Algebras with cycle-finite derived categories
Let A be a finite-dimensional algebra (associative, with an identity) over an algebraically closed field, and let \(D^ b(A)\) denote the derived category of bounded complexes of finite-dimensional right modules. We show that, if A is connected, then every cycle of \(D^ b(A)\) lies in a tube if and only if \(D^ b(A)\) is equivalent, as a triangulated category, to \(D^ b(C)\), where C either is hereditary of Dynkin or Euclidean type, or is a tubular canonical algebra (in the sense of Ringel). Moreover, we prove that this is the case if and only if A can be obtained from C by a finite sequence of tilts and cotilts. In the representation-infinite case, we also give a complete description of such an algebra A in terms of its bound quiver.
derived category of bounded complexes of finite-dimensional right modules
scientific article; zbMATH DE number 4000158
scientific article