Last update: 31 January 2012
An inverse system, or projective system, indexed by is a collection such that
Let and be inverse systems indexed by . A morphism from to is a collection of module morphisms such that
Let be a projective system indexed by . The inverse limit, or projective limit, of is the module of coherent sequences in with the morphisms
be projective systems indexed by . Let
be a morphism of projective systems.
The inverse limit, or projective limit, of is given by
Let be a morphism of inverse systems and let Show that is the unique morphism such that
The universal property of
Let be an inverse 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: for morphisms and of inverse systems.
Show that is a left exact functor.
Give an example which shows that is not an exact functor:
Let , and
is exact, but
is not exact.
A partially ordered set is left filtered if satisfies Assume is a left filtered partially ordered set. Let be an module and let be the inverse system given by for with Show that the diagonal in
Let be a partially ordered set with equality as the partial order. Let be an inverse system on . Show that
Let be a partially ordered set. Show that there exists an inverse system indexed by such that
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.