# Abstract Nonsense

## Exact Sequences and Homology (Pt. I)

Point of Post: In this post we discuss how to define exactness for chains in a general abelian category and then discuss the homology objects associated to a chain.

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Motivation

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Last time we discussed the notions of chain complexes in abelian categories (amongst other things). This time we are going to discuss the notion of exact sequences which, in essence, are the “best kind” of exact sequences one can reasonably expect from a general chain complex. We should have a pretty good idea about what exactness means in our favorite categories like $\mathbf{Ab}$–it’s just the old image equals kernel routine. Of course, going from our favorite abelian category to general ones is a task which, by now, should be obvious isn’t always quite easy or obvious. Indeed, how exactly do we define “image equals kernel” when a) our objects aren’t necessarily sets, b) kernels are objects defined only up to isomorphism, and so even if they were sets there is no reason that kernel has to be literally contained inside image, c) . As has been a theme in our development of abelian categories we can replace the notion of “literal equality” in our more standard, tame categories with the notion of “canonical isomorphism”. Though, we shall see that while $\text{im }f=\ker g$ shall be meaningless in a general abelian category, that there will be a canonical maps $\text{im }f\to\ker g$ whose invertibility shall be equivalent to being exact. Here is where it shall be extremely important that we are dealing with abelian categories. Namely, we shall see that in general we shall only get a canonical map $\text{coim }f\to\ker g$ and it’s the fact that there is a natural isomorphism $\text{coim }f\xrightarrow{\approx}\text{im }f$ that allows us to construct our canonical map $\text{im }f\to \ker g$.

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Once we have defined exact sequences we shall define the homology objects of a given chain. Roughly, these will be measures of how far away a given point in a chain is from being exact. What this shall mean is that for each chain $\mathbf{C}$ in $\mathbf{Ch}(\mathscr{A})$ we shall associated objects $H_n(\mathbf{C})$ for $n\in\mathbb{Z}$ such that $\mathbf{C}$ will be exact at $C_n$ if and only if $H_n(\mathbf{C})=0$.

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Ok, so I think it’s about time that I tried (no doubt, to no avail) to explain the reasons that I have come to understand homology is important and why we care about it.

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Very often in mathematics we are curious about whether structure $X$ has property $P$–stupid, right? That said, we often don’t just care whether or not $X$ has property $P$ but, if it does not, to what “degree” does it fail to have property $P$. Allow me to give an example. Suppose we are doing something like finite group theory. Given a structure in this subject, a finite group $G$ we can start asking questions about whether or not $G$ has certain properties–is it simple, is it of finite exponent, is it indecomposable, etc. One question we could ask is whether or not $G$ is abelian, an honest enough question. Of course, it is pretty naive to hope that $G$ will be abelian, even “most” of the time. So, a pretty obvious question is how “badly” does $G$ fail to be abelian? One can start exploring this by discussing the lower central series

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$G\;\unrhd\gamma_2(G)\;\unrhd\cdots$

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Then, a group should be “kind of abelian” if the lower central series eventually terminates (i.e. if $G$ is nilpotent), and if it does eventually terminate the length of the series (how many steps it should take to terminate) is an even finer instrument to measure “failure to be abelian”. So, for example, if we denote the length of the lower central series for $G$ by $\ell(G)$ (where $\ell(G)=\infty$ is an option) then we see that $G$ is abelian if and only if $\ell(G)=1$, and how large $\ell(G)$ is shows how un-abelian $G$ is.

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Another example of this can be thought of as coming up in studying finite field extensions of some field $F$. Namely, we’d like to know if a given extension $k/F$ is Galois. Of course, just as in the case of hoping a group will be abelian, it’s naive to think that this is going to happen often. So, an answer “Yes it is Galois” or “No, it is not Galois” is not quite as helpful. What would be better is if we could get some measurement of how “un-Galois” it is. Here’s a possibility. We know that for finite extensions $|\text{Gal}(k/F)|\leqslant [k:F]$ with equality precisely when $k/F$ is Galois. In fact, more sharply than the above inequality we know that $|\text{Gal}(k/F)| \mid [k:F]$ and thus to an extension $k/F$ we could associate the number $\displaystyle G_{k/F}=\frac{[k:F]}{|\text{Gal}(k/F)|}$. We see then that $G_{k/F}=1$ if and only if $k/F$ is Galois, and moreover how large $G_{k/F}$ is should be an indication of how “un-Galois” $k/F$ is.

