The direct 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 $\underrightarrow{\lim }$

A partially ordered set $I$ is right filtered if $I$ satisfies $$\begin{array}{c}if\; x,y\in I \; then\; there\; exists\; \gamma \in I \; such\; that\\ \gamma \ge x \; and\; \gamma \ge y.\end{array}$$

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 $$\begin{array}{c}if\; \alpha ,\beta \in I \; and\; \alpha \le \beta \; then\; g_{\beta \alpha }\circ u_{\alpha }=u_{\beta }\circ f_{\beta \alpha }\end{array}$$

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 $$\gamma \ge \alpha \; and\; \gamma \ge \beta \; and\; f_{\gamma \alpha }\left(x\right)=f_{\gamma \beta }\left(y\right).$$

The direct limit, or inductive limit, of $\left(E_{\alpha },f_{\beta \alpha }\right)$ is $$\underrightarrow{\lim }=\frac{⨁E_{\alpha }}{\sim }$$ with morphisms $$\begin{array}{rrcl}{f}_{\alpha }:& E_{\alpha }& \to & \underrightarrow{\lim }{E}_{\alpha }\\ & x& \mapsto & \left[x\right],\end{array}$$ where $\left[x\right]$ is the equivalence class of $x$ in $\frac{(⨁E_{\alpha })}{\sim }.$

Let $u_{\alpha }:E_{\alpha }\to F_{\alpha }$ be a morphism of direct systems. The direct limit, or inductive limit, of $u_{\alpha }$ is $$u=\underrightarrow{\lim }{u}_{\alpha }$$ given by $$\begin{array}{rrcl}u:& \underrightarrow{\lim }{E}_{\alpha }& \to & \underrightarrow{\lim }{F}_{\alpha }\\ & \left[x\right]& \mapsto & \left[u_{\alpha }\right(x\left)\right],\end{array}$$ for $x\in E_{\alpha }$.

Homework questions

HW 1:

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

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 $$if\; \alpha ,\beta \in I \; and\; \alpha \le \beta \; then\; v_{\beta }\circ f_{\beta \alpha }=v_{\alpha },$$ then there is a unique morphism $v:\underrightarrow{\lim }{E}_{\alpha }\to F$ such that $$if\; \alpha \in I \; then\; v_{\alpha }=u\circ f_{\alpha }.$$

HW 2:

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

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

HW 3:

$\underrightarrow{\lim }$ 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

References?

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