Algebraic characterization of finite (branched) coverings

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. Introduction. The aim of this paper is to characterize finite coverings X → S between topological spaces by means of the algebraic properties of the induced homomorphism C(S) → C(X) between the algebras of realvalued continuous functions. Our starting point is the well-known result which states that, in the realm of realcompact spaces, every space X is determined by the algebra C(X) of all real-valued continuous functions defined on it, and that continuous maps between such spaces are in one-to-one correspondence with homomorphisms between their algebras of continuous functions. In view of this equivalence, it seems natural to expect that every property concerning topological spaces and continuous maps may be characterized in terms of the algebras of continuous functions. By a finite (branched) covering we mean an open and closed continuous map π : X → S with finite fibres. Note that a finite covering is not necessarily a local homeomorphism (for example, π : C→ C, π(z) = z2). The set of points in X at which π is not a local homeomorphism is called the branch set of π. In [15] the author gives an algebraic characterization of this branch set in terms of finite C(S)-subalgebras A of C(X) that separate points in X and the module ΩA/C(S) of its Kähler differentials. 1991 Mathematics Subject Classification: 54C10, 54C40, 13C11.