Nonvanishing Higher Derived Limits without $w\diamondsuit_{\omega_1}$

TL;DR

This paper demonstrates that under certain set-theoretic hypotheses, the higher derived limits of a specific inverse system of abelian groups do not vanish, providing counterexamples to previous conjectures in various models of set theory.

Contribution

It refines existing theorems to show nonvanishing of higher derived limits under broader conditions, disproving a conjecture about their vanishing in the Miller model.

Findings

Higher derived limits of the inverse system do not vanish under certain hypotheses.

Nonvanishing of the second derived limit in many models of set theory.

Disproof of the conjecture that these limits vanish in the Miller model.

Abstract

We prove a common refinement of theorems of Bergfalk and of Casarosa and Lambie-Hanson, showing that under certain hypotheses, the higher derived limits of a certain inverse system of abelian groups $\mathbf{A}$ do not vanish. The refined theorem has a number of interesting corollaries, including the nonvanishing of the second derived limit of $\mathbf{A}$ in many of the common models of set theory of the reals and in the Mitchell model. In particular, we disprove a conjecture of Bergfalk, Hru\v{s}\'ak, and Lambie-Hanson that higher derived limits of $\mathbf{A}$ vanish in the Miller model.