The Unapologetic Mathematician

Mathematics for the interested outsider

Exact Sequences

A sequence of groups is just a list of groups with homomorphisms going down the list: ...\rightarrow G_{n-1}\rightarrow G_n\rightarrow G_{n+1}\rightarrow .... We’ll use d_n to refer to the homomorphism from G_n to G_{n+1}. We say that a sequence is “exact” if the image of d_{n-1} is the kernel of d_n for each n.

What does this mean? First off, if we start with an element of G_{n-1} and hit it with d_{n-1} we get something in {\rm Im}(d_{n-1}). Since this is the same as {\rm Ker}(d_n), if we now apply d_n we go to the identity element of G_{n+1}. That is, the composition d_n\circ d_{n-1} is always the trivial homomorphism sending all of G_{n-1} to the identity in G_{n+1}.

On the other hand, if we have an element in the kernel of d_n it’s also in the image of d_{n-1}. This means that if an element gets sent to the identity in the next group, it must have come from an element in the previous group.

So why should we care? Well, a number of different things can be very nicely said with exact sequences. If we write \mathbf1 for the group containing only one element, we can set up a sequence: {\mathbf1}\rightarrow G\rightarrow^fH. What does it mean for this sequence to be exact? Well there’s only one homomorphism from \mathbf1 to any group, and its image is just the identity element in G. So the kernel of f is trivial — f is a monomorphism.

Now let’s flip the diagram over to G\rightarrow^fH\rightarrow{\mathbf1}. There’s only one homomorphism possible from any group to \mathbf1, and its kernel is the whole domain. This means that the image of f has to be all of Hf is an epimorphism.

Let’s put these two together to get a sequence {\mathbf1}\rightarrow N\rightarrow^\iota G\rightarrow^\pi H\rightarrow{\mathbf1}. Exactness at N means that \iota is a monomorphism, which we can think of as describing a copy of N sitting inside G. Exactness at H means that \pi is an epimorphism. What does exactness at G mean? The image of \iota is that copy of N, which has to also be the kernel of \pi. That is, H is (isomorphic to) G/N. We call any sequence of this form a “short exact sequence”.

Remember that the First Isomorphism Theorem tells us that we can factor any homomorphism into an epimorphism from the domain onto a quotient, followed by a monomorphism putting that quotient into the codomain. We can use that here to weave any exact sequence out of short exact sequences. Here is the (really cool) diagram:
Weaving a Long Exact Sequence
Each of the diagonal lines is a short exact sequence, and as it says each (nontrivial) group off the main line is the image of one of the homomorphisms on the line and the kernel of the next.

We can also write an exact sequence {\mathbf1}\rightarrow G\rightarrow H\rightarrow{\mathbf1}. This just says that the homomorphism between G and H is an isomorphism. It’s really nice when this shows up in the middle of a longer exact sequence. If we can show that G_{n-1} and G_{n+2} are both trivial the sequence looks like ...\rightarrow{\mathbf1}\rightarrow G_n\rightarrow G_{n+1}\rightarrow{\mathbf1}\rightarrow ..., so G_n and G_{n+1} are immediately isomorphic.

Another way exact sequences show up is in describing the structure of a group. We know that every group is a quotient of a free group. That is, there is some free group F_1 so that F_1\rightarrow G\rightarrow{\mathbf1} is exact. Then the kernel of this projection is another group, so it’s the quotient of another free group F_2. Now the sequence F_2\rightarrow F_1\rightarrow G\rightarrow{\mathbf1} is exact. This is the presentation of G by generators and relations. But the homomorphism from F_2 to F_1 might have a nontrivial kernel — there might be relations between the relations. In that case we can describe those relations as the quotent of another free group F_3: F_3\rightarrow F_2\rightarrow F_1\rightarrow G\rightarrow{\mathbf1} is exact. We can keep going like this to construct an exact sequence called a “free resolution of G“. It’s particularly nice if the process terminates at some point, giving a sequence {\mathbf1}\rightarrow F_n\rightarrow F_{n-1}\rightarrow ...\rightarrow F_2\rightarrow F_1\rightarrow G\rightarrow{\mathbf1}. A free resolution of a group that has only finitely many terms gives a lot of information about the structure of G.

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March 6, 2007 - Posted by | Algebra, Group Homomorphisms, Group theory

3 Comments »

  1. And welcome to my own area of research!

    These free resolutions are pretty near the objects I study. I prefer free resolutions with group algebras, and not groups. Easier to think about. :)

    Comment by Mikael Johansson | March 7, 2007 | Reply

  2. Oh surely. But of course I haven’t said what an algebra is.. yet!

    If nothing else, this whole project is reminding me how much I do know. I should be able to keep this going for quite a while yet.

    Comment by John Armstrong | March 7, 2007 | Reply

  3. i enjoy the interactive session but im a masters student currently studying on delay differential equations.please i will be glad to have any basic knowledge and materials that would help in my study.thanks

    Comment by Esther Oluwaseun | April 25, 2012 | Reply


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