Non-vanishing higher derived limits

Abstract: In the study of strong homology Marde\v{s}i\'c and Prasolov isolated a certain inverse system of abelian groups $\mathbf A$ indexed by elements of $\omega<sup>\omega$. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits $\limⁿ \mathbf A$ must vanish, for $n&gt;0$. They also proved that under the Continuum Hypothesis $\lim¹ \mathbf A \neq 0$. The question whether $\limⁿ \mathbf A$ vanishes, for $n&gt;0$, has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under PFA $\lim¹ \mathbf A =0$. Bergfalk show that it is consistent that $\lim^2\mathbf A$ does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that $\limⁿ \mathbf A =0$, for all $n$. The large cardinal assumption was recently removed by Bergfalk, Hru\v{s}ak and Lambie-Henson. We complete the picture by showing that, for any $n&gt;0$, it is relatively consistent with ZFC that $\limⁿ \mathbf A \neq 0$.