Description 
1 online resource (xxi, 401 pages) 
Series 
Springer Monographs in Mathematics, 14397382 

Springer monographs in mathematics

Contents 
Introduction to Serre's conjecture: 19551976  Foundations  The "classical" results on Serre's conjecture  The basic calculus of unimodular rows  Horrock's theorem  Quillen's methods  K₁analogue of Serre's conjecture  The quadratic analogue of Serre's conjecture  New developments (since 1977) 
Summary 
"Serre's Conjecture", for the most part of the second half of the 20th century,  ferred to the famous statement made by J.P. Serre in 1955, to the effect that one did not know if?nitely generated projective modules were free over a polynomial ring k[x, . . ., x], where k is a?eld. This statement was motivated by the fact that 1 n the af?ne scheme de?ned by k[x, . . ., x] is the algebrogeometric analogue of 1 n the af?ne nspace over k. In topology, the nspace is contractible, so there are only trivial bundles over it. Would the analogue of the latter also hold for the nspace in algebraic geometry? Since algebraic vector bundles over Speck[x, . . ., x] corre 1 n spond to?nitely generated projective modules over k[x, . . ., x], the question was 1 n tantamount to whether such projective modules were free, for any base?eld k. ItwasquiteclearthatSerreintendedhisstatementasanopenproblemintheshe theoretic framework of algebraic geometry, which was just beginning to emerge in the mid1950s. Nowhere in his published writings had Serre speculated, one way or another, upon the possible outcome of his problem. However, almost from the start, a surmised positive answer to Serre's problem became known to the world as "Serre's Conjecture". Somewhat later, interest in this "Conjecture" was further heightened by the advent of two new (and closely related) subjects in mathematics: homological algebra, and algebraic Ktheory 
Notes 
Revised edition of author's "Serre's conjecture," 1978 
Bibliography 
Includes bibliographical references (pages 271276 and pages 363393) and index 
Notes 
Online resource; title from digital title page (ProQuest Ebook Central, viewed September 4, 2020) 
Subject 
Projective modules (Algebra)


Commutative algebra.


Geometry, Algebraic.


Commutative algebra.


Geometry, Algebraic.


Projective modules (Algebra)


SerreVermutung


Anéis e álgebras comutativos.

Form 
Electronic book

ISBN 
9783540345756 

3540345752 

9783642062353 

3642062350 
