Complexity of trivial extensions of iterated tilted algebras. (Q2909797)

Complexities of modules over a selfinjective algebra have close relation with the representation type of the algebra. Although it seems that there is some problem with \textit{J. Rickard}'s proof [Bull. Lond. Math. Soc. 22, No. 6, 540-546 (1990; Zbl 0742.16007)], it is believed that if a selfinjective algebra has a module of complexity at least \(3\), then the algebra has wild representation type. If this is true, tame representation type implies that complexities of modules should be only \(0\), \(1\) or \(2\); it is known that finite representation type implies that complexities of modules are at most one. One would like to verify the above folklore conjecture by computing complexities of modules for certain classes of algebras. This paper under review verifies it for trivial extensions of iterated tilted algebras.NEWLINENEWLINE The method is as follows: Let \(A\) be an iterated tilted algebra of type \(H\). Then \(A\) and \(H\) are related by a sequence of tilting modules hence are derived equivalent, so their trivial extensions are derived equivalent and stably equivalent (in fact stably equivalent of Morita type). As a ``good'' stable equivalence preserves complexities of modules (Theorem 4.8), one reduces the problem to trivial extensions of hereditary algebras. The author then uses a result of \textit{H. Tachikawa} [Representation theory II, Proc. 2nd int. Conf., Ottawa 1979, Lect. Notes Math. 832, 579-599 (1980; Zbl 0451.16019)] which describes the Auslander-Reiten quiver of the trivial extension of a hereditary algebra. As modules in the same component of the Auslander-Reiten quiver have the same complexities (Lemma 2.3), one then makes use of known results about Coxeter transformation on modules over hereditary algebras [\textit{N. A'Campo}, Invent. Math. 33, 61-67 (1976; Zbl 0406.20041); \textit{O. Kerner}, CMS Conf. Proc. 19, 65-107 (1996; Zbl 0863.16010); \textit{C. M. Ringel}, Math. Ann. 300, No. 2, 331-339 (1994; Zbl 0819.15008)].