---
date: 2022-04-18 12:20
modification date: Monday 18th April 2022 12:20:27
title: "flat morphism"
aliases: [flat morphism]
---

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

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

---

# flat morphism

```ad-summary
Constant-dimensional fibers.
```



- For $\mcf\in \mods{\OO_X}$ then $\mathcal{F}$ is a **flat sheaf relative to $Y$** at a point $x \in X$ if $\mcf_{x}\in \mods{\OO_{Y, f(x)}}^\flat$ is [flat](Unsorted/faithfully%20flat.md) as a module.
	- A sheaf $\mathcal{F}$ is a **flat sheaf** if it is flat at every point of $X$.
	  
- A morphism $X \mapsvia f Y$ is a **flat morphism at a point** $x \in X$ if $\OO_{X, x}\in \mods{\OO_{Y, f(x)}}^\flat$ is [flat](Unsorted/faithfully%20flat.md) as a module.
	- A morphism $f$ is a **flat morphism** if $f$ is flat at every point $x \in X$. 
	  
- A scheme $X$ is **flat** if its structure sheaf $\OO_X$ is a flat sheaf.