# finite morphism

## Affine schemes

Let $X$ and $Y$ be affine schemes, so that $X=\mathrm{Spec}A$ and $Y=\mathrm{Spec}B$. Let $f:X\to Y$ be a morphism^{}, so that it induces a
homomorphism^{} of rings $g:B\to A$.

The homomorphism $g$ makes $A$ into a $B$-algebra^{}. If $A$ is
finitely-generated as a $B$-algebra, then $f$ is said to be a morphism
of *finite type*.

If $A$ is in fact finitely generated^{} as a $B$-module, then $f$ is said
to be a *finite* morphism.

For example, if $k$ is a field, the scheme ${\mathbb{A}}^{n}(k)$ has a natural morphism to $\mathrm{Spec}k$ induced by the ring homomorphism $k\to k[{X}_{1},\mathrm{\dots},{X}_{n}]$. This is a morphism of finite type, but if $n>0$ then it is not a finite morphism.

On the other hand, if we take the affine scheme $$, it has a natural morphism to ${\mathbb{A}}^{1}$ given by the ring homomorphism $$. Then this morphism is a finite morphism. As a morphism of schemes, we see that every fiber is finite.

## General schemes

Now, let $X$ and $Y$ be arbitrary schemes, and let $f:X\to Y$
be a morphism. We say that $f$ is of *finite type* if there exist an
open cover of $Y$ by affine schemes $\{{U}_{i}\}$ and a finite open cover
of each ${U}_{i}$ by affine schemes $\{{V}_{ij}\}$ such that ${f|}_{{V}_{ij}}$
is a morphism of finite type for every $i$ and $j$. We say that $f$
is *finite* if there exists an open cover of $Y$ by affine
schemes $\{{U}_{i}\}$ such that each inverse image, ${V}_{i}={f}^{-1}({U}_{i})$ is
itself affine, and such that ${f|}_{{V}_{i}}$ is a finite morphism of affine
schemes.

### Example.

Let $X={\mathbb{P}}^{1}(k)$ and $Y=\mathrm{Spec}k$. We cover $X$ by two copies of ${\mathbb{A}}^{1}$ and consider the natural morphisms from each of these copies to $\mathrm{Spec}k$. Both of these affine morphisms are of finite type, but are not finite. The covering morphisms patch together to give a morphism from ${\mathbb{P}}^{1}$ to $\mathrm{Spec}k$. The overall morphism is of finite type, but again is not finite.

## References.

D. Eisenbud and J. Harris, The Geometry of Schemes, Springer.

Title | finite morphism |
---|---|

Canonical name | FiniteMorphism |

Date of creation | 2013-03-22 12:51:47 |

Last modified on | 2013-03-22 12:51:47 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 9 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 14A10 |

Classification | msc 14-00 |

Classification | msc 14A15 |

Defines | affine morphism |

Defines | finite type |