# Flat Modules

Recall from proposition 3 here: for an *A*-module *M*, is a right-exact functor.

Definition.We say M is

flatover A (orA-flat) if is an exact functor, equivalently, ifA

flat A-algebrais an A-algebra which is flat as an A-module.We say a ring homomorphism is

flatif the resulting A-algebra structure on B is flat.

Flat modules and algebras are a little hard to get a handle on, so we will first go through some properties before offering a few examples.

Proposition 1.If is a collection of A-flat modules, then is A-flat.

Conversely, if is flat, so is .

**Proof**

This follows from the natural isomorphism in *N*:

♦

Proposition 2.The localized ring is A-flat. In particular, A is A-flat.

**Proof**

We have the natural isomorphism in *M* given by . Then use the fact that localization functors are exact (theorem 1 here). ♦

Corollary 1.A free module is flat. More generally, a projective module is flat.

**Proof**

Since *A* is flat, by proposition 1, any free module is also flat.

If *P* is projective, is free for some *A*-module *Q*. Thus is flat. By proposition 1 again, *P* is flat. ♦

Proposition 3.If B is a flat A-algebra and M is a flat B-module, then M is a flat A-module.

In particular, if C is a flat B-algebra and B is a flat A-algebra, then C is a flat A-algebra.

**Proof**

Indeed, we have a natural isomorphism for any *A*-module *N*:

♦

**Exercise A**

Prove the following.

- If
*M*,*N*are*A*-flat, so is . - If M is a flat A-module, so is .
- If
*M*is*A*-flat, then is*B*-flat for any*B*-algebra.

As a special case of the last property, we have:

for any multiplicative subset and ideal .

Summary.The above properties can be summarized as follows.

- Flatness is preserved by direct sum and decomposition.
- Localization is flat.
- Projective modules are flat.
- Flatness is transitive.

**Note**

The property of flatness may seem mysterious and obscure for the first-time reader. Roughly speaking, flat algebras can be interpreted geometrically as follows: if is a flat ring homomorphism, the fibres *maintain some consistency over various .* We will see some concrete examples in the next article.

# Flatness is Local

Before we state and prove the local properties, first recall that tensor product commutes with localization:

Also we need the following.

Lemma 1.If is an -module and we treat it as an A-module, then as -modules.

**Proof**

Easy exercise. ♦

## Local Property for *A*-Modules

Proposition 4.Flatness is a local property, i.e. M is A-flat if and only if for each maximal ideal , is -flat.

**Proof**

(⇒) Let *M* be *A*-flat; to show is -flat let be an injective map of -modules. Now

Since *M* is *A*-flat, is injective, and hence so is .

For (⇐) let be an injective homomorphism of *A*-modules. From corollary 2 here, is injective if and only if is injective for each . But this map is just:

obtained by . Since is -flat and is injective, we are done. ♦

## Local Property for *A*-Algebras

Next, we have an even better result for *A*-algebras.

Proposition 5.Let B be an A-algebra. B is A-flat if and only if for any maximal ideal and , the localization is -flat.

**Note **

is indeed an algebra over since any is invertible in .

**Proof**

(⇒) is flat over . Now is a further localization of so it is flat over . Apply transitivity of flatness.

(⇐) Now suppose is flat over for all maximal and . Let be an injective map of *A*-modules and let *K* be the kernel of so we have an exact sequence of *B*-modules

For any maximal ideal we obtain an exact sequence

But (exercise). And since is flat over and the latter is flat over *A*, by proposition 3 is flat over *A*. Thus for all maximal and . ♦

# Ideals and Submodules

Here are some results which are useful for determining when a module is not flat.

Lemma 2.If M is A-flat, we get an isomorphism by multiplication.

**Proof**

Take the exact sequence of *A*-modules. Since tensoring is right-exact we get an exact sequence

,

where the first map is multiplication. If *M* is *A*-flat, the first map is injective; its image is clearly . ♦

**Note**

In fact, the converse is true: if *M* is such that for all *finitely generated ideals* , then *M* is *A*-flat. But we will need more tools than what we have at the moment to prove this.

Now for any *A*-module *M*, we obtain a map as follows.

Such a map satisfies a few nice properties for all modules. Examples:

1. The zero ideal (resp. (1)) maps to the zero submodule (resp. *M*).

2. If , then .

3. We have .

4. For collection of ideals , we have .

When *M* is *A*-flat, we get many more. Lemma 2 tells us multiplication gives an isomorphism . And since is exact, we can apply many of the nice properties of exact functors.

5. For any ideals , we have .

6. If is an *A*-linear map of ideals, we obtain an induced *A*-linear , .

7. Let ; then .

The above correspondence is neither injective nor surjective in general, even when *M* is *A*-flat. [Exercise: find a counter-example for each case.]

As a preview of (much) later chapters, when *M* is faithfully flat, the correspondence is injective.

**Exercise B**

Find an *A*-linear map of ideals of *A* and an *A*-module *M*, such that the *A*-linear map , is *not *well-defined. Thus property 6 is non-trivial.

What can we say about the kernel of addition ?