Recall from proposition 3 here: for an A-module M, is a right-exact functor.
We say M is flat over A (or A-flat) if is an exact functor, equivalently, if
A flat A-algebra is an A-algebra which is flat as an A-module.
We say a ring homomorphism is flat if 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.
If is a collection of A-flat modules, then is A-flat.
Conversely, if is flat, so is .
This follows from the natural isomorphism in N:
The localized ring is A-flat. In particular, A is A-flat.
We have the natural isomorphism in M given by . Then use the fact that localization functors are exact (theorem 1 here). ♦
A free module is flat. More generally, a projective module is flat.
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. ♦
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.
Indeed, we have a natural isomorphism for any A-module N:
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 .
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.
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.
If is an -module and we treat it as an A-module, then as -modules.
Easy exercise. ♦
Local Property for A-Modules
Flatness is a local property, i.e. M is A-flat if and only if for each maximal ideal , is -flat.
(⇒) 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.
Let B be an A-algebra. B is A-flat if and only if for any maximal ideal and , the localization is -flat.
is indeed an algebra over since any is invertible in .
(⇒) 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.
If M is A-flat, we get an isomorphism by multiplication.
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 . ♦
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.
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 ?