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created: 2023-04-10T20:40
updated: 2023-04-10T20:40
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---
date: 2022-01-15 21:49
modification date: Friday 28th January 2022 18:14:48
title: faithfully flat
aliases: ["flat", "flat module", "flat algebra", "flat morphism", "faithfully flat", "faithfully flat module"]

---

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


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- Tags 
	- #AG 
- Refs:
	- #todo/add-references
- Links:
	- [scheme](scheme.md)
	- [Unsorted/stacks MOC](Unsorted/stacks%20MOC.md)
	- [descent](Unsorted/descent.md)
	- [deformation](deformation.md)
	- [Serre-Swan](Serre-Swan.md)
	- [faithfully flat descent](Unsorted/descent.md)

---

# faithfully flat

![](2023-04-10-5.png)

# Classical flatness

Idea: modules are generalized bundles, and flatness is local triviality.
Generally flats are colimits of free modules.

## Flat modules
![](attachments/Pasted%20image%2020220123175437.png)

## Flat Morphisms
![](attachments/Pasted%20image%2020220123175621.png)

# Faithful flatness

## Faithfully flat modules

Idea: faithfully flat iff tensoring reflects exactness, i.e. $\xi$ is an exact SES iff $\xi\tensor M$ is a SES.
![Pasted image 20220114185822.png](attachments/Pasted%20image%2020220114185822.png)
![](attachments/Pasted%20image%2020220123174337.png)![](attachments/Pasted%20image%2020220123174111.png)![Pasted image 20220115150943.png](attachments/Pasted%20image%2020220115150943.png)
Ideas: 
	- If $F$ is an $R\dash$algebra with structure morphism $f:R\to F$, then $F$ is flat over $R$ when $\spec F\to \spec R$ is a [submersion](submersion), i.e. the induced topology on $\spec R$ is a [quotient topology](quotient%20topology).
	- Open covers are often faithfully flat (surjective) morphisms.
	- Can prove some statement about algebras/schemes are a faithfully flat [base change](base%20change.md).

## Faithfully flat [descent](descent.md)

- A special case of faithfully flat descent: [Zariski descent](Zariski%20descent.md).

- For $S\in \Alg\slice R$ and $M\in \mods{R}$, there is a base-change functor $\Alg\slice R\to \Alg\slice S$ where $X\mapsto X\tensor_R S$ that preserves many properties: e.g. if $M\in \mods{R}^\fg$ then $M\tensor_R S \in \mods{S}^\fg$.

- The reverse implication will hold if $S\slice R$ is faithfully flat.
![Pasted image 20220115181832.png](attachments/Pasted%20image%2020220115181832.png)

See [Unsorted/descent](Unsorted/descent.md).

One can reformulate faithfully flat descent as saying the pseudofunctor QCoh mapping $U$ to quasicoherent sheaves on $U$ is a fpqc stack over Aff $S$. Consequence: the map taking $U$ to the category of affine morphisms $V \rightarrow U$ is a fpqc stack.

# Flat Schemes

![](attachments/Pasted%20image%2020220201135354.png)
![](attachments/Pasted%20image%2020220407234655.png)

# Derived Flatness
![](attachments/Pasted%20image%2020220209190114.png)
# Exercises 

#examples/exercises

- [ ] Show that if $R$ is Noetherian and $M\in \rmod^\fg$ then $R$ is flat iff $R$ is [locally free](locally%20free).
- [ ] Show that $\zmod^\flat$ consists of torsionfree abelian groups.
- [ ] Show that the completion of a local ring $R$ at its maximal ideal is a faithfully flat $R\dash$algebra.
- [ ] Show that localizations can generally fail to be faithfully flat.
- [ ] Show that every field extension is faithfully flat.
 - [ ] Show that a nonzero finitely generated [flat](Unsorted/faithfully%20flat.md) module over a local ring is [faithfully flat](faithfully%20flat.md).
 - [ ] Show that $R\to R[x]$ is always a [faithfully flat](faithfully%20flat.md) morphism.
 - [ ] Show that the category of flat $R\dash$modules is closed under direct sums and summands.
 - [ ] Show that free implies flat.
 - [ ] Show that nonzero vector spaces are faithfully flat.
 - [ ] Show that projections $A\times B\to A$ are flat.
	 - [ ] Show that if $A$ is Noetherian, then every flat quotient map is a projection of this form.
 - [ ] Show that if $A\to B$ is a faithfully flat morphism of rings, then for $M\in\mods{A}$, $M$ is (faithfully) flat iff $B\tensor_A M$ is (faithfully) flat.
 - [ ] Find a flat but not faithfully flat $\ZZ\dash$algebra.
 - [ ] Show that a flat morphism of local rings is faithfully flat.
 - [ ] If $f:R\to S$ is flat, show that $R\plocalize{ (f\inv(\mfp)) } \to S\plocalize{\mfp}$ is faithfully flat.
 - [ ] For $R\to S$ faithfully flat, show that $S$ Noetherian implies $R$ is Noetherian, but the converse is not true.
 - [ ] Show that if $f:R\to S$ is faithfully flat then $f^*: \spec S\to \spec R$ is surjective.
	 - [ ] Conversely, show that if $f$ is flat and $f^*$ is surjective, then $f$ is faithfully flat.
	 - [ ] Show that if $f$ is faithfully flat, then $f^*$ is a quotient map of topological spaces.
 - [ ] Show that being fiathfully flat is preserved under [base change](base%20change.md).
- [ ] ![](attachments/Pasted%20image%2020220404005723.png)