Simultaneously vanishing higher derived limits without large cardinals

0570763 - MÚ 2024 RIV SG eng J - Journal Article
Bergfalk, J. - Hrušák, M. - Lambie-Hanson, Christopher
Simultaneously vanishing higher derived limits without large cardinals.
Journal of Mathematical Logic. Roč. 23, č. 1 (2023), č. článku 2250019. ISSN 0219-0613. E-ISSN 1793-6691
Institutional support: RVO:67985840
Keywords : Cohen forcing * Delta system lemma * derived limit * nontrivial coherence
OECD category: Pure mathematics
Impact factor: 0.9, year: 2022
Method of publishing: Limited access
https://doi.org/10.1142/S0219061322500192

A question dating to Mardešić and Prasolov's 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725-744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits limn (n > 0) of a certain inverse system A indexed by ωω to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all n-coherent families of functions indexed by ωω to be trivial. In this paper, we prove that, in any forcing extension given by adjoining ω-many Cohen reals, limnA vanishes for all n > 0. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional Δ-system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of limnA for all n > 0.
Permanent Link: https://hdl.handle.net/11104/0342099