Ideals and reality : projective modules and number of generators of ideals

The authors study projective modules over commutative rings and, partly in connection with these, numbers of generators of modules and ideals. Especially they treat the question when affine curves are ideal or set theoretic complete intersections. In two parts, at the beginning and in the middle, the book gives a comprehensive introduction to Commutative Algebra. This enables students who know only the fundamentals of Algebra to follow the book. A chapter on topological vector bundles is included where the authors describe their wellknown connection with projective modules in a recent form, due to Vaserstein. TOC:Basic Commutative Algebra, Spectrum, Modules, Localization, Multiplicatively Closed Subsets, Rings and Modules of Fractions, Localization Technique, Prime Ideals of a Localized Ring, Integral Ring Extensions, Integral Elements, Integrality and Primes, Direct Sums and Products, The Tensor Product, Definition, Functoriality, Exactness, Flat Algebras, Exterior Powers, Introduction to Projective Modules, Generalities on Projective Modules, Rank, Special Residue Class Rings, Projective Modules of Rank 1, Stably Free Modules, Generalities, Localized Polynomial Rings, Action of GLn (R) on Umn (R), Elementary Action on Unimodular Rows, Examples of completable Vectors, Stable Freeness over Polynomial Rings, Schanuel's Lemma, Proof of Stable Freeness, Serre's Conjecture, Elementary Divisors, Horrocks' Theorem, Quillen's Local Global Principle, Suslin's Proof, Vaserstein's Proof, Continuous Vector Bundles, Categories and Functors, Vector Bundles, Vector Bundles and Projective Modules, Examples, Vector Bundles and Grassmannians, The Direct Limit and Infinite Matrices, Metrization of the Set of Continuous Maps, Correspondence of Vector Bundles and Classes of Maps, Projective Modules over Topological Rings, Basic Commutative Algebra II, Noetherian Rings and Modules, Irreducible Sets, Dimension of Rings, Artinian Rings, Small Dimension Theorem, Noether Normalization, Affine Algebras, Hilbert's Nullstellensatz, Dimension of a Polynomial ring, Splitting Theorem and Lindel's Proof, Serre's Splitting Theorem, Lindel's Proof, Regular Rings, Definition, Regular Residue Class Rings, Homological Dimension, Associated Prime Ideals, Homological Characterization, Dedekind Rings, Examples, Modules over Dedekind Rings, Finiteness of Class Numbers, Number of Generators, The Problems, Regular sequences, Forster-Swan Theorem, Varieties as Intersections of n Hypersurfaces, Curves as Complete Intersection, A Motivation of Serre's Conjecture, The Conormal Module, Local Complete Intersection Curves, Cowsik - Nori Theorem, A Projection Lemma, Proof of Cowsik-Nori, Classical EE - Estimates, Examples of Set Theoretical Complete Intersection Curves

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• ISBN: 9783540230328 ; 3540230327
• Dimensions: 24 cm
• Extent: XIV, 336 S.
• Language: Englisch
• Notes: graph. Darst. ; Literaturverz. S. 325 - 331

Classification and Topics

• Classification: Mathematik
• Keyword: Kommutative Algebra ; Projektiver Modul ; Serre-Vermutung ; Algebraische K-Theorie ; Vollständiger Durchschnitt

Contributors, Places and Time

• Event: Veröffentlichung
• (where): Berlin, Heidelberg, New York
• (who): Springer
• (when): 2005
• Creator: Ischebeck, Friedrich ; Rao, Ravi A.

Further information

• Table of contents: https://d-nb.info/972000364/04
• Rights: Bei diesem Objekt liegt nur das Inhaltsverzeichnis digital vor. Der Zugriff darauf ist unbeschränkt möglich.
• Last update: 11.06.2025, 2:14 PM CEST