Short exact sequences
I am claiming the following is not a short exact sequence
I am thinking of it this way: If it was a short exact sequence, then i would be injective and j would be onto. Also, would be isomorphic to .
Now for some . Now, is it true that
? And that ?If yes, how do you show this is not isomorphic to ? If not, what am I doing wrong and how do you prove my claim?
That sequence can't be exact. If you have 0->A->B->C->0, then C is isomorphic to B/A', where A' is the image of A under the map A->B. Because of exactness, that map is injective, so that A' is isomorphic to A. You can't take Z^3, mod out by something isomorphic to Z, and get Z.
Ok, that makes sense, but what would be a good argument to show is not isomorphic to ?
Can we say, has 3 generators, now is isomorphic to which has at least 2 generators, but only has one generator?
You have to be careful when talking about the number of generators, because it's not fixed--for instance, the integers are generated by (1) or by (2,3). I think the free rank of an abelian group might be helpful here.
Or just say it's not exact by definition. To be exact, we would need .
Of course, the other issue is still a good question in and of itself, so don't let this stop that discussion.
I am not sure I understand how helps show is not isomorphic to . Can you elaborate on that?
Also @ Tinyboss: How, do I use the free rank of an abelian group to prove that? My algebra is very rusty, and I tried to look it up, but I don't think I understand how to apply the definition here...
Sorry, what I thought would work, doesn't. I asked a buddy and he said to tensor the sequence with Q, yielding the same sequence except with Q's in place of Z's. Since Q is a field, there is a well-defined notion of dimension, and so that sequence can't be exact. And because tensoring preserves exact sequences, this implies the original sequence wasn't.
I don't understand it 100%, but I'm putting it here since my first guess was wrong.
I don't know that it does. I was just giving another way of answering the original question, about the sequence not being exact.
Originally Posted by math8