\(J\)-adic filtration of orders with application to orders of finite representation type. (Q1411716)

For an order \(\Lambda\) over a complete discrete valuation domain \(R\) with residue class field \(k\), the author defines a filtering overorder as a hereditary overorder \(\Gamma\) of \(\Lambda\) such that \(\text{Rad}^n\Lambda=\Lambda\cap\text{Rad}^n\Gamma\) holds for \(n\in\mathbb{N}\). If a filtering overorder exists, it is uniquely determined. The main theorem states that \(\Lambda\) has a filtering overorder if and only if the graded ring \(\text{Gr\,}\Lambda\) with respect to the radical filtration is an order (in a semisimple algebra) over a complete discrete valuation domain. Orders with a filtering overorder form an interesting class of orders deserving a further study in their own right. Most notably, however, the author shows that Auslander orders admit a filtering overorder. This leads to the concept of filtering functor, with relationships to Grothendieck groups of orders and additive functions on the Auslander-Reiten quiver. In particular, it follows that any representation-finite order \(\Lambda\) has a uniquely determined hereditary overorder which yields a graded order \(\Lambda'\) over a power series ring \(k[\![x]\!]\) such that \(\Lambda\) and \(\Lambda'\) have the same Auslander-Reiten quiver.