Let k be an algebraically closed field and H a connected abelian k-category. We assume that H is hereditary, that is the Yoneda Ext~2 (-,-) vanishes, and we assume that H has finite dimensional homomorphism and extension spaces. In addition H has a tilting object, that is some object T with Ext_H~1(T,T) = 0 such that Hom_H(T,X) = 0 = Ext_H~1(T,X) implies X = 0. This concept was introduced in [HRS] to obtain a common treatment of both the class of tilted algebras (compare [HRi]) and the class of canonical algebras (compare [R2], [GL] or [LP]). This common treatment lead to the definition of a quasitilted algebra. A quasitilted algebra is the endomorphism algebra End_HT of a tilting object T ∈ H. In [HRS] quasitilted algebras are characterized by the following homological property. This class coincides with the class of finite dimensional k-algebras of global dimension at most 2 whose finitely generated indecomposable modules have either projective or injective dimension at most 1.
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