Symmetric algebras stably equivalent to the trivial extensions of tubular algebras (Q1913941)

Let \(K\) be an algebraically closed field, \(A\) a finite dimensional \(K\)-algebra and \(T(A)\) the trivial extension of \(A\) by its minimal cogenerator bimodule \(DA=\text{Hom}_K(A,K)\). Denote by mod-\(A\) the category of finitely generated right \(A\)-modules, and by \(\underline{\text{mod}}\)-\(A\) the stable category of mod-\(A\). The main result of the paper is the following theorem. Assume that \(A\) is a tubular algebra and \(\Lambda\) a symmetric algebra. Then \(\Lambda\) is stably equivalent to \(T(A)\) if and only if \(\Lambda\) is stably equivalent to \(T(E)\), where \(E\) is a tubular algebra which is tilting-cotilting equivalent to \(A\). Two corollaries are also derived. The proof relies on the characterization of tubular algebras as a cycle finite algebra with many families of sincere tubes, given by the second author [J. Pure Appl. Algebra 103, No. 1, 105-116 (1995; Zbl 0841.16020)], and on the investigation, made in the first part of the paper, of the structure of \(\underline{\text{mod}}\)-\(T(C)\), where \(C\) is a canonical tubular algebra.