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The Resource Integral closure of ideals, rings, and modules, Craig Huneke, Irena Swanson

Integral closure of ideals, rings, and modules, Craig Huneke, Irena Swanson

Label
Integral closure of ideals, rings, and modules
Title
Integral closure of ideals, rings, and modules
Statement of responsibility
Craig Huneke, Irena Swanson
Creator
Contributor
Subject
Language
eng
Member of
Cataloging source
Z@L
http://library.link/vocab/creatorName
Huneke, C.
Dewey number
512.44
Government publication
government publication of a state province territory dependency etc
Illustrations
illustrations
Index
index present
Literary form
non fiction
Nature of contents
bibliography
http://library.link/vocab/relatedWorkOrContributorName
Swanson, Irena
Series statement
London Mathematical Society lecture note series
Series volume
336
http://library.link/vocab/subjectName
  • Integral closure
  • Ideals (Algebra)
  • Commutative rings
  • Modules (Algebra)
Label
Integral closure of ideals, rings, and modules, Craig Huneke, Irena Swanson
Instantiates
Publication
Bibliography note
Includes bibliographical references (pages [405]-421) and index
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • 1.2
  • 5.5
  • Defining equations of Rees algebras
  • 104
  • 5.6
  • Blowing up
  • 108
  • 6
  • Valuations
  • 113
  • 6.1
  • Integral closure via reductions
  • Valuations
  • 113
  • 6.2
  • Value groups and valuation rings
  • 115
  • 6.3
  • Existence of valuation rings
  • 117
  • 6.4
  • More properties of valuation rings
  • 5
  • 119
  • 6.5
  • Valuation rings and completion
  • 121
  • 6.6
  • Some invariants
  • 124
  • 6.7
  • Examples of valuations
  • 130
  • 1.3
  • 6.8
  • Valuations and the integral closure of ideals
  • 133
  • 6.9
  • The asymptotic Samuel function
  • 138
  • 7
  • Derivations
  • 143
  • 7.1
  • Integral closure of an ideal is an ideal
  • Analytic approach
  • 143
  • 7.2
  • Derivations and differentials
  • 147
  • 8
  • Reductions
  • 150
  • 8.1
  • Basic properties and examples
  • 6
  • 150
  • 8.2
  • Connections with Rees algebras
  • 154
  • 8.3
  • Minimal reductions
  • 155
  • 8.4
  • Reducing to infinite residue fields
  • 159
  • 1.4
  • 8.5
  • Superficial elements
  • 160
  • 8.6
  • Superficial sequences and reductions
  • 165
  • 8.7
  • Non-local rings
  • 169
  • 8.8
  • Monomial ideals
  • Sally's theorem on extensions
  • 171
  • 9
  • Analytically unramified rings
  • 177
  • 9.1
  • Rees's characterization
  • 178
  • 9.2
  • Module-finite integral closures
  • 9
  • 180
  • 9.3
  • Divisorial valuations
  • 182
  • 10
  • Rees valuations
  • 187
  • 10.1
  • Uniqueness of Rees valuations
  • 187
  • 1.5
  • 10.2
  • A construction of Rees valuations
  • 191
  • 10.4
  • Properties of Rees valuations
  • 201
  • 10.5
  • Rational powers of ideals
  • 205
  • 11
  • Table of basic properties
  • Integral closure of rings
  • Multiplicity and integral closure
  • 212
  • 11.1
  • Hilbert-Samuel polynomials
  • 212
  • 11.2
  • Multiplicity
  • 217
  • 11.3
  • Rees's theorem
  • 13
  • 222
  • 11.4
  • Equimultiple families of ideals
  • 225
  • 12
  • The conductor
  • 234
  • 12.1
  • A classical formula
  • 235
  • 1.6
  • 12.2
  • One-dimensional rings
  • 235
  • 12.3
  • The Lipman-Sathaye theorem
  • 237
  • 13
  • The Briancon-Skoda Theorem
  • 244
  • 13.1
  • How integral closure arises
  • Tight closure
  • 245
  • 13.2
  • Briancon-Skoda via tight closure
  • 248
  • 13.3
  • The Lipman-Sathaye version
  • 250
  • 13.4
  • General version
  • 14
  • 253
  • 14
  • Two-dimensional regular local rings
  • 257
  • 14.1
  • Full ideals
  • 258
  • 14.2
  • Quadratic transformations
  • 263
  • 1.7
  • 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
  • Dedekind-Mertens formula
  • Simple integrally closed ideals
  • 277
  • 15
  • Computing integral closure
  • 281
  • 15.1
  • Method of Stolzenberg
  • 282
  • 15.2
  • Some computations
  • 17
  • 286
  • 15.3
  • General algorithms
  • 292
  • 15.4
  • Monomial ideals
  • 295
  • 16
  • Integral dependence of modules
  • 302
  • 2
  • 16.2
  • Using symmetric algebras
  • 304
  • 16.3
  • Using exterior algebras
  • 307
  • 16.4
  • Properties of integral closure of modules
  • 309
  • 16.5
