Invariability of repetitive algebras of tilted algebras under stable equivalence (Q1340001)

Let \(A\) be a finite dimensional \(K\)-algebra over an algebraically closed field \(K\). The trivial extension \(A \ltimes D(A)\) of \(A\) by the injective cogenerator \(D(A) = \Hom_K (A,K)\) admits a canonical Galois covering \(\widehat {A} \to A \ltimes D(A)\) with infinite cyclic group, where \(\widehat {A}\) is the repetitive (locally bounded, Frobenius) algebra of \(A\). If \(A\) is of finite global dimension, then by a result due to \textit{D. Happel} [Comment. Math. Helv. 62, 339--389 (1987; Zbl 0626.16008)] the stable module category \(\underline {\text{mod}} \widehat {A}\) is equivalent, as a triangulated category, to the derived category \(D^b (A)\) of bounded complexes of finite dimensional \(A\)-modules. In the paper it is shown that if \(\mathcal A\) is a Frobenius locally bounded algebra and there is an equivalence \(\underline \mod \mathcal A} \simeq D^b (K \Delta)\), for some finite quiver \(\Delta\) without oriented cycles, then \({\mathcal A} \simeq \widehat {A}\) for some tilted algebra \(A\) of type \(\Delta\). For \(\Delta\) non-Dynkin, the above algebra \(A\) can be chosen to be representation-infinite. As a consequence one obtains that if \(B\) is an iterated tilted algebra of type \(\Delta\) then there exists a tilted algebra \(A\) of type \(\Delta\) such that \(\widetilde {B} \ltimes \widetilde {A}\) and \(B \ltimes D(B) \simeq A \ltimes D(A)\). Moreover, it follows that a finite dimensional \(K\)-algebra \(B\) is iterated tilted of type \(\Delta\) if and only if \(B\) can be obtained from a tilted algebra \(A\) of type \(\Delta\) by a sequence of reflections in the sense of \textit{D. Hughes} and \textit{J. Waschbüsch} [Proc. Lond. Math. Soc. (3) 46, 347--364 (1983; Zbl 0488.16021)]. In the Dynkin case, the above results were proved by Hughes-Waschbüsch and Assem-Happel-Roldan, and in the Euclidean case by Assem-Nehring-Skowroński, Happel, Skowroński.