Tilted special biserial algebras
Let \(k\) be an algebraically closed field, and \(A\) be a finite dimensional basic \(k\)-algebra. In this paper, the authors assume that \(A\) is special biserial (in the sense of \textit{A. Skowroński} and \textit{J. Waschbüsch} [J. Reine Angew. Math. 345, 172-181 (1983; Zbl 0511.16021)]), and give combinatorial criteria allowing to verify whether \(A\) is tilted or not. They had already considered [in ``Tilted string algebras'' (preprint)] the case where \(A\) is a string algebra, they may thus assume that it is not. Then, \(A\) is tilted if and only if it is quasi-tilted. The proof rests on using the (well-known) description of the indecomposable \(A\)-modules in order to obtain combinatorial conditions for an indecomposable module to have projective, or injective, dimension at most one. The authors then obtain necessary and sufficient conditions for the global dimension of \(A\) to be at most two. They deduce their main results from the homological characterisation of quasi-tilted algebras.
scientific article; zbMATH DE number 1340567
scientific article