Last update: 31 January 2012
A partially ordered set is right filtered if satisfies
Let be a ring and let be a right filtered partially ordered set.
A direct system, or inductive system, indexed by is a collection such that
Let and be direct systems indexed by .
A morphism from to is a collection of module morphisms such that
Let be an inductive system indexed by . Let be the relation on the direct sum (disjoint union for sets) given by
The direct limit, or inductive limit, of is with morphisms where is the equivalence class of in
Let be a morphism of direct systems. The direct limit, or inductive limit, of is given by for .
The universal property of
Let be a directed system. Let be an module with morphisms such that then there is a unique morphism such that
is a functor.
Show that is a covariant functor:
is an exact functor.
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.