The inverse limit functor

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 31 January 2012

The functor $\underleftarrow{\lim }$

Let $A$ be a ring. Let $I$ be a partially ordered set.

An inverse system, or projective system, indexed by $I$ is a collection ${\left(E_{\alpha },f_{\alpha \beta }\right)}_{\underset{\alpha \le \beta }{\alpha ,\beta \in I}}$ such that

1. $E_{\alpha }$ are $A-$modules
2. $f_{\alpha \beta }:E_{\beta }\to E_{\alpha }$ are morphisms of $A-$modules

such that

1. If $\alpha ,\beta ,\gamma \in I$ and $\alpha \le \beta \le \gamma$ then $f_{\alpha \gamma }=f_{\alpha \beta }\circ f_{\beta \gamma }$
2. If $\alpha \in I$ then $f_{\alpha \alpha }={\mathrm{id}}_{E_{\alpha }}.$

Let $\left(E_{\alpha },f_{\alpha \beta }\right)$ and $\left(F_{\alpha },g_{\alpha \beta }\right)$ be inverse systems indexed by $I$. A morphism from $\left(E_{\alpha }\right)$ to $\left(F_{\alpha }\right)$ is a collection ${\left(u_{\alpha }\right)}_{\alpha \in I}$ of $A-$module morphisms $u_{\alpha }:E_{\alpha }\to F_{\alpha }$ such that $$if\; \alpha ,\beta \in I \; and\; \alpha \le \beta \; then\; u_{\alpha }\circ f_{\alpha \beta }=g_{\alpha \beta }\circ u_{\beta }$$

Let $\left(E_{\alpha },f_{\alpha \beta }\right)$ be a projective system indexed by $I$. The inverse limit, or projective limit, of $E_{\alpha }$ is the $A-$module of coherent $I-$sequences in $\prod _{\alpha }{E}_{\alpha },$ $$\underleftarrow{\lim }{E}_{\alpha }= {x=\left(x_{\alpha }\right)\in \prod _{\alpha }{E}_{\alpha }|\begin{array}{l}If\; \alpha ,\beta \in I \; and\; \alpha \le \beta \; then\\ {\mathrm{pr}}_{\alpha }x=f_{\alpha \beta }{\mathrm{pr}}_{\beta }x\end{array}}$$ with the morphisms $$\begin{array}{rrcl}{f}_{\alpha }:& \underleftarrow{\lim }{E}_{\alpha }& \to & E_{\alpha }\\ & x=\left(x_{\alpha }\right)& \mapsto & x_{\alpha }.\end{array}$$

Let $\left(E_{\alpha },f_{\alpha \beta }\right)$ and $\left(F_{\alpha },g_{\alpha \beta }\right)$ be projective systems indexed by $I$. Let $u_{\alpha }:E_{\alpha }\to F_{\alpha }$ be a morphism of projective systems.

The inverse limit, or projective limit, of $u_{\alpha }$ is $u=\underleftarrow{\lim }{u}_{\alpha }$ given by $$\begin{array}{rrcl}u:& \underleftarrow{\lim }{E}_{\alpha }& \to & \underleftarrow{\lim }{F}_{\alpha }\\ & \left(y_{\alpha }\right)& \mapsto & \left(u_{\alpha }\left(y_{\alpha }\right)\right).\end{array}$$

Homework questions

HW 1:

Let $u_{\alpha }:E_{\alpha }\to F_{\alpha }$ be a morphism of inverse systems and let $u=\underleftarrow{\lim }{u}_{\alpha }.$ Show that $u:\underleftarrow{\lim }{E}_{\alpha }\to \underleftarrow{\lim }{F}_{\alpha }$ is the unique morphism such that $$if\; \alpha \in I \; then\; g_{\alpha }\circ u=u_{\alpha }\circ f_{\alpha }, \; where\; E=\underleftarrow{\lim }{E}_{\alpha } \; and\; F=\underleftarrow{\lim }{F}_{\alpha }.$$

HW 2:

The universal property of $\underleftarrow{\lim }{E}_{\alpha }.$

Let $\left(E_{\alpha },f_{\alpha \beta }\right)$ be an inverse system. Let $F$ be an $A-$module with morphisms $v_{\alpha }:F\to E_{\alpha }$ such that $$if\; \alpha ,\beta \in I \; and\; \alpha \le \beta \; then\; f_{\alpha \beta }\circ v_{\beta }=v_{\alpha }.$$ Then there is a unique morphism $v:F\to \underleftarrow{\lim }{E}_{\alpha }$ such that $$if\; \alpha \in I \; then\; v_{\alpha }=f_{\alpha }\circ v.$$

HW 3:

$\underleftarrow{\lim }$ is a functor.

