Quick question about a kind of morphism between algebraic varieties

Question

I'm reading a proof where the author needs to use the Stein's factorization theorem. Reading this theorem, i've found the term finite morphism. What does it mean? Can someone tell me a topological definition?

Answer 1

Fredrik Meyer basically answered your question, but then his answer vanished for some reason. Anyway I think the best one can do in the direction you want is:

finite = proper + quasi-finite

where proper means universally closed, and quasi-finite means that fibres are finite.

Now you may argue that the definition of universally closed requires some knowledge of schemes. Well, sort of, but it's still a topological notion: see the Wikipedia article on proper map for the connection to the classical topological notion of properness.

Answer 2

Let $f: X \rightarrow Y$ ba a morphism between schemes. Then the following conditions are equivalent:

(1). There exists a covering of $Y$ by open affine subsets $V_i =$ Spec$B_i$, such that for each $i$, $f^{-1}(V_i)$ is affine, equal to Spec$A_i$, where $A_i$ is a $B_i$-algebra which is a finitely generated $B_i$-module.

(2). For each open affine subset $V =$ Spec$B$ of $Y$, $f^{-1}(V)$ is affine, equal to Spec$A$, where $A$ is a $B$-algebra which is a finitely generated $B$-module.

If any (and hence both) of these properties are satisfied, $f$ is called a $finite$ morphism.