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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.Read more
- First book to collect the material on integral closures into a unified treatment
- Ideal for graduate students and researchers in commutative algebra or ring theory, with many worked examples and exercises
- Provides a one-stop shop for newcomers and experts
Reviews & endorsements
"...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
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- Date Published: October 2006
- format: Paperback
- isbn: 9780521688604
- length: 448 pages
- dimensions: 229 x 152 x 25 mm
- weight: 0.65kg
- contains: 6 b/w illus. 346 exercises
- availability: Available
Table of Contents
Table of basic properties
Notation and basic definitions
1. What is the integral closure
2. Integral closure of rings
4. Noetherian rings
5. Rees algebras
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
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