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 jdchrist@mit.edu