Higher-dimensional orders, graph-orders, and derived equivalences

Green-orders (tree-orders) in the classical one-dimensional case are the setting, to understand p-adic blocks with cyclic defect of finite groups. Blocks with “cyclic defect” of Hecke orders however, are Green-orders over two-dimensional rings. Hecke orders of dihedral groups of order divisible by 4 are even defined over a three-dimensional ring. We extend the notion of Green-orders to orders associated to a locally embedded graph instead of a tree, and to general complete regular local noetherian ground rings of finite dimension. We extend the result, that classical tree-orders are derived equivalent to star-orders. We then use these results to clarify the derived equivalence classes of tame algebras of Dihedral type.