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.
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Table of ContentsI. Commutative Arithmetical Graded Rings.- I.1. Graded Krull Domains.- I.2. Valuations and the structure of Gr-Dedekind Rings.- II. Graded Rings and Orders.- II.1. Graded Rings over Goldie Rings.- II.2. Divisorially Graded Rings and Generalized Rees Rings.- II.3. Graded Rings Satisfying Polynomial Identities.- II.4. Orders and Graded Orders.- II.5. Some General Techniques.- III. Arithmetically Graded Rings over Orders.- III.1. Orders Graded by Torsion Free Abelian Groups.- III.2. Orders Graded by Finite Groups.- III.3. Generalized Crossed Product Azumaya Algebras.- III.4. Extensions of Tame Orders and Related Sequences.- IV. Regular Orders.- IV.1. Moderated Gorenstein and Regular Orders.- IV.2. Orders of Finite Representation Type.- IV.3. Regularizable Domains.- IV.4. Smooth Tame Orders.- IV.5. Weak Brauer-Severi Schemes.- V. Two Dimensional Tame and Maximal Orders of Finite Representation Type.- V.1. Introduction.- V.2. Rational Double Points Associated to Two-dimensional Tame Orders of Finite Representation Type.- V.3. Skew Group Rings and Projective Representations Associated to Two-dimensional Tame Orders of Finite Representation Type.- V.4. Classification of Two-dimensional Maximal Orders of Finite Representation Type.- V.5. Appendix: Recent Work of I. Reiten and the Second Author.- References.