  The stable module category has been realized as a subcategory of the
unbounded homotopy category of projective modules by Kato. We construct the
triangulated hull of this subcategory inside the homotopy category. This can
also be used to characterize self-injective algebras. Moreover, we extend this
construction to a subcategory closed under an induced Nakayama functor. Both of
these categories are shown to be preserved by stable equivalences of Morita
type. As an application, we study the Grothendieck group of this triangulated
hull and compare it with the stable Grothendieck group.
