The direct limit functor

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 31 January 2012

The functor lim

A partially ordered set I is right filtered if I satisfies if   x,yI   then there exists   γI   such that γx   and   γy.

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 Eα fβα α,βI αβ such that

  1. Eα are A-modules
  2. fβα: Eα Eβ are morphisms of A-modules
such that
  1. If α,β,γI and αβγ then fγα = fγβ fβα
  2. If αI then fαα = idEα .

Let Eα fβα and Fα gβα be direct systems indexed by I.

A morphism from Eα to Fα is a collection uα αI of A-module morphisms uα: Eα Fα such that if   α,βI   and   αβ   then   gβαuα = uβ fβα

Eα Fα Eβ Fβ uα uβ fβα gβα

Let &Eα fβα be an inductive system indexed by I. Let be the relation on the direct sum α Eα (disjoint union α Eα for sets) given by

  1. if xEα and yEβ then
  2. xy if there exists γI such that γα   and   γβ   and   fγα x = fγβ y .

The direct limit, or inductive limit, of Eα fβα is lim = Eα with morphisms fα: Eα lim Eα x [x], where [x] is the equivalence class of x in (Eα) .

Let uα: Eα Fα be a morphism of direct systems. The direct limit, or inductive limit, of uα is u=limuα given by u: lim Eα lim Fα [x] [uα(x)], for xEα.

Homework questions

HW 1:

The universal property of lim Eα.

Let Eα fβα be a directed system. Let F be an A-module with morphisms vα: EαF such that if   α,βI   and   αβ   then   vβ fβα =vα, then there is a unique morphism v: lim EαF such that if   αI   then   vα = ufα.

HW 2:

lim is a functor.

Show that lim : direct systems of   A-modules A-modules is a covariant functor: lim (vα uα) = ( lim vα ) ( lim uα ).

HW 3:

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.



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