Simultaneously nonvanishing higher derived limits

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 $\mathfrak{d}=\ddot{I}{‰}_n$, then ${\lim }^n\mathbf{A}[H]\ne 0$ holds for $H={\mathbb{Z}}^{(\ddot{I}{‰}_n)}$ (i.e. the direct sum of $\ddot{I}{‰}_n$-many copies of $\mathbb{Z}$). The same holds for $H=\mathbb{Z}$ under the assumption that $\mathrm{w}♢(S_k^{k+1})$ holds for all $k<n$. In particular, this shows that if ${\lim }^n\mathbf{A}[H]=0$ holds for all $n\ge 1$ and all abelian groups $H$, then $2^{{\aleph }_0}\ge {\aleph }_{\ddot{I}‰+1}$, 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 ${\bigwedge }_{2\le k<\ddot{I}‰}{\lim }^k\mathbf{A}\ne 0$.

34 pages