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1998 | 158 | 2 | 165-180
Tytuł artykułu

Algebraic characterization of finite (branched) coverings

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Języki publikacji
EN
Abstrakty
EN
Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S) → C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched) covering, i.e., an open and closed map whose fibres are finite, if and only if the induced homomorphism C(S) → C(X) is integral and flat.
Twórcy
autor
  • Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain, mamulero@unex.es
Bibliografia
  • [1] M. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
  • [2] R. L. Blair and A. W. Hagger, Extensions of zero-sets and of real-valued functions, Math. Z. 136 (1974), 41-52.
  • [3] N. Bourbaki, Algèbre Commutative, Chs. 1 and 2, Hermann, 1961.
  • [4] V. I. Danilov, Algebraic varieties and schemes, in: Algebraic Geometry I, I. R. Shafarevich (ed.), Encyclopaedia Math. Sci. 23, Springer, 1994.
  • [5] R. Engelking, General Topology, Heldermann, 1989.
  • [6] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976.
  • [7] K. R. Goodearl, Local isomorphisms of algebras of continuous functions, J. London Math. Soc. (2) 16 (1977), 348-356.
  • [8] A. Grothendieck, Éléments de Géométrie Algébrique IV, Inst. Hautes Études Sci. Publ. Math. 28 (1966).
  • [9] T. Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455-480.
  • [10] L. F. McAuley and E. E. Robinson, Discrete open and closed maps on generalized continua and Newman's property, Canad. J. Math. 36 (1984), 1081-1112.
  • [11] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966.
  • [12] W. S. Massey, Algebraic Topology: An Introduction, Springer, 1967.
  • [13] H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1986.
  • [14] M. A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), 55-66.
  • [15] M. A. Mulero, Rings of continuous functions and the branch set of a covering, Proc. Amer. Math. Soc. 126 (1998), 2183-2189.
  • [16] J. C. Tougeron, Idéaux de Fonctions Différentiables, Springer, 1972.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-fmv158i2p165bwm
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