A triangulated hull and a Nakayama closure of the stable module category\n inside the homotopy category

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