# Abstract Nonsense

## Direct Limit of Modules (Pt. II)

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

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Ok, fine, so now we have defined what it means to take a direct limit of a directed system, let’s get our hands a little dirty and find direct limits for each of the examples of directed systems we gave.

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As claimed in the introduction I claim that if we take the described sequence $R^n$ and the obvious inclusions $\iota_{i,j}:R^i\to R^j$  that $R^{\oplus\mathbb{N}}\cong\varinjlim R^n$ with the morphisms $\varphi_n:R^n\to R^{\oplus\mathbb{N}}$ being the obvious inclusions (i.e. padding a $n$-tuple with infinitely many zeros at the tail of the $n$-tuple). Indeed, to prove this we first must show that $\varphi_j\circ \iota_{i,j}=\varphi_i$. To do this we note that

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$\varphi_j(\iota_{i,j}(x_1,\cdots,x_i))=\varphi_j\underbrace{(x_1,\cdots,x_i,0\cdots,0)}_{\in R^j}=\underbrace{(x_1,\cdots,x_i,0,\cdots)}_{\in R^{\oplus\mathbb{N}}}=\varphi_i(x_1,\cdots,x_i)$

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we now need to show that given any set of maps $g_n:R^n\to N$ such that $g_j\circ f_{i,j}=g_i$ there exists a unique $f:R^{\oplus\mathbb{N}}\to N$ such that $f\circ \varphi_n=g_n$. But, this is clear enough. Namely, given some $(x_n)\in R^{\oplus\mathbb{N}}$ choose some $N\in\mathbb{N}$ for which $x_n=0$ for $n\geqslant N$ and define $f((x_n))=g_N((x_1,\cdots,x_N))$. Note that since $g_j\circ\iota_{i,j}=g_i$ this map is well-defined (i.e. independent of choice of $N$). Note that this map satisfies

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$f(\varphi_n(x_1,\cdots,x_n))=f((x_1,\cdots,x_n,0,\cdots))=g_n(x_1,\cdots,x_n)$

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moreover it’s clear that this map is unique since any element of $R^{\oplus\mathbb{N}}$ is of the form $\varphi_n(x_1,\cdots,x_n)$ for some $n\in\mathbb{N}$. Thus, the claim follows.

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Somewhat surprisingly, given a set $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ of morphisms where we have defined the trivial directed system then the coproduct $\displaystyle \bigoplus_{\alpha\in\mathcal{A}}M_\alpha$ is a direct limit. Indeed, if one notes that since the only $f_{\alpha,\beta}$ map defined is $f_{\alpha,\alpha}=\text{id}_{M_\alpha}$ for each $\alpha\in\mathcal{A}$ we see that we may ignore most of the properties a direct limit $(M,\{\varphi_\alpha\})$ must follow. Indeed, with this revelation we see that $(M,\{\varphi_\alpha\})$ is a direct limit precisely when it satisfies the following universal property: given any set of maps $g_\alpha:M_\alpha\to N$ there exists a unique map $j:M\to N$ such that $j\circ\varphi_\alpha=g_\alpha$. We see that this is precisely the universal characterization for a coproduct of the set $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ of modules. We also see that this gives us some intuition about how “big” a direct limit can be. Indeed, since the relations between the $M_\alpha$ are trivial (obviously) in the trivial directed system we see that the coproduct represents the “freest” possible gluing of the $\{M_\alpha\}$ and so we can expect any other directed system with underlying set of modules $\{M_\alpha\}$ to have a “smaller” direct limit. This is intuition rings fairly true, as shall be made clear soon enough.

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Let us now see if we can find a nice direct limit of the directed system consisting of $\{C^0(U)\}_{U\in\mathcal{U}_x}$ and $\{\text{res}_{U,V}:C^0(U)\to C^0(V)\}$. Intuitively a direct limit of this system should be a collection of functions who fundamentally don’t agree near $x$. To be particular, since we are taking the limit of this system over a shrinking set of neighborhoods it should be clear (at least intuitively) that if two functions are to agree on a neighborhood of $x$ then they will be “equivalent” in the direct limit. Cashing in on this intuition we can define the set $\mathcal{G}_x$ of germs at $x$ to be the set $C^0(X)/\sim$ where $f\sim g$ if and only if there exists some $U\in\mathcal{U}_x$ with $f_{\mid U}=g_{\mid U}$. It’s easy to verify that $\mathcal{G}_x$ is a $\mathbb{R}$-space. It is not hard to see that $\varinjlim C^0(U)\cong\mathcal{G}_x$ where the morphisms $C^0(U)\to \mathcal{G}_x$ are given by $f\mapsto [f]$.

