Direct Limits of Rings
Let be a directed system of rings. Regard them as a directed system of abelian groups (i.e. ℤ-modules) and take the direct limit A.
The abelian group A has a natural structure of a commutative ring.
General philosophy of the direct limit: “if something happens at index i and another thing happens at index j, then by picking k greater than i and j, both things may be assumed to happen at the same index”. The cautious reader is advised to fill in the gaps in the following proof.
Sketch of Proof
For each , let be the canonical map.
Given , by proposition 2 here there exists and such that , . Now define multiplication in A by . This does not depend on our choice of j and . Indeed, if we have another index and such that , , by proposition 2 here again pick greater than i and j such that
This gives us the desired equality
Clearly product in A is commutative. To show associativity, given , pick and such that , and . Then .
To define , we pick any index and set . ♦
1. Prove that in the above proof is well-defined, and for all .
2. Prove that the resulting ring A with the canonical gives the direct limit of in the category of rings.
3. Prove that if the direct limit of rings is zero, then for some i. [ Hint: a ring is zero if and only if 1 = 0. ]
4. Suppose is an arbitrary collection of A-algebras. For each finite subset , define . Define a directed system of over the directed set of all finite subsets of I, ordered by inclusion . The tensor product of over A is defined to be the direct limit of this system. Prove that this gives the coproduct of in the category A-algebras.
Since direct limits are denoted by , we will write for the earlier limits and call them inverse limits. For most cases of interest, inverse limits will be taken over J such that is directed.
Even over directed sets, taking the inverse limit is not exact. A useful criterion for determining exactness is given by the Mittag-Lefler condition, which we will not cover (for now).
We have seen many constructions which commute and some which do not. In the following examples, is an arbitrary collection of modules; M and N are modules, B is an A-algebra and is a multiplicative subset.
Case 1 (proposition 1 here): but in general.
Case 2 (exercise A here): but .
Case 4 (exercise B here): for any collection of submodules of M, we have but in general.
Case 5 (exercise A here): if M is a flat A-module, then is a flat B-module.
Case 6 (proposition 1 here): we have .
Case 7: more generally, we have if is a diagram of A-modules of type J.
Case 8: hence we have an isomorphism of B-modules; in particular .
Case 9 (proposition 3 here): if is surjective, then is surjective; however, if f is injective is not injective in general.
Prove that there is always a canonical map between and . Find an example where the map is not an isomorphism.
If M is a projective A-module, must be a projective B-module?
Remembering all the above relations may seem like a pain: in general if we have n constructions, we have about relations to learn. It turns out most of these constructions can be classified as either “left-adjoint-like” or “right-adjoint-like”, which saves us a whole lot of effort in remembering them.
In the following table, constructions on the same side tend to commute or have consistent properties. Constructions on different sides may commute under specific additional conditions (e.g. finiteness, noetherianness).
|Sum of submodules||Intersection of submodules|
|Right-exact functors||Left-exact functors|
|Pushouts||Pullbacks, fibre products|
|Direct sum of modules||Direct product of modules|
|Tensor products of modules||Hom modules|
|HomA(-, M)||HomA(M, -)|
|Colimits / Direct limits||Limits / Inverse limits|
|Injective maps||Surjective maps|
|Induced modules .||(Coinduced modules)|
|Projective / free modules||(Injective modules)|
|Multiplying ideal by module:|
Do consider this table as a very rough guide. For example, if is a short exact sequence of A-modules, we do not get a right-exact sequence . [ Take and . ]
Also note that the terms in brackets have not been defined yet.
1. We have .
2. The functor takes a right-exact sequence to a left-exact sequence.
3. Recall that the colimit of a diagram of A-modules was constructed by taking a quotient of the direct sum of these modules. Dually, its limit can be constructed by taking a submodule of the direct product.
4. The tensor product was constructed by taking a quotient of a free (hence projective) module.
Exercise C (Coinduced Modules)
1. Let B be an A-algebra. Prove that for an A-module M, , the set of all A-linear maps , has a natural structure of a B-module.
2. Prove that we get a functor
such that there is a natural bijection
for any B-module N. Thus is right-adjoint to the forgetful functor .
We call the coinduced B-module from M.