Geodesic
Algebraic characterization of finite (branched) coverings
Fundamenta Mathematicae, Tome 158 (1998) no. 2, pp. 165-180

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DOI

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.
DOI : 10.4064/fm-158-2-165-180
Keywords: branched covering, open and closed map, ring of continuous functions, flat homomorphism, integral homomorphism 
M. A. Mulero. Algebraic characterization of finite (branched) coverings. Fundamenta Mathematicae, Tome 158 (1998) no. 2, pp. 165-180. doi: 10.4064/fm-158-2-165-180
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