Abstract Nonsense

Exact Sequences of Modules

Point of Post: In this post we discuss the notion of exact sequences of modules, short exact sequences, characterizations of when short exact sequences split, etc.

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Motivation

We now come to what shall be one of the most important linguistic tools in our discussion of modules (and eventually much more general objects)–exact sequences. So, what precisely are exact sequences? Well, often times when we are discussing algebraic objects we care quite a bit about quotient objects and subobjects of some given object (e.g. quotient groups and subgroups, quotient rings and subrings ,etc.) An important characteristic of subobjects and quotient objects is that they both have equipped morphisms attached to the initial object. To be specific, if we have an abelian group $G$ and a  subgroup $N\leqslant G$ we have the natural maps $N\hookrightarrow G$ and $G\twoheadrightarrow G/N$. Or, “lining things up” we have $N\hookrightarrow G\twoheadrightarrow G/N$. Note the interesting thing though, is that if we attempt to do what’s natural given a series of maps like this, we compose them, we get nothing more than the zero map. In fact, the kernel of the map out of $G$ is precisely the image of the map going into $G$! Ok, so what? What if we had instead started with abelian groups $A,B,C$ and suppose we had a similar situation $A\overset{f}{\rightarrowtail}B\overset{g}{\twoheadrightarrow}C$ with $\text{im }f=\ker g$. We see then (by the first isomorphism theorem)  that $B/\ker g\cong C$ and so $B/\text{im }f\cong C$. But, since $f$ is a monomorphism we can morally regard $\text{im }f$ as being equal to $A$ and so this says that, for all intents and purposes, $A$ is a subgroup of $B$ and $B/A\cong C$. From this it’s not hard to see that a lot of more non-trivial examples of abelian groups and maps satisfy this property. For example, $A\rightarrowtail A\oplus B\twoheadrightarrow B$ (where the first map is the canonical injection and the second is) and more generally $A\rightarrowtail A\rtimes B\twoheadrightarrow B$. Thus, we see that, the idea of sequences $A\rightarrowtail B\twoheadrightarrow$, with $\ker\left(B\twoheadrightarrow C\right)=\text{im}\left(A\rightarrowtail\right)$ tell us that “roughly” $G$ is “made up” of $A,C$ (how though isn’t at all apparent). Thus, if we are able to put abelian groups, or in our more general case modules, in sequences of maps of this form we can tell a lot about the modules in question.  More generally, we shall see that putting maps in “chains” of the form $\cdots\to A\to B\to C\to\cdots$ where the kernel of the map out of each object coincides with the image of the map into each object tell us a lot about the modules involved. Consequently, if we hope to be able to put modules into this form, it makes sense to say precisely what being in that form gives us–i.e. given that we are able to put a sequence of modules into such a mapping scheme what can we say about the modules or the maps involved? What if we impose more strict information on the maps involved, or the modules themselves? This is precisely the subject of this post, to glean these properties.

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Exact Sequences

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Given a ring $R$ and left $R$-modules $M,N,L$ we say that the “chain” of $R$-morphisms $M\xrightarrow{f}N\xrightarrow{g}L$ is exact at $N$ if $\ker g=\text{im }f$. A “chain” of maps and modules

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$\cdots M_{-1}\xrightarrow{f_0}M_0\xrightarrow{f_1}M_1\xrightarrow{f_2}M_2\to\cdots$

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is called exact if it’s exact at each module. There is a particular class of exact sequences that interest us. Before we define them we note that there is only one meaning for $0\to M$ and $M\to 0$ where $0$ is the zero module and $M$ is some left $R$-module (i.e. $0$ is a “zero object” in the category $R\text{-}\bold{Mod}$) with $0\to M$ being the map sending $0\mapsto 0$ and $M\to 0$ is given by $m\mapsto 0$. So, interesting things start to happen if you start putting this zero module into exact sequences. For example, it’s clear by definition that $0\to M\xrightarrow{f} N$ is exact if and only if $f$ is a monomorphism. Similarly, $M\xrightarrow{f}N\to 0$ is exact if and only if $f$ is an epimorphism. Combining these gives $0\to M\xrightarrow{f}N\to 0$ is exact if and only if $f$ is an isomorphism.  Thus, when we have a sequence of modules starting and ending with $0$ with only two modules in between, the theory is fairly easy. What shall interest us (which is the special case we discussed in the motivation) o short exact sequences which are exact sequences of the form

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$0\to M\xrightarrow{f}N\xrightarrow{g}L\to 0$

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Of course, we could equally well write a short exact sequence as $M\overset{f}{\rightarrowtail}N\overset{g}{\twoheadrightarrow}L$ from where the motivation gives us the following two short exact sequences:

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$0\to N\to G\to G/N\to0\;\text{ and }\; 0\to A\to A\oplus B\to B\to 0$

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where $A,B,G,N$ are abelian groups (or more generally any left $R$-module, as we shall soon see).

