Higher limits of wider systems

2025/07/07 by Jeffrey Bergfalk, Matteo Casarosa, Bergfalk, Jeffrey +1
Computer Science · Mathematics · #03E35 #03E75 #18G10 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #K-Theory and Homology (math.KT) #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2507.05471

openalex publication_date 2025/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Write A_λ for what might be described as the most elementary nontrivial inverse system of abelian groups indexed by the functions from the cardinal λ to the set of natural numbers. The question of whether for any fixed n the derived limit limⁿ A_λ may vanish for only a nonempty subset of the class of infinite cardinals λ is recorded in both [Be17] and [Ban23], and bears closely on several related further ones. We answer this question in the affirmative; in fact, we show the maximal possibility, namely that this can simultaneously happen in every degree n>1.

Citations

• Infinitary combinatorics in condensed math and strong homology
• Simultaneously nonvanishing higher derived limits
• Nonvanishing derived limits without scales
• Additivity of derived limits in the Cohen model
• A descriptive approach to higher derived limits
• Non-vanishing higher derived limits
• Simutaneously vanishing higher derived limits without large cardinals
• On the additivity of strong homology for locally compact separable metric spaces
• Strong Homology, Derived Limits, and Set Theory

Related

• Non-vanishing higher derived limits
• Simutaneously vanishing higher derived limits without large cardinals
• Nonvanishing Higher Derived Limits without w\diamondsuit_ω1
• Simultaneously nonvanishing higher derived limits
• Nonvanishing derived limits without scales