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Algebraic characterization of finite (branched) coverings

M. Mulero

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 2, page 165-180
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Mulero, M.. "Algebraic characterization of finite (branched) coverings." Fundamenta Mathematicae 158.2 (1998): 165-180. <http://eudml.org/doc/212309>.

@article{Mulero1998,
abstract = {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.},
author = {Mulero, M.},
journal = {Fundamenta Mathematicae},
keywords = {branched covering; open and closed map; ring of continuous functions; flat homomorphism; integral homomorphism},
language = {eng},
number = {2},
pages = {165-180},
title = {Algebraic characterization of finite (branched) coverings},
url = {http://eudml.org/doc/212309},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Mulero, M.
TI - Algebraic characterization of finite (branched) coverings
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 2
SP - 165
EP - 180
AB - 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.
LA - eng
KW - branched covering; open and closed map; ring of continuous functions; flat homomorphism; integral homomorphism
UR - http://eudml.org/doc/212309
ER -

References

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  1. [1] M. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, 1969. 
  2. [2] R. L. Blair and A. W. Hagger, Extensions of zero-sets and of real-valued functions, Math. Z. 136 (1974), 41-52. Zbl0264.54011
  3. [3] N. Bourbaki, Algèbre Commutative, Chs. 1 and 2, Hermann, 1961. 
  4. [4] V. I. Danilov, Algebraic varieties and schemes, in: Algebraic Geometry I, I. R. Shafarevich (ed.), Encyclopaedia Math. Sci. 23, Springer, 1994. Zbl0682.14002
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  6. [6] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976. Zbl0327.46040
  7. [7] K. R. Goodearl, Local isomorphisms of algebras of continuous functions, J. London Math. Soc. (2) 16 (1977), 348-356. Zbl0381.54005
  8. [8] A. Grothendieck, Éléments de Géométrie Algébrique IV, Inst. Hautes Études Sci. Publ. Math. 28 (1966). 
  9. [9] T. Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455-480. Zbl0149.40501
  10. [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. Zbl0552.54005
  11. [11] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966. 
  12. [12] W. S. Massey, Algebraic Topology: An Introduction, Springer, 1967. 
  13. [13] H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1986. Zbl0603.13001
  14. [14] M. A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), 55-66. Zbl0840.54020
  15. [15] M. A. Mulero, Rings of continuous functions and the branch set of a covering, Proc. Amer. Math. Soc. 126 (1998), 2183-2189. Zbl0893.54008
  16. [16] J. C. Tougeron, Idéaux de Fonctions Différentiables, Springer, 1972. Zbl0251.58001

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