# Abstract Nonsense

## A Different Way of Looking at Direct Limits

Point of Post: In this post we discuss a perhaps more intuitive way to think about direct limits.

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Motivation

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So, we have already discussed what it means to have a “directed system” of modules and what a “direct limit” of this directed system may look like. To recant,  we can think of a directed system of modules as being a “lattice” (I mean this only in the visualization sense) diagram of modules with the property that whenever one module $M_\alpha$ “sits below” another module $M_\beta$ there is a map $f_{\alpha,\beta}:M_\alpha\to M_\beta$. Intuitively, we thought about these maps as being embeddings, so what we could view $M_\alpha$ as sitting inside $M_\beta$ (while this is true most of the time, we see that there is no actual need to require these maps be embeddings). What we were then curious about (especially in the case that our indexing set was directed) was what the “limiting behavior” of this diagram looked like. Namely, while the diagram may not have a “highest member” (i.e. while our indexing set might not have a largest member, so that there is a module sitting at the “top” of the diagram) we see that, at least intuitively, going up the diagram gives us modules that are getting closer together. Indeed, assume for a second that our indexing set is directed. Then, at least intuitively going high enough allows one to put any two modules “close together”, since if we are dealing with $M_\alpha,M_\beta$ by assumption that our preordered set is directed we can find some $\gamma\geqslant\alpha,\beta$ and so $M_\alpha\to M_\gamma$ and $M_\beta\to M_\gamma$ putting them “close together”. Thus, we see that going up the lattice diagram gives us modules that seem to be “approximating” some module, but most often this module isn’t part of the diagram–the direct limit is this module, the ideal “top module” which ‘should’ be there. One can completely analogize this situation to completing a metric space so that all Cauchy sequences converge. To make the translation more apparent, we add in the direct limit $\varinjlim M_\alpha$ to our lattice diagram to “complete it”, so that that our “Cauchy sequence” of modules actually has a limit–the direct limit.

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That said, even with all of this intuition about what the direct limit “does” it’s hard to get a firm grasp on what it should “look like”. That is the goal of this post, to describe direct limits over directed systems of modules (over directed sets, with injective put-in maps) as “directed unions”. What are directed unions? And, why do they help us understand direct limits? Well, directed unions are just, in essence, unions of submodules, but the collection of submodules (over which we are unioning) is such that we actually get a submodule. I think this will be more clear if one recalls that for two submodules $L,N\leqslant M$ one has that $L\cup N$ is a submodule of $M$ if and only if, without loss of generality, $L\subseteq N$. Thus, we shall see that having a set $\{N_\alpha\}$ of submodules of some fixed module $M$ is exactly the condition one needs so that $\displaystyle \bigcup_{\alpha\in\mathcal{A}}N_\alpha$ is actually a submodule of $N$–moreover (and we have actually already proven this) the union is precisely the direct limit of the directed system of submodules! Now, this construction is easy to picture, it’s just a union. The point of this post shall be to show that (for sufficiently nice examples) all direct limits are of this form! Roughly, once we put our directed system of modules “inside the same module” we will see that the direct limit of this system really is just the directed union of these submodules.

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Direct Limits as Directed Unions

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We begin by defining precisely what we mean by a directed system of submodules and what we mean by their directed union. In particular, let $R$ be some ring and $M$ some fixed left $R$-module. We then define a set $\Omega$ of submodules of $M$ to be directed if we can find a directed set $\left(\mathcal{A},\leqslant\right)$ such that $\Omega$ may be indexed as $\Omega=\left\{N_\alpha\right\}_{\alpha\in\mathcal{A}}$ such that it becomes a  directed system over $\left(\mathcal{A},\leqslant\right)$ with the inclusion maps $\iota_{\alpha,\beta}:N_\alpha\to N_\beta$. Or, said more concretely, $\Omega$ will be directed if and only if given any $N,N'\in \Omega$ one can find some $N''\in \Omega$ with $N\cup N'\subseteq N''$. If $\Omega$ is a directed system of submodules we call the union over $\Omega$ the directed union of $\Omega$. What we claim now is the following:

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Theorem: Let $M$ be a fixed module. Then, a directed set$\Omega$ of submodules of $M$ has the property that its union is also a submodule.

