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,. . .
Erscheinungsdatum: 24.11.2004

lgrsnf/Ischebeck F., Rao R.A. Ideals and Reality.. Projective Modules and Number of Generators of Ideals (Springer, 2005)(ISBN 3540230327)(T)(O)(339s)_MAa_.djvu

Ischebeck, Friedrich, Rao, Ravi A.

Springer Spektrum. in Springer-Verlag GmbH

Steinkopff. in Springer-Verlag GmbH

Springer monographs in mathematics, Berlin ; New York, ©2005

Springer Nature, Berlin, 2005

1 edition, January 12, 2005

Germany, Germany

Contents
1 Basic Commutative Algebra 1
1.1 The Spectrum
1.2 Modules
1.3 Localization
1.3.1 Multiplicatively Closed Subsets
1.3.2 Rings and Modules of Fractions
1.3.3 Localization Technique
1.3.4 Prime Ideals of a Localized Ring
1.4 Integral Ring Extensions
1.4.1 Integral Elements
1.4.2 Integrality and Primes
1.5 Direct Sums and Products
1.6 The Tensor Product
1.6.1 Definition
1.6.2 Functoriality
1.6.3 Exactness
1.6.4 Flat Algebras
1.6.5 Exterior Powers
2 Introduction to Projective Modules
2.1 Generalities on Projective Modules
2.2 Rank
2.3 Special Residue Class Rings
2.4 Projective Modules of Rank 1
3 Stably Free Modules
3.1 Generalities
3.2 Localized Polynomial Rings
3.3 Action of GL[sub(n)](R) on Um[sub(n)](R)
3.4 Elementary Action on Unimodular Rows
3.5 Examples of Completable Rows
3.6 Direct sums of a stably free module
3.7 Stable Freeness over Polynomial Rings
3.7.1 Schanuel’s Lemma
3.7.2 Proof of Stable Freeness
4 Serre’s Conjecture
4.1 Elementary Divisors
4.2 Horrocks’ Theorem
4.3 Quillen’s Local Global Principle
4.4 Suslin’s Proof
4.5 Vaserstein’s Proof
5 Continuous Vector Bundles
5.1 Categories and Functors
5.2 Vector Bundles
5.3 Vector Bundles and Projective Modules
5.4 Examples
5.5 Vector Bundles and Grassmannians
5.5.1 The Direct Limit and Infinite Matrices
5.5.2 Metrization of the Set of Continuous Maps
5.5.3 Correspondence of Vector Bundles and Classes of Maps
5.6 Projective Spaces
5.7 Algebraization of Vector Bundles
5.7.1 Projective Modules over Topological Rings
5.7.2 Projective Modules as Pull-Backs
5.7.3 Construction of a Noetherian Subalgebra
6 Basic Commutative Algebra II
6.1 Noetherian Rings and Modules
6.2 Irreducible Sets
6.3 Dimension of Rings
6.4 Artinian Rings
6.5 Small Dimension Theorem
6.6 Noether Normalization
6.7 Affine Algebras
6.8 Hilbert’s Nullstellensatz
6.9 Dimension of a Polynomial ring
7 Splitting Theorem and Lindel’s Proof
7.1 Serre’s Splitting Theorem
7.2 Lindel’s Proof
8 Regular Rings
8.1 Definition
8.2 Regular Residue Class Rings
8.3 Homological Dimension
8.4 Associated Prime Ideals
8.5 Homological Characterization
8.6 Dedekind Rings
8.7 Examples
8.8 Modules over Dedekind Rings
8.9 Finiteness of Class Numbers
9 Number of Generators
9.1 The Problems
9.2 Regular sequences
9.3 Forster-Swan Theorem
9.3.1 Basic elements
9.3.2 Basic elements and the Forster-Swan theorem
9.3.3 Forster-Swan theorem via K-theory
9.4 Eisenbud-Evans theorem
9.5 Varieties as Intersections of n Hypersurfaces
9.6 The Eisenbud-Evans conjectures
10 Curves as Complete Intersection
10.1 A Motivation of Serre’s Conjecture
10.2 The Conormal Module
10.3 Local Complete Intersection Curves
10.4 Cowsik-Nori Theorem
10.4.1 A Projection Lemma
10.4.2 Proof of Cowsik-Nori
A Normality of E[sub(n)] in GL[sub(n)]
B Some Homological Algebra
B.1 Extensions and Ext[sup(1)]
B.2 Derived functors
C Complete intersections and Connectedness
D Odds and Ends
E Exercises
References
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
P
R
S
T
V
W
Z

The authorsstudy projective modules over commutative rings and, partly in connection with these, numbers of generators of modules and ideals. Especially they treat the question when affine curves are ideal or set theoretic complete intersections. In two parts, at the beginning and in the middle,the book gives a comprehensive introduction to Commutative Algebra. This enables students who know only the fundamentals of Algebra to follow the book.A chapter ontopological vector bundles is included wherethe authorsdescribe their wellknown connection with projective modules in a recent form, due to Vaserstein. TOC: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

<p><p>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, Borat&#253;nsk. <p>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 <p>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.<p>even This Advanced Material, Presented Comprehensively, Keeps In Mind The Young Student As Potential Reader Besides The Specialists Of The Subject.</p>

2024-07-31