# Abstract Nonsense

## Exact Sequences and Homology (Pt. III)

Point of Post: This is a continuation of this post.

Homology Objects

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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 $\mathbf{C}$  is exact at a given term, say at $C_n$.

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Well, by the mere fact that $C_{n+1}\xrightarrow{\partial_{n+1}}C_n\xrightarrow{\partial_n}C_{n-1}$ is exact gives an exact sequence

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$0\to \text{im }\partial_{n+1}\to \ker \partial_n\quad\mathbf{(1)}$

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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, $R-\text{-}\mathbf{Mod}$ is balanced for every ring $R$ since if an $R$-map $M\to N$ is both a mono and an epi (i.e. injective and surjective) then there exists an inverse $R$-map $N\to M$, and thus the map $M\to N$ 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.

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Thus, armed with the fact that abelian categories are balanced, we can safely say that the natural map in $\mathbf{(1)}$ 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

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$0\to\text{im }\partial_{n+1}\to\ker \partial_n\to \text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n)\to 0$

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and, in fact, we know that $\text{im }\partial_{n+1}\to\ker\partial_n$ will be an epimorphism if and only if we have the following equality: $\text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n)=0$.

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For notational convenience let us denote $\text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n)$ as $H_n(\mathbf{C})$ and call it the $n^{\text{th}}$ homology object of $\mathbf{C}$. Thus, we see that $\mathbf{C}$ is exact at $C_n$ if and only if $H_n(\mathbf{C})$ is zero, and thus $\mathbf{C}$ is exact if $H_n(\mathbf{C})$ is zero for all $n\in\mathbb{Z}$.

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These homology objects are precisely the measures of how far a given chain complex is from being exact.

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If our category $\mathscr{A}$ admits abritrary coproducts (as is often assumed), such as in $R\text{-}\mathbf{Mod}$ for some ring $R$, we can define the total homology object $H(\mathbf{C})$ of $\mathbf{C}$ to be $\displaystyle \bigoplus_{n\in\mathbb{Z}}H_n(\mathbf{C})$. One can then see that $\mathbf{C}$ is exact if and only if $H(\mathbf{C})$ is zero.

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For modules there is a simple description of $H_n(\mathbf{C})$. Indeed, we know that in this case $\text{im }\partial_{n+1}$ is just the honest to God image of $\partial_{n+1}$ sitting inside $C_n$ and that $\ker\partial_n$ is the similarly honest literal kernel, also sitting inside $C_n$. We also know then (since $\mathbf{C}$ is a chain complex) that $\text{im }\partial_{n+1}\subseteq\ker\partial_n$ and that the map $\text{im }\partial_{n+1}\to\ker\partial_n$ is just the inclusion. We see then that the cokernel $\text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n)$ is nothing more than the quotient module $\ker\partial_n/\text{im }\partial_{n+1}$.

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Our first order of business is to prove that $n^{\text{th}}$ homology is actually an additive functor $\mathbf{Ch}(\mathscr{A})\to\mathscr{A}$. To do this though we need to specify what the action of $H_n$ should be on chain maps between complexes. To do this let $\mathbf{C}\xrightarrow{\{f_n\}}\mathbf{D}$ be a chain map.

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Let’s start with the case when everything is in $R\text{-}\mathbf{Mod}$ and see if we can extrapolate this to the general case. So,  to prove we get a well-defined map $H_n(\mathbf{C})\to H_n(\mathbf{D})$ we need to show that we can get a well-defined map $\ker\partial_n\to\ker d_n$ and that the map sends $\text{im }\partial_{n+1}$ to $\text{im }d_{n+1}$. Of course we get a map $\ker\partial_n\to D_n$ just by restricting $f_n$ to $\ker\partial_n$. Thus, the first step is to check that $f_n(\ker\partial_n)\subseteq\ker d_n$. To see this we merely note that for each $x\in\text{im }\partial_{n+1}$, $d_n(f_n(x))=f_{n-1}(\partial_n(x))=0$ where the last step is because $\text{im }\partial_{n+1}\subseteq\ker\partial_n$. Thus, we see that $f_n$ restricts to a well-defined map $\ker\partial_n\to\ker d_n$ and so evidently, by composing with the canonical projection, a map $\ker\partial_n\to H_n(\mathbf{D})$. To show that this descends to a well-defined map $H_n(\mathbf{C})\to H_n(\mathbf{D})$ what we must show is that $f_n$ sends $\text{im }\partial_{n+1}$ to zero, or equivalently that $f_n(\text{im }\partial_{n+1})\subseteq \text{im }d_{n+1}$. Once again, the fact that $f_n$ is a chain map saves us since $f_n(\partial_{n+1}(x))=d_{n+1}(f_{n+1}(x))$ from where the desired containment follows. Thus, $f_n$ restricts and then descends to a well-defined map $H_n(\mathbf{C})\to H_n(\mathbf{D})$ which takes $x+\text{im }\partial_{n+1}$ to $f_n(x)+\text{im }d_{n+1}$.

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References

[1] Weibel, Charles A. An Introduction to Homological Algebra. Cambridge [England: Cambridge UP, 1994. Print.

[2] Schapira, Pierre. “Categories and Homological Algebra.” Web. <http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf&gt;.

[3] Rotman, Joseph. An Introduction to Homological Algebra. Dordrecht: Springer, 2008. Print.