2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126212In this paper we give a sufficient condition of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations. We also introduce property (W$^{k}$). As an application of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations we give a connection between the finitistic dimension conjecture, the Auslander-Reiten conjecture and property (W$^{k}$).19 pages. To appear in Communications in Algebra. The orginal title is: $W^t$-approximation representations and homological conjecturesRings and AlgebrasRepresentation Theory16E10; 16E30; 16G10text