# Finite morphism

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

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

such that for each ${\displaystyle i}$,

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

is an open affine subscheme ${\displaystyle {\mbox{Spec}}\;A_{i}}$, and the restriction of f to ${\displaystyle U_{i}}$, which induces a map of rings

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

## Properties of finite morphisms

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

## Morphisms of finite type

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

${\displaystyle y^{3}=x^{4}-z}$

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

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

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

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