Alfred Wiedemann

Auslander-reiten quivers of schurian orders

We shall consider orders ⋀ over a complete discrete rank one valuation ring R in a split full matrix ring containing a complete sex of primitive orthogonal idempot ents. In case s of finite lattice type and R is the power series ring in one variable over its residexe class field k , we give a description of its index composable lattices and its Auslander-Reiten quiver in terms of representations of partially ordered sets. By a model theoretic argument, this implies a description of all indecomposable lattices if R is the completion of the ring of integers in an algebraic number field at all but possibly a finite number of primes.

Global dimension two orders are quasi-hereditary

Motivated by results of Cline, Parshall and Scott on quasi-hereditary algebras, in [8] the concept of a quasi-hereditary order is introduced in integral representation theory. In this note we show that the results of Dlab and Ringel on quasi-hereditary semiprimary rings and hereditary artinian rings presented in [6] have integral analogues in the theory of orders. In particular, we prove as our main result the followingTheorem: An order of global dimension at most two over a complete Dedekind domain R in a separable algebra over the quotient field of R is quasi-hereditary.

The existence of covering functors for orders and consequences for stable components with oriented cycles

The starting point for the covering theory for algebras of finite type over algebraically closed fields was Riedtmann’s paper, “Algebren, Darstellungskocher, uberlagerungen und zurtick” [ 14 1. Her main result, from today’s point of view, is the existence of covering functors, i.e., the existence of a bijection between the morphisms between indecomposable modules and purely combinatorial defined vector spaces generated by paths in a translation quiver (cf. (1.1) [ 141). This bijection has recently turned out to be more and more important in the representation theory of algebras. It was the first step leading to the use of Auslander-Reiten quivers as a tool for the classification of algebras of linite type [ 15, 81. The main purpose of this paper is to initiate an analog for classical orders over complete Dedekind domains. We shall formulate our results and proofs only for orders, but it is always obvious that everything carries immediately over to the algebra situation. Let R be a complete Dedekind domain with quotient field K, and let n be an R-order in a separable K-algebra A [ 161: n is a subring of A with the same identity, and moreover A is a full R-sublattice, which means that /i is an R-lattice and contains a K-basis of A. We denote by r the Auslander-Reiten quiver of/i, i.e., vertices of r are the isomorphism classes of indecomposable n-lattices, and two vertices are joined by an arrow provided there exists an irreducible /i-map between the corresponding A-lattices. Moreover, we denote by f the universal cover of I[8], in particular, the vertices of r” are homotopy classes of walks in I-, with covering morphism F: F-P r [8]. For each arrow x + y in F we define

Auslander-Reiten quivers of local, orders of finite lattice type

,, as the space of irreducible maps Irr(Fx, Fy) [4, 171 from Fx to Fy. Moreover, for indecomposable /i-lattices IV, N let r’(M, N) c Hom,(M, N) be the ith functorial radical, i.e., r’(M, N) = ri(-, N)(M), where ri(-, N) is the ith power of the radical of the presentable fun&or (-, N) = Hom,((, N) [ 21. 292 0021-8693/85

A geometrically motivated investigation of the Tor-groups of quasi-hereditary algebras

3.00

A remark on the structure of the Auslander-Reiten quiver of orders, blocks with cyclic defect two and the Dynkin diagram\(\mathbb{E}_6 \)

The classification of the Auslander-Reiten quivers of the local orders of finite lattice type is completed. For this purpose, it is shown—using the results of [7]—that to the list of the known Auslander-Reiten quivers of the local Bäckström orders of finite lattice type [11], [14] and of the local Gorenstein orders of finite type and their minimal overorders [18] one has to add two remaining types of valued translation quivers to obtain a complete list of all Auslander-Reiten quivers of the local orders of finite lattice type.

Orders with loops in their auslander-reiten graph

This paper presents part of the outcome of joint discussions with Leonard L. Scott to whom I am very grateful for his introducing me to the subject of derived categories of module categories of quasi-hereditary algebras and perverse sheaves. The exactness of some functors for sheaves in connection with a recollement diagram of derived categories [6], namely the “extension-by-zerofunctor” and the “restriction-functor,” suggests that the corresponding functors between the module categories of quasi-hereditary algebras should also be in some sense exact, at least on reasonably good subcategories. More precisely, let A be a quasi-hereditary algebra over an algebraically closed field, and let e be an idempotent such that J= AeA is an ideal occurring in a defining heredity chain of A [2, 31. Then we show that the functor

Path orders of global dimension two

Introduction. In [5] we determined up to stable equivalence the structure of the Auslander-Reiten quiver of a block ~ of the group ring RG, G a finite group, R an unramified extension of the p-adic integers, having a cyclic defect group of order p2. It was not possible using the well-known structure theorem on stable Auslander-Reiten quivers [3] or group theoretical concepts as Green correspondence to rule out the unpleasant case that for p = 3, the tree class of the stable Auslander-Reiten quiver [3] of ~ might be the Dynkin diagram E 6. The purpose of this paper is to exclude this case. So we may reformulate our main result in [5, 4.4 Theorem (ii)] as follows: