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The Bass-Milnor-Serre theorem for nonstandard models in Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

Anatole Khelif
Anatole Khelif*
Affiliation:
Equipe de Logique Mathématique, Université Paris-VII, 75251 Paris, France, E-mail: [email protected]
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Extract

The aim of this paper is to extend the Bass-Milnor-Serre theorem to the nonstandard rings associated with nonstandard models of Peano arithmetic, in brief to Peano rings.

First, we recall the classical setting. Let k be an algebraïc number field, and let θ be its ring of integers. Let n be an integer ≥ 3, and let G be the group Sln(θ) of (n, n) matrices of determinant 1 with coefficients in θ.

The profinite topology in G is the topology having as fundamental system of open subgroups the subgroups of finite index.

Congruence subgroups of finite index of G are the kernels of the maps Sln(θ) → Sln(θ/I) for which all ideals I of θ are of finite index. By taking these subgroups as a fundamental system of open subgroups, one obtains the congruence topology on G. Every open set for this topology is open in the profinite topology.

We denote by Ḡ (resp., Ĝ) the completion of G for the congruence (resp., profinite) topology.

The Bass-Milnor-Serre theorem [1] consists of the two following statements:

(A) If k admits a real embedding, then we have an exact sequence

That is, Ĝ and Ḡ are isomorphic.

(B) If k is totally imaginary, then one has an exact sequence

where μ(k)is the group of the roots of unity of k.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 4 , December 1993 , pp. 1451 - 1458
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Bass, H., Milnor, J., and Serre, J. P., Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2), Institut des Hautes Études Scientifiques Publícations Mathématiques, vol. 33 (1967), pp. 59–137.CrossRefGoogle Scholar
[2]Selberg, Atle, An elementary proof of Dirichlet's theorem about primes, Annals of Mathematics, vol. 50(1949), pp. 297–304.CrossRefGoogle Scholar
[3]Lang, S., Algebraic number theory, Addison-Wesley Publishing Company, Menlo Park, California, 1970.Google Scholar
[4]Carter, D. and Keller, Gordon, Elementary generation of SLn(θ), American Journal of Mathematics, vol. 105 (1983), pp. 673–686.CrossRefGoogle Scholar
[5]Macintyre, Angus, Residue fields of models of P, Logic, methodology and philosophy of science VI (Hannover 1979), North-Holland Publishing Company and PWN-Polish Scientific Publishers, 1982, pp. 193–206.Google Scholar
[6]Ax, J., The elementary theory of finite fields, Annals of Mathematics, vol. 2 (1968), pp. 239–271.CrossRefGoogle Scholar
[7]Khelif, Anatole, Dirichlet's theorem about primes in nonstandard models of Peano, in preparation.Google Scholar