Simultaneously nonvanishing higher derived limits
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929603712385024 |
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| author | Casarosa, Matteo Lambie-Hanson, Chris |
| author_facet | Casarosa, Matteo Lambie-Hanson, Chris |
| contents | 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 $\mathfrak{d}=ω_n$, then $\lim^n \mathbf{A}[H] \neq 0$ holds for $H=\mathbb{Z}^{(ω_n)}$ (i.e. the direct sum of $ω_n$-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 $2^{\aleph_0} \geq \aleph_{ω+1}$, 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 $\bigwedge_{2 \leq k < ω} \lim^k \mathbf{A} \neq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simultaneously nonvanishing higher derived limits Casarosa, Matteo Lambie-Hanson, Chris Logic Algebraic Topology Category Theory 03E35, 03E05, 03E17, 03E75, 18G10 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 $\mathfrak{d}=ω_n$, then $\lim^n \mathbf{A}[H] \neq 0$ holds for $H=\mathbb{Z}^{(ω_n)}$ (i.e. the direct sum of $ω_n$-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 $2^{\aleph_0} \geq \aleph_{ω+1}$, 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 $\bigwedge_{2 \leq k < ω} \lim^k \mathbf{A} \neq 0$. |
| title | Simultaneously nonvanishing higher derived limits |
| topic | Logic Algebraic Topology Category Theory 03E35, 03E05, 03E17, 03E75, 18G10 |
| url | https://arxiv.org/abs/2411.15856 |