We consider the functor from the stable module category to the homotopy category constructed by Kato. This functor gives an equivalence between the stable module category and a full subcategory L of the unbounded homotopy category of projective modules. Moreover, the functor induces a correspondence between distinguished triangles in the homotopy category and perfect exact sequences in the module category. In general, the stable module category and the category L are not triangulated. We

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... a description of a triangulated hull of L inside the homotopy category and discuss its Grothendieck group. We also construct a larger subcategory which is shown to be characteristic inside the homotopy category under suitable assumptions. Both subcategories coincide with L if and only if the algebra is self-injective. Furthermore, stable equivalence of Morita type are shown to preserve both subcategories. Another focus is put on the relationship between stable equivalences and perfect exact sequences. On the one hand, we give sufficient conditions for a stable equivalence to preserve perfect exact sequences up to projective direct summands. A stable equivalence which preserves perfect exact sequences in this way is shown to induce a triangulated equivalence between the categories of stable Gorenstein-projective modules. On the other hand, given a stable equivalence that is induced by an exact functor, we provide various sufficient conditions under which the equivalence is a stable equivalence of Morita type. In particular, stable equivalences of Morita type arise from equivalences that are given by tensoring with an arbitrary bimodule on the level of the category L. Finally, we give a description of all algebras that can be obtained by deleting or inserting nodes via stable equivalences constructed by Koenig and Liu.