Simultaneously vanishing higher derived limits

Abstract

In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ 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 $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ 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 $\lim ^n \textbf {A}=0$ 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 $\lim ^n\textbf {A}=0$ 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 generalise the classical and well-studied case of $n=1$ .

A descriptive approach to higher derived limits

“…Relevant background material on homological algebra and on higher derived limits can be found in Mardešić's [27]. Finally, many of the proofs in the present paper have their roots in [1,3].…”

“…The arguments of [3] and [1] both invoke strong combinatorial properties of the weakly compact Hechler model of Theorem 5.1 below. In this and the following section, we show that these properties may be more simply regarded as the assertion that PH n holds for every n ∈ ω.…”

“…Corollary 4.5 admits the following interpretation: the families Φ and Ψ of Definition 4.3 correspond to cocycles and coboundaries of the standard cochain complex K(B/A) for computing the derived limits of a well-studied inverse system termed A in the literature (see [19,2,3]). Combining Corollary 4.5 with Theorem 2.2, we have the following result.…”

“…This question was answered in [1] by developing the aforementioned implications into a more general circuit of equivalences, or near-equivalences, between (1) the commutativity of the lim n and colim functors, (2) the additivity of strong homology, and (3) the triviality of higher-dimensional coherent families of functions, each on suitably restricted domains. The third of these items had recently been shown to be consistent relative to a weakly compact cardinal in [3]; it followed that item (2) on the category of locally compact separable metric spaces and, equivalently, item (1) for countable discrete diagrams of inverse sequences of finitelygenerated abelian groups (in the category of pro-abelian groups) were both consistent relative to a weakly compact cardinal as well. The solution, in short, consisted of a reduction of the questions of ( 1) and ( 2) to the more combinatorial question of (3), together with a solution to the latter.…”

“…The present work takes this reduction one step further: here we introduce a family of purely set-theoretic hypotheses PH n on partitions of powers of ω ω and show these hypotheses to lie at the heart of the rather technical arguments of [1] and [3]. More precisely, we distill those arguments into an initial step establishing the simultaneous consistency of the principles PH n for all n ∈ ω, followed by a ZFC deduction from the latter of conditions (1) through (3) above.…”