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Über dieses Buch

In a clear, well-developed presentation this book provides the first systematic treatment of structure results for algebras which are graded by a goup. The fruitful method of constructing graded orders of special kind over a given order, culminating in applications of the construction of generalized Rees rings associated to divisors, is combined with the theory of orders over graded Krull domains. This yields the construction of generalized Rees rings corresponding to the central ramification divisor of the orders and the algebraic properties of the constructed orders. The graded methods allow the study of regularity conditions on order.

The book also touches upon representation theoretic methods, including orders of finite representation type and other aspects of this theory applicable to the classification of orders. The final chapter describes the ring theoretical approach to the classification of orders of global dimension two, originally carried out by M. Artin using more geometrical methods.

Since its subject is important in many research areas, this book will be valuable reading for all researchers and graduate students with an interest in non-commutative algebra.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
Tracing back the Nile to its origin must be about as difficult as tracing back the origins of our interest in the theory of orders. At many junctions one has to choose in an almost arbitrary way which is the Nile and which is the other river joining it, wondering whether in such problems one should stick to the wider or to the deeper stream. Perhaps a convenient solution is to recognize that there are many sources and then to list just a few. Those inspired by number theory will certainly think first about the theory of maximal orders over Dedekind domains in number fields, the representation theory-based algebraist will refer to integral group rings, an algebraic geometer will perhaps point to orders over normal domains, and the ring theorist might view orders in central simple algebras as his favorite class of P.I. rings. In these topics graded orders and orders over graded rings appear not only as natural examples, but also as important basic ingredients: crossed products for finite groups, group rings considered as graded rings, orders over projective varieties, rings of generic matrices, trace rings, etc. On these observations we founded our belief that the application of methods from the theory of graded rings to the special case of orders may lead to some interesting topics for research, new points of view, and results. The formulation of this intent alone creates several problems of choice.
L. le Bruyn, M. Van den Bergh, F. Van Oystaeyen

I. Commutative Arithmetical Graded Rings

Abstract
The graded rings encountered in this chapter are the ones that will appear as the centres of the graded orders considered in this book. An order graded by an arbitrary group need not have a graded centre, but when the grading group is abelian this property does hold. Because we exclusively consider orders over domains it makes sense to restrict attention to commutative rings which are graded by torsion free abelian groups, in particular where Krull domains are concerned. On the other hand, constructions over gr-Dedekind rings appear as examples or in the constructive methods for studying class groups, hence it will be sufficient to develop the basic facts about gr-Dedekind rings and the related valuation theory in the ℤ-graded case only.
L. le Bruyn, M. Van den Bergh, F. Van Oystaeyen

II. Graded Rings and Orders

Abstract
The rings considered in consequent chapters will allways be P.I. rings, nevertheless in the present section the restriction to the P.I. case is really superfluous and it would serve no aim at all since none of the proofs given in this section would simplify if one restricts attention to P.I. rings. Hence, the fact that we prefer to include the following results in a generality exceding that of the main body of this book is a choice inspired by an esthetic evaluation rather than by pragmatic arguments.
L. le Bruyn, M. Van den Bergh, F. Van Oystaeyen

III. Artihmetically Graded Rings over Orders

Abstract
Throughout this section, Λ will be a prime p.i. ring, graded by a torsion free Abelian group G. We want to prove under mild conditions on the grading (in particular, Λ has to be divisorially graded) that arithmetical properties of Λ e , e being the neutral element of G, extend to Λ. Further, we include counterexamples in order to show that the converse implications do not hold.
L. le Bruyn, M. Van den Bergh, F. Van Oystaeyen

IV. Regular Orders

Abstract
In this first section we will recall some basic results on orders having finite global dimension. Since we will freely use results from homological algebra, the reader is referred to [13] for more details.
L. le Bruyn, M. Van den Bergh, F. Van Oystaeyen

V. Two-dimensional Tame and Maximal Orders of Finite Representation Type

Abstract
In Chapter IV we have already touched upon the matter of orders of finite representation type. In this chapter we show how the methods developed there, maybe used to rederive some of the results in [3]. In particular, a variant of Theorem V.2.10 tells us that the representation theory of two dimensional tame orders is, in the case of finite repre­sentation type, determined by a rational double point together with a cyclic group action on its module category. This allows one to compute the ranks of the Cohen Macaulay modules. We then prove a struc­ture theorem for tame orders with the above hypothesis. Each such tame order is of the form Λ = k[[x, y]] * c G where G acts linearly on the two dimensional space V = kx + ky and c takes values in k. We then relate the C.M modules of A to the projective representations of G corresponding to c.
L. le Bruyn, M. Van den Bergh, F. Van Oystaeyen

Backmatter

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