Higher limits of wider systems

Abstract

Write ${\mathbf{A}}_{\hat{I}}»$ for what might be described as the most elementary nontrivial inverse system of abelian groups indexed by the functions from the cardinal $\hat{I}»$ to the set of natural numbers. The question of whether for any fixed $n$ the derived limit ${\lim }^n{\mathbf{A}}_{\hat{I}}»$ may vanish for only a nonempty subset of the class of infinite cardinals $\hat{I}»$ is recorded in both [Be17] and [Ban23], and bears closely on several related further ones. We answer this question in the affirmative; in fact, we show the maximal possibility, namely that this can simultaneously happen in every degree $n>1$.

25 pages; comments welcome