The direct limit functor

Last update: 31 January 2012

The functor $\underset{\to }{\mathrm{lim}}$

A partially ordered set $I$ is right filtered if $I$ satisfies

Let $A$ be a ring and let $I$ be a right filtered partially ordered set.

A direct system, or inductive system, indexed by $I$ is a collection ${\left({E}_{\alpha },{f}_{\beta \alpha }\right)}_{\underset{\alpha \le \beta }{\alpha ,\beta \in I}}$ such that

1. ${E}_{\alpha }$ are $A-$modules
2. ${f}_{\beta \alpha }:{E}_{\alpha }\to {E}_{\beta }$ are morphisms of $A-$modules
such that
1. If $\alpha ,\beta ,\gamma \in I$ and $\alpha \le \beta \le \gamma$ then ${f}_{\gamma \alpha }={f}_{\gamma \beta }\circ {f}_{\beta \alpha }$
2. If $\alpha \in I$ then ${f}_{\alpha \alpha }={\mathrm{id}}_{{E}_{\alpha }}.$

Let $\left({E}_{\alpha },{f}_{\beta \alpha }\right)$ and $\left({F}_{\alpha },{g}_{\beta \alpha }\right)$ be direct 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 $\left({\mathrm{&E}}_{\alpha },{f}_{\beta \alpha }\right)$ be an inductive system indexed by $I$. Let $\sim$ be the relation on the direct sum $\underset{\alpha }{⨁}{E}_{\alpha }$ (disjoint union $\underset{\alpha }{⨆}{E}_{\alpha }$ for sets) given by

1. if $x\in {E}_{\alpha }$ and $y\in {E}_{\beta }$ then
2. $x\sim y$ if there exists $\gamma \in I$ such that

The direct limit, or inductive limit, of $\left({E}_{\alpha },{f}_{\beta \alpha }\right)$ is $lim → = ⨁Eα ∼$ with morphisms $fα: Eα → lim → Eα x ↦ [x],$ where $\left[x\right]$ is the equivalence class of $x$ in $\frac{\left(⨁{E}_{\alpha }\right)}{\sim }.$

Let ${u}_{\alpha }:{E}_{\alpha }\to {F}_{\alpha }$ be a morphism of direct systems. The direct limit, or inductive limit, of ${u}_{\alpha }$ is $u=lim→uα$ given by $u: lim → Eα → lim → Fα [x] ↦ [uα(x)],$ for $x\in {E}_{\alpha }$.

Homework questions

HW 1:

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

Let $\left({E}_{\alpha },{f}_{\beta \alpha }\right)$ be a directed system. Let $F$ be an $A-$module with morphisms ${v}_{\alpha }:{E}_{\alpha }\to F$ such that then there is a unique morphism $v:\underset{\to }{\mathrm{lim}}{E}_{\alpha }\to F$ such that

HW 2:

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

Show that is a covariant functor: $lim → (vα∘ uα) = ( lim → vα ) ∘ ( lim → uα ).$

HW 3:

$\underset{\to }{\mathrm{lim}}$ is an exact functor.

Notes and References

The basic theory of inductive limits appears in [Bou, Ens. Ch III §7], [Bou, Alg Ch I §10], [Bou, Alg Ch II §6] and [AM, Ch 2, Ex 14-19].

In particular, the solution to HW1 is given in [Bou, Ens. Ch III §7 No.6 Prop 6], the solution to HW2 is given in [Bou, Ens. Ch III §7 No.6 Cor 2] and the solution to HW3 is given in [Bou, Alg Ch II §6 No.2 Prop 3] (see also [AM, Ch 2, Ex 19]). All of these are routine enough that the mathematician experienced at writing proofs should have no need to refer to the solutions.

References?