<p>The derived functors <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lim ^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mo movablelimits="true">lim</mo> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">A</mi> </math></EquationSource> </InlineEquation> of abelian groups indexed over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${}^\omega \omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mrow /> <mi>ω</mi> </mmultiscripts> <mi>ω</mi> </mrow> </math></EquationSource> </InlineEquation>. So far, consistency results for nonvanishing derived limits of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">A</mi> </math></EquationSource> </InlineEquation> have always assumed the existence of a scale (i.e. a linear cofinal subset of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$({}^\omega \omega , \le ^*)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mmultiscripts> <mrow /> <mrow /> <mi>ω</mi> </mmultiscripts> <mi>ω</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, or equivalently that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathfrak {b} = \mathfrak {d} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">b</mi> <mo>=</mo> <mi mathvariant="fraktur">d</mi> </mrow> </math></EquationSource> </InlineEquation>). Here we eliminate that assumption and prove that nonvanishing derived limits, and hence the non-additivity of strong homology, are consistent with any value of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\aleph _1 \le \mathfrak {b} \le \mathfrak {d} < \aleph _\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> <mo>≤</mo> <mi mathvariant="fraktur">b</mi> <mo>≤</mo> <mi mathvariant="fraktur">d</mi> <mo><</mo> <msub> <mi>ℵ</mi> <mi>ω</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, thus giving a partial answer to a question of Bannister.</p>

Nonvanishing derived limits without scales

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2025年11月4日

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Matteo Casarosa

摘要

The derived functors $\lim ^n$ ${lim}^{n}$ 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 $\textbf{A}$ $A$ of abelian groups indexed over ${}^\omega \omega $ $^{ω} ω$ . So far, consistency results for nonvanishing derived limits of $\textbf{A}$ $A$ have always assumed the existence of a scale (i.e. a linear cofinal subset of $({}^\omega \omega , \le ^*)$ $(^{ω} ω, \leq^{*})$ , or equivalently that $\mathfrak {b} = \mathfrak {d} $ $b = d$ ). 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 $\aleph _1 \le \mathfrak {b} \le \mathfrak {d} < \aleph _\omega $ $ℵ_{1} \leq b \leq d < ℵ_{ω}$ , thus giving a partial answer to a question of Bannister.