Integral Closure of Ideals, Rings, and Modules, Volume 13

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Cambridge University Press, Oct 12, 2006 - Mathematics - 431 pages
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.
 

Contents

What is integral closure of ideals?
1
Integral closure of rings
23
Separability
47
Noetherian rings
56
Rees algebras
93
Valuations
113
Derivations
143
Reductions
150
The BrianconSkoda Theorem
244
Twodimensional regular local rings
257
Computing integral closure
281
Integral dependence of modules
302
Joint reductions
331
Adjoints of ideals
360
Normal homomorphisms
378
Appendix A Some background material
392

Analytically unramified rings
177
Rees valuations
187
Multiplicity and integral closure
212
The conductor
234
References
405
Index
422
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About the author (2006)

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

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