# Ideals and Reality

## Projective Modules and Number of Generators of Ideals

Part of the Springer Monographs in Mathematics book series (SMM)

Part of the Springer Monographs in Mathematics book series (SMM)

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.

Classical algebraic K-theory Complete intersections Finite K-theory Numbers of generators Projective modules Serre's conjecture Vector bundles algebra commutative algebra geometry proof theorem

- DOI https://doi.org/10.1007/b137484
- Copyright Information Springer-Verlag Berlin Heidelberg 2005
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-540-23032-8
- Online ISBN 978-3-540-26370-8
- Series Print ISSN 1439-7382
- About this book