The Resource Integral closure of ideals, rings, and modules, Craig Huneke, Irena Swanson, (electronic resource)

Integral closure of ideals, rings, and modules, Craig Huneke, Irena Swanson, (electronic resource)

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
Genre
Language
  • eng
  • eng
Summary
Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure
Member of
Cataloging source
MiAaPQ
http://library.link/vocab/creatorName
Huneke, C.
Dewey number
512.44
Illustrations
illustrations
Index
index present
Language note
English
LC call number
QA251.3
LC item number
.H86 2006
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
http://library.link/vocab/relatedWorkOrContributorName
Swanson, Irena
Series statement
  • London Mathematical Society lecture note series
  • 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, (electronic resource)
Instantiates
Publication
Note
Description based upon print version of record
Bibliography note
Includes bibliographical references (p. [405]-421) and index
Carrier category
online resource
Carrier category code
  • cr
Content category
text
Content type code
  • txt
Contents
  • Cover; Series Page; Title; Copyright; Contents; Table of basic properties; Notation and basic definitions; Preface; 1 What is integral closure of ideals?; 1.1. Basic properties; 1.2. Integral closure via reductions; 1.3. Integral closure of an ideal is an ideal; 1.4. Monomial ideals; 1.5. Integral closure of rings; 1.6. How integral closure arises; 1.7. Dedekind-Mertens formula; 1.8. Exercises; 2 Integral closure of rings; 2.1. Basic facts; 2.2. Lying-Over, Incomparability, Going-Up, Going-Down; 2.3. Integral closure and grading; 2.4. Rings of homomorphisms of ideals; 2.5. Exercises
  • 3 Separability3.1. Algebraic separability; 3.2. General separability; 3.3. Relative algebraic closure; 3.4. Exercises; 4 Noetherian rings; 4.1. Principal ideals; 4.2. Normalization theorems; 4.3. Complete rings; 4.4. Jacobian ideals; 4.5. Serre's conditions; 4.6. Affine and Z-algebras; 4.7. Absolute integral closure; 4.8. Finite Lying-Over and height; 4.9. Dimension one; 4.10. Krull domains; 4.11. Exercises; 5 Rees algebras; 5.1. Rees algebra constructions; 5.2. Integral closure of Rees algebras; 5.3. Integral closure of powers of an ideal; 5.4. Powers and formal equidimensionality
  • 5.5. Defining equations of Rees algebras5.6. Blowing up; 5.7. Exercises; 6 Valuations; 6.1. Valuations; 6.2. Value groups and valuation rings; 6.3. Existence of valuation rings; 6.4. More properties of valuation rings; 6.5. Valuation rings and completion; 6.6. Some invariants; 6.7. Examples of valuations; 6.8. Valuations and the integral closure of ideals; 6.9. The asymptotic Samuel function; 6.10. Exercises; 7 Derivations; 7.1. Analytic approach; 7.2. Derivations and differentials; 7.3. Exercises; 8 Reductions; 8.1. Basic properties and examples; 8.2. Connections with Rees algebras
  • 8.3. Minimal reductions8.4. Reducing to infinite residue fields; 8.5. Superficial elements; 8.6. Superficial sequences and reductions; 8.7. Non-local rings; 8.8. Sally's theorem on extensions; 8.9. Exercises; 9 Analytically unramified rings; 9.1. Rees's characterization; 9.2. Module-finite integral closures; 9.3. Divisorial valuations; 9.4. Exercises; 10 Rees valuations; 10.1. Uniqueness of Rees valuations; 10.2. A construction of Rees valuations; 10.3. Examples; 10.4. Properties of Rees valuations; 10.5. Rational powers of ideals; 10.6. Exercises; 11 Multiplicity and integral closure
  • 11.1. Hilbert-Samuel polynomials11.2. Multiplicity; 11.3. Rees's theorem; 11.4. Equimultiple families of ideals; 11.5. Exercises; 12 The conductor; 12.1. A classical formula; 12.2. One-dimensional rings; 12.3. The Lipman-Sathaye theorem; 12.4. Exercises; 13 The Briançon-Skoda Theorem; 13.1. Tight closure; 13.2. Briançon-Skoda via tight closure; 13.3. The Lipman-Sathaye version; 13.4. General version; 13.5. Exercises; 14 Two-dimensional regular local rings; 14.1. Full ideals; 14.2. Quadratic transformations; 14.3. The transform of an ideal; 14.4. Zariski's theorems
  • 14.5. A formula of Hoskin and Deligne
Dimensions
unknown
Extent
1 online resource (448 p.)
Form of item
online
Isbn
9781107095557
