  In this paper we study the finiteness of global Gorenstein AC-homological
dimensions for rings, and answer the questions posed by Becerril, Mendoza,
P\'{e}rez and Santiago. As an application, we show that any left (or right)
coherent and left Gorenstein ring has a projective and injective stable
homotopy category, which improves the known result by Beligiannis.
