# Limits Are Left-Exact

By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in is a right-exact functor, but there is a rather huge issue here: the functors are between and , the category of diagrams in while we only defined exactness of functors between categories of *modules*. The proper way to do this is to introduce the framework of *abelian categories* and extend our concept of additive functors and exact functors there. However, doing this will take us too far afield so we will prove it directly (which is, admittedly, a bit of a cop out).

Proposition 1.Let J be an index category, and be diagrams of type J. For concreteness, write these diagrams as

where and . Let be morphisms, written as a collection of over . Then

**Note**

*In summary, taking the limit is left-exact while taking the colimit is right-exact.*

**Proof**

We prove the second claim, leaving the first as an exercise. By proposition 1 here, is concretely described as follows. Take the quotient of by all , where is an arrow in *J*, and are identified with their images in .

With this description, clearly is surjective. Also, composing is the zero map so . Now write for and for .

Conversely, let represent an element in the kernel of . Thus is a finite sum of . Since is surjective, we can write such a term as

for some . Since is a finite sum of , we can replace *x* by another representative such that . Then for some . ♦

Neither the limit nor the colimit functor is exact in general. For the colimit case, consider the following commutative diagram of *A*-modules

where all maps are identities. The rows are short exact sequences and the squares all commute, but taking the colimit of the columns gives

which is not exact.

**Exercise A**

Find an example for the case of limits.

# Direct Limits

*We will describe a special case where taking the colimit is exact.*

Given a poset , we recall the category whose objects are elements of *S*, and between any , with equality if and only if . Composition is the obvious one.

Definition.A poset is called a

directed setif for any , there is a such that and .

*In other words, a poset is directed if every finite set has an upper bound.*

Definition.If J is an index category obtained from for some directed set S, then a diagram in of type J is called a

directed system. The colimit of is called thedirect limitand denoted by.

*In other words, direct limit = colimit over directed set*. We will abuse notation a little and regard *J* as the directed set itself.

To avoid set-theoretic difficulties, the directed set *J* is always assumed to be non-empty.

**Example**

In exercise C.3 here, for a multiplicative and *A*-module *M*, we have an isomorphism of *A*-modules

where if *g* is a multiple of *f*. Since *S* is multiplicative, any {*f*, *g*} has an upper bound *fg*. Hence is the *direct limit* of over :

.

Similarly, we have the following direct limit in the category of rings:

.

Next we will discuss the general direct limit in the categories ** A-Mod** and

**Ring**.

# Direct Limit of Modules

Let *A* be a fixed ring; the following holds for direct limits in the category of *A*-modules.

Proposition 2.Suppose is a directed system of A-modules over a directed set J. Let

, with canonical for each .

Then for each , there exists an and such that .

Also if satisfies , then there exists such that .

**Note**

The philosophy is that “whatever happens in the direct limit happens in for some sufficiently large index *j*“.

**Proof**

By proposition 1 here, the colimit *M* is described concretely by taking the quotient of (with canonical ) by relations of the form

Hence any can be written as for . But *J* is a directed set, so we can pick index such that ; then

, where ,

proving the first claim.

For the second claim, if then is a finite sum of the above relations. Pick an index larger than *i* and all indices *k*, *l* in the sum; then is the sum of the images of these relations in . But each such relation has image in , so as desired. ♦

Corollary 1.If is a directed system of A-modules such that are all injective, then

is also injective for each .

Finally we have:

Proposition 3.Let and be directed systems of A-modules and be a morphism of the directed systems, i.e. for any , we have .

If each is injective, so is .

*Since taking the colimit is right-exact by proposition 1, we see that taking the direct limit is exact. *

**Proof**

Write and for the canonical maps.

Suppose for . By proposition 2, we have for some ; then

so by proposition 2 again, there exists such that , so . Since is injective we have so . ♦

**Exercise B**

Describe the direct limit of sets over *J*. State and prove an analogue of proposition 2.