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Invariability of repetitive algebras of tilted algebras under stable equivalence. (English) Zbl 0824.16014

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 \(D(A) = \operatorname{Hom}_K (A,K)\) 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 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 D. Hughes and 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.

MSC:

16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16G20 Representations of quivers and partially ordered sets
18E30 Derived categories, triangulated categories (MSC2010)
16D90 Module categories in associative algebras
16L60 Quasi-Frobenius rings
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