# Simultaneously vanishing higher derived limits

## TL;DR

This paper demonstrates, assuming a weakly compact cardinal, that it is consistent with ZFC that all higher derived limits of a specific inverse system vanish, impacting the understanding of strong homology's additivity.

## Contribution

It proves the consistency of the vanishing of all higher derived limits of a particular inverse system under ZFC, using a finite support iteration of Hechler forcings.

## Key findings

- Under certain set-theoretic assumptions, all higher derived limits vanish.
- The result is achieved via a specific forcing extension with Hechler forcings.
- The triviality of certain coherent families of functions is established.

## Abstract

In 1988, Sibe Marde\v{s}i\'{c} and Andrei Prasolov isolated an inverse system $\mathbf{A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim^n\mathbf{A}$ (the $n^{\text{th}}$ derived limit of $\mathbf{A}$) vanishes for every $n >0$. Since that time, the question of whether it is consistent with the $\mathsf{ZFC}$ axioms that $\lim^n \mathbf{A}=0$ for every $n >0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that, assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf{ZFC}$ axioms that $\lim^n \mathbf{A}=0$ for all $n >0$. We show this via a finite support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration a condition equivalent to $\lim^n\mathbf{A}=0$ will hold for each $n>0$. This condition is of interest in its own right; namely, it is the triviality of every coherent $n$-dimensional family of certain specified sorts of partial functions $\mathbb{N}^2\to\mathbb{Z}$ which are indexed in turn by $n$-tuples of functions $f:\mathbb{N}\to\mathbb{N}$. The triviality and coherence in question here generalize the well-studied case of $n=1$.