Show that $$\underleftarrow{\lim }: {\begin{array}{c}inverse\; systems\\ of\; A-modules\end{array}} \to {\begin{array}{c}A-modules\end{array}}$$ is a covariant functor: $$\underleftarrow{\lim }(v_{\alpha }\circ u_{\alpha })=\left(\underleftarrow{\lim }{v}_{\alpha }\right)\circ \left(\underleftarrow{\lim }{u}_{\alpha }\right)$$ for morphisms $\left(v_{\alpha }\right)$ and $\left(u_{\alpha }\right)$ of inverse systems.

HW 4:

Show that $\underleftarrow{\lim }$ is a left exact functor.

HW 5:

Give an example which shows that $\underleftarrow{\lim }$ is not an exact functor:

		Example solution.
	Let $I={\mathbb{Z}}_{>0}$, $E_n=\mathbb{Z}$ and $$\begin{array}{rrcl}{f}_{nm}:& \mathbb{Z}& \to & \mathbb{Z}\\ & x& \mapsto & 3^{m-n}x\end{array}for\; n\le m.$$ Let $$\begin{array}{rrcl}{u}_n:& E_n& \to & \frac{\mathbb{Z}}{2\mathbb{Z}}.\end{array}$$ Then $\underleftarrow{\lim }\frac{\mathbb{Z}}{2\mathbb{Z}}=\frac{\mathbb{Z}}{2\mathbb{Z}},$ $\underleftarrow{\lim }0=0$ and $$E_n\stackrel{u_n}{\to }\frac{\mathbb{Z}}{2\mathbb{Z}}\to 0$$ is exact, but $$\underleftarrow{\lim }{E}_n\stackrel{u}{\to }\underleftarrow{\lim }\frac{\mathbb{Z}}{2\mathbb{Z}}\to \underleftarrow{\lim } 0$$ is not exact. $\square$

HW 6:

A partially ordered set $I$ is left filtered if $I$ satisfies $\begin{array}{c}if\; \alpha ,\beta \in I \; then\; there\; exists\; \lambda \in I \; such\; that\\ \lambda \le \alpha \; and\; \lambda \le \beta .\end{array}$ Assume $I$ is a left filtered partially ordered set. Let $F$ be an $A-$module and let $\left(E_{\alpha },f_{\alpha \beta }\right)$ be the inverse system given by $$E_{\alpha }=F \; and\; f_{\alpha \beta }={\mathrm{id}}_F$$ for $\alpha ,\beta \in I$ with $\alpha \le \beta .$ Show that $$\underleftarrow{\lim }{E}_{\alpha }= {x=\left(x_{\alpha }\right)\in \prod _{\alpha }{E}_{\alpha }|if\; \alpha ,\beta \in I \; then\; x_{\alpha }=x_{\beta }} ,$$ the diagonal in $\prod _{\alpha }{E}_{\alpha }.$

HW 7:

Let $I$ be a partially ordered set with equality as the partial order. Let $\left(E_{\alpha },f_{\alpha \beta }\right)$ be an inverse system on $I$. Show that $$\underleftarrow{\lim }{E}_{\alpha }=\prod _{\alpha }{E}_{\alpha }.$$

HW 8:

Let $I$ be a partially ordered set. Show that there exists an inverse system $\left(E_{\alpha },f_{\alpha \beta }\right)$ indexed by $I$ such that

1. if $\alpha \in I$ then $E_{\alpha }\ne \varnothing$,
2. if $\alpha ,\beta \in I$ and $\alpha \le \beta$ then $f_{\alpha \beta }:E_{\beta }\to E_{\alpha }$ is surjective,
3. $\underleftarrow{\lim }{E}_{\alpha }=\varnothing .$

Notes and References

The basic theory of projective limits appears in [Bou, Ens, Ch III §7], [Bou, Alg Ch I §10], [Bou, Alg Ch II §6], [Bou, Top Gen Ch I §4.4], [Bou, Top Gen Ch II §2.7 and §3.5], [Bou, Top Gen Ch III §7] and [AM, Ch 10 p.103].

In particular the solution to HW1 is given in [Bou, Ens. Ch III §7 No.2 Cor 1] which shows that HW2 should really be done before HW1. The solution to HW2 is given in [Bou, Ens. Ch III §7 No.2 Prop 1], the solution to HW3 is given in [Bou, Ens. Ch III §7 No.2 Cor 2], the solution to HW4 is given in [Bou, Alg Ch II §6 Prop 1] and [AM, Prop 10.2], and the example in HW5 is taken from [Bou, Alg Ch II §6 Ex 1] (an alternative is in [AM, Ch 10 Ex 2]). HW6 is taken from [Bou, Ens. Ch III §7 No.1 Example 2], HW7 is taken from [Bou, Ens. Ch III §7 No.1 Example 1], and the solution to HW8 is found in [Bou, Ens. Ch III §7 Ex 4]. All of these, except perhaps HW8, are routine enough that the mathematician experienced at writing proofs should have no need to refer to the solutions.

References

References?

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