Exact Sequences and Homology (Pt. III)
Point of Post: This is a continuation of this post.
As we have stated before, there is absolutely no reason to believe that a given chain complex is going to be exact and so we’d like to have a measure of how far a chain complex is from being exact. So, how exactly should we go about doing this? Let’s start off with a slightly easier problem by figuring out we can tell if a chain is exact at a given term, say at .
Well, by the mere fact that is exact gives an exact sequence
The question is, when is this an isomorphism? The key to when this is an isomorphism lies within a very interesting property of abelian categories. Let us call a category balanced if every monic and epi arrow is an isomorphism. For example, is balanced for every ring since if an -map is both a mono and an epi (i.e. injective and surjective) then there exists an inverse -map , and thus the map is an isomorphism. Well, as one can probably guess from the buildup, every abelian category is balanced. While there are more elementary ways to see this, it follows quite immediately from the metaprinciple afforded to us by the embedding theorem.
Thus, armed with the fact that abelian categories are balanced, we can safely say that the natural map in is going to be an isomorphism if and only if it is an epimorphism. But, we know from the basic theory that we have the exact sequencece
and, in fact, we know that will be an epimorphism if and only if we have the following equality: .
For notational convenience let us denote as and call it the homology object of . Thus, we see that is exact at if and only if is zero, and thus is exact if is zero for all .
These homology objects are precisely the measures of how far a given chain complex is from being exact.
If our category admits abritrary coproducts (as is often assumed), such as in for some ring , we can define the total homology object of to be . One can then see that is exact if and only if is zero.
For modules there is a simple description of . Indeed, we know that in this case is just the honest to God image of sitting inside and that is the similarly honest literal kernel, also sitting inside . We also know then (since is a chain complex) that and that the map is just the inclusion. We see then that the cokernel is nothing more than the quotient module .
Our first order of business is to prove that homology is actually an additive functor . To do this though we need to specify what the action of should be on chain maps between complexes. To do this let be a chain map.
Let’s start with the case when everything is in and see if we can extrapolate this to the general case. So, to prove we get a well-defined map we need to show that we can get a well-defined map and that the map sends to . Of course we get a map just by restricting to . Thus, the first step is to check that . To see this we merely note that for each , where the last step is because . Thus, we see that restricts to a well-defined map and so evidently, by composing with the canonical projection, a map . To show that this descends to a well-defined map what we must show is that sends to zero, or equivalently that . Once again, the fact that is a chain map saves us since from where the desired containment follows. Thus, restricts and then descends to a well-defined map which takes to .
 Weibel, Charles A. An Introduction to Homological Algebra. Cambridge [England: Cambridge UP, 1994. Print.
 Schapira, Pierre. “Categories and Homological Algebra.” Web. <http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf>.
 Rotman, Joseph. An Introduction to Homological Algebra. Dordrecht: Springer, 2008. Print.