# Abstract Nonsense

## Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. II)

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

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Ok, fine. So we know now that $\varinjlim$ serves as a functor $\mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)\to R\text{-}\mathbf{Mod}$, but how “nice” of a functor is it? Well, while we have yet to discuss the categorical machinery to precisely say how nice it is, we can nonetheless prove that it has this property (e.g. we lack the language to put the following theorem on the proper theoretical ground, but we can prove and use the theorem nonetheless). Roughly what we’d like to show is that the functor $\varinjlim$ perserves exactness. Of course, we have yet to define what exactness should mean for a chain of directed systems, but this isn’t too hard. Namely, we call a chain $\mathscr{A}\xrightarrow{\{v_\alpha\}}\mathscr{B}\xrightarrow{\{w_\alpha\}}\mathscr{C}$, where $\mathscr{A}=\left(\{M_\alpha\},\{f_{\alpha,\beta}\}\right)$, $\mathscr{B}=\left(\{N_\alpha\},\{g_{\alpha,\beta}\}\right)$, and $\mathscr{C}=\left(\{L_\alpha\},\{h_{\alpha,\beta}\}\right)$exact if the chains of $R$-maps $M_\alpha\xrightarrow{v_\alpha}N_\alpha\xrightarrow{w_\alpha}L_\alpha$ are exact for each $\alpha\in\mathcal{A}$. Our meaning then is clear, we’d like to show that $\varinjlim$ carries exact sequences to exact sequences. In fact, to be honest we need to assume that $\mathcal{A}$ is, in fact, directed, but since most real-life examples of preordered sets are directed, this is really no loss. In particular:

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Theorem: Let $\mathscr{A,B,C}$ be elements of $\mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)$ listed as above, with limits cones $\{\eta_\alpha\}$, $\{\varphi_\alpha\}$, and $\{\psi_\alpha\}$ respectively, and assume $\mathcal{A}$ is directed). Suppose further that we have an exact sequence $\mathscr{A}\xrightarrow{\{v_\alpha\}}\mathscr{B}\xrightarrow{\{w_\alpha\}}\mathscr{C}$, then $\varinjlim M_\alpha\xrightarrow{\varinjlim v_\alpha}\varinjlim N_\alpha\xrightarrow{\varinjlim w_\alpha}\varinjlim L_\alpha$ is exact.

Proof: We begin by showing that $\ker\varinjlim w_\alpha\subseteq\text{im }\varinjlim v_\alpha$, to this end suppose that $x\in\ker\varinjlim w_\alpha$. Recalling our characterization of direct limits over directed sets we know that there exists some $\alpha\in\mathcal{A}$ and some $y\in N_\alpha$ such that $x=\varphi_\alpha(y)$. But, we then have that $0=\varinjlim w_\alpha(x)=(\varinjlim w_\alpha\circ \varphi_\alpha)(y)=(\psi_\alpha\circ w_\alpha)(y)$. Thus, we see that $w_\alpha(y)\in\ker\psi_\alpha$, but using the same characterization of direct limits over directed sets as we previously mentioned, there exists some $\beta\geqslant\alpha$ such that $0=h_{\alpha,\beta}(w_\alpha(y))=w_\beta(g_{\alpha,\beta}(y))$. Thus, $g_{\alpha,\beta}(y)\in\ker w_\beta$, but by exactness we may then conclude that there exists $z\in M_\alpha$ such that $v_\beta(z)=g_{\alpha,\beta}(y)$. We see then that

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$\left(\varinjlim v_\alpha\circ\eta_\beta\right)(z)=\varphi_\beta(g_{\alpha,\beta}(y))=\varphi_\alpha(y)=x$

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and so $x\in\text{im }\varinjlim v_\alpha$. Conversely, to prove that $\text{im }\varinjlim v_\alpha\subseteq\ker\varinjlim w_\alpha$ it suffices to show that $\varinjlim w_\alpha\circ\varinjlim v_\alpha=0$. But, by definition of $\varinjlim M_\alpha$ it suffices to show that $\varinjlim w_\alpha\circ\varinjlim v_\alpha\circ\eta_\alpha=0$ for all $\alpha\in\mathcal{A}$. But, this is easy since

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$\varinjlim w_\alpha\circ\varinjlim v_\alpha\circ \eta_\alpha=\varinjlim w_\alpha\circ \varphi_\alpha\circ v_\alpha=\psi_\alpha\circ w_\alpha\circ v_\alpha=\psi_\alpha\circ0=0$

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From these two inclusions we may conclude that $\text{im }\varinjlim v_\alpha=\ker\varinjlim w_\alpha$, and so exactness follows. $\blacksquare$

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Since obviously $\varinjlim$ takes the zero module to the zero module, and the zero map to the zero map we get the following corollary:

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Corollary: Let $\mathscr{A}$ and $\mathscr{B}$ be directed systems such that each $M_\alpha\xrightarrow{v_\alpha}N_\alpha$ is injective, then $\varinjlim v_\alpha:\varinjlim M_\alpha\to\varinjlim N_\alpha$ is injective. Similarly, if each $M_\alpha\xrightarrow{v_\alpha}N_\alpha$ is surjective, then $\varinjlim v_\alpha:\varinjlim M_\alpha\to\varinjlim N_\alpha$ is surjective.

Proof: Since each $v_\alpha$ is injective we have that each $0\to M_\alpha\xrightarrow{v_\alpha}N_\alpha$ is exact, and so by the above theorem and the fact mentioned in the previous paragraph we may conclude that $0\to\varinjlim M_\alpha\xrightarrow{\varinjlim v_\alpha}\varinjlim N_\alpha$ is also exact, and so $\varinjlim v_\alpha$ is injective. The case for surjectivity follows similarly by noting that the exact sequences $M_\alpha\xrightarrow{v_\alpha}N_\alpha\to0$ get sent to the exact sequence $\varinjlim M_\alpha\xrightarrow{\varinjlim v_\alpha}\varinjlim N_\alpha\to0$. $\blacksquare$

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Ok, let’s get more down to earth and look at some examples of how this functor plays out, in some specific cases. Suppose we start with two sets $\{M_\alpha\}$ and $\{N_\alpha\}$  of $R$-modules and we considered it as a directed system over $\left(\mathcal{A},\leqslant_{\text{triv}}\right)$ where $\leqslant_\text{triv}$ is, as usual, the trivial directed system. Suppose further that we have have any set of maps $\left\{v_\alpha:M_\alpha\to N_\alpha\right\}$. We note then that $\{v_\alpha\}$ trivially (by default even) is a morphism in the category $\mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)$ since the only thing we must check is that $\text{id}_{M_\alpha}\circ v_\alpha=v_\alpha\circ \text{id}_{M_\alpha}$ (this is because the only comparable things in $\mathcal{A}$ are elements to themselves). From this we conclude that we may apply the functor $\varinjlim$ to get an $R$-map $\varinjlim M_\alpha\xrightarrow{\varinjlim v_\alpha}\varinjlim N_\alpha$. Note though that by previous discussion $\varinjlim M_\alpha$ and $\varinjlim N_\alpha$ are nothing more than the coproducts $\displaystyle \bigoplus M_\alpha$ and $\displaystyle \bigoplus N_\alpha$ respectively. What we now claim is that $\varinjlim v_\alpha$ is the map often denoted $\displaystyle \bigoplus v_\alpha$ which sends $(x_\alpha)$ to $(v_\alpha(x_\alpha))$. To see this we merely note that $\displaystyle \bigoplus v_\alpha\circ \eta_\alpha=\varphi_\alpha\circ v_\alpha$ where $\eta_\alpha,\varphi_\alpha$ are the natural inclusions $\displaystyle M_\alpha\hookrightarrow \bigoplus M_\alpha$ and $\displaystyle N_\alpha\hookrightarrow \bigoplus N_\alpha$ respectively. But, note that $\{\eta_\alpha\}$ and $\{\varphi_\alpha\}$ are limit cones for $\displaystyle \bigoplus M_\alpha$ and $\displaystyle \bigoplus N_\alpha$, and we know by construction that $\varinjlim v_\alpha$ satisfies this same equation. Thus, by the definition of direct limit we may conclude that $\displaystyle \bigoplus v_\alpha=\varinjlim v_\alpha$.

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Thus, we see from this and the above general theory that if we have a set of exact chains $M_\alpha\xrightarrow{v_\alpha}N_\alpha\xrightarrow{w_\alpha}L_\alpha$ that we may conclude that the chain $\displaystyle \bigoplus M_\alpha\xrightarrow{\bigoplus v_\alpha}\bigoplus N_\alpha\xrightarrow{\bigoplus w_\alpha}\bigoplus L_\alpha$ is also exact (this of course also implies that taking direct sums preserves injectivity and surjectivity).  The most common case of this situation is when we have a single exact sequence $M\xrightarrow{v}N\xrightarrow{w}L$ and we take the $\lambda$-fold coproduct of the chain to conclude that $M^{\oplus\lambda}\xrightarrow{v^{\oplus\lambda}}N^{\oplus\lambda}\xrightarrow{w^{\oplus\lambda}}L^{\oplus\lambda}$ is exact.

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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.

December 28, 2011 -