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Simultaneously nonvanishing higher derived limits
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abstract
The derived functors $\lim^n$ of the inverse limit find many applications in algebra and topology. In particular, the vanishing of certain derived limits $\lim^n \mathbf{A}[H]$, parametrized by an abelian group $H$, has implications for strong homology and condensed mathematics. In this paper, we prove that if $\mathfrak{d}=\omega_n$, then $\lim^n \mathbf{A}[H] \neq 0$ holds for $H=\mathbb{Z}^{(\omega_n)}$ (i.e. the direct sum of $\omega_n$-many copies of $\mathbb{Z}$). The same holds for $H=\mathbb{Z}$ under the assumption that $\mathrm{w}\diamondsuit(S^{k+1}_k)$ holds for all $k < n$. In particular, this shows that if $\lim^n \mathbf{A}[H] = 0$ holds for all $n \geq 1$ and all abelian groups $H$, then $2^{\aleph_0} \geq \aleph_{\omega+1}$, thus answering a question of Bannister. Finally, we prove some consistency results regarding simultaneous nonvanishing of derived limits, again in the case of $H = \mathbb{Z}$. In particular, we show the consistency, relative to $\mathsf{ZFC}$, of $\bigwedge_{2 \leq k < \omega} \lim^k \mathbf{A} \neq 0$.
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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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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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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