Higher-dimensional orders, graph-orders, and derived equivalences
In an earlier paper Roggenkamp defined and studied a class of classical orders, which he called Green orders. These are \(p\)-adic analogues for Brauer tree algebras, and indeed, blocks of finite group rings over a complete discrete valuation ring of characteristic 0 with residue field of characteristic \(p\) turn out to be Green orders. In the present paper the authors generalize the construction of Green orders in two directions. They generalize to coefficient domains being commutative complete regular local Noetherian rings of finite dimension. Moreover they allow more general (finite) graphs instead of trees as underlying combinatorial structure. The authors characterize these orders in intrinsic terms. The proof follows the line of the corresponding proof for Green orders. Moreover, the authors classify these orders up to derived equivalence. The result is more subtle than the corresponding result for Green orders. As a byproduct the authors manage to distinguish the derived equivalence class of two tame algebras of dihedral type which was left open in an earlier classification given by Thorsten Holm.
scientific article; zbMATH DE number 1563012
scientific article