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Simultaneously vanishing higher derived limits

Part of:Set theoryAbelian categoriesHomology and cohomology theories

Published online by Cambridge University Press:  14 June 2021

Jeffrey Bergfalk and
Chris Lambie-HansonOpen the ORCID record for Chris Lambie-Hanson[Opens in a new window]
Jeffrey Bergfalk
Affiliation:
Universität Wien, Institut für Mathematik, Kurt Gödel Research Center, Kolingasse 14-16, 1010 Wien, Austria;.Centro de Ciencas Matemáticas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacán, 58089, México; E-mail: jeffrey.bergfalk@univie.ac.at.
Chris Lambie-Hanson
Affiliation:
Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, USA; E-mail: cblambiehanso@vcu.edu.

Abstract

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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 $n=1$.

MSC classification

Primary:03E05: Other combinatorial set theory
Secondary:03E75: Applications of set theory03E55: Large cardinals55N07: Steenrod-Sitnikov homologies55N40: Axioms for homology theory and uniqueness theorems18E25: Derived functors and satellites

Information

Type
Foundations
Information
Forum of Mathematics, Pi , Volume 9 , 2021, e4
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press.

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