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A Course in Commutative Algebra / by Gregor Kemper
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Graduate Texts in Mathematics ; 256 | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2011Description: Online-Ressource (XI, 246p, digital)ISBN:- 9783642035456
- 512.44
- 516.35
- QA564-609
Contents:
Summary: Introduction -- Part I The Algebra Geometry Lexicon: 1 Hilbert's Nullstellensatz; 2 Noetherian and Artinian Rings; 3 The Zariski Topology; 4 A Summary of the Lexicon -- Part II Dimension: 5 Krull Dimension and Transcendence Degree; 6 Localization; 7 The Principal Ideal Theorem; 8 Integral Extensions -- Part III Computational Methods: 9 Gröbner Bases; 10 Fibers and Images of Morphisms Revisited; 11 Hilbert Series and Dimension -- Part IV Local Rings: 12 Dimension Theory; 13 Regular Local Rings; 14 Rings of Dimension One -- References -- Notation -- Index.Summary: This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text.PPN: PPN: 1650743955Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""A Course in Commutative Algebra ""; ""Preface""; ""Contents""; ""Introduction""; ""Part I The Algebra�Geometry Lexicon""; ""Chapter 1: Hilbert's Nullstellensatz""; ""1.1 Maximal Ideals""; ""1.2 Jacobson Rings""; ""1.3 Coordinate Rings""; ""Exercises for Chapter 1 ""; ""Chapter 2: Noetherian and Artinian Rings""; ""2.1 The Noether and Artin Properties for Rings and Modules""; ""2.2 Noetherian Rings and Modules""; ""Exercises for Chapter 2 ""; ""Chapter 3: The Zariski Topology""; ""3.1 Affine Varieties""; ""3.2 Spectra""; ""3.3 Noetherian and Irreducible Spaces""
""Exercises for Chapter 3 """"Chapter 4: A Summary of the Lexicon""; ""4.1 True Geometry: Affine Varieties""; ""4.2 Abstract Geometry: Spectra""; ""Exercises for Chapter 4 ""; ""Part II Dimension""; ""Chapter 5: Krull Dimension and Transcendence Degree""; ""Exercises for Chapter 5 ""; ""Chapter 6: Localization""; ""Exercises for Chapter 6 ""; ""Chapter 7: The Principal Ideal Theorem""; ""7.1 Nakayama's Lemma and the Principal Ideal Theorem""; ""7.2 The Dimension of Fibers""; ""Exercises for Chapter 7 ""; ""Chapter 8: Integral Extensions""; ""8.1 Integral Closure""
""8.2 Lying Over, Going Up, and Going Down""""8.3 Noether Normalization""; ""Exercises for Chapter 8 ""; ""Part III Computational Methods""; ""Chapter 9: Gröbner Bases""; ""9.1 Buchberger's Algorithm""; ""9.2 First Application: Elimination Ideals""; ""Exercises for Chapter 9 ""; ""Chapter 10: Fibers and Images of Morphisms Revisited""; ""10.1 The Generic Freeness Lemma""; ""10.2 Fiber Dimension and Constructible Sets""; ""10.3 Application: Invariant Theory""; ""Exercises for Chapter 10 ""; ""Chapter 11: Hilbert Series and Dimension""; ""11.1 The Hilbert�Serre Theorem""
""11.2 Hilbert Polynomials and Dimension""""Exercises for Chapter 11 ""; ""Part IV Local Rings""; ""Chapter 12: Dimension Theory""; ""12.1 The Length of a Module""; ""12.2 The Associated Graded Ring""; ""Exercises for Chapter 12 ""; ""Chapter 13: Regular Local Rings""; ""13.1 Basic Properties of Regular Local Rings""; ""13.2 The Jacobian Criterion""; ""Exercises for Chapter 13 ""; ""Chapter 14: Rings of Dimension One""; ""14.1 Regular Rings and Normal Rings""; ""14.2 Multiplicative Ideal Theory""; ""14.3 Dedekind Domains""; ""Exercises for Chapter 14 ""; ""Solutions of Some Exercises""
""References""""Notation""; ""Index""
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