröhlich, which is defined by means of the Artin root numbers of the irreducible symplectic representations of G.”
For further details, see the paper.
Reviewer: R.W.van der Waall

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R32 Galois theory
16S34 Group rings

Citations:

Zbl 0564.12016
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