# Abstract Nonsense

## Exact Sequences and Homology (Pt. II)

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

Exact Sequences

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Now that we have given ample motivation let’s discuss some actual technical details.

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So here’s the setup, we have some abelian category $\mathscr{A}$ and some chain of morphisms $X\xrightarrow{f}Y\xrightarrow{g}Z$ (where the rest of the objects in the chain are zero). We can then set up the following diagram

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$\begin{array}{ccccc}\text{coim }f & & & & \\ ^{s}\big\uparrow & & & & \\ X & \xrightarrow{f} & Y & \xrightarrow{g} & Z\\ ^{k_1}\big\uparrow & & ^{k_2}\big\uparrow & &\\ \ker f& & \ker g & & \end{array}$

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Now, note that since $g\circ f=0$ (by assumption that it’s a chain) we get that $f$ factors through $\ker g$ uniquely to give some arrow $X\xrightarrow{u}\ker g$ making the following diagram commute

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$\begin{array}{ccccc}\text{coim }f & & & & \\ ^{s}\big\uparrow & & & & \\ X & \xrightarrow{f} & Y & \xrightarrow{g} & Z\\ ^{k_1}\big\uparrow & ^u\searrow & ^{k_2}\big\uparrow & &\\ \ker f& & \ker g & & \end{array}$

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Note though that $k_2\circ g\circ k_1=f\circ k_1=0$ and since $k_2$ is monic (being the kernel map) we may conclude that $g\circ k_1=0$. Thus, we get an arrow $\text{coim }f\xrightarrow{e}\ker g$ making the following diagram commute

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$\begin{array}{ccccc}\ker f & \xrightarrow{k_1} & X & \xrightarrow{s} & \text{coim }f\\ & & ^{u}\big\downarrow & \swarrow^{e} & \\ & & \ker g & & \end{array}$

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This is what we’ll call the canonical arrow $\text{coim }f\to\ker g$. Now, since $\mathscr{A}$ is abelian we have a canonical arrow $\text{im }f\xrightarrow{\approx}\text{coim }f$ (the inverse of the usual canonical arrow $\text{coim }f\xrightarrow{\approx}\text{im }f$ which is the “first isomorphism theorem” which holds in $\mathscr{A}$). Composing these two arrows gives the canonical arrow $\text{im }f\to \ker g$. We shall call $X\xrightarrow{f}Y\xrightarrow{g}Z$ exact if $\text{im }f\xrightarrow{\approx}\ker g$ (i.e. if the arrow is invertible).

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Ok, to keep ourselves grounded let’s work through this exactly when we are dealing with $\mathscr{A}=R\text{-}\mathbf{Mod}$ for some ring $R$. Let’s start by reworking our diagram to account for what we know kernels and coimages look like

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$\begin{array}{ccccc}X/\ker f & & & & \\ ^{\pi}\big\uparrow & & & & \\ X & \xrightarrow{f} & Y & \xrightarrow{g} & Z\\ \big\uparrow & & \big\uparrow & &\\ \ker f& & \ker g & & \end{array}$

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Where $\pi$ is the projection map and the unmarked maps with the kernels are inclusion. The first step was constructing the map $u:X\to \ker g$. But this map came from the fact that $g\circ f=0$ (using the universal characterization of kernels and so, in fact, $u=f$–in other words our map is just $f:X\to \ker g$ which is just tantamount to the fact that $\text{im }f\subseteq \ker g$. The next step was then to consider the map we get $X/\ker f\to \ker g$ via the fact that $f$ precomposed with the inclusion $\ker f\hookrightarrow X$ was zero. But, if one recalls this maps just is given by $x+\ker f\mapsto f(x)$. Now, the first isomorphism theorem gives us an isomorphism $\text{im }f\to X/\ker f$ with $f(x)\mapsto x+\ker f$. We see then that the natural map $\text{im }f\to \ker g$ is the composition of these maps which is $f(x)\mapsto x+\ker f\mapsto f(x)$. Thus, we see that the natural map is just the inclusion $\ker f\hookrightarrow\ker g$ and since this map is always injective it will be an isomorphism if and only if it’s surjective which is true if and only if $\ker f=\ker g$.

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The above shows that we aren’t making the stuff up. The general notion of exactness really is the natural generalization of “normal exactness” in the categories $R\text{-}\mathbf{Mod}$

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More generally if we have a chain complex $\mathbf{C}$ we say that $\mathbf{C}$ is exact at $C_n$ if $C_{n}\xrightarrow{\partial_n}C_{n-1}\xrightarrow{\partial_{n-1}}C_{n-2}$ is exact. We call $\mathbf{C}$ exact if it’s exact at $C_n$ for all $n\in\mathbb{Z}$.

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Some obvious things are obvious. For example, since (as we have previously proven) a morphism is mono if and only if it has zero kernel, and is an epi if and only if it has zero cokernel it’s easy to see that the chain $0\to X\xrightarrow{f}Y\xrightarrow{g}Z\to0$ is exact if and only if it is exact at $Y$, $f$ is a mono, and $g$ is an epi (I leave this to you). Such a sequence is called short exact.

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Another obvious fact is that every morphism $X\xrightarrow{f}Y$ gives rise to the following short exact sequences $0\to\ker f\to X\to\text{coim }f\to0$ and $0\to\text{im }f\to Y\to\text{coker }f\to 0$.

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Now, let’s give some examples of some short exact sequences.  Of course short exact sequences abound in categories such as $\mathbf{Ab}$. For example, one always has sequences such as $2\mathbb{Z}\hookrightarrow\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}$, or more generally $B\hookrightarrow A\to A/B$ where $B\leqslant A$. We also have sequences such as $\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ where the first map is multiplication by $2$. Another rich source of exact sequences (and one which shall be very important for us soon enough) is the presentation of a given abelian group.

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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.