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

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.