Serre's Problem on Projective Modules

Couverture
Springer Science & Business