## The inverse limit functor

Last update: 31 January 2012

## The functor $\underset{←}{\mathrm{lim}}$

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

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 $\prod _{\alpha }{E}_{\alpha },$ with the morphisms $fα: lim ← Eα → Eα x=xα ↦ xα.$

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 $u=\underset{←}{\mathrm{lim}}{u}_{\alpha }$ given by $u: lim ← Eα → lim ← Fα yα ↦ uα yα .$

## Homework questions

#### HW 1:

Let ${u}_{\alpha }:{E}_{\alpha }\to {F}_{\alpha }$ be a morphism of inverse systems and let $u=\underset{←}{\mathrm{lim}}{u}_{\alpha }.$ Show that $u:\underset{←}{\mathrm{lim}}{E}_{\alpha }\to \underset{←}{\mathrm{lim}}{F}_{\alpha }$ is the unique morphism such that

#### HW 2:

The universal property of $\underset{←}{\mathrm{lim}}{E}_{\alpha }.$

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 Then there is a unique morphism $v:F\to \underset{←}{\mathrm{lim}}{E}_{\alpha }$ such that

#### HW 3:

$\underset{←}{\mathrm{lim}}$ is a functor.

Show that is a covariant functor: $lim ← (vα∘ uα) = lim ← vα ∘ lim ← uα$ for morphisms $\left({v}_{\alpha }\right)$ and $\left({u}_{\alpha }\right)$ of inverse systems.

#### HW 4:

Show that $\underset{←}{\mathrm{lim}}$ is a left exact functor.

#### HW 5:

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

 Example solution. Let $I={ℤ}_{>0}$, ${E}_{n}=ℤ$ and Let $un: En → ℤ 2ℤ .$ Then $\underset{←}{\mathrm{lim}}\frac{ℤ}{2ℤ}=\frac{ℤ}{2ℤ},$ $\underset{←}{\mathrm{lim}}0=0$ and $En →un ℤ 2ℤ →0$ is exact, but is not exact. $\square$

#### HW 6:

A partially ordered set $I$ is left filtered if $I$ satisfies 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 for $\alpha ,\beta \in I$ with $\alpha \le \beta .$ Show that 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 $lim ← Eα = ∏ α Eα.$

#### 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. $\underset{←}{\mathrm{lim}}{E}_{\alpha }=\varnothing .$

## 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?