Finite morphism

In algebraic geometry, a morphism f: XY of schemes is a finite morphism if Y has an open cover by affine schemes

${\displaystyle V_{i}={\mbox{Spec}}\;B_{i}}$

such that for each i,

${\displaystyle f^{-1}(V_{i})=U_{i}}$

is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism

${\displaystyle B_{i}\rightarrow A_{i},}$

makes Ai a finitely generated module over Bi.[1] One also says that X is finite over Y.

In fact, f is finite if and only if for every open affine open subscheme V = Spec B in Y, the inverse image of V in X is affine, of the form Spec A, with A a finitely generated B-module.[2]

For example, for any field k, ${\displaystyle {\text{Spec}}(k[t,x]/(x^{n}-t))\to {\text{Spec}}(k[t])}$ is a finite morphism since ${\displaystyle k[t,x]/(x^{n}-t)\cong k[t]\oplus k[t]\cdot x\oplus \cdots \oplus k[t]\cdot x^{n-1}}$ as ${\displaystyle k[t]}$ -modules. Geometrically, this is obviously finite since this 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. (Indeed, the Laurent polynomial ring k[y, y−1] is not finitely generated as a module over k[y].) 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: XY is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y ZZ is finite. This corresponds to the following algebraic statement: if A and C are (commutative) B-algebras, and A is finitely generated as a B-module, then the tensor product AB C is finitely generated as a C-module. Indeed, the generators can be taken to be the elements ai ⊗ 1, where ai are the given generators of A as a B-module.
• Closed immersions are finite, as they are locally given by AA/I, where I is the ideal corresponding to the closed subscheme.
• Finite morphisms are closed, hence (because of their stability under base change) proper.[3] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
• Finite morphisms have finite fibers (that is, they are quasi-finite).[4] 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: XY, X and Y have the same dimension.
• By Deligne, a morphism of schemes is finite if and only if it is proper and quasi-finite.[5] This had been shown by Grothendieck if the morphism f: XY is locally of finite presentation, which follows from the other assumptions if Y is Noetherian.[6]
• Finite morphisms are both projective and affine.[7]

Morphisms of finite type

For a homomorphism AB 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 finite A-algebra, which means that B is finitely generated as an A-module. For example, for any commutative ring A and natural number n, the polynomial ring A[x1, ..., xn] 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 ${\displaystyle \mathbb {C} [t]\to \mathbb {C} [t][x,y]/(y^{2}-x^{3}-t)}$ .

The analogous notion in terms of schemes is: a morphism f: XY of schemes is of finite type if Y has a covering by affine open subschemes Vi = Spec Ai such that f−1(Vi) 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 (that is, over Spec 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 scheme X of finite type over a field k has a finite surjective morphism to affine space An over k, where n is the dimension of X. Likewise, every projective scheme X over a field has a finite surjective morphism to projective space Pn, where n is the dimension of X.

Notes

1. Hartshorne (1977), section II.3.
2. Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
3. Grothendieck, EGA IV, Part 3, Théorème 8.11.1.