Ideals and Reality

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, Boratýnski.

"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.

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.