Abstract Nonsense

Crushing one theorem at a time

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

Point of Post: In this post we define the category of directed/inverse system of modules over a given ring and then discuss how the construction of the direct/inverse limit is a functor from this category to the category of R-modules. We will then show how this functor is “exact” in the sense that it preserves exact sequences.

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We’d now like to take a more theoretical approach to looking at direct/inverse systems, and their limits. Roughly what we we shall see is that direct systems over a fixed preordered set form a nice little category and limits shall be functors on this category. Moreover, what we shall see is that there is a natural notion of exactness for chains of direct/inverse systems, and we shall see that the limit functor preserves exactness when passed into the target category (which will be just R\text{-}\mathbf{Mod}). The reason for this abstraction is nothing more than a desire to put phrase a common construction in the convenient language of category theory. In fact,  not only will we see that this phrasing will be useful in and of itself, but it shall serve as a prime example of a more general categorical construct in the future, and this functor shall even occupy much of our time once we start talking about homological algebra (or the left derivation of this functor).

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The Category of Directed Systems and the \varinjlim Functor

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Suppose for a second that we have fixed some ring R and some preordered set \left(\mathcal{A},\leqslant\right), which for the duration of this post we shall just denote as \mathcal{A} (foregoing the \leqslant). There is then a natural category \mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right) whose objects are directed systems \left(\{M_\alpha\},\{f_{\alpha,\beta}\}\right) of left R-modules over \mathcal{A}, and whose morphisms, say between (\{M_\alpha\} ,\{f_{\alpha,\beta}\})\to(\{N_\alpha\},\{g_{\alpha,\beta}\}) are collections of R-maps \{w_\alpha\}_{\alpha\in\mathcal{A}} with w_\alpha:M_\alpha\to N_\alpha such that the square

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\begin{matrix} M_\alpha & \overset{w_\alpha}{\longrightarrow} & N_\alpha\\ _{f_{\alpha,\beta}}\big\downarrow & & \big\downarrow _{g_{\alpha,\beta}}\\ M_\beta & \overset{w_\beta}{\longrightarrow} & N_\beta\end{matrix}

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commutes whenever \alpha\leqslant\beta. We compose these morphisms “component-wise” in the sense that \{w_\alpha\}\circ\{v_\alpha\}=\{w_\alpha\circ v_\alpha\}. To prove that this is a category it suffices to prove that the composition of morphisms, as defined, yields a morphism since clearly if so the identity for a given directed system is \{1_{M_\alpha}\} and associativity follows from function composition associativity. So, suppose that we have three direct systems \mathscr{A}=(\{M_\alpha\},\{f_{\alpha,\beta}\}), \mathscr{B}=(\{N_\alpha\},\{g_{\alpha,\beta}\}), and \mathscr{C}= (\{L_\alpha\},\{h_{\alpha,\beta}\}) and morphisms \mathscr{A}\xrightarrow{\{v_\alpha\}}\mathscr{B}\xrightarrow{\{w_\alpha\}}\mathscr{C}. Since w_\alpha\circ v_\alpha:M_\alpha\to L_\alpha is an R-map for each \alpha it suffices to check that it satisfies the commutativity diagram. But, this amounts to showing that h_{\alpha,\beta}\circ (w_\alpha\circ v_\alpha)=(w_\beta\circ v_\beta)\circ f_{\alpha,\beta}. But, by assumption

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\begin{aligned}h_{\alpha,\beta}\circ (w_\alpha\circ v_\alpha) &=(h_\alpha\circ w_\alpha)\circ v_\alpha\\ &=(w_\beta\circ g_{\alpha,\beta})\circ v_\alpha\\ &= w_\beta\circ (g_{\alpha,\beta}\circ v_\alpha)\\ &= w_\beta\circ(v_\beta\circ f_{\alpha,\beta})\\ &= (w_\beta\circ v_\beta)\circ f_{\alpha,\beta}\end{aligned}

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and so the composition of two morphisms, really is a morphism, thus allowing us to conclude that \mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right) is a category as defined.

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As hinted at in the motivation, lurking in the definition of this category seems to be a built-in functor \varinjlim:\mathbf{DS}_\mathcal{A}(R\text{-}\mathbf{Mod})\to R\text{-}\mathbf{Mod}, or at least a built in map between the objects. The only puzzle-piece left to fit in, is exactly what \varinjlim should do to morphisms. But, this is easily answerable. Indeed, suppose that we have two directed systems \left(\{M_\alpha\},\{f_{\alpha,\beta}\}\right) and \left(\{N_\alpha\},\{g_{\alpha,\beta}\}\right) and some morphism \{w_\alpha\} between them. We would now like to define a map \varinjlim w_\alpha:\varinjlim M_\alpha\to\varinjlim N_\alpha, but how? The key comes in the way we defined our \{w_\alpha\}, namely consider the set of maps \psi_\alpha\circ w_\alpha:M_\alpha\to\varinjlim N_\alpha. By assumption, for each \alpha\leqslant\beta we have that

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\begin{aligned}(\psi_\beta\circ w_\beta)\circ f_{\alpha,\beta} &=\psi_\beta\circ(w_\beta\circ f_{\alpha,\beta})\\ &=\psi_\beta\circ(g_{\alpha,\beta\circ}w_\alpha)\\ &= (\psi_\beta\circ g_{\alpha,\beta})\circ w_\alpha\\ &= \psi_\alpha\circ w_\alpha\end{aligned}

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and thus by the definition of direct limits we are guaranteed a unique map j:\varinjlim M_\alpha\to\varinjlim N_\alpha such that j\circ \varphi_\alpha=\psi_\alpha\circ w_\alpha. We then define \varinjlim w_\alpha to be equal to this j.

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Thus, we now have a map \varinjlim:\mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)\to R\text{-}\mathbf{Mod} on objects and morphisms, and to verify that it’s a functor we must therefore only check that it respects identities and compositions. The fact that identities are respected is fine, for if we take the w_\alpha=1_{M_\alpha} then we see that \varinjlim 1_{M_\alpha}\circ\varphi_\alpha=\varphi_\alpha for all \alpha, and since 1_{\varinjlim M_\alpha} also satisfies this, by uniqueness we may conclude that \varinjlim 1_{M_\alpha}=1_{\varinjlim M_\alpha}. Suppose now that we have a third inverse system (\{L_\alpha\},\{h_{\alpha,\beta}\}) and we have that \varinjlim L_\alpha has limit cone \{\zeta_\alpha\}. Furthermore, suppose that we have some morphism \{v_\alpha\} from (\{L_\alpha\},\{h_{\alpha,\beta}\})\to(\{M_\alpha\},\{f_{\alpha,\beta}\})  note then that \varinjlim (w_\alpha\circ v_\alpha) satisfies

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\begin{aligned}\varinjlim (w_\alpha\circ v_\alpha)\circ\zeta_\alpha &=\psi_\alpha\circ(w_\alpha\circ v_\alpha)\\ &=(\psi_\alpha\circ w_\alpha)\circ v_\alpha\\ &=\left(\varinjlim w_\alpha\circ \varphi_\alpha\right)\circ v_\alpha\\ &=\varinjlim w_\alpha\circ\left(\varphi_\alpha\circ v_\alpha\right)\\ &=\varinjlim w_\alpha\circ\left(\varinjlim v_\alpha\circ\zeta_\alpha\right)\\ &=\left(\varinjlim w_\alpha\circ\varinjlim v_\alpha\right)\circ\zeta_\alpha\end{aligned}

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and so by uniqueness we may conclude that \varinjlim(w_\alpha\circ v_\alpha)=\varinjlim w_\alpha\circ\varinjlim v_\alpha. Thus, all of these computations considered we may conclude that \varinjlim:\mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)\to R\text{-}\mathbf{Mod} is a functor as desired.

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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.

[5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.


December 27, 2011 - Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , ,


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