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, Boratinsk. 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.

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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