# Abstract Nonsense

## Direct Limit of Modules (Pt. I)

Point of Post: In this post we discuss the notion of the direct limit of modules, giving some particular examples of direct limits.

$\text{ }$

Motivation

$\text{ }$

In this post we shall discuss one of the most useful constructions in the entirety of module theory–direct limits. Intuitively direct limits allow us to define a gluing process which makes rigorous sense of things such as the following: “$R^{\oplus\mathbb{N}}$ is the limit of the set of all finite products of $R^n$“. To be more specific we wish to look at the case when we have a “chain” (really, we shall be discussing a more general notion, but chain gives the right idea) of modules and a chain of morphisms

$\text{ }$

$M_1\underset{\leqslant}{\xrightarrow{f_1}} M_2\underset{\leqslant}{\xrightarrow{f_2}}\cdots$

$\text{ }$

and we wish to glue the chain upwards in a way the respects the morphisms–in other words we’d like to, at least intuitively, glue them to get some object $M_\infty$ for which there are always maps $M_i\to M_\infty$ which respect the mappings $M_i\to M_j$ with $i\leqslant j$ in the sense that $M_i\to M_\infty$ should be equal to $M_i\to M_j\to M_\infty$. Thus, in a sense we are really taking a limit of the $M_i,f_i$, allowing us to often realize certain objects as the limit of certain finitary objects. We shall end up seeing that this limit is a fairly faithful representation of the individual $M_i$ in the sense that coproducts are good representation of the factor modules–this shouldn’t be surprising since coproducts are themselves direct limits. All in all, the intuitive idea of direct limits are that they are a two-step process consisting of gluing a set of modules together and then identifying the elements of the gluing which are “eventually equal” (in the sense of the limit).

$\text{ }$

Directed Sets and Directed Systems

$\text{ }$

We define a directed set to be a pair $\left(\mathcal{A},\leqslant\right)$ where $\leqslant$ is some preorder with the property that any two elements of $\mathcal{A}$ possess an upper bound in $\mathcal{A}$ (i.e. for any $a,b\in\mathcal{A}$ there exists some $c\in\mathcal{A}$ such that $a\leqslant c$ and $b\leqslant c$). Directed sets can be seen as a generalization of total orderings (i.e. chains) since while not every two elements of $\mathcal{A}$ are related, we see that there is some interaction between them. For example, given a set $X$ the poset $\left(2^X,\subseteq\right)$ is not totally ordered in general (in fact, it’s only totally ordered if $X$ is a singleton) but it is a directed set since $X$ is always an upper bound for any two elements of $2^X$.

$\text{ }$

Given a preordered set (most often times, it shall be a directed set) $\left(\mathcal{A},\leqslant\right)$ a directed system of module over $\left(\mathcal{A},\leqslant\right)$ is a set of left $R$-modules $\left\{M_\alpha\right\}_{\alpha\in\mathcal{A}}$ and a set of $R$-maps $\left\{f_{\alpha,\beta}:M_\alpha\to M_\beta\right\}$ such that $f_{\alpha,\alpha}=\text{id}_{M_\alpha}$ and $f_{\gamma,\alpha}=f_{\beta,\gamma}\circ f_{\alpha,\beta}$. Intuitively, this second condition means that if we take the natural directed graph for $\left(\mathcal{A},\leqslant\right)$ (i.e. the directed graph with vertices $\mathcal{A}$ and $(\alpha,\beta)$ a directed edge if and only if $\alpha\leqslant\beta$) and replace each vertex $\alpha$ by the corresponding module $M_\alpha$ and each directed line $\alpha\to\beta$ and replace it with the connecting morphisms $M_\alpha\xrightarrow{f_{\alpha,\beta}}M_\beta$ one gets a commutative diagram.

$\text{ }$

Let’s take a look at some examples of directed systems of modules. Take a ring $R$ and consider the $R$-modules $R^n$ for each $n\in\mathbb{N}$. We have the obvious inclusions $\iota_{i,j}:R^i\to R^j$ for $i\leqslant j$ given by $(r_1,\cdots,r_i)\mapsto (r_1,\cdots,r_n,0,\cdots,0)$. Clearly then $\iota_{i,i}=\text{id}_{R^i}$ and $\iota_{i,k}=\iota_{j,k}\circ\iota_{i,j}$ so that $\left\{R^n\right\}_{n\in\mathbb{N}}$ and $\{\iota_{i,j}:R^i\to R^j\}_{i,j\in\mathbb{N},\; i\leqslant j}$ forms a directed system of modules over the directed system (in fact, totally ordered set) $\left(\mathbb{N},\leqslant\right)$.