Media category
computer
Media type code
  • c
Specific material designation
remote
System control number
  • (CKB)2550000001095225
  • (EBL)1179159
  • (OCoLC)850149065
  • (SSID)ssj0000915879
  • (PQKBManifestationID)12387962
  • (PQKBTitleCode)TC0000915879
  • (PQKBWorkID)10869918
  • (PQKB)10937181
  • (MiAaPQ)EBC1179159
  • (EXLCZ)992550000001095225
Label
Integral closure of ideals, rings, and modules, Craig Huneke, Irena Swanson, (electronic resource)
Publication
Note
Description based upon print version of record
Bibliography note
Includes bibliographical references (p. [405]-421) and index
Carrier category
online resource
Carrier category code
  • cr
Content category
text
Content type code
  • txt
Contents
  • Cover; Series Page; Title; Copyright; Contents; Table of basic properties; Notation and basic definitions; Preface; 1 What is integral closure of ideals?; 1.1. Basic properties; 1.2. Integral closure via reductions; 1.3. Integral closure of an ideal is an ideal; 1.4. Monomial ideals; 1.5. Integral closure of rings; 1.6. How integral closure arises; 1.7. Dedekind-Mertens formula; 1.8. Exercises; 2 Integral closure of rings; 2.1. Basic facts; 2.2. Lying-Over, Incomparability, Going-Up, Going-Down; 2.3. Integral closure and grading; 2.4. Rings of homomorphisms of ideals; 2.5. Exercises
  • 3 Separability3.1. Algebraic separability; 3.2. General separability; 3.3. Relative algebraic closure; 3.4. Exercises; 4 Noetherian rings; 4.1. Principal ideals; 4.2. Normalization theorems; 4.3. Complete rings; 4.4. Jacobian ideals; 4.5. Serre's conditions; 4.6. Affine and Z-algebras; 4.7. Absolute integral closure; 4.8. Finite Lying-Over and height; 4.9. Dimension one; 4.10. Krull domains; 4.11. Exercises; 5 Rees algebras; 5.1. Rees algebra constructions; 5.2. Integral closure of Rees algebras; 5.3. Integral closure of powers of an ideal; 5.4. Powers and formal equidimensionality
  • 5.5. Defining equations of Rees algebras5.6. Blowing up; 5.7. Exercises; 6 Valuations; 6.1. Valuations; 6.2. Value groups and valuation rings; 6.3. Existence of valuation rings; 6.4. More properties of valuation rings; 6.5. Valuation rings and completion; 6.6. Some invariants; 6.7. Examples of valuations; 6.8. Valuations and the integral closure of ideals; 6.9. The asymptotic Samuel function; 6.10. Exercises; 7 Derivations; 7.1. Analytic approach; 7.2. Derivations and differentials; 7.3. Exercises; 8 Reductions; 8.1. Basic properties and examples; 8.2. Connections with Rees algebras
  • 8.3. Minimal reductions8.4. Reducing to infinite residue fields; 8.5. Superficial elements; 8.6. Superficial sequences and reductions; 8.7. Non-local rings; 8.8. Sally's theorem on extensions; 8.9. Exercises; 9 Analytically unramified rings; 9.1. Rees's characterization; 9.2. Module-finite integral closures; 9.3. Divisorial valuations; 9.4. Exercises; 10 Rees valuations; 10.1. Uniqueness of Rees valuations; 10.2. A construction of Rees valuations; 10.3. Examples; 10.4. Properties of Rees valuations; 10.5. Rational powers of ideals; 10.6. Exercises; 11 Multiplicity and integral closure
  • 11.1. Hilbert-Samuel polynomials11.2. Multiplicity; 11.3. Rees's theorem; 11.4. Equimultiple families of ideals; 11.5. Exercises; 12 The conductor; 12.1. A classical formula; 12.2. One-dimensional rings; 12.3. The Lipman-Sathaye theorem; 12.4. Exercises; 13 The Briançon-Skoda Theorem; 13.1. Tight closure; 13.2. Briançon-Skoda via tight closure; 13.3. The Lipman-Sathaye version; 13.4. General version; 13.5. Exercises; 14 Two-dimensional regular local rings; 14.1. Full ideals; 14.2. Quadratic transformations; 14.3. The transform of an ideal; 14.4. Zariski's theorems
  • 14.5. A formula of Hoskin and Deligne
Dimensions
unknown
Extent
1 online resource (448 p.)
Form of item
online
Isbn
9781107095557
Media category
computer
Media type code
  • c
Specific material designation
remote
System control number
  • (CKB)2550000001095225
  • (EBL)1179159
  • (OCoLC)850149065
  • (SSID)ssj0000915879
  • (PQKBManifestationID)12387962
  • (PQKBTitleCode)TC0000915879
  • (PQKBWorkID)10869918
  • (PQKB)10937181
  • (MiAaPQ)EBC1179159
  • (EXLCZ)992550000001095225

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