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A notion of mutation pairs of subcategories in an abelian category is defined in this article. For an extension closed subcategory and a rigid subcategory ⊂ , the subfactor category / is also a triangulated category whenever forms amutation pair. Moreover, if and satisfy certain conditions in mod , the category of finitely generated -modules over an artin algebra , the triangulated category / has a Serre functor.

In this short article, we prove that a finite dimensional algebra is hereditary if and only if there is no loop in its ordinary quiver and every -tilting module is tilting.