# Abstract Nonsense

## Inverse Limits of Modules (Pt. I)

Point of Post: In this post we introduce the notion of inverse limits, show uniqueness, and give some examples.

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Motivation

As often happens when defining universal objects (i.e. those which can be defined (up to isomorphism) by some mapping properties) the notion of direct limits have a “dual notion”.Very explicitly one can define inverse limits as being the construction one gets by taking the universal characterization of direct limits and reversing all the arrows. We have already seen this kind of duality between products and coproducts, and in fact this shall serve as the main kind of duality between direct and inverse limits since intuitively direct limits are generalized coproducts and inverse limits are generalized products. So, what is the intuition for inverse limits? Roughly one thinks about inverse limits as “zooming in”, in a sense, whereas direct limits are “blowing up”. Inverse limits can be analogized to generalized intersections, the same way direct limits can be analogized to generalized unions.

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Inverse Systems

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We begin with the notion of an inverse system which can be thought as the “dual notion” of an inverse system. Namely, suppose we have some preordered set $\left(\mathcal{A},\leqslant\right)$, then an ordered pair $\left(\{M_\alpha\}_{\alpha\in\mathcal{A}},\{f_{\alpha,\beta}:M_\beta\to M_\alpha\}_{\alpha,\beta\in\mathcal{A},\; \alpha\leqslant\beta}\right)$ (where $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ is a set of left $R$-modules and the $f_{\alpha,\beta}$ are $R$-maps) (note the reversal of the roles of $\alpha,\beta$ as in the case of a directed system) is called an inverse system over $(\mathcal{A},\leqslant)$ (or just an inverse system when the context is clear) if $f_{\alpha,\gamma}=f_{\alpha,\beta}\circ f_{\beta,\gamma}$ and $f_{\alpha,\alpha}=\text{id}_{M_\alpha}$  for all $\alpha\leqslant\beta\leqslant\gamma$ in $\mathcal{A}$.

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Examples of inverse systems of modules abound as much (if not, in some sense, more often) than directed systems. Consider for example the following:

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Let $\{\alpha,\beta,\gamma\}=\mathcal{A}$ be given the order $\alpha\geqslant\gamma$ and $\beta\geqslant\gamma$, then an inverse system over this $\left(\mathcal{A},\leqslant\right)$ set is a diagram of modules of the form

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$M_\alpha\xrightarrow{f_{\gamma,\alpha}}M_\gamma\overset{f_{\gamma,\beta}}{\longleftarrow}M_\beta$

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Let $M$ be some fixed module and let $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ be a set of submodules. Define a preorder on $\mathcal{A}$ by $\alpha\leqslant\beta$ if and only if $M_\beta\subseteq M_\alpha$. We then define maps $f_{\alpha,\beta}:M_\beta\to M_\alpha$ to be the inclusions $M_\beta\hookrightarrow M_\alpha$. This clearly defines an inverse system

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Consider any set $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ of modules and define the trivial direct system (as defined before), this is trivially an inverse system over $\left(\mathcal{A},\leqslant\right)$.

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Let $M$ be a left $R$-module and $\mathfrak{a}$ be a left ideal in $R$ . Define for each $n\in\mathbb{N}$ the module $M_n$ to be equal to $M/\mathfrak{a}^nM$ (where this notion has been defined before). We define mappings $f_{n,m}:M_m\to M_n$ for each $n\leqslant m$ by the rule $x+\mathfrak{a}^mM\mapsto x+\mathfrak{a}^nM$. Note that this actually makes sense since $\mathfrak{a}^mM\subseteq\mathfrak{a}^nM$ (since $\mathfrak{a}^m\subseteq\mathfrak{a}^n$). It’s trivially then that these maps work nicely with composition, i.e. that $f_{n,m}\circ f_{m,k}=f_{n,k}$ for $n\leqslant m\leqslant k$. Thus, this defines an inverse system as desired.

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Inverse Limits

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Now that we have defined the notion of inverse systems, we may continue to define the dual notion to direct limits–inverse limits. Namely, let $(\mathcal{A},\leqslant)$ be a preordered set and $\left(\{M_\alpha\}_{\alpha\in\mathcal{A}}\{,f_{\alpha,\beta}:M_\beta\to M_\alpha\}_{\alpha,\beta\in\mathcal{A},\; \alpha\leqslant\beta}\right)$ be an inverse system over $(\mathcal{A},\leqslant)$. An inverse limit over this inverse system is an ordered pair $(M,\{\pi_\alpha\})$ where $M$ is some fixed module and $\pi_\alpha:M\to M_\alpha$ is a set, called an (inverse) cone, of homomorphisms with $\pi_\alpha=f_{\alpha,\beta}\circ\pi_\beta$ whenever $\alpha\leqslant\beta$ and universal with respect to this property. In other words, given any set of maps $g_\alpha:N\to M_\alpha$ such that $g_\alpha=f_{\alpha,\beta}\circ g_\beta$ there exists a unique $g:N\to M$ with $\pi_\alpha\circ g=g_\alpha$.

