Finite morphism

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In algebraic geometry, a branch of mathematics, a morphism f: X \rightarrow Y of schemes is a finite morphism if Y has an open cover by affine schemes

V_i = \mbox{Spec} \; B_i

such that for each i,

f^{-1}(V_i) = U_i

is an open affine subscheme \mbox{Spec} \; A_i, and the restriction of f to U_i, which induces a map of rings

B_i \rightarrow A_i,

makes A_i a finitely generated module over B_i.

Properties of finite morphisms[edit]

In the following, f : XY denotes a finite morphism.

  • The composition of two finite maps is finite.
  • Any base change of a finite morphism is finite, i.e. if g: Z \rightarrow Y is another (arbitrary) morphism, then the canonical morphism X \times_Y Z \rightarrow Z is finite. This corresponds to the following algebraic statement: if A is a finitely generated B-module, then the tensor product A \otimes_B C is a finitely generated C-module, where C \rightarrow B is any map. The generators are a_i \otimes 1, where a_i are the generators of A as a B-module.
  • Closed immersions are finite, as they are locally given by A \rightarrow A / I, 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 BA. 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.

Morphisms of finite type[edit]

There is another finiteness condition on morphisms of schemes, morphisms of finite type, which is much weaker than being finite.

Morally, a morphism of finite type corresponds to a set of polynomial equations with finitely many variables. For example, the algebraic equation

y^3 = x^4 - z

corresponds to the map of (affine) schemes \mbox{Spec} \; \mathbb Z [x, y, z] / \langle y^3-x^4+z \rangle \rightarrow \mbox{Spec} \; \mathbb Z or equivalently to the inclusion of rings \mathbb Z \rightarrow \mathbb Z [x, y, z] / \langle y^3-x^4+z \rangle . This is an example of a morphism of finite type.

The technical definition is as follows: let \{V_i = \mbox{Spec} \; B_i\} be an open cover of Y by affine schemes, and for each i let \{U_{ij} = \text{Spec} \; A_{ij}\} be an open cover of f^{-1}(V_i) by affine schemes. The restriction of f to U_{ij} induces a morphism of rings B_i \rightarrow A_{ij}. The morphism f is called locally of finite type, if A_{ij} is a finitely generated algebra over B_i (via the above map of rings). If in addition the open cover f^{-1}(V_i) = \bigcup_j U_{ij} can be chosen to be finite, then f is called of finite type.

For example, if k is a field, the scheme \mathbb{A}^n(k) has a natural morphism to \text{Spec} \; k induced by the inclusion of rings k \to k[X_1,\ldots,X_n]. This is a morphism of finite type, but if n \ge 1 then it is not a finite morphism.

On the other hand, if we take the affine scheme {\mbox{Spec}} \; k[X,Y]/ \langle Y^2-X^3-X \rangle, it has a natural morphism to \mathbb{A}^1 given by the ring homomorphism k[X]\to k[X,Y]/ \langle Y^2-X^3-X \rangle. Then this morphism is a finite morphism.

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