Tilted symmetric orders are symmetric orders (Q1305337)

Let \(R\) be a Dedekind domain with quotient field \(K\), and \(\Lambda\) an \(R\)-order, that is, an \(R\)-algebra with \(_R\Lambda\) finitely generated and projective such that \(K\otimes_R\Lambda\) is semisimple. \(\Lambda\) is said to be symmetric if there is a bimodule isomorphism \(_\Lambda\Lambda_\Lambda\cong\Hom_R(\Lambda,R)\). It is proved that if \(\Lambda\) is derived equivalent to an \(R\)-algebra \(\Gamma\), then \(\Gamma\) is again a symmetric \(R\)-order. This improves a result of \textit{J. Rickard} [J. Lond. Math. Soc., II. Ser. 43, No. 1, 37-48 (1991; Zbl 0683.16030)].