Consistency of additivity of derived limits for all _kappa systems

Determine whether it is consistent with ZFC that, for every cardinal κ and every integer n ≥ 0, derived inverse limits are additive for all _κ systems of abelian groups; that is, for every such system G, the canonical map ⊕_{α<κ} lim^n G_α → lim^n G is an isomorphism.

Background

The main results relate vanishing of derived limits of A_κ to additivity of strong homology and, via Theorem 1.3, to additivity of derived limits for _κ systems with finitely generated groups. The open problem asks for a uniform consistency result extending additivity to all _κ systems, not only special cases or under additional finiteness assumptions.

This would represent a broad strengthening of existing consistency results linking set theory, inverse systems, and homological properties.