Abstract Nonsense

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.

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