Integral Closure of Ideals, Rings, and Modules, Volume 13Integral 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 |
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algebra analytically assume assumption Chapter characteristic claim Clearly complete compute condition construction contained contracted contradiction Corollary corresponding defined Definition denote dimension elements equals equivalent example Exercise exists exists an integer extension field field of fractions finitely follows formally equidimensional gives graded height Hence holds homogeneous homomorphism implies induction infinite integral closure integral dependence integral domain isomorphic joint reduction Lemma Let R,m loss m-primary ideal m–primary Math maximal ideal minimal prime ideal module module–finite multiplicity natural necessarily Noetherian local ring Noetherian ring non–zero normal Note particular polynomial polynomial ring positive integer Proof Proposition prove R–module rank reduction Rees algebra Rees valuations regular local ring residue field respect satisfies satisfying separable sequence shows superficial Suppose Theorem theory unique unit valuation ring variables Write zero