# Nonvanishing derived limits without scales

## TL;DR

This paper shows that nonvanishing derived limits can exist without assuming a scale, which has implications for strong homology additivity.

## Contribution

The paper removes the assumption of a scale in proving nonvanishing derived limits, allowing broader consistency results.

## Key findings

- Nonvanishing derived limits are consistent without assuming a scale.
- This result applies to any value of ℵ₁ ≤ b ≤ d < ℵ_ω.
- It partially answers a question posed by Bannister.

## Abstract

The derived functors \documentclass[12pt]{minimal}
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\begin{document}$$\lim ^n$$\end{document}limn 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 \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{A}$$\end{document}A of abelian groups indexed over \documentclass[12pt]{minimal}
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\begin{document}$${}^\omega \omega $$\end{document}ωω. So far, consistency results for nonvanishing derived limits of \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{A}$$\end{document}A have always assumed the existence of a scale (i.e. a linear cofinal subset of \documentclass[12pt]{minimal}
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\begin{document}$$({}^\omega \omega , \le ^*)$$\end{document}(ωω,≤∗), or equivalently that \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {b} = \mathfrak {d} $$\end{document}b=d). Here we eliminate that assumption and prove that nonvanishing derived limits, and hence the non-additivity of strong homology, are consistent with any value of \documentclass[12pt]{minimal}
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\begin{document}$$\aleph _1 \le \mathfrak {b} \le \mathfrak {d} < \aleph _\omega $$\end{document}ℵ1≤b≤d<ℵω, thus giving a partial answer to a question of Bannister.