---
date: 2022-04-24 17:48
modification date: Sunday 24th April 2022 17:48:36
title: "Artinian algebra"
aliases: [Artinian algebra, thickening, nilpotent thickening, "square-zero extension"]
created: 2023-03-31T12:25
updated: 2023-03-31T12:25
---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [formally smooth](Unsorted/formally%20smooth.md)
	- [deformation theory](deformation%20theory.md)

---

# Artinian algebra

![](2023-03-31-15.png)

Definition:
![](attachments/2021-10-03_18-01-54.png)

Use in schemes:
![](attachments/Pasted%20image%2020220407234023.png)
![](2023-03-31-16.png)

The following are local Artinian algebras: $\mathbb{K}[t] /\left(t^n\right)$ with $n>1, \mathbb{K}[[t]]$ and the fields.

## Square-zero extensions

![](attachments/2021-10-03_18-02-35.png)