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Lastly, we had the example where we considered some fixed module $M$ and defined $\mathcal{F}$ to be its set of finitely generated submodules ordered by inclusion. We noted then that if we consider $\{A\}_{A\in\mathcal{F}}$ and the inclusions $\{\iota_{A,B}:A\hookrightarrow B\}_{A,B\in\mathcal{F},\; A\subseteq B}$ formed a direct system. We claim then that $\underset{^\mathcal{F}}{\varinjlim}A\cong M$. Indeed, define $\varphi_A:A\to M$ to be the usual inclusion $A\hookrightarrow M$. We note then that evidently $\varphi_B\circ\iota_{A,B}=\varphi_A$. Moreover, given any set of mappings $g_A:A\to N$ for every $A\in\mathcal{F}$ with $g_B\circ \iota_{A,B}=g_A$ we can merely define $j:M\to N$ by $j(x)=g_{\langle x\rangle}(x)$ where $\langle x\rangle$ denotes, as per usual, the submodule generated by $x$ and claim that $j$ satisfies the needed properties. To see this we just have to note that $j(\varphi_A(x))=j(x)=g_{\langle x\rangle}(x)=g_A\iota_{\langle x\rangle,A}(x)=g_A(x)$ (since we know that since $A$ is a submodule containing $x$ that $\langle x\rangle\hookrightarrow A$). Moreover, it’s clear that $j$ is the unique such function, and so the claim follows. Thus, we may state this result as

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Theorem: Every module is a direct limit of its finitely generated submodules.

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Ok, fine, so for a bunch of specific examples we have seen what direct limits look like. This is good, it gives us intuition about what direct limits are, what they do, and why they might be interesting. It is now in our general interest though to prove that direct limits don’t exist in general, to verify that we aren’t just fantastically lucky pickers and that direct limits always exist given a directed system of modules. Indeed the following not only verifies that direct limits always exist (for given directed systems) but moreover that our intuition about directed limits being, in general, smaller than coproducts (they are, in fact, always a homomorphic image of a coproduct):

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Theorem: Let $R$ be a ring, $\left(\mathcal{A},\leqslant\right)$ a preordered 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. Consider the direct sum $\displaystyle M=\bigoplus_{\alpha\in\mathcal{A}}M_\alpha$ and consider the submodule

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$I=\left\langle \iota_\beta f_{\alpha,\beta}(x_\alpha)-\iota_\alpha(x_\alpha):\alpha\leqslant\beta,\; x_\alpha\in M_\alpha\right\rangle$

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where $\iota_\alpha$ are the natural injections. Then, $\displaystyle \left(M/I,\{\varphi_\alpha\}\right)$ (where $\varphi_\alpha$ is the composition of $\iota_\alpha$ postcomposed with the canonical projection $\pi:M\to M/I$) is a direct limit of the directed system.

Proof: We must first show that if $\alpha\leqslant\beta$ then $\varphi_\beta\circ f_{\alpha,\beta}=\varphi_\alpha$ but this is immediate since writing $\varphi_\gamma=\pi\circ\iota_\gamma$ and using linearity we have that

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$\varphi_\beta(f_{\alpha,\beta}(x_\alpha))-\varphi_\alpha(x_\alpha)=\pi(\iota_\beta(f_{\alpha,\beta}(x_\alpha))-\iota_\alpha(x_\alpha))=0$

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where the last equality holds since $\iota_\beta(f_{\alpha,\beta}(x_\alpha))-\iota_\alpha(x_\alpha)\in I$. We now need to prove that given any set of maps $g_\alpha:M_\alpha\to N$ there exists a unique map $j:M/I\to N$ such that $j\circ\varphi_\alpha=g_\alpha$. To do this we define $j:M/I\to N$ by the rule $\displaystyle \sum_{\alpha\in\mathcal{A}}(x_\alpha)+I\mapsto \sum_{\alpha\in\mathcal{A}}g_\alpha(x_\alpha)$. Now, note that it’s not apriori clear that $j$ is well-defined. To do this we merely note that if $(x_\alpha)+I=(y_\alpha)+I$ then their difference $(x_\alpha-y_\alpha)\in I$ but then that means that we can write $(x_\alpha-y_\alpha)$ as a finite sum of the form $\displaystyle \sum_{i}a_i((\varphi_\beta(f_{\alpha_i,\beta_i}(x_\alpha))-\varphi_\alpha(x_\alpha))$ and so trivially by design we see that the image of this under $j$ is $\displaystyle \sum_{\alpha}a_i(g_\beta(f_{\alpha,\beta}(x_\alpha))-g_\alpha(x_\alpha))=0$ by the assumption on the $g_\alpha,g_\beta$‘s. Thus, $j$ is well-defined as desired. We are thus left to prove that this was the only conceivable way to define $j$ if we want it to satisfy the golden properties. But, this is clear for clearly how any such $j$ acts on $M/I$ will be determined by how it acts on each element of the form $\varphi_\alpha$ (since the union over each image of $M_\alpha$ under $\varphi_\alpha$ generates $M/I$) and we already are forced by the condition on $j$ to define this in the way we already have. It follows from this that $(M/I,\{\varphi_\alpha\})$ is a direct limit of the direct system as desired.  $\blacksquare$

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This constructed module being an example of a direct limit, and thus the unique direct limit up to isomorphism, shall serve as our standard go-to (the way that the usual direct sum serves as our usual representation for a coproduct). Thus, when we now write $\varinjlim M_\alpha$ we shall mean, unless otherwise stated, the module constructed in the previous theorem.

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This theorem, giving us a stereotypical direct product, indulges our understanding that the direct limit can be thought of as a two-step process: 1) gluing together (the coproduct) and 2) identifying things which are “eventually equal” in the sequence (the modding out by $I$).

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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.

November 30, 2011 -

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