Finite Morphism - Properties of Finite Morphisms

Properties of Finite Morphisms

In the following, f : X → Y denotes a finite morphism.

• The composition of two finite maps is finite.
• Any base change of a finite morphism is finite, i.e. if is another (arbitrary) morphism, then the canonical morphism is finite. This corresponds to the following algebraic statement: if A is a finitely generated B-module, then the tensor product is a finitely generated C-module, where is any map. The generators are, where are the generators of A as a B-module.
• Closed immersions are finite, as they are locally given by, where I is the ideal corresponding to the closed subscheme.
• Finite morphisms are closed, hence (because of their stability under base change) proper. Indeed, replacing Y by the closure of f(X), one can assume that f is dominant. Further, one can assume that Y=Spec B is affine, hence so is X=Spec A. Then the morphism corresponds to an integral extension of rings B ⊂ A. Then the statement is a reformulation of the going up theorem of Cohen-Seidenberg.
• Finite morphisms have finite fibres (i.e. they are quasi-finite). This follows from the fact that any finite k-algebra, for any field k is an Artinian ring. Slightly more generally, for a finite surjective morphism f, one has dim X=dim Y.
• Conversely, proper, quasi-finite locally finite-presentation maps are finite. (EGA IV, 8.11.1.)
• Finite morphisms are both projective and affine.