Piecewise hereditary algebras
In some sense, this article continues research about quasi-tilted algebras [see: ``Tilting in Abelian Categories'', Mem. Am. Math. Soc. 575 (1996; Zbl 0849.16011) by the same authors, and ``Piecewise hereditary algebras'', by \textit{D. Happel, J. Rickard} and \textit{A. Schofield}, Bull. Lond. Math. Soc. 20, No. 1, 23-28 (1988; Zbl 0638.16018)]. The reading of the paper will be facilitated by some knowledge of Happel's book on applications of derived and triangulated categories and of papers of Lenzing-Geigle and others on representations of sheaves. A piecewise hereditary algebra of type \(\mathcal H\) is an \(R\)-Artin algebra \(\Lambda\) such that \({\mathcal D}^b(\Lambda)\) is triangle equivalent to \({\mathcal D}^b({\mathcal H})\), where \(\mathcal H\) is a locally finite, hereditary, abelian category. The main theorem of the paper, stated below, gives a description of those algebras for the important cases where \(\mathcal H\) is \(\text{mod }H\) (\(H\) a hereditary \(R\)-Artin algebra) or \(\text{coh }{\mathbf X}(p_1,\dots,p_r)\) (\(\text{coh }{\mathbf X}(p_1,\dots,p_r)\) a weighted projective curve). Theorem. Let \(\Lambda\) be a piecewise hereditary algebra of type \(\text{mod }H\) or \(\text{coh }{\mathbf X}(p_1,\dots,p_r)\). Then, there exists a chain \(\Lambda=\Lambda_0,\dots,\Lambda_i=\text{End}_{\Lambda_{i-1}}(T_{i-1})^{\text{op}},\dots\) (\(i=1,2,\dots,m\)) where each \(T_i\) is a tilting or a cotilting \(\Lambda_i\)-module and where \(\Lambda_m\) is a quasitilted algebra. Actually, the \(\text{mod }H\) case of this theorem had been proved already in the above cited paper by Happel, Reiten and Smalø, and its proof is obtained by reducing it to this earlier argument.
scientific article; zbMATH DE number 938523
scientific article