Subject: vanishing of derived functors of inverse limit

From: categories <[email-masked]>
Date: Tue, 6 Feb 1996 21:36:17 -0400 (AST)

Date: Tue, 6 Feb 1996 20:08:27 -0500 (EST)
From: Dan Christensen <[email-masked]>

Dear category theorists,

  I have been working on a problem in stable homotopy theory that led
to a question about the vanishing of the higher derived functors of
the inverse limit functor.  I was wondering if anyone on this list
can answer it.

  Let G be any abelian group and let {G_\alpha} be the filtered
  diagram of all the finitely generated subgroups of G.  Apply a
  contravariant additive functor T : Ab --> Ab to the diagram to
  produce {T G_\alpha}.  Is it always true that lim^i T G_\alpha 
  is zero for i at least 2?

If G is countable, then this is true, and if T takes finitely generated
groups to finitely generated groups, then it is true.  But is it
true in general?

  I should say that I have certain examples of T that I care about,
namely TA = [HA,HB]_k (k and B fixed), the abelian group of maps from
HA, the Eilenberg-Mac Lane spectrum with group A in dimension 0, to
\Sigma^k HB, the E-M spectrum with group B in dimension k.  However, I
was hoping that it would be enough to use the abstract information we
have.  For example, the shape of the inverse system is "Noetherian" in
a certain sense (because it comes from a diagram of finitely generated
subgroups of a group) and while the objects in the diagram need not be
finitely generated, they are finite sums of abelian groups chosen from
a countable list.

  Thanks for any help or references you can provide.  I have looked
through several things written by Roos, Jensen and others and nothing
answers the question either positively or negatively.

Dan Christensen
[email-masked]