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Nonvanishing derived limits without scales
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abstract
The derived functors $\lim^n$ of the inverse limit are widely studied for their topological applications, among which are some repercussions on the additivity of strong homology. Set theory has proven useful in dealing with these functors, for instance in the case of the inverse system $\mathbf{A}$ of abelian groups indexed by ${}^\omega \omega$. So far, consistency results for nonvanishing derived limits of $\mathbf{A}$ have always assumed the existence of a scale (i.e. a linear cofinal subset of $({}^\omega \omega, \leq^\ast )$, or equivalently that $\mathfrak{b} = \mathfrak{d} $). Here we do away with that assumption and prove that nonvanishing derived limits, and hence the non-additivity of strong homology, are consistent with any value of $\aleph_1 \leq \mathfrak{b} \leq \mathfrak{d} < \aleph_\omega$, thus giving a partial answer to a question of Bannister.
Forward citations
Cited by 3 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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Nonvanishing Higher Derived Limits without $w\diamondsuit_{\omega_1}$
Under hypotheses d=ℵ_n plus weak diamonds, the nth derived limit of a natural inverse system of abelian groups is nonzero, and the second derived limit is nonzero in the Miller and Mitchell models.
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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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