Algebraic characterization of finite (branched) coverings

• Autorzy: M. A. Mulero
• 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.

Słowa kluczowe

EN

branched covering; open and closed map; ring of continuous functions; flat homomorphism; integral homomorphism

Kategorie tematyczne

• 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
• 54C40: Algebraic properties of function spaces
• 13C10: Projective and free modules and ideals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1998
• Tom: 158
• Numer: 2
• Strony: 165-180

Daty

wydano

1998

otrzymano

1997-08-19

poprawiono

1998-06-22

Twórcy

autor

M. A. Mulero

• Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain,

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.