On the periodicity of Coxeter transformations and the non-negativity of their Euler forms

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also give a simple, direct proof, that certain products of reflections, defined for any square matrix A with 2 on its main diagonal, and in particular the Coxeter transformation corresponding to a generalized Cartan matrix, can be expressed as $-A_{+}^{-1} A_{-}^t$, where A_{+}, A_{-} are closely associated with the upper and lower triangular parts of A.Comment: 12 pages, (v2) revision, to appear in Linear Algebra and its Application

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.Comment: v3: 21 pages. Slight revision, to appear in the Journal of the London Mathematical Society; v2: 20 pages. Minor changes, pictures and references adde