Orders in non-semisimple algebras

Abstract

In several branches of representation theory, the existence of Auslander-Reiten sequences has led to new structural insights. for example, in the module theory of artinian algebras [11, 6], in the theory of lattices over classical orders [18, 2] over a complete discrete valuation domain R, and for the corresponding derived categories [12, 17]. For an R-order ⋀ in a finite dimentional algebra A over the quotient field K of R, Auslander and Reiten [2, 5] have charaterized the non-projective indecomposable ⋀-lattice E for which an Auslander-Reiten sequence (AR-sequence for short) L → H → E exists as those ⋀-lattices A-module KE is projective. In the present paper, we shall introduce a modified version of AR-sequences in the category ⋀-lat of ⋀-lattices which behave similar to AR-sequence of modules over artinian algebras. In fact, there will be a close relationship to AR-sequences in ⋀-mod, where ⋀ : = ⋀/(RadR)⋀. This relationship extends to AR-sequences in A-mod if ⋀ is hereditary (e.g. for a path order ⋀ = R△ of a quiver △ without oriented cycles.) Our investigation is inspired by recent work of W. Crawley-Boevey [9] who determined the lattices E with Extras (E, E = 0 over a path order R△.