×

On the self-duality of a ring of integers as a Galois module. (English) Zbl 0381.12007

In this paper a conjecture of Fröhlich is proved, namely: Let \(E/K\) be a Galois extension of number fields with \(\mathrm{Gal}(E/K) = \Gamma\), \(\theta_E\) be the ring of integers of \(E\) and \(D_E\) the absolute different of \(E\). If the primes of \(K\) which divide \([E:K]\) are unramified in \(E\) then \(\theta_E\) is stably isomorphic to \(D_k^{-1}\) as a module over the integral group ring \(\mathbb Z\Gamma\). Using the involutory automorphism of \(\mathrm{Cl}(\mathbb Z\Gamma): (M) \to (\bar M)\) obtained from the action of complex conjugation on the characters of \(\Gamma\) [cf. A. Fröhlich, J. Reine Angew. Math. 286/287, 380–440 (1976; Zbl 0385.12004)], the theorem can be stated as follows: \[ (\theta_E)(\bar{\theta}_E) =1. \] The proof is based on the following two facts:
1. The class group \(\mathrm{Cl}(\mathbb Z\Gamma)\) can be represented as a group of certain functions on the characters of \(\Gamma\) [cf. the cited paper]. The class \((\theta_E)(\bar{\theta}_E)\) then becomes \(x \to Nf(x)^{-1}\), the inverse of the absolute norm of the Artin conductor of \(\chi\).
2. \(Nf(x)^{-1}\) represents also the class of a direct sum of certain modules induced by Swan modules of the inertia groups of primes of \(E\). But now since the extension is tame the inertia groups are cyclic so the Swan modules are free and therefore \((\theta_E)(\bar{\theta}_E) =1\).

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers

Citations:

Zbl 0385.12004
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [F1] Fröhlich, A.: Arithmetic and Galois module structure for tame extensions, J. Reine Angew. Math.286/287, 380-440 (1976) · Zbl 0385.12004 · doi:10.1515/crll.1976.286-287.380
[2] [F2] Fröhlich, A.: Locally free modules over arithmetic orders, J. Reine Angew. Math.274/5, 112-124 (1975) · Zbl 0316.12013 · doi:10.1515/crll.1975.274-275.112
[3] [Se 1] Serre, J.-P.: Représentations linéaires des groupes finis, 2e edition, Paris 1971
[4] [Se2] Serre, J.-P.: Corps Locaux, 2e edition, Paris: Hermann 1968
[5] [Sw] Swan, R.G.: Periodic resolutions for finite groups, Ann. of Math.72, 267-291 (1960) · Zbl 0096.01701 · doi:10.2307/1970135
[6] [U] Ullom, S.V.: Non-trivial lower bounds for classgroups, J. of Mathematics (Illinois)20, 361-371 (1976) · Zbl 0334.20005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.