  • Integral closure of rings
  • 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
  • ix
  • 23
  • 325
  • 17
  • Joint reductions
  • 331
  • 17.1
  • Definition of joint reductions
  • 331
  • 17.2
  • Superficial elements
  • 333
  • 2.2
  • 17.3
  • Existence of joint reductions
  • 335
  • 17.4
  • Mixed multiplicities
  • 338
  • 17.5
  • More manipulations of mixed multiplicities
  • 344
  • 17.6
  • Lying-Over, Incomparability, Going-Up, Going-Down
  • 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
  • 30
  • 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
  • 2.3
  • 18.4
  • Adjoints of monomial ideals
  • 366
  • 18.5
  • Adjoints in two-dimensional regular rings
  • 369
  • 18.6
  • Mapping cones
  • 372
  • 18.7
  • Integral closure and grading
  • Analogs of adjoint ideals
  • 375
  • 19
  • Normal homomorphisms
  • 378
  • 19.1
  • Normal homomorphisms
  • 379
  • 19.2
  • Locally analytically unramified rings
  • 34
  • 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
  • 2.4
  • Appendix
  • A Some background material
  • 392
  • A.1
  • Some forms of Prime Avoidance
  • 392
  • A.2
  • Caratheodory's theorem
  • 392
  • A.3
  • Rings of homomorphisms of ideals
  • Grading
  • 393
  • A.4
  • Complexes
  • 394
  • A.5
  • Macaulay representation of numbers
  • 396
  • Appendix B
  • Height and dimension formulas
  • 39
  • 397
  • B.1
  • Going-Down, Lying-Over, flatness
  • 397
  • B.2
  • Dimension and height inequalities
  • 398
  • B.3
  • Dimension formula
  • 399
  • 1
  • 3
  • B.4
  • Formal equidimensionality
  • 401
  • B.5
  • Dimension Formula
  • 403
  • Separability
  • 47
  • 3.1
  • Algebraic separability
  • 47
  • 3.2
  • General separability
  • 48
  • 3.3
  • What is integral closure of ideals?
  • Relative algebraic closure
  • 52
  • 4
  • Noetherian rings
  • 56
  • 4.1
  • Principal ideals
  • 56
  • 4.2
  • Normalization theorems
  • 1
  • 57
  • 4.3
  • Complete rings
  • 60
  • 4.4
  • Jacobian ideals
  • 63
  • 4.5
  • Serre's conditions
  • 70
  • 1.1
  • 4.6
  • Affine and Z-algebras
  • 73
  • 4.7
  • Absolute integral closure
  • 77
  • 4.8
  • Finite Lying-Over and height
  • 79
  • 4.9
  • Basic properties
  • Dimension one
  • 83
  • 4.10
  • Krull domains
  • 85
  • 5
  • Rees algebras
  • 93
  • 5.1
  • Rees algebra constructions
  • 2
  • 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
Control code
73458763
Dimensions
23 cm
Extent
xiv, 431 pages
Isbn
9780521688604
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other control number
9780521688604
Other physical details
illustrations
System control number
(OCoLC)73458763
Label
Integral closure of ideals, rings, and modules, Craig Huneke, Irena Swanson
Publication
Bibliography note
Includes bibliographical references (pages [405]-421) and index
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • 1.2
  • 5.5
  • Defining equations of Rees algebras
  • 104
  • 5.6
  • Blowing up
  • 108
  • 6
  • Valuations
  • 113
  • 6.1
  • Integral closure via reductions
  • Valuations
  • 113
  • 6.2
  • Value groups and valuation rings
  • 115
  • 6.3
  • Existence of valuation rings
  • 117
  • 6.4
  • More properties of valuation rings
  • 5
  • 119
  • 6.5
  • Valuation rings and completion
  • 121
  • 6.6
  • Some invariants
  • 124
  • 6.7
  • Examples of valuations
  • 130
  • 1.3
  • 6.8
  • Valuations and the integral closure of ideals
  • 133
  • 6.9
  • The asymptotic Samuel function
  • 138
  • 7
  • Derivations
  • 143
  • 7.1
  • Integral closure of an ideal is an ideal
  • Analytic approach
  • 143
  • 7.2
  • Derivations and differentials
  • 147
  • 8
  • Reductions
  • 150
  • 8.1
  • Basic properties and examples
  • 6
  • 150
  • 8.2
  • Connections with Rees algebras
  • 154
  • 8.3
  • Minimal reductions
  • 155
  • 8.4
  • Reducing to infinite residue fields
  • 159
  • 1.4
  • 8.5
  • Superficial elements
  • 160
  • 8.6
  • Superficial sequences and reductions
  • 165
  • 8.7
  • Non-local rings
  • 169
  • 8.8
  • Monomial ideals
  • Sally's theorem on extensions
  • 171
  • 9
  • Analytically unramified rings
  • 177
  • 9.1
  • Rees's characterization
  • 178
  • 9.2
  • Module-finite integral closures
  • 9
  • 180
  • 9.3
  • Divisorial valuations
  • 182
  • 10
  • Rees valuations
  • 187
  • 10.1