Proof: Assume first that $\Omega$ is directed and let $U$ denote the union over $\Omega$. Let $r\in R$ and $x,y\in U$ be arbitrary. To see that $rx\in U$ merely note that $x\in N$ for some $N\in\Omega$ and since $N\leqslant M$ one has that $rx\in N\subseteq U$. To see that $x-y\in U$ note that by assumption we can find $N,N'\in \Omega$ with $x\in N$ and $y\in N'$. But, since $\Omega$ is directed we can find some $N''\in\Omega$ such that $x,y\in N''$ and since $N''\leqslant M$ this implies that $x-y\in N''\subseteq U$. It follows that $U\subseteq M$ as desired. $\blacksquare$

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For what comes next we need to define the general notion of a cone for a directed system. Namely, given a directed system $\left(\{M_\alpha\}_{\alpha\in\mathcal{A}},\{f_{\alpha,\beta}:M_\alpha\to M_\beta\}_{\alpha,\beta\in\mathcal{A},\;\alpha\leqslant\beta}\right)$ over some preordered set $\left(\mathcal{A},\leqslant\right)$ and some module $N$ we say a cone is a set of maps $\{\varphi_\alpha:M_\alpha\to N\}$ with the property that $\varphi_\beta\circ f_{\alpha,\beta}=\varphi_\alpha$. Thus, we see that limit cones are special cases of cones. The following theorem characterizes when a cone is, in fact, a limit cone:

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Theorem: Let $\left(\mathcal{A},\leqslant\right)$ be a directed set and $\left(\{M_\alpha\}_{\alpha\in\mathcal{A}},\{f_{\alpha,\beta}:M_\alpha\to M_\beta\}_{\alpha,\beta\in\mathcal{A},\; \alpha\leqslant\beta}\right)$ a directed system over $\left(\mathcal{A},\leqslant\right)$. Then, a cone $\{\varphi_\alpha:M_\alpha\to N\}$ is a limit cone if and only if the following conditions are satisfies the following conditions

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\displaystyle \begin{aligned}&\mathbf{(1)}\quad N=\bigcup_{\alpha\in\mathcal{A}}\text{im }\varphi_\alpha\\ &\mathbf{(2)}\quad \ker\varphi_\alpha=\bigcup_{\beta\geqslant\alpha}\ker f_{\alpha,\beta}\end{aligned}

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If $\{\varphi_\alpha\}$ is a limit cone then we further have that $\varphi_\alpha(x)=\varphi_\beta(y)$ if and only if $f_{\alpha,\gamma}(x)=f_{\beta,\gamma}(y)$ for some $\gamma\geqslant\alpha,\beta$.

Proof: Suppose first that $\{\varphi_\alpha:M_\alpha\to N\}$ is a limit cone. We begin by noting that $\left\{\text{im }\varphi_\alpha\right\}_{\alpha\in\mathcal{A}}$ is a directed set of submodules of $N$. To see this we merely need to show that given any $\text{im }\varphi_\alpha,\text{im }\varphi_\beta$ we can find some $\text{im }\varphi_\gamma$ that contains them both. To see this, we merely note that since $\mathcal{A}$ is directed we can find a $\gamma\in\mathcal{A}$ such that $\alpha,\beta\leqslant\gamma$. To see that this implies $\text{im }\varphi_\gamma\supseteq\text{im }\varphi_\alpha,\text{im }\varphi_\beta$ we merely note that by assumption that $\{\varphi_\alpha\}$ is a cone that for any $x\in M_\alpha$ and $y\in M_\beta$ we have that $\varphi_\alpha(x)=\varphi_\gamma(f_{\alpha,\gamma}(x))$ and $\varphi_\beta(y)=\varphi_\gamma(f_{\beta,\gamma}(y))$ from where the desired inclusions clearly follow. From this, and the above theorem concerning directed sets of submodules, we know that the union over $\{\text{im }\varphi_\alpha\}$, call it $U$, is a submodule of $N$. Our goal now is to show that $U=N$. To do this note that we have the obvious maps $\displaystyle \varphi_\alpha:M_\alpha\to U$ which evidently satisfy the compatibility requirement, and since $\{\varphi_\alpha\}$ is a limit cone we know we get a map $j:N\to U$ such that $j\circ\varphi_\alpha=\varphi_\alpha$. Note then that if we consider the inclusion $\iota:U\to N$ we have that $(\iota\circ j)\circ\varphi_\alpha=\varphi_\alpha$ for all $\alpha$, but since $\text{id}_N:N\to N$ also satisfies this property, by uniqueness we may conclude that $\iota\circ j=\text{id}_N$. But, this implies that, as a map, $\iota$ has a right inverse, and thus is surjetive, or equivalently $U=N$. Now, fix some $\alpha\in\mathcal{A}$, we wish to show $\mathbf{(2)}$ holds. But, note that since $N$ is a direct limit of this directed system we have a natural isomorphism $N\cong\varinjlim M_\alpha$, which is such that it suffices to prove this result holds for $latex\ varinjlim M_\alpha$, but this is trivial.

