The Euler Characteristic of an Exact Sequence Vanishes
Naturally, one kind of linear map we’re really interested in is an isomorphism. Such a map has no kernel and no cokernel, and so its index is definitely zero. If it weren’t clear enough already, this shows that isomorphic vector spaces have the same dimension!
First, remember that a linear map whose kernel vanishes looks like this in terms of exact sequences:
And one whose cokernel vanishes looks like this:
So an isomorphism is just an exact sequence like this:
And then we have the equation
Yes, I’m writing the negative of the index here, but there’s a good reason for it.
Now what if we have a segment of an exact sequence:
Considering the map allows us to break up as (since short exact sequences split). On the other hand, considering the map allows us to break up as . Exactness tells us that , which gives us the isomorphism .
Now the rank-nullity theorem says that , and similarly for all other linear maps. So we get — which expresses as the direct sum of one subspace of and one of . And each of those vector spaces has another part to hand off to the vector space on its other side, and so on!
What does this mean? It says that if we look at every other term of an exact sequence and take their direct sum, the result is the same whether we look at the odd or the even terms. More explicitly, let’s say we have a long exact sequence
Then we can decompose each term as either or — one for the even terms and the other for the odd terms. Then direct-summing everything up we have an isomorphism
which tells us that
But since direct sums add dimensions this means
And now we can just combine these sums:
Which generalizes the formula we wrote above in the case of an isomorphism. This alternating sum we call the “Euler characteristic” of a sequence, and we’ll be seeing a lot more of that sort of thing in the future. But here the major result is that for exact sequences we always get the value zero.
In fact, this amounts to a far-reaching generalization of the rank-nullity theorem. And that theorem, of course, is essential to the proof. Yet again we see this pattern of “bootstrapping” our way from a special case to a larger theorem. Despite some mathematicians being enamored of reductio ad absurdum, this induction from special to general has to be one of the most useful tools we keep running across.