$\text{ }$

Any set $\left\{M_\alpha\right\}_{\alpha\in\mathcal{A}}$ of modules can be made, in a trivial way, into a directed system by defining the trivial preorder $\leqslant_{\text{triv}}$ on $\mathcal{A}$ (i.e. $\alpha\leqslant\alpha$ and nothing else is comparable) and defining $f_{\alpha,\alpha}=\text{id}_{M_\alpha}$.

$\text{ }$

Consider the following slightly more interesting example. Let $X$ be some topological space and fix some $x\in X$. Let $\mathcal{U}_x$ denote the neighborhood system of $x$ and consider for each $U\in\mathcal{U}_x$ the $\mathbb{R}$-algebra $C^0(U)$ of continuous maps $U\to\mathbb{R}$. Define the obvious preorder $\leqslant$ on $\mathcal{U}_x$ by $U\leqslant V$ if $V\subseteq U$ (so the $\subseteq_{\text{op}}$ for those who have done any category theory or order theory) . This is clearly a directed set since given any two $U,V\in\mathcal{U}_x$ we know that $U\cap V\in\mathcal{U}_x$ and $U,V\leqslant U\cap V$. Consider now the restriction homomorphisms $\text{res}_{U,V}:C^0(U)\to C_0(V)$ given by $\text{res}_{U,V}(f)=f_{\mid V}$ (i.e. restriction of a mapping on $U$ to a mapping on $V$). It’s not hard to see then that $\left\{C^0(U)\right\}_{U\in\mathcal{U}_x}$ and $\{\text{res}_{U,V}:C^0(U)\to C^0(V)\}_{U,V\in\mathcal{U}_x,\; U\leqslant V}$ forms a directed system over the directed set $\left(\mathcal{U}_x,\leqslant\right)$.

$\text{ }$

As a last example consider for a second some fixed $R$-module $M$ and let $\mathcal{F}$ denote the set of all finitely generated submodules of $M$. Clearly  $(\mathcal{F},\subseteq)$ is a directed set. Moreover, it’s clear that if we define for $A\subseteq B$ the inclusion maps $\iota_{A,B}:A\to B$ then it’s not hard to see that $\{A\}_{A\in\mathcal{F}}$ and $\{\iota_{A,B}:A\to B\}_{A,B\in\mathcal{F},\; A\subseteq B}$ is a directed system over $\left(\mathcal{F},\subseteq\right)$.

$\text{ }$

$\text{ }$

Direct Limits

$\text{ }$

Now that we have successfully defined the directed system of modules we can define direct limits over such directed systems. Indeed, given a preordered set $\left(\mathcal{A},\leqslant\right)$ and a directed system consisting of modules $\left\{M_\alpha\right\}_{\alpha\in\mathcal{A}}$ and morphisms $\{f_{\alpha,\beta}:M_\alpha\to M_\beta\}_{\alpha,\beta\in\mathcal{A},\; \alpha\leqslant\beta}$ we define, if one exists, a direct limit of the system to be an ordered pair $(M,\{\varphi_\alpha\})$ where $M$ is a module $M$ and a $\{\varphi_\alpha\}$ a set, called a limit cone, of $R$-maps $M_\alpha\to M\}$ such that $\varphi_\beta\circ f_{\alpha,\beta}=\varphi_\alpha$ whenever $\alpha\leqslant \beta$ and which satisfies the following universal property: given morphism $g_\alpha:M_\alpha\to N$ (where $N$ is some left $R$-module) such that $g_\beta\circ f_{\alpha,\beta}=g_\alpha$ then there exists a unique morphisms $j:M\to N$ such that $j\circ\varphi_\alpha=g_\alpha$.

$\text{ }$

If we have a preordered set $\left(\mathcal{A},\leqslant\right)$ and a directed system consisting of $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ and $\{f_{\alpha,\beta}:M_\alpha\to M_\beta\}_{\alpha,\beta\in\mathcal{A},\; \alpha\leqslant\beta}$ we may denote that a module $M$ is a direct limit of this directed system by writing $\displaystyle M\cong\underset{\mathcal{A}\ni\alpha}{\varinjlim}M_\alpha$, or when no confusion will arise, $\varinjlim M_\alpha$.

$\text{ }$

This object can be thought of as a natural way to complete a particular diagram. Indeed, as was mentioned above one can think of getting a certain commutative diagram $\mathscr{D}$ from our directed system by replacing the vertices $\alpha$ in the natural directed graph induced by $\left(\mathcal{A},\leqslant\right)$ by their associated modules $M_\alpha$, and given any edge $(\alpha,\beta)$ replacing it by the connecting homomorphism $f_{\alpha,\beta}:M_\alpha\to M_\beta$. We see then that $M$ is precisely the “outward mapping completion” of $\mathscr{D}$ in the sense that if we add to $\mathscr{D}$ the module $M$ and the (mapping out of $\mathscr{D}$!)  elements of the limit cone we get a new diagram $\overline{\mathscr{D}}$ which is still commutative.