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We shall see that the notion of exact sequences makes the idea of diagram completion very easy. For example:

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Theorem: Let $R$ be a ring and consider the following a diagram of left $R$-modules and $R$-morphisms

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$\displaystyle \begin{matrix} & & & & A & & & \\ & & & & \big\downarrow{^h} & & \\ 0 & \longrightarrow & X & \underset{f}{\longrightarrow} & Y & \underset{g}{\longrightarrow} & Z\end{matrix}$

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where the bottom row is exact and $g\circ h=0$. Then, the diagram can be commutatively completed with a map $t:A\to X$, and moreover this $t$ is unique.

Proof: Since $0\to X\xrightarrow{f} Y$ is exact we know that $f$ is a monomorphism. Moreover, if $h(x)\in\text{im }h$ then $g(h(x))=0$ so that $h(x)\in\ker g=\text{im }f$. Thus, $\text{im }h\subseteq\text{im }f$. The rest follows from our discussion of triangle completions. $\blacksquare$

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What this theorem is saying, in simple terms, is that if we have an embedding of some module $M$ into the module $N$ (call its image $\overline{M}$), and if we have a mapping $g:N\to L$ with kernel $\overline{M}$ then any time we have a fourth module $P$ and a map $P\to N$ with $\text{im }P\subseteq\overline{M}$ then we can factor the mapping through $M$ (obviously, because for all intents and purposes $M=\overline{M}$).

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The last thing we’d like to mention are the ‘morphisms’ between exact sequences (or more general chains, but that doesn’t concern us right now). Namely, we’d like to define a notion of a kind of map between exact sequences that tells us that they are “similar”. So, what precisely should this mean? Suppose we are given two chains (exact sequences for now) $(A_n,f_n)$ and $(B_n,g_n)$ (where it’s clear what this is short-hand for). A morphism between them $c:(A_n,f_n)\to (B_n,f_n)$ should affect each of the individual modules and each of the individual morphisms. To be particular, $c$ should descend to morphisms $c_n:A_n\to B_n$, but that these morphisms should respect the $f_n,g_n$‘s (in the sense that the morphisms should tell us that the $f_n$ and $g_n$ are related). So, how should this pan out, what should this relationship between $f_n$ and $g_n$ be? Well, considering the context we are in, it’s not surprising that the condition we seek is that the resulting square diagrams (shown below) should all commute

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$\displaystyle \begin{matrix}\cdots & A_{n-1} & \xrightarrow{f_n} & A_n & \xrightarrow{f_{n+1}} & A_{n+1} & \xrightarrow{f_{n+2}} & A_{n+2} & \cdots\\ & ^{c_{n-1}}\big\downarrow & & ^{c_n}\big\downarrow & & {^{c_{n+1}}}\big\downarrow & & ^{c_{n+2}}\big\downarrow & \\ \cdots & B_{n-1} & \xrightarrow{g_{n}} & B_n & \xrightarrow{g_{n+1}} & B_{n+1} & \xrightarrow{g_{n+2}} & B_{n+2} & \cdots\end{matrix}$

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Intuitively this is telling us that we can think of $B_{n+1}$ as “being $A_{n+1}$” (in a loose sense) and that under this way of thinking $f_{n+1}$ and $g_{n+1}$ induce similar maps $A_n\to B_{n+1}$. So that, in a perfect world, where everything is an isomorphism this roughly tells us that $A_n$ and $B_n$ are exactly the same and that the way $f_{n+1}$ transforms $A_n\to B_{n+1}$ is exactly the same as the way $g_{n+1}$ transforms $B_n\to B_{n+1}$ except they had the misfortune of being born into different, ostensibly non-interacting universes (of course the $c_n$‘s are the bridge between these universes). We call such a collection of maps $c=\{c_n\}_{n\in\mathbb{N}}$ a chain map between $(A_n,f_n)$ and $(B_n,g_n)$ and are liable to write $c:(A_n,f_n)\to (B_n,g_n)$. If each $c_n$ is an isomorphism of modules we are apt to call $c$ a chain isomorphism.

References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. Module Theory. Clarendon, 1990. Print.

November 3, 2011 -

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