Integral closure of ideals, rings, and modules /

Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Huneke, C. (Craig)
Otros Autores: Swanson, Irena
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge, UK : Cambridge University Press, 2006.
Colección:London Mathematical Society lecture note series ; 336.
Temas:
Integral closure.
Ideals (Algebra)
Commutative rings.
Modules (Algebra)
Fermeture intégrale.
Idéaux (Algèbre)
Anneaux commutatifs.
Modules (Algèbre)
MATHEMATICS > Algebra > Intermediate.
Commutative rings
Integral closure
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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