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Conversely, suppose that $\{\varphi_\alpha\}$ is a cone satisfying these two properties. Suppose we are given a set of maps $g_\alpha:M_\alpha\to L$ (where $L$ is some fixed left $R$-module) such that $g_\beta\circ f_{\alpha,\beta}=g_\alpha$. Define a map $j:N\to L$ by taking $n\in N$ to $g_\alpha(x)$ where $n=\varphi_\alpha(x)$ (by $\mathbf{(1)}$ we know that we can find such a representation). To see that this map is well-defined suppose that $n=\varphi_\beta(y)$ as well. Since $\mathcal{A}$ is directed we can choose $\gamma\geqslant \alpha,\beta$ and so we have by assumption that $\varphi_\gamma(f_{\alpha,\gamma}(x))=\varphi_\gamma(f_{\beta,\gamma}(y))$ or $f_{\alpha,\gamma}(x)-f_{\beta,\gamma}(y)\in\ker\varphi_\gamma$. But, by $\mathbf{(2)}$ this allows us to choose $\tau\geqslant\gamma$ such that $f_{\gamma,\tau}(f_{\alpha,\gamma}(x)-f_{\beta,\gamma}(y))=0$ or $f_{\alpha,\tau}(x)=f_{\beta,\tau}(y)$.  Thus,

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$g_\alpha(x)=g_\tau(f_{\alpha,\tau}(x))=g_\tau(f_{\beta,\tau}(y))=g_\beta(y)$

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and so $j$ is well-defined. Note as well that $j(\varphi_\alpha(x))=g_\alpha(x)$ for any $\alpha\in\mathcal{A}$ and $x\in M_\alpha$ so that $j$ satisfies the compatibility relations.  It’s evident that this $j$ is unique by $\mathbf{(1)}$ since if $k$ is another map $N\to L$ satisfying the compatibility relations then $j(\varphi_\alpha(x))=g_\alpha(x)=k(\varphi_\alpha(x))$, and since every element of $N$ is of the form $\varphi_\alpha(x)$ it follows that $j=k$. Thus, $\{\varphi_\alpha\}$ is a limit cone as claimed.

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This last claimed property was proven to be true (in the previous paragraph) if the cone satisfies $\mathbf{(1)}\&\mathbf{(2)}$ and since (by the first paragraph) all limit cones satisfy these properties this follows.$\blacksquare$

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This gives us the satisfying result we’ve been looking for. Namely, if we assume that each $f_{\alpha,\beta}$ is an embedding (as is most natural) then given any direct limit $(M,\{\varphi_\alpha\})$ (where $\mathcal{A}$ is directed) we see by $\mathbf{(2)}$ above that each $\varphi_\alpha$ is an embedding and moreover from the theorem we also see that $M$ is the just the direct union of the $\text{im }\varphi_\alpha\cong M_\alpha$. Thus, direct limits in this case are nothing more than just direct unions where the “factors” had the unfortunate turn of fate to not be born into the same module!

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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.

[5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.