# Abstract Nonsense

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

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

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What we’d now like to show is that, similar to the case of $\varinjlim$, $\varprojlim$ is an “exact” functor, where we define exactness of inverse systems the exact same way we define it for directed systems (e.g. the individuals sequences are exact), but unfortunately things don’t work out quite as well as they did for direct limits. Namely, we can really only prove that the inverse limit functor is “left exact” which means it takes exact sequences, where the leftmost map is injective  to exact sequences where the left most map is injective.

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Theorem: Let

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$0\to\left(\{L_\alpha\},\{h_{\alpha,\beta}\}\right)\xrightarrow{\{v_\alpha\}}\left(\{M_\alpha\},\{f_{\alpha,\beta}\}\right)\xrightarrow{\{w_\alpha\}}\left(\{N_\alpha\},\{g_{\alpha,\beta}\}\right)$

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be an exact sequence of inverse systems over some directed set $\left(\mathcal{A},\leqslant\right)$ (where $\varprojlim L_\alpha,\varprojlim M_\alpha$, and $\varprojlim N_\alpha$ have limit cones $\{\psi_\alpha\},\{\eta_\alpha\}$, and $\{\varphi_\alpha\}$ respectively) then

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$0\to \varprojlim L_\alpha\xrightarrow{\varprojlim v_\alpha}\varprojlim M_\alpha\xrightarrow{\varprojlim w_\alpha}\varprojlim N_\alpha$

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is an exact sequence of $R$-maps.

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We postpone the proof of this, so that we may give a cool proof of it using abstract nonsense concerning adjoint functors.

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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.