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Volume 58 - Issue 4 - December 1993

Contents

Page 2 of 2

Research Article

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

  • Anatole Khelif
  • Published online by Cambridge University Press:
    12 March 2014, pp. 1451-1458

  • 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.

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Front matter

  • JSL volume 58 issue 4 Cover and Front matter

  • Published online by Cambridge University Press:
    12 March 2014, pp. f1-f2
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Back matter

  • JSL volume 58 issue 4 Cover and Back matter and Errata

  • Published online by Cambridge University Press:
    12 March 2014, pp. b1-b10
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Page 2 of 2