Simultaneously vanishing higher derived limits without large cardinals

Abstract

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 [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-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 [Formula: see text] for all [Formula: see text].