Higher limits of wider systems

TL;DR

This paper proves that for the inverse systems of abelian groups indexed by functions from infinite cardinals to natural numbers, the higher derived limits can simultaneously vanish in all degrees greater than one across all infinite cardinals.

Contribution

It establishes the maximal possibility that the higher derived limits of these inverse systems vanish in every degree greater than one for all infinite cardinals.

Findings

Higher limits can vanish in all degrees n>1 across all infinite cardinals.

Answers an open question about the behavior of derived limits in inverse systems.

Shows the maximal vanishing scenario is achievable.

Abstract

Write $\mathbf{A}_\lambda$ for what might be described as the most elementary nontrivial inverse system of abelian groups indexed by the functions from the cardinal $\lambda$ to the set of natural numbers. The question of whether for any fixed $n$ the derived limit $\mathrm{lim}^n\,\mathbf{A}_\lambda$ may vanish for only a nonempty subset of the class of infinite cardinals $\lambda$ 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 $n>1$.