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PIERIN, T. C. m-Quasitilted and m-almost hereditary algebras. Tese (Doutorado) Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2015. Based on [HRS96b] we present a generalization of the classes of quasitilted and almost hereditary algebras, which we call m-quasitilted and m-almost hereditary algebras, where m ≥ 1. For the latter one, we can obtain a trisection of their module categories determined by the following subcategories Lm = {X indecomposable; dp Y ≤ m, for each predecessor Y of X} and R = {X indecomposable; di Y ≤ 1 for each successor Y of X}. Moreover, if there exists a module Em such that {X ; Hom(Em,τX) = 0} = {X ; dp X ≤ m} then Em ∈ add R and any path of indecomposable modules starting in a module E ∈ add Em and ending in a projective module can be refined to a path of irreducible morphisms, which is also sectional. This result on paths allow us to obtain a characterization for m-almost hereditary algebras in terms of their projective modules. It is also possible to prove that any m-quasitilted algebra with global dimension m+1 is a m-almost hereditary algebra and as a consequence we can obtain that any piecewise hereditary algebra of type mod H, for some hereditary algebra H, and with global dimension m+ 1 is m-almost hereditary. We present an example of a 2-almost hereditary which is not 2-quasitilted, which entails that the converse of the above mentioned result does not hold true. Thus we seek for conditions which can ensure that a given 2-almost hereditary is 2-quasitilted and, in particular, a piecewise hereditary algebra. For this, we use the Φ−1 correspondence, obtained as an adaptation of results of Happel, Reiten and Smalφ , which under certain assumptions shows that an algebra is an endomorphism algebra of a tilting object. It is shown that a 2-almost hereditary algebra with some other properties and satisfying (H1), (H2), (H3) is 2-quasitilted.

Álgebras m-quase inclinadas e m-quase hereditárias