---
date: 2022-01-15 21:49
modification date: Saturday 29th January 2022 22:45:07
title: finite type
aliases: [finite type, "locally of finite type"]

---


Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #AG/schemes 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# finite type

Ideas:

- Finite type: fibers of finite dimension.
- Finite: proper and finite type, so like a [branched cover](branched%20cover)

## For rings/algebras

Recall that $f\in \CRing(B, A) \leadsto A\in \Alg\slice B$.

- $A\in \Alg\slice B^\ft$ is an algebra of **finite type** iff $A$ is finitely generated *as an algebra* over $B$.
- $A\in \Alg\slice B^\fin$ is a **finite** algebra iff $A$ is finitely generated *as an module* over $B$.

So define a morphism of rings $B\to A$ to be **finite type** if it makes $A$ a finite type algebra and **finite** if it makes $A$ a finite algebra.

- Note that being a *finite* algebra is stronger than being *finite type* algebra.

## For schemes

A morphism $f\in \Sch(X, Y)$ is **locally of finite type** iff for every open $\spec B \subseteq Y$, there is a cover $\ts{\spec A_i}_{i\in I}\covers f\inv(\spec B)$ where the induced ring morphisms $f_i^*\in \CRing(B, A_i)$ are finite type as above. 
A morphism $f$ is **finite** if the $f_i^*$ are finite ring morphisms, or equivalently $f$ is locally of finite type and the above cover can be chosen to be finite.

- So $f\in \Sch(X, Y)$ is 
	- **Locally of finite type** iff $f_i^*\in \CRing(B, A_i)$ are finite type iff $A_i \in \Alg\slice B^\ft$ (finite generation as *algebras*)
	- **Finite** iff $f_i^*\in \CRing(B, A_i)$ are finite iff  $A_i\in \Alg\slice B^\fg$ (finite generation as *modules*)
- Some equivalent characterizations of locally of finite type morphisms:
	- The ring morphism $\OO_X(V) \to \OO_X(U)$ is finite type for every $U \subseteq X, V\subseteq Y$ with $f(U) \subseteq V$

- Equivalently: a relative scheme $f:X\to Y$ is finite type iff there is an affine open cover of $Y$ by $U_i = \spec S_i$ where each $f\inv(U_i)$ admits a finite open cover by affine schemes $\spec R_{ij}$ where each $R_{ij}$ is a finitely-generated $S_i\dash$algebra.

![](attachments/Pasted%20image%2020220418115730.png)

# Examples

- There are finite type but non-finite algebras:
$$
\kxn \in \Alg\slice k^\ft\sm \Alg\slice k^\fin
$$
	So $\AA^n\slice k$ is finite type over $k$ but not finite.

- Any [quasiprojective](quasiprojective) object in $\Sch\slice k$ is finite type over $k$.
- [Noether normalization](Unsorted/Noether%20normalization.md): if $X\in \Aff\Sch\slice k^\ft$, then there is a finite surjective morphism $X\surjects \AA^d\slice k$ where $d\da \dim X$.
![](attachments/Pasted%20image%2020220417014434.png)