Additivity of derived limits in the Cohen model

TL;DR

This paper demonstrates that in models with many Cohen reals, derived limits exhibit additivity on broad classes of systems, extending previous work and connecting to strong homology.

Contribution

It generalizes prior results on derived limits in Cohen models and identifies a key partition principle influencing their behavior.

Findings

Derived limits are additive in Cohen models for large classes of systems.

A partition principle responsible for vanishing derived limits is isolated.

Additivity of derived limits implies similar results for strong homology.

Abstract

We show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. This generalizes the work of Jeffrey Bergfalk, Michael Hru\v s\'ak, and Chris Lambie-Hanson which focuses on the system $\mathbf{A}$. In the process, we isolate a partition principle responsible for the vanishing of derived limits on collections of Cohen reals and reframe the propagating trivializations results of Bergfalk, Hru\v s\'ak and Lambie-Hanson as a theorem of ZFC. In light of results of the author, Jeffrey Bergfalk, and Justin Moore, the additivity of derived limits also implies additivity results for strong homology.