Ideals and Reality Projective Modules and Number of Generators of Ideals 1st Edition by Friedrich Ischebeck, Ravi Rao 3642061958 9783642061950

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ISBN 10: 3642061958

ISBN 13: 9783642061950

Author: Friedrich Ischebeck, Ravi A. Rao

Besides giving an introduction to Commutative Algebra – the theory of c- mutative rings – this book is devoted to the study of projective modules and the minimal number of generators of modules and ideals. The notion of a module over a ring R is a generalization of that of a vector space over a field k. The axioms are identical. But whereas every vector space possesses a basis, a module need not always have one. Modules possessing a basis are called free. So a finitely generated free R-module is of the form Rn for some n E IN, equipped with the usual operations. A module is called p- jective, iff it is a direct summand of a free one. Especially a finitely generated R-module P is projective iff there is an R-module Q with P @ Q S Rn for some n. Remarkably enough there do exist nonfree projective modules. Even there are nonfree P such that P @ Rm S Rn for some m and n. Modules P having the latter property are called stably free. On the other hand there are many rings, all of whose projective modules are free, e. g. local rings and principal ideal domains. (A commutative ring is called local iff it has exactly one maximal ideal. ) For two decades it was a challenging problem whether every projective module over the polynomial ring k[X1,. . .

Ideals and Reality Projective Modules and Number of Generators of Ideals 1st Table of contents:

Basic Commutative Algebra

Spectrum, Modules, and Localization

Multiplicatively Closed Subsets

Rings and Modules of Fractions

Localization Technique

Prime Ideals of a Localized Ring

Integral Ring Extensions

Integral Elements and Their Properties

Integrality and Prime Ideals

Module Theory

Direct Sums and Products

The Tensor Product: Definition, Functoriality, and Exactness

Flat Algebras and Exterior Powers

Introduction to Projective Modules

Projective Modules and Stability

Rank and Special Residue Class Rings

Projective Modules of Rank 1

Stably Free Modules and Localized Polynomial Rings

Action of GLn(R)GL_n (R)GLn​(R) on Umn(R)Um_n (R)Umn​(R)

Elementary Action on Unimodular Rows

Completable Vectors and Stable Freeness Over Polynomial Rings

Schanuel’s Lemma and Proof of Stable Freeness

Serre’s Conjecture and Related Topics

Elementary Divisors and Horrocks’ Theorem

Quillen’s Local-Global Principle

Proofs by Suslin and Vaserstein

Vector Bundles and Topology

Continuous Vector Bundles

Categories and Functors in Vector Bundle Theory

Grassmannians and the Correspondence with Projective Modules

Direct Limits and Infinite Matrices

Metrization of Continuous Maps

Projective Modules Over Topological Rings

Advanced Commutative Algebra

Noetherian Rings and Modules

Irreducible Sets and Dimension of Rings

Artinian Rings and the Small Dimension Theorem

Noether Normalization and Affine Algebras

Hilbert’s Nullstellensatz and Dimension of a Polynomial Ring

Splitting and Regularity

Splitting Theorem and Lindel’s Proof

Serre’s Splitting Theorem

Regular Rings and Dedekind Rings

Definition and Regular Residue Class Rings

Homological Dimension and Associated Prime Ideals

Dedekind Rings and Their Modules

Finiteness of Class Numbers and Number of Generators

Intersection Theory and Applications

Regular Sequences and the Forster-Swan Theorem

Varieties as Intersections of Hypersurfaces

Complete Intersection Curves and the Conormal Module

Local Complete Intersection Curves

Cowsik-Nori Theorem and Its Proof

Classical EE-Estimates and Examples of Set-Theoretical Complete Intersection Curves

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