A question dating to Mardesic and Prasolov's 1988 work [S. Mardesic 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 limt' (n > 0) of a certain inverse system A indexed by (omega)omega 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 (omega)omega to be trivial. In this paper, we prove that, in any forcing extension given by adjoining (sic)(omega)-many Cohen reals, limt(n) A vanishes for all n > 0. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated ap-plications of higher-dimensional Delta-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 lim(n) A for all n > 0.
展开▼
