A Reduction Functor, Tameness, and Tits Form for a Class of Orders Original Research Article

Let D be a complete discrete valuation domain which is an algebra over an algebraically closed field K. We study in the paper a class of suborders Λ of tiled D-orders by means of a rational quadratic form qΛ associated to Λ and a finite poset I*+Λ having exactly two maximal elements * and +. Criteria for finite lattice type and for tame lattice type of Λ are given in terms of the form qΛ and of the category I*+Λ-spr of socle projective K-linear representations of the poset I*+Λ. The shape of Auslander-Reiten quiver Γ(latt(Λ)) is described in Corollary 3.2. A reduction functor image: latt(Λ) → I*+Λ-spr preserving representation types is constructed. It is shown in Corollary 2.10 that for any poset I having exactly two maximal elements there exists a D-order in our class, an additive functor latt(Λ) → I-spr preserving representation types and a poset isomorphism I congruent with I*+Λ.