$\text{ }$

As usual with these universal constructions we have that direct limits, if they exists, are unique up to unique isomorphism.  Indeed:

$\text{ }$

Theorem: Let $\left(\mathcal{A},\leqslant\right)$ be preordered set and $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ with $\{f_{\alpha,\beta}:M_\alpha\to M_\beta\}_{\alpha\beta\in\mathcal{A},\; \alpha\leqslant\beta}$ a directed system. Then, if $(M,\varphi_\alpha)$ and $(N,\psi_\alpha)$ are two direct limits of this direct system then $M\cong N$.

Proof: Since $\varphi_\alpha:M_\alpha\to M$ are a set of maps such that $\varphi_\beta\circ f_{\alpha,\beta}=\varphi_\alpha$ we know by the universal property of $(N,\{\psi_\alpha\})$ that there exists a unique map $k:N\to M$ such that $k\circ\psi_\alpha=\varphi_\alpha$. Similarly, we are afforded a map $j:M\to N$ such that $j\circ\varphi_\alpha=\psi_\alpha$. We see then that $\varphi_\alpha=k\circ\psi_\alpha=k\circ j\circ\varphi_\alpha$ and $\psi_\alpha=j\circ\varphi_\alpha=j\circ k\circ\psi_\alpha$. Now, since clearly the identities $\text{id}_M$ and $\text{id}_N$ clearly satisfy the respective properties we have by uniqueness that $k\circ j=\text{id}_M$ and $j\circ k=\text{id}_N$ from where it follows that $j:M\xrightarrow{\approx}N$ is an isomorphism. $\blacksquare$

$\text{ }$

$\text{ }$

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 29, 2011 -

1. […] Direct Limit of Modules (Pt. II) Point of Post: This is a continuation of this post. […]

Pingback by Direct Limit of Modules (Pt. II) « Abstract Nonsense | November 30, 2011 | Reply

2. […] of the examples we have discussed up until this point concerning direct limits have involved directed systems not over the bare minimum preordered set–no most of the time […]

Pingback by Direct Limits of Directed Sets « Abstract Nonsense | December 1, 2011 | Reply

3. […] we can take the direct limit of rings in much the same way we can take the direct limit of modules. This reflects that the notion of a direct limit can truly be defined in any category, though we […]

Pingback by Direct Limit of Rings (Pt. I) « Abstract Nonsense | December 4, 2011 | Reply

4. […] we have already discussed what it means to have a “directed system” of modules and what a “direct […]

Pingback by A Different Way of Looking at Direct Limits (Pt. I) « Abstract Nonsense | December 9, 2011 | Reply

5. […] (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 […]

Pingback by Inverse Limits of Modules (Pt. I) « Abstract Nonsense | December 9, 2011 | Reply

6. […] one has that and whenenver . Thus, we see that we have constructed an inverse system over the directed set […]

Pingback by Inverse Limit of Rings (Pt. I) « Abstract Nonsense | December 9, 2011 | Reply

7. […] that we have properly defined notions of direct  and inverse limits of modules there are several natural questions we can ask, relating to […]

Pingback by Relationship Between Hom and Limits (Modules) « Abstract Nonsense | December 26, 2011 | Reply

8. […] note that if we consider the trivial direct or inverse system then the induced inverse system (in both cases) is trivial from where we […]

Pingback by Relationship Between Hom and Limits (Modules)(Pt. II) « Abstract Nonsense | December 26, 2011 | Reply

9. […] post we shall just denote as (foregoing the ). There is then a natural category whose objects are directed systems of left -modules over , and whose morphisms, say between are collections of -maps with such […]

Pingback by Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor « Abstract Nonsense | December 27, 2011 | Reply

10. […] sets and  of -modules and we considered it as a directed system over where is, as usual, the trivial directed system. Suppose further that we have have any set of maps . We note then that trivially (by default even) […]

Pingback by Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. II) « Abstract Nonsense | December 28, 2011 | Reply

11. […] this should look astonishingly familiar! Indeed, if is this is just the definition of a directed system of modules over ! So, we can basically identify directed systems of modules over some preordered set as […]

Pingback by Preordered Sets as Categories, and their Functor Categories « Abstract Nonsense | January 10, 2012 | Reply

12. […] that we have a directed system of right -bimodules  over some directed set where, considering our recent discussion of directed systems as functors, […]

Pingback by Tensor Products Naturally Commute with Direct Limits « Abstract Nonsense | January 19, 2012 | Reply

13. […] be a PID. The quotient of injective -modules is injective. Moreover, the direct limit of injective -modules is […]

Pingback by Injective Modules (Pt. III) « Abstract Nonsense | April 28, 2012 | Reply

14. […] be a directed system of flat left -modules where is a directed set. Then, the direct limit  is a flat left […]

Pingback by Flat Modules (Pt. I) « Abstract Nonsense | May 4, 2012 | Reply