All you need is $\mathbf{A}_κ$

TL;DR

The paper proves that the vanishing of all higher derived limits $\\lim^n\\mathbf{A}_\\kappa$ is equivalent to the additivity and compact supports of strong homology on locally compact metric spaces of weight at most $\\kappa$, thereby closing the circle of implications with earlier results. The main technique reduces arbitrary $\\Omega_\\kappa$-systems to $\\mathbf{A}_\\kappa$-like and essentially $\\mathbf{A}_\\kappa$ systems and deduces the required isomorphisms via exact sequences and the five-lemma. This yields a general theorem: for every cardinal $\\kappa$, strong homology is additive and has compact supports on the class of locally compact metric spaces of weight at most $\\kappa$ iff $\\lim^n\\mathbf{A}_\\kappa=0$ for all $1\\le n<\\omega$, extending the established $\\kappa=\\omega$ case. The results connect set-theoretic derived-limit phenomena with topological homology properties, guiding future consistency analyses and clarifying the relationship between derived limits and additive strong homology.

Abstract

We show that the vanishing of higher derived limits of the system $\mathbf{A}_κ$ implies the additivity of strong homology on the class of locally compact metric spaces of weight at most $κ$, thereby establishing a converse to a theorem of Mardešić and Prasolov.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2: B. ADLCM
• Theorem 3.1
• Definition 3.2
• Lemma 3.3
• proof
• proof
• Definition 3.4