# Algebraic characterization of finite (branched) coverings

Fundamenta Mathematicae (1998)

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

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topMulero, 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

top- [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. Zbl0264.54011
- [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. Zbl0682.14002
- [5] R. Engelking, General Topology, Heldermann, 1989.
- [6] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976. Zbl0327.46040
- [7] K. R. Goodearl, Local isomorphisms of algebras of continuous functions, J. London Math. Soc. (2) 16 (1977), 348-356. Zbl0381.54005
- [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. Zbl0149.40501
- [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] 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. Zbl0603.13001
- [14] M. A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), 55-66. Zbl0840.54020
- [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] J. C. Tougeron, Idéaux de Fonctions Différentiables, Springer, 1972. Zbl0251.58001

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