The Algebraic Structure of Crossed ProductsBy
- G. Karpilovsky
In the past 15 years, the theory of crossed products has enjoyed a period of vigorous development. The foundations have been strengthened and reorganized from new points of view, especially from the viewpoint of graded rings.The purpose of this monograph is to give, in a self-contained manner, an up-to-date account of various aspects of this development, in an effort to convey a comprehensive picture of the current state of the subject. It is assumed that the reader has had the equivalent of a standard first-year graduate course, thus familiarity with basic ring-theoretic and group-theoretic concepts and an understanding of elementary properties of modules, tensor products and fields. A chapter on algebraic preliminaries is included, which briefly surveys topics required later in the book.
North-Holland Mathematics Studies
Published: May 1987
- Preface. Chapters: 1. Preliminaries. Notation and Terminology. Projective, Injective and Flat Modules. Artinian and Noetherian Modules. Group Actions. Cohomology Groups and Group Extensions. Some Properties of Cohomology Groups. Matrix Rings and Related Results. 2. Group-Graded Algebras and Crossed Products: General Theory. Definitions and Elementary Properties. Equivalent Crossed Products. Some Ring-Theoretic Results. The Centre of Crossed Products over Simple Rings. Projective Crossed Representations. Graded and G-Invariant Ideals. Induced Modules. Montgomery's Theorem. 3. The Classical Theory of Crossed Products. Central Simple Algebras. The Brauer Group. Classical Crossed Products and the Brauer Group. 4. Clifford Theory for Graded Algebras. Graded Modules. Restriction to A1. Graded Homomorphism Modules. Extension from A1. Induction from A1. 5. Primitive and Prime Ideals of Crossed Products. Primitive, Prime and Semiprime Ideals. Primitive Ideals in Crossed Products. Prime Coefficient Rings. Incomparability and Going Down. A Going Up Theorem. Chains of Prime and Primitive Ideals. 6. Semiprime and Prime Crossed Products. Coset Calculus. &Dgr;-Methods. The Main Theorem and Its Applications. Sufficient Conditions for Semiprimeness. Twisted Group Algebras. Bibliography. Notation. Index.