内容

目次

Basic Commutative Algebra, Spectrum, Modules, Localization,Multiplicatively Closed Subsets, Rings and Modules of Fractions, LocalizationTechnique, Prime Ideals of a Localized Ring, Integral Ring Extensions,Integral Elements, Integrality and Primes, Direct Sums and Products, TheTensor Product, Definition, Functoriality, Exactness, Flat Algebras, ExteriorPowers, Introduction to Projective Modules, Generalities on ProjectiveModules, 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 CompletableVectors, Stable Freeness over Polynomial Rings, Schanuel's Lemma, Proof ofStable 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, VectorBundles and Projective Modules, Examples, Vector Bundles and Grassmannians,The Direct Limit and Infinite Matrices, Metrization of the Set of ContinuousMaps, Correspondence of Vector Bundles and Classes of Maps, ProjectiveModules over Topological Rings, Basic Commutative Algebra II, NoetherianRings and Modules, Irreducible Sets, Dimension of Rings, Artinian Rings,Small Dimension Theorem, Noether Normalization, Affine Algebras, Hilbert'sNullstellensatz, Dimension of a Polynomial Ring, Splitting Theorem andLindel's Proof, Serre's Splitting Theorem, Lindel's Proof, Regular Rings,Definition, Regular Residue Class Rings, Homological Dimension, AssociatedPrime Ideals, Homological Characterization, Dedekind Rings, Examples, Modulesover Dedekind Rings, Finiteness of Class Numbers, Number of Generators, TheProblems, Regular Sequences, Forster-Swan Theorem, Varieties as Intersectionsof n Hypersurfaces, Curves as Complete Intersection, A Motivation of Serre'sConjecture, 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.