Ideals and Reality: Projective Modules and Number of Generators of Ideals

ISBN-13: 9783642061950 / Angielski / Miękka / 2010 / 336 str.

Friedrich Ischebeck; Ravi A. Rao

Opis

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, . .

Recenzje

From the reviews:

"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 book gives a comprehensive introduction to basic commutative 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 ... ." (Bulletin Bibliographique, Vol. 51 (1-2), 2005)

"The book under review deals with projective modules and the minimal number of generators of ideals and modules over a Noetherian ring. This book is written in a style accessible to a graduate student and fairly self-contained. It has a collection of interesting exercises at the end ... . It also has an extensive bibliography, supplemented by yet another bibliography giving only the Math. Review numbers. ... I highly recommend this book to anyone interested in problems related to complete intersections and projective modules." (N. Mohan Kumar, Zentralblatt MATH, Vol. 1075, 2006)

"This study of projective modules begins with an introduction to commutative algebra, followed by an introduction to projective modules. Stably-free modules are considered in some detail ... . This ... unusual mixture provides a coherent presentation of many important ideas." (Mathematika, Vol. 52, 2005)

"This is a rather ambitious undertaking, but the authors do an admirable job. ... There are several remarkable things about this book. The two biggest are the density and the efficiency. ... And it's done very concisely. It is accessible to most graduate students with at least some experience in algebra. ... it can be used to bring these students 'up to speed' with many of the contemporary ideas of algebra. ... And algebraists will find it to be a handy reference." (Donald L. Vestal, MathDL, May, 2005)

Spis treści

Basic Commutative Algebra 1.- to Projective Modules.- Stably Free Modules.- Serre’s Conjecture.- Continuous Vector Bundles.- Basic Commutative Algebra II.- Serre’s Splitting Theorem and Lindel’s Proof of Serre’s Conjecture.- Regular Rings.- Bounds on the Number of Generators.- Curves as Complete Intersection.

Blurb

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.

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.