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Additivity of derived limits in the Cohen model
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abstract
We show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. This generalizes the work of Jeffrey Bergfalk, Michael Hru\v s\'ak, and Chris Lambie-Hanson which focuses on the system $\mathbf{A}$. In the process, we isolate a partition principle responsible for the vanishing of derived limits on collections of Cohen reals and reframe the propagating trivializations results of Bergfalk, Hru\v s\'ak and Lambie-Hanson as a theorem of ZFC. In light of results of the author, Jeffrey Bergfalk, and Justin Moore, the additivity of derived limits also implies additivity results for strong homology.
Forward citations
Cited by 4 Pith papers
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Higher limits of wider systems
Under GCH plus diamond principles, and in Gödel's constructible universe, the higher derived limits lim^n A_λ are nonzero for every cardinal λ where Goblot's vanishing theorem does not force them to zero.
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Simultaneously nonvanishing higher derived limits
If the dominating number equals aleph_n, then the n-th derived limit lim^n A[Z^(aleph_n)] is nonzero; and it is consistent that lim^k A is nonzero for every finite k >= 2, with continuum aleph_{omega+2}.
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Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions
It is consistent that b=d=ω_n and lim^k A ≠ 0 for all 1≤k≤n, and that b=d=ω_{ω+2} with lim^k A ≠ 0 for every k≥1, by new forcings for lim^1 A ≠ 0 compatible with prior nonvanishing methods.
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Infinitary combinatorics in condensed math and strong homology
Higher derived limits of the systems A_kappa_lambda are shown to control non-fullness, non-additivity of strong homology, and non-compactness of products of compact projective condensed anima.
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