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As a last example (last example before the punchline that is) let’s give an analysis example which should be much more “obvious”. Suppose I am looking at $C[X,\mathbb{R}]$ (the space of continuous maps $X\to\mathbb{R}$) for some relatively well-behaved space $X$. Hell, let’s take $X=[0,1]$ so that we are dealing with $C[0,1]$. A fairly natural (albeit, possibly stupid to someone who hasn’t examined such questions before) is to whether $f\in C[0,1]$ is the zero function–as to whether $f\equiv 0$. Well, just as in both of the previous cases there is always a yes or no answer (the function is zero or it isn’t) but its naive to think that all non-zero functions should be equally well considered for this problem. For example, it seems fairly sensible to say that $f(x)=1$ is “less zero” than $f(x)=\frac{1}{2}$. So, what would be nice is if we could find some kind of measurement that indicates how badly a function $f\in C[0,1]$ fails to be zero. Of course, as anyone who’s taken a basic analysis course should know, there are many, many such measurements. For example, define $\|f\|_\infty$ to be equal to $\displaystyle \sup_{x\in[0,1]}|f(x)|$. We see then that $f\equiv 0$ if and only if $\|f\|_\infty=0$ and one could argue that how large $\|f\|_\infty$ gets is an indication of how badly $f$ is non-zero. Another possibility is to define

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$\displaystyle \|f\|_2=\left(\int_0^1 |f(x)|^2\;dx\right)^{\frac{1}{2}}$

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Note then that $\|f\|_2=0$ if and only if $f\equiv 0$ (it’s important that $f$ is continuous here!) and one could argue, equally well to the case of $\|\cdot\|_\infty$, that $\|f\|_2$‘s size is a measure of how non-zero $f$ is. This example highlights the important fact that there is not, nor should one expect there to be, a single God-given to make these “measurements”.

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The basic idea in all of the above cases was that instead of just asking how badly does $X$ fail to be $P$ we assigned some statistic $X\mapsto S(X)$ such that $X$ is $P$ if and only if $S(X)$ is _____ and how “large” $S(X)$ is indicates how far $X$ deviates from being perfect–i.e. from satisfying $P$.

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So, what exactly does this all have to do with exact sequences/homological algebra? Well, believe it or not, the statistic of choice for a lot of objects/scenarios is chain complexes in some abelian category. Indeed, often times the statement “$X$ has property $P$” can be reprhased “The chain $\mathbf{S}(X)$ is exact” for some chain $\mathbf{S}(X)$ in some abelian category $\mathscr{A}$. For example, it’s the founding idea of algebraic topology that a topological space $X$ has “no holes” if it’s associated singular complex

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$\cdots\to \Delta_3(X)\xrightarrow{\partial_3}\Delta_2(X)\xrightarrow{\partial_2}\Delta_1(X)\xrightarrow{\partial_1}\Delta_0(X)\to0\to0\cdots$

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is exact. That said, what we’d really like to do is have some kind of statistic that allows us to measure “how far” a structure $X$ is from satisfying $P$, so we really haven’t done that–we’ve just turned a binary question (it is, or it isn’t question) into another one. That said, if we can find  in general,a measure $M$ of how far a chain complex is from being exact we’ll be golden. Because, then anytime we want to measure anything, in any subject we can just  turn it into a question about some chain complex $\mathbf{C}$ being exact and then apply $M$ to $\mathbf{C}$.

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This measurement shall be turn out to be the homology objects of a chain complex–they are objects that shall (all) be zero when the chain is exact and how badly they fail to zero shall be an indication of how badly the chain complex fails to be zero. Note though that while this last part may seem crazy “how do we measure how far an object in a general abelian category is from being “zero”??” the categories often times support is with their own built in measurements. So, for example, if we had a structure $X$ and we turned the statement “$X$ satisfies property $P$” into the statement “The chain complex $\mathbf{S}(X)$ in $\mathbf{Ch}(\mathbf{Vect}_\mathbb{Q})$ is exact” we can easily measure how badly $\mathbf{S}(X)$ fails to be exact since the homology objects turn this into a problem of seeing how badly objects $V$ in $\mathbf{Vect}_\mathbb{Q}$ fail to be zero, but $\mathbf{Vect}_\mathbb{Q}$ has a great measurement theory to do this: the dimension of a vector space. For example, applying this idea to the case of singular homology for a topological space $X$ gives rise to the Betti numbers of the space.

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Ok, so let’s see if we can sum up everything we have said above. In general, we have some structure $X$ and some property $P$ for which we’d like to decide if $X$ enjoys, and if it does not, how large is the obstruction to this delicious enjoyment. We have discussed a general methodology to do this, namely find a way to rephrase “$X$ has property $P$” to a statement of the form “$\mathbf{S}(X)$ is exact” where $\mathbf{S}(X)$ is some chain complex in some abelian category $\mathscr{A}$. We can then apply our general theory of homology to turn the question of the exactness of $\mathbf{S}(X)$ into a question of how far certain objects of $\mathscr{A}$ are from being zero. Graphically what we’ve done is the following

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$\begin{matrix}\mathcal{C} & \to & \;\;\mathscr{A} & \to & \text{stastics}\\ & \searrow & \nearrow\\ & & \mathbf{Ch}(\mathscr{A}) & \end{matrix}$

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where $\mathcal{C}$ is the class of objects we want to measure how far they deviate from having property $P$. The reason we do all of this, or why we’d ever want to, is that it’s often much easier to go $\mathcal{C}\to\mathbf{Ch}(\mathscr{A})$ than it is to go straight $\mathcal{C}\to\mathscr{A}$, and once we figure out $\mathcal{C}\to\mathbf{Ch}(\mathscr{A})$ we can apply our general techniques of homology to figure out $\mathbf{Ch}(\mathscr{A})\to\mathscr{A}$–this arrow is what homological algebra is, the study of the homology functor.

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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.