We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Mardešić and Prasolov [1] showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author [2] which establishes the consistency, relative to a weakly compact cardinal, of lim s A = 0 for all s ≥ 1 for a certain pro-abelian group A; we show that that work's arguments carry implications for the vanishing and additivity of the lim s functors over a substantially more general class of pro-abelian groups indexed by N N .

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We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set Ω of functions from N to N and prove that cocycles of this complexity are images of cochains of the roughly the same complexity. In the course of this work, we isolate a partition principle for powers D n+1 of products of directed sets D and show that whenever this principle holds, the corresponding derived limit lim n is additive; vanishing results for this limit are the typical corollary. The formulation of this partition hypothesis synthesizes and clarifies several recent advances in this area.

We generalize the work of Jeffrey Bergfalk, Michael Hrušák, to show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. In the process, we isolate a partition principle responsible for the vanishing of derived limits on collections of Cohen reals as well as reframe the propagating trivializations result of [5] as a theorem of ZFC. The results also build on the work of the author, Jeffrey Bergfalk, and Justin Moore [2] to show additvity results for strong homology.