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As to be expected at this point, inverse limits are unique up to isomorphism:

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Theorem: Let $(\mathcal{A},\leqslant)$ be a preordered set and $(M,\{\pi_\alpha\})$ and $(N,\{\eta_\alpha\})$ two direct limits of some inverse system $\left(\{M_\alpha\}_{\alpha\in\mathcal{A}},\{f_{\alpha,\beta}:M_\beta\to M_\alpha\}_{\alpha,\beta\in\mathcal{A},\; \alpha\leqslant \beta}\right)$, then $N\cong M$.

Proof: By virtue of the existence of maps $\pi_\alpha:M\to M_\alpha$ and $\eta_\alpha:N\to M_\alpha$ which satisfy the universal properties we get maps $g:M\to N$ and $f:N\to M$ such that $\eta_\alpha\circ g=\pi_\alpha$ and $\pi_\alpha\circ f=\eta_\alpha$. From this we deduce that $\eta_\alpha\circ (g\circ f)=\eta_\alpha$ and $\pi_\alpha\circ (f\circ g)=\pi_\alpha$. But, since $\text{id}_N$ and $\text{id}_M$ are also maps satisfying this, we may conclude by uniqueness that $g\circ f=\text{id}_N$ and $f\circ g=\text{id}_M$ respectively. Thus,  $f$ is an isomorphism and the conclusion follows. $\blacksquare$

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Examples of Inverse Limits

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We now look at what the inverse limits are of the inverse systems we previously defined.

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For our first example consider the module $P=\left\{(a,b)\in M_\alpha\oplus M_\beta:f_{\gamma,\alpha}(a)=f_{\gamma,\beta}(b)\right\}$. It’s easy to see that $P$ with the natural projections $\pi_\alpha:P\to M_\alpha,\pi_\beta:P\to M_\beta$, and the map $\pi_\gamma:P\to M_\gamma:(a,b)\mapsto f_{\gamma,\alpha}(a)$ is an inverse limit for this inverse system. This is called a pull-back of the diagram associated to the inverse system. These, and their direct limit analogues, will have their own post soon enough–they are of the utmost importance.

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Suppose now that we have an inverse system of submodules $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ as we had in our second example. We claim that $U=\displaystyle \bigcap_{\alpha\in\mathcal{A}}M_\alpha$ is an inverse limit of this system with the usual inclusions $\displaystyle \pi_\alpha:U\hookrightarrow M_\beta$. Indeed, it’s clear that $f_{\alpha,\beta}\circ\pi_\beta=\pi_\alpha$ since this merely says that including $U\hookrightarrow M_\beta$ and then including $M_\beta\hookrightarrow M_\alpha$ is the same thing as including $U\hookrightarrow M_\alpha$, which is trivially true. Thus, to show that $U$ really is an inverse limit we need to show that given any left $R$-module $N$ and any set $g_\alpha:N\to M_\alpha$ such that $f_{\alpha,\beta}\circ g_\beta=g_\alpha$ then there exists a unique $j:N\to U$ such that $\pi_\alpha\circ j=g_\alpha$. But, it’s clear that this is true. Namely, we can define $j:N\to U$ by taking $j(n)=g_\alpha(n)$ where $\alpha$ is any index such that $n\in M_\alpha$.

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What we claim is that when we put the trivial preorder on a collection of modules, the product of the modules is an inverse limit. But, this is clear enough since, by definition, we have the projection maps $\displaystyle \pi_\beta:\prod_{\alpha\in\mathcal{A}}\to M_\beta$ and since the only $f_{\alpha,\beta}$ maps to check compatibility against are $f_{\alpha,\alpha}=\text{id}_{M_\alpha}$ we obviously have that the $\pi_\beta$‘s are compatible. The universality then follows by the universality of the product of a set of modules.

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This last example we leave empty, for we shall discuss a result when we discuss the inverse limit of rings that is better.

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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 9, 2011 -

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