Integral Closure Of Ideals, Rings, And Modules

Description:
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 Briancon-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.

Review:
'...the appearance of this wonderful text, which gives an extensive, detailed and pretty well complete account of this whole area up to the present day, is very welcome. As is expected of these authors, the treatment is impressively well-informed, wide-ranging and convincing in its treatment of all aspects of the subject. The mathematics is presented elegantly and efficiently, helpful motivation is put in place in a perfectly judged manner, striking and informative asides are mentioned throughout, and the wealth of background and general culture carried by the many and varied exercises invaluable. ...A huge amount of material scattered throughout the classical and mroe up-to-date literature has been brought together in detailed, coherent and thoroughly worked-through form.' - Liam O'Carroll, Mathematical Reviews Clippings

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 Briancon-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.

Author Biography:
Irena Swanson is a Professor in the Department of Mathematics at Reed College, Portland. Craig Huneke is the Henry J. Bischoff Professor in the Department of Mathematics, University of Kansas.