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Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.
Integral closure has played a role in number theory and algebraic geometry since the nineteenth century, but a modern formulation of the concept for ideals perhaps began with the work of Krull and Zariski in the 1930s. It has developed into a tool for the analysis of many algebraic and geometric problems. This book collects together the central notions of integral closure and presents a unified treatment. Techniques and topics covered include: behavior of the Noetherian property under integral closure, analytically unramified rings, the conductor, field separability, valuations, Rees algebras, Rees valuations, reductions, multiplicity, mixed multiplicity, joint reductions, the Briançon-Skoda theorem, Zariski's theory of integrally closed ideals in two-dimensional regular local rings, computational aspects, adjoints of ideals and normal homomorphisms. With many worked examples and exercises, this book will provide graduate students and researchers in commutative algebra or ring theory with an approachable introduction leading into the current literature.
Table of contents:
Table of basic properties; Notation and basic definitions; Preface; 1. What is the integral closure; 2. Integral closure of rings; 3. Separability; 4. Noetherian rings; 5. Rees algebras; 6. Valuations; 7. Derivations; 8. Reductions; 9. Analytically unramified rings; 10. Rees valuations; 11. Multiplicity and integral closure; 12. The conductor; 13. The Briançon-Skoda theorem; 14. Two-dimensional regular local rings; 15. Computing the integral closure; 16. Integral dependence of modules; 17. Joint reductions; 18. Adjoints of ideals; 19. Normal homomorphisms; Appendix A. Some background material; Appendix B. Height and dimension formulas; References; Index.
Integral closure has played a role in number theory and algebraic geometry since the nineteenth century, but a modern formulation of the concept for ideals perhaps began with the work of Krull and Zariski in the 1930s. It has developed into a tool for the analysis of many algebraic and geometric problems. This book collects together the central notions of integral closure and presents a unified treatment. Techniques and topics covered include: behavior of the Noetherian property under integral closure, analytically unramified rings, the conductor, field separability, valuations, Rees algebras, Rees valuations, reductions, multiplicity, mixed multiplicity, joint reductions, the Briançon-Skoda theorem, Zariski's theory of integrally closed ideals in two-dimensional regular local rings, computational aspects, adjoints of ideals and normal homomorphisms. With many worked examples and exercises, this book will provide graduate students and researchers in commutative algebra or ring theory with an approachable introduction leading into the current literature.
Table of contents:
Table of basic properties; Notation and basic definitions; Preface; 1. What is the integral closure; 2. Integral closure of rings; 3. Separability; 4. Noetherian rings; 5. Rees algebras; 6. Valuations; 7. Derivations; 8. Reductions; 9. Analytically unramified rings; 10. Rees valuations; 11. Multiplicity and integral closure; 12. The conductor; 13. The Briançon-Skoda theorem; 14. Two-dimensional regular local rings; 15. Computing the integral closure; 16. Integral dependence of modules; 17. Joint reductions; 18. Adjoints of ideals; 19. Normal homomorphisms; Appendix A. Some background material; Appendix B. Height and dimension formulas; References; Index.