Abstract Nonsense

Crushing one theorem at a time

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.

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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

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

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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).

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Directed Sets and Directed Systems

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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.

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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.

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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).

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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}.

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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).

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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).

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Direct Limits

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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.

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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.

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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.

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

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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

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[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.

[4] Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.


November 29, 2011 - Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , ,


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