Type: Article

Simultaneously vanishing higher derived limits without large cardinals

Bergfalk, Jeffrey; Hrušák, Michael; Lambie-Hanson, Chris;

Abstract:

A question dating to Mardeši? and Prasolov’s 1988 work [S. Mardeši? 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 ?? 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 ?? to be trivial. In this paper, we prove that, in any forcing extension given by adjoining ??-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 ?-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.