In algebraic geometry, a morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes such that for each i, is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism makes Ai a finitely generated module over Bi. One also says that X is finite over Y. In fact, f is finite if and only if for every open affine open subschemeV = Spec B in Y, the inverse image of V in X is affine, of the form Spec A, with A a finitely generatedB-module. For example, for any fieldk, is a finite morphism since as -modules. Geometrically, this is obviously finite since this is a ramified n-sheeted cover of the affine line which degenerates at the origin. By contrast, the inclusion of A1 − 0 into A1 is not finite. This restricts our geometric intuition to surjective families with finite fibers.
Properties of finite morphisms
The composition of two finite morphisms is finite.
Any base change of a finite morphism f: X → Y is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×YZ → Z is finite. This corresponds to the following algebraic statement: if A and C are B-algebras, and A is finitely generated as a B-module, then the tensor productA ⊗BC is finitely generated as a C-module. Indeed, the generators can be taken to be the elementsai ⊗ 1, where ai are the given generators of A as a B-module.
Closed immersions are finite, as they are locally given by A → A/I, where I is the ideal corresponding to the closed subscheme.
Finite morphisms have finite fibers. This follows from the fact that for a field k, every finite k-algebra is an Artinian ring. A related statement is that for a finite surjective morphism f: X → Y, X and Y have the same dimension.
For a homomorphism A → B of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finiteA-algebra, which means that B is finitely generated as an A-module. For example, for any commutative ringA and natural numbern, the polynomial ringA is an A-algebra of finite type, but it is not a finite A-algebra unless A = 0 or n = 0. Another example of a finite-type morphism which is not finite is. The analogous notion in terms of schemes is: a morphism f: X → Y of schemes is of finite type if Y has a covering by affine open subschemes Vi = Spec Ai such that f−1 has a finite covering by affine open subschemes Uij = Spec Bij with Bij an Ai-algebra of finite type. One also says that X is of finite type over Y. For example, for any natural number n and field k, affine n-space and projective n-space over k are of finite type over k, while they are not finite over k unless n = 0. More generally, any quasi-projective scheme over k is of finite type over k. The Noether normalization lemma says, in geometric terms, that every affine schemeX of finite type over a field k has a finite surjective morphism to affine spaceAn over k, where n is the dimension of X. Likewise, every projective schemeX over a field has a finite surjective morphism to projective spacePn, where n is the dimension of X.
