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Unramified class field theory for orders


Author: Peter Stevenhagen
Journal: Trans. Amer. Math. Soc. 311 (1989), 483-500
MSC: Primary 11R37; Secondary 11R54
DOI: https://doi.org/10.1090/S0002-9947-1989-0978366-3
MathSciNet review: 978366
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Abstract: The main theorem of unramified class field theory, which states that the class group of the ring of integers of a number field $ K$, is canonically isomorphic to the Galois group of the maximal totally unramified abelian extension of $ K$ over $ K$, is generalized and proved for all infinite commutative rings with unit that, like rings of integers, are connected and finitely generated as a module over $ {\mathbf{Z}}$. Modulo their nilradical, these rings are exactly the connected orders in products of number fields.


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  • [1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • [2] Hyman Bass, Algebraic 𝐾-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • [3] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1972.
  • [4] Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548
  • [5] Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
  • [6] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [7] Philip J. Higgins, Notes on categories and groupoids, Van Nostrand Reinhold Co., London-New York-Melbourne, 1971. Van Nostrand Rienhold Mathematical Studies, No. 32. MR 0327946
  • [8] David Hilbert, Über die Theorie der relativ-Abel’schen Zahlkörper, Acta Math. 26 (1902), no. 1, 99–131 (German). MR 1554953, https://doi.org/10.1007/BF02415486
  • [9] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [10] H. W. Lenstra, Jr., Galois theory for schemes, Mathematisch Instituut, Universiteit van Amsterdam, 1985.
  • [11] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • [12] John Milnor, Introduction to algebraic 𝐾-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72. MR 0349811
  • [13] J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Institute of Fundamental Research, Bombay, 1967. Notes by S. Anantharaman; Tata Institute of Fundamental Research Lectures on Mathematics, No 40. MR 0302650
  • [14] P. Stevenhagen, Generalized unramified class field theory, Mathematisch Instituut, Universiteit van Amsterdam, report 85-13, 1985.
  • [15] Edwin Weiss, Algebraic number theory, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR 0159805

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0978366-3
Keywords: Artin isomorphism, Grothendieck group, algebraic fundamental group, van Kampen theorem
Article copyright: © Copyright 1989 American Mathematical Society

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