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Selected Works of Maurice Auslander
Edited by: Idun Reiten, Sverre O. Smalø, and Øyvind Solberg, Norwegian University of Science and Technology, Trondheim, Norway
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Collected Works
1999; 1638 pp; hardcover
Volume: 10
ISBN-10: 0-8218-0679-3
ISBN-13: 978-0-8218-0679-1
List Price: US$334
Member Price: US$267.20
Order Code: CWORKS/10
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In view of Maurice Auslander's important contributions to many parts of algebra, there is great interest in the present volume. This book features a broad selection of the core of his work, including commutative algebra, singularity theory, the theory of orders, and the representation theory of artin algebras.

Although Auslander worked in many areas, there are characteristics common to most of his research. Of particular note is his use of homological methods, including functor categories. While his early work was concerned mostly with commutative rings and his later work mainly with artin algebras, he was always interested in finding common features and common settings.

The broad range and impact of Auslander's contributions are reflected clearly in this volume. The editors have included background material, interrelationships between papers and indications of further developments. A paper of note and one that is not available readily is included: the Queen Mary College Notes on "Representation Dimension of Artin Algebras". This book is of interest for the historical development of algebra over a 40-year period and for the use of homological methods in algebra, covering both commutative ring theory and artin algebra theory.

Readership

Graduate students, researchers, and historians interested in commutative rings and algebras.

Reviews

"One should thank the editors, all of whom were longstanding collaborators of Auslander, for the work involved in editing these volumes. They are a must for any algebraist working in the areas mentioned above to see all the important contributions by Auslander, most of them very timely even today, and some of them more timely than when they were written."

-- Mathematical Reviews

Table of Contents

Part 1. Chapter I
  • Homological dimension and local rings
  • On the dimension of modules and algebras. III: Global dimension
  • Commutator subgroups of free groups
  • On the dimension of modules and algebras. VI: Comparison of global and algebra dimension
  • On regular group rings
  • Homological dimension in local rings
  • Homological dimension in noetherian rings. II
  • Codimension and multiplicity
  • Codimension and multiplicity (corrections)
  • Unique factorization in regular local rings
  • A remark on a paper of M. Hironaka
Chapter II
  • Ramification theory
  • On ramification theory in noetherian rings
  • Maximal orders
  • The Brauer group of a commutative ring
  • Modules over unramified regular local rings
  • On the purity of the branch locus
  • Ramification index and multiplicity
  • Modules over unramified regular local rings
  • Brauer groups of discrete valuation rings
  • Galois actions on rings and finite Galois coverings
Chapter III
  • Functors
  • Coherent functors
  • Stable equivalence of artin algebras
  • Stable equivalence of dualizing \(R\)-varieties
  • A functorial approach to representation theory
  • Adjoint functors and an extension of the Green correspondence for group representations
  • \(D\) Tr-periodic modules and functors
Chapter IV
  • Almost split sequences and artin algebras
  • Representation dimension of artin algebras
  • A characterization of orders of finite lattice type
  • Representation theory of artin algebras. I
  • Representation theory of artin algebras. II
  • Representation theory of artin algebras. III: Almost split sequences
  • Large modules over artin algebras
  • Representation theory of artin algebras. IV: Invariants given by almost split sequences
  • Representation theory of artin algebras. V: Methods for computing almost split sequences and irreducible morphisms
  • Representation theory of artin algebras. VI: A functorial approach to almost split sequences
  • Representation theory of hereditary artin algebras
  • Almost split sequences whose middle term has at most two indecomposable summands
  • Relations for Grothendieck groups of artin algebras
Chapter V
  • Some topics in representation theory
  • On a generalized version of the Nakayama conjecture
  • Modules with waists
  • Modules determined by their composition factors
  • Almost split sequences and group rings
  • On a theorem of E. Green on the dual of the transpose
Part 2. Chapter VI
  • Lattices over general orders
  • Functors and morphisms determined by objects
  • Applications of morphisms determined by modules
  • A survey of existence theorems for almost split sequences
Chapter VII
  • Tilting theory and homologically finite subcategories
  • Coxeter functors without diagrams
  • Preprojective modules over artin algebras
  • Almost split sequences in subcategories
  • Applications of contravariantly finite subcategories
  • Homological theory of idempotent ideals
Chapter VIII
  • Almost split sequences and commutative rings
  • Isolated singularities and existence of almost split sequences
  • Rational singularities and almost split sequences
  • Almost split sequences for rational double points
  • The Cohen-Macaulay type of Cohen-Macaulay rings
  • Almost split sequences for Cohen-Macaulay modules
  • The what, where, and why of almost split sequences
  • Cohen-Macaulay modules for graded Cohen-Macaulay rings and their completions
  • Graded modules and their completions
Chapter IX
  • Grothendieck groups and Cohen-Macaulay approximations
  • Grothendieck groups of algebras and orders
  • Grothendieck groups of algebras with nilpotent annihilators
  • The homological theory of maximal Cohen-Macaulay approximations
  • Liftings and weak liftings of modules
Chapter X
  • Relative theory and syzygy modules
  • Relative homology and representation theory. I: Relative homology and homologically finite subcategories
  • Relative homology and representation theory. II: Relative cotilting theory
  • \(k\)-Gorenstein algebras and syzygy modules
  • Syzygy modules for noetherian rings
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