  • Uniqueness of Rees valuations
  • 187
  • 1.5
  • 10.2
  • A construction of Rees valuations
  • 191
  • 10.4
  • Properties of Rees valuations
  • 201
  • 10.5
  • Rational powers of ideals
  • 205
  • 11
  • Table of basic properties
  • Integral closure of rings
  • Multiplicity and integral closure
  • 212
  • 11.1
  • Hilbert-Samuel polynomials
  • 212
  • 11.2
  • Multiplicity
  • 217
  • 11.3
  • Rees's theorem
  • 13
  • 222
  • 11.4
  • Equimultiple families of ideals
  • 225
  • 12
  • The conductor
  • 234
  • 12.1
  • A classical formula
  • 235
  • 1.6
  • 12.2
  • One-dimensional rings
  • 235
  • 12.3
  • The Lipman-Sathaye theorem
  • 237
  • 13
  • The Briancon-Skoda Theorem
  • 244
  • 13.1
  • How integral closure arises
  • Tight closure
  • 245
  • 13.2
  • Briancon-Skoda via tight closure
  • 248
  • 13.3
  • The Lipman-Sathaye version
  • 250
  • 13.4
  • General version
  • 14
  • 253
  • 14
  • Two-dimensional regular local rings
  • 257
  • 14.1
  • Full ideals
  • 258
  • 14.2
  • Quadratic transformations
  • 263
  • 1.7
  • 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
  • Dedekind-Mertens formula
  • Simple integrally closed ideals
  • 277
  • 15
  • Computing integral closure
  • 281
  • 15.1
  • Method of Stolzenberg
  • 282
  • 15.2
  • Some computations
  • 17
  • 286
  • 15.3
  • General algorithms
  • 292
  • 15.4
  • Monomial ideals
  • 295
  • 16
  • Integral dependence of modules
  • 302
  • 2
  • 16.2
  • Using symmetric algebras
  • 304
  • 16.3
  • Using exterior algebras
  • 307
  • 16.4
  • Properties of integral closure of modules
  • 309
  • 16.5
  • Integral closure of rings
  • 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
  • ix
  • 23
  • 325
  • 17
  • Joint reductions
  • 331
  • 17.1
  • Definition of joint reductions
  • 331
  • 17.2
  • Superficial elements
  • 333
  • 2.2
  • 17.3
  • Existence of joint reductions
  • 335
  • 17.4
  • Mixed multiplicities
  • 338
  • 17.5
  • More manipulations of mixed multiplicities
  • 344
  • 17.6
  • Lying-Over, Incomparability, Going-Up, Going-Down
  • 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
  • 30
  • 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
  • 2.3
  • 18.4
  • Adjoints of monomial ideals
  • 366
  • 18.5
  • Adjoints in two-dimensional regular rings
  • 369
  • 18.6
  • Mapping cones
  • 372
  • 18.7
  • Integral closure and grading
  • Analogs of adjoint ideals
  • 375
  • 19
  • Normal homomorphisms
  • 378
  • 19.1
  • Normal homomorphisms
  • 379
  • 19.2
  • Locally analytically unramified rings
  • 34
  • 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
  • 2.4
  • Appendix
  • A Some background material
  • 392
  • A.1
  • Some forms of Prime Avoidance
  • 392
  • A.2
  • Caratheodory's theorem
  • 392
  • A.3
  • Rings of homomorphisms of ideals
  • Grading
  • 393
  • A.4
  • Complexes
  • 394
  • A.5
  • Macaulay representation of numbers
  • 396
  • Appendix B
  • Height and dimension formulas
  • 39
  • 397
  • B.1
  • Going-Down, Lying-Over, flatness
  • 397
  • B.2
  • Dimension and height inequalities
  • 398
  • B.3
  • Dimension formula
  • 399
  • 1
  • 3
  • B.4
  • Formal equidimensionality
  • 401
  • B.5
  • Dimension Formula
  • 403
  • Separability
  • 47
  • 3.1
  • Algebraic separability
  • 47
  • 3.2
  • General separability
  • 48
  • 3.3
  • What is integral closure of ideals?
  • Relative algebraic closure
  • 52
  • 4
  • Noetherian rings
  • 56
  • 4.1
  • Principal ideals
  • 56
  • 4.2
  • Normalization theorems
  • 1
  • 57
  • 4.3
  • Complete rings
  • 60
  • 4.4
  • Jacobian ideals
  • 63
  • 4.5
  • Serre's conditions
  • 70
  • 1.1
  • 4.6
  • Affine and Z-algebras
  • 73
  • 4.7
  • Absolute integral closure
  • 77
  • 4.8
  • Finite Lying-Over and height
  • 79
  • 4.9
  • Basic properties
  • Dimension one
  • 83
  • 4.10
  • Krull domains
  • 85
  • 5
  • Rees algebras
  • 93
  • 5.1
  • Rees algebra constructions
  • 2
  • 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
Control code
73458763
Dimensions
23 cm
Extent
xiv, 431 pages
Isbn
9780521688604
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other control number
9780521688604
Other physical details
illustrations
System control number
(OCoLC)73458763

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