Ideals and Reality: Projective Modules and Number of Generators of Ideals / Edition 1

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This monograph tells the story of a philosophy of J-P. Serre and his vision of relating that philosophy to problems in affine algebraic geometry. It gives a lucid presentation of the Quillen-Suslin theorem settling Serre's conjecture. The central topic of the book is the question of whether a curve in $n$-space is as a set an intersection of $(n-1)$ hypersurfaces, depicted by the central theorems of Ferrand, Szpiro, Cowsik-Nori, Mohan Kumar, Boratýnsk.
The book gives a comprehensive introduction to basic commutative algebra, together with the related methods from homological algebra, which will enable students who know only the fundamentals of algebra to enjoy the power of using these tools. At the same time, it also serves as a valuable reference for the research specialist and as potential course material, because the authors present, for the first time in book form, an approach here that is an intermix of classical algebraic K-theory and complete intersection techniques, making connections with the famous results of Forster-Swan and Eisenbud-Evans. A study of projective modules and their connections with topological vector bundles in a form due to Vaserstein is included. Important subsidiary results appear in the copious exercises.
Even this advanced material, presented comprehensively, keeps in mind the young student as potential reader besides the specialists of the subject.
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Editorial Reviews

From the Publisher
From the reviews:

"This monograph tells the story of a philosophy of J.-P. Serre and his vision of relating that philosophy to problems in affine algebraic geometry. It gives a lucid presentation of the Quillen-Suslin theorem settling Serre’s conjecture. … The book gives a comprehensive introduction to basic commutative algebra … which will enable students who know only the fundamentals of algebra to enjoy the power of using these tools. At the same time, it also serves as a valuable reference for the research specialist and as potential course material … ." (Bulletin Bibliographique, Vol. 51 (1-2), 2005)

"The book under review deals with projective modules and the minimal number of generators of ideals and modules over a Noetherian ring. This book is written in a style accessible to a graduate student and fairly self-contained. It has a collection of interesting exercises at the end … . It also has an extensive bibliography, supplemented by yet another bibliography giving only the Math. Review numbers. … I highly recommend this book to anyone interested in problems related to complete intersections and projective modules." (N. Mohan Kumar, Zentralblatt MATH, Vol. 1075, 2006)

"This study of projective modules begins with an introduction to commutative algebra, followed by an introduction to projective modules. Stably-free modules are considered in some detail … . This … unusual mixture provides a coherent presentation of many important ideas." (Mathematika, Vol. 52, 2005)

"This is a rather ambitious undertaking, but the authors do an admirable job. … There are several remarkable things about this book. The two biggest are the density and the efficiency. … And it’s done very concisely. It is accessible to most graduate students with at least some experience in algebra. … it can be used to bring these students ‘up to speed’ with many of the contemporary ideas of algebra. … And algebraists will find it to be a handy reference." (Donald L. Vestal, MathDL, May, 2005)

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Product Details

  • ISBN-13: 9783540230328
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 1/12/2005
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 2005
  • Edition number: 1
  • Pages: 336
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.81 (d)

Table of Contents

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.
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