Abelian Groups and Representations of Finite Partially Ordered Sets / Edition 1 available in Hardcover
- Pub. Date:
- Springer New York
A recurring theme in a traditional introductory graduate algebra cours e is the existence and consequences of relationships between different algebraic structures. This is also the theme of this book, an exposit ion of connections between representations of finite partially ordered sets and abelian groups. Emphasis is placed throughout on classificat ion, a description of the objects up to isomorphism, and computation o f representation type, a measure of when classification is feasible.
Table of Contents
1 Representations of Posets over a Field.- 1.1 Vector Spaces with Distinguished Subspaces.- 1.2 Representations of Posets and Matrix Problems.- 1.3 Finite Representation Type.- 1.4 Tame and Wild Representation Type.- 1.5 Generic Representations.- 2 Torsion-Free Abelian Groups.- 2.1 Quasi-isomorphism and Isomorphism at p.- 2.2 Near-isomorphism of Finite Rank Groups.- 2.3 Stable Range Conditions for Finite Rank Groups.- 2.4 Self-Small Groups and Endomorphism Rings.- 3 Butler Groups.- 3.1 Types and Completely Decomposable Groups.- 3.2 Characterizations of Finite Rank Groups.- 3.3 Quasi-isomorphism and-Representations of Posets.- 3.4 Countable Groups.- 3.5 Quasi-Generic Groups.- 4 Representations over a Discrete Valuation Ring.- 4.1 Finite and Rank-Finite Representation Type.- 4.2 Wild Modulo p Representation Type.- 4.3 Finite Rank Butler Groups and Isomorphism at p.- 5 Almost Completely Decomposable Groups.- 5.1 Characterizations and Properties.- 5.2 Isomorphism at p and Representation Type.- 5.3 Uniform Groups.- 5.4 Primary Regulating Quotient Groups.- 6 Representations over Fields and Exact Sequences.- 6.1 Projectives, Injectives, and Exact Sequences.- 6.2 Coxeter Correspondences.- 6.3 Almost Split Sequences.- 6.4 A Torsion Theory and Localizations.- 7 Finite Rank Butler Groups.- 7.1 Projectives, Injectives, and Exact Sequences.- 7.2 Endomorphism Rings.- 7.3 Bracket Groups.- 8 Applications of Representations and Butler Groups.- 8.1 Torsion-Free Modules over Discrete Valuation Rings.- 8.2 Finite Valuated Groups.- References.- List of Symbols.- Index of Terms.