Abstract Nonsense

Crushing one theorem at a time

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


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

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The Category of Inverse Systems and the \varprojlim Functor

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We have discussed the category of directed systems and the associated direct limit functor, and so we now move in a parallel direction to define the category of inverse systems and the associated inverse limit functor.

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We suppose now, similar to the case of \mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right), that we have fixed some preordered set \left(\mathcal{A},\leqslant\right) (from hereon out denoted \mathcal{A}) and some ring R. We then define the category \mathbf{IS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right) of inverse systems in R\text{-}\mathbf{Mod} over (\mathcal{A},\leqslant) to be such that the objects of the category are inverse systems \left(\{M_\alpha\},\{f_{\alpha,\beta}\}\right) of left R-modules over (\mathcal{A},\leqslant) and whose morphisms  \left(\{M_\alpha\},\{f_{\alpha,\beta}\}\right)\xrightarrow{\{w_\alpha\}}\left(\{N_\alpha\},\{g_{\alpha,\beta}\}\right) are sets \{w_\alpha\} of morphisms w_\alpha:M_\alpha\to N_\alpha such that following square commutes whenever \alpha\leqslant\beta

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

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It is completely routine to verify, and in fact is almost isomorphic to the case of directed systems, that this, in fact, a category.

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We’d now like to define the inverse limit functor \varprojlim:\mathbf{IS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)\to R\text{-}\mathbf{Mod}. We already have an object-map \mathbf{IS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)\to R\text{-}\mathbf{mod} defined by (\{M_\alpha\},\{f_{\alpha,\beta}\})\mapsto\varprojlim M_\alpha and so we must only construct, given a morphism (\{M_\alpha\},\{f_{\alpha,\beta}\})\xrightarrow{\{w_\alpha\}}(\{N_\alpha\},\{g_{\alpha,\beta}\}) a corresponding morphism \varprojlim M_\alpha\xrightarrow{\varprojlim w_\alpha}\varprojlim N_\alpha. We proceed as we did for the case of directed systems by picking out limit cones \{\eta_\alpha\} and \{\varphi_\alpha\} for \varprojlim M_\alpha and \varprojlim N_\alpha respectively. We then note that for each \alpha\in\mathcal{A} we have maps w_\alpha\circ\eta_\alpha:\varprojlim M_\alpha\to N_\alpha, and by assumption that w_\alpha is a morphism we check that

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

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Thus, by the definition of the inverse limit we are guaranteed a unique morphism j:\varprojlim M_\alpha\to\varprojlim N_\alpha such that \varphi_\alpha\circ j=w_\alpha\circ\eta_\alpha. We define \varprojlim w_\alpha to be this j.

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So, to check that \varprojlim is, in fact, a functor we must merely check that it respects identities and compositions. To check that it respects identities we merely note that if we take each w_\alpha to be the identity 1_{M_\alpha} then our functional equation for \varprojlim w_\alpha says that \varphi_\alpha\circ \varprojlim 1_{M_\alpha}\circ=\varphi_\alpha for every \alpha\in\mathcal{A}. But, since 1_{\varprojlim M_\alpha} also satisfies this equation, we may conclude by the definition of the inverse limit that \varprojlim 1_{M_\alpha}=1_{\varprojlim M_\alpha}. Similarly, to check that \varprojlim respects compositions we suppose that we have a third inverse system \left(\{L_\alpha\},\{h_{\alpha,\beta}\}\right), a morphism \left(\{L_\alpha\},\{h_{\alpha,\beta}\}\right)\xrightarrow{\{v_\alpha\}}\left(\{M_\alpha\},\{f_{\alpha,\beta}\}\right), and that \varprojlim L_\alpha has a limit cone \{\psi_\alpha\}. We see then that for each \alpha\in\mathcal{A}

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\displaystyle \begin{aligned}\varphi_\alpha\circ\varprojlim(w_\alpha\circ v_\alpha) &= (w_\alpha\circ v_\alpha)\circ\psi_\alpha\\ &= w_\alpha\circ(v_\alpha\circ \psi_\alpha)\\ &=w_\alpha\circ(\eta_\alpha\circ\varprojlim v_\alpha)\\ &= (w_\alpha\circ\eta_\alpha)\circ\varprojlim v_\alpha\\ &= (\varphi_\alpha\circ\varprojlim w_\alpha)\circ\varprojlim v_\alpha\\ &= \varphi_\alpha\circ(\varprojlim w_\alpha\circ\varprojlim v_\alpha)\end{aligned}

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and by definition of the inverse limit we may then conclude that \varprojlim (w_\alpha\circ v_\alpha)=\varprojlim w_\alpha\circ\varprojlim v_\alpha. Thus, \varprojlim really is a functor.

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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.

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

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

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December 28, 2011 - Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , ,

2 Comments »

  1. […] Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. IV) Point of Post: This is a continuation of this post. […]

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

  2. […] pair of indexed sets of objects/arrows. It’s not then hard to check that the category  of inverse systems of modules over  is the same thing as . We can thus define, in general, the category of inverse systems in over […]

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


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