| Description |
1 online resource (xiv, 431 pages) : illustrations |
| Series |
London Mathematical Society lecture note series ; 336 |
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London Mathematical Society lecture note series ; 336.
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| Contents |
Table of basic properties ix -- 1 What is integral closure of ideals? 1 -- 1.1 Basic properties 2 -- 1.2 Integral closure via reductions 5 -- 1.3 Integral closure of an ideal is an ideal 6 -- 1.4 Monomial ideals 9 -- 1.5 Integral closure of rings 13 -- 1.6 How integral closure arises 14 -- 1.7 Dedekind-Mertens formula 17 -- 2 Integral closure of rings 23 -- 2.2 Lying-Over, Incomparability, Going-Up, Going-Down 30 -- 2.3 Integral closure and grading 34 -- 2.4 Rings of homomorphisms of ideals 39 -- 3 Separability 47 -- 3.1 Algebraic separability 47 -- 3.2 General separability 48 -- 3.3 Relative algebraic closure 52 -- 4 Noetherian rings 56 -- 4.1 Principal ideals 56 -- 4.2 Normalization theorems 57 -- 4.3 Complete rings 60 -- 4.4 Jacobian ideals 63 -- 4.5 Serre's conditions 70 -- 4.6 Affine and Z-algebras 73 -- 4.7 Absolute integral closure 77 -- 4.8 Finite Lying-Over and height 79 -- 4.9 Dimension one 83 -- 4.10 Krull domains 85 -- 5 Rees algebras 93 -- 5.1 Rees algebra constructions 93 -- 5.2 Integral closure of Rees algebras 95 -- 5.3 Integral closure of powers of an ideal 97 -- 5.4 Powers and formal equidimensionality 100 -- 5.5 Defining equations of Rees algebras 104 -- 5.6 Blowing up 108 -- 6 Valuations 113 -- 6.1 Valuations 113 -- 6.2 Value groups and valuation rings 115 -- 6.3 Existence of valuation rings 117 -- 6.4 More properties of valuation rings 119 -- 6.5 Valuation rings and completion 121 -- 6.6 Some invariants 124 -- 6.7 Examples of valuations 130 -- 6.8 Valuations and the integral closure of ideals 133 -- 6.9 The asymptotic Samuel function 138 -- 7 Derivations 143 -- 7.1 Analytic approach 143 -- 7.2 Derivations and differentials 147 -- 8 Reductions 150 -- 8.1 Basic properties and examples 150 -- 8.2 Connections with Rees algebras 154 -- 8.3 Minimal reductions 155 -- 8.4 Reducing to infinite residue fields 159 -- 8.5 Superficial elements 160 -- 8.6 Superficial sequences and reductions 165 -- 8.7 Non-local rings 169 -- 8.8 Sally's theorem on extensions 171 -- 9 Analytically unramified rings 177 -- 9.1 Rees's characterization 178 -- 9.2 Module-finite integral closures 180 -- 9.3 Divisorial valuations 182 -- 10 Rees valuations 187 -- 10.1 Uniqueness of Rees valuations 187 -- 10.2 A construction of Rees valuations 191 -- 10.4 Properties of Rees valuations 201 -- 10.5 Rational powers of ideals 205 -- 11 Multiplicity and integral closure 212 -- 11.1 Hilbert-Samuel polynomials 212 -- 11.2 Multiplicity 217 -- 11.3 Rees's theorem 222 -- 11.4 Equimultiple families of ideals 225 -- 12 The conductor 234 -- 12.1 A classical formula 235 -- 12.2 One-dimensional rings 235 -- 12.3 The Lipman-Sathaye theorem 237 -- 13 The Briancon-Skoda Theorem 244 -- 13.1 Tight closure 245 -- 13.2 Briancon-Skoda via tight closure 248 -- 13.3 The Lipman-Sathaye version 250 -- 13.4 General version 253 -- 14 Two-dimensional regular local rings 257 -- 14.1 Full ideals 258 -- 14.2 Quadratic transformations 263 -- 14.3 The transform of an ideal 266 -- 14.4 Zariski's theorems 268 -- 14.5 A formula of Hoskin and Deligne 274 -- 14.6 Simple integrally closed ideals 277 -- 15 Computing integral closure 281 -- 15.1 Method of Stolzenberg 282 -- 15.2 Some computations 286 -- 15.3 General algorithms 292 -- 15.4 Monomial ideals 295 -- 16 Integral dependence of modules 302 -- 16.2 Using symmetric algebras 304 -- 16.3 Using exterior algebras 307 -- 16.4 Properties of integral closure of modules 309 -- 16.5 Buchsbaum-Rim multiplicity 313 -- 16.6 Height sensitivity of Koszul complexes 319 -- 16.7 Absolute integral closures 321 -- 16.8 Complexes acyclic up to integral closure 325 -- 17 Joint reductions 331 -- 17.1 Definition of joint reductions 331 -- 17.2 Superficial elements 333 -- 17.3 Existence of joint reductions 335 -- 17.4 Mixed multiplicities 338 -- 17.5 More manipulations of mixed multiplicities 344 -- 17.6 Converse of Rees's multiplicity theorem 348 -- 17.7 Minkowski inequality 350 -- 17.8 The Rees-Sally formulation and the core 353 -- 18 Adjoints of ideals 360 -- 18.1 Basic facts about adjoints 360 -- 18.2 Adjoints and the Briancon-Skoda Theorem 362 -- 18.3 Background for computation of adjoints 364 -- 18.4 Adjoints of monomial ideals 366 -- 18.5 Adjoints in two-dimensional regular rings 369 -- 18.6 Mapping cones 372 -- 18.7 Analogs of adjoint ideals 375 -- 19 Normal homomorphisms 378 -- 19.1 Normal homomorphisms 379 -- 19.2 Locally analytically unramified rings 381 -- 19.3 Inductive limits of normal rings 383 -- 19.4 Base change and normal rings 384 -- 19.5 Integral closure and normal maps 388 -- Appendix A Some background material 392 -- A.1 Some forms of Prime Avoidance 392 -- A.2 Caratheodory's theorem 392 -- A.3 Grading 393 -- A.4 Complexes 394 -- A.5 Macaulay representation of numbers 396 -- Appendix B Height and dimension formulas 397 -- B.1 Going-Down, Lying-Over, flatness 397 -- B.2 Dimension and height inequalities 398 -- B.3 Dimension formula 399 -- B.4 Formal equidimensionality 401 -- B.5 Dimension Formula 403 |
| Bibliography |
Includes bibliographical references (pages 405-421) and index |
| Notes |
Print version record |
| Subject |
Integral closure.
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Ideals (Algebra)
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Commutative rings.
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Modules (Algebra)
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MATHEMATICS -- Algebra -- Intermediate.
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Commutative rings
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Ideals (Algebra)
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Integral closure
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Modules (Algebra)
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| Form |
Electronic book
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| Author |
Swanson, Irena
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| ISBN |
9781107089303 |
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1107089301 |
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