---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "finitely presented"
aliases: [finitely presented, locally of finite presentation, of finite presentation]
---

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

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
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# finitely presented 

# Ring morphisms

A ring map $R\to A$ is **finitely presented** iff $A\cong R[x_1,\cdots, x_n]/\gens{f_1,\cdots, f_m}\in \ralg$.
![](attachments/Pasted%20image%2020220116002438.png)


# Scheme morphisms

- For affines: a morphism $f\in \Sch(\spec A, \spec B)$ is **of finite presentation** iff the induced ring morphism $B\to A$ should be of finite presentation.
- For arbitrary schemes: $f\in \Sch(X, Y)$ is **of finite presentation** iff 
	- $f$ is locally of finite presentation, so there are affine open covers $\mcu\covers X, \mcv \covers Y$ with $f(U) \subseteq V$ and the induced ring morphism $U = \spec B \to V=\spec B$ is of finite presentation, and
	- $f$ is [quasicompact](Unsorted/quasicompact.md) and [quasiseparated](Unsorted/separated.md).
	  
- In terms of [compact](Unsorted/compact%20object%20of%20a%20category.md) objects:

![](attachments/Pasted%20image%2020220421221355.png)