On derived equivalences of lines, rectangles and triangles

We present a method to construct new tilting complexes from existing ones using tensor products, generalizing a result of Rickard. The endomorphism rings of these complexes are generalized matrix rings that are “componentwise” tensor products, allowing us to obtain many derived equivalences that have not been observed by using previous techniques. Particular examples include algebras generalizing the ADE-chain related to singularity theory, incidence algebras of posets and certain Auslander algebras or more generally endomorphism algebras of initial preprojective modules over path algebras of quivers. Many of these algebras are fractionally Calabi-Yau and we explicitly compute their CY dimensions. Among the quivers of these algebras one can find shapes of lines, rectangles and triangles.