All you need is 𝐀_{𝜅}

We show that the vanishing of higher derived limits of the system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A Subscript kappa"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> κ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {A}_{\kappa }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies the additivity of strong homology on the class of locally compact metric spaces of weight at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi> κ </mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , thereby establishing a converse to a theorem of Mardešić and Prasolov [Trans. Amer. Math. Soc. 307 (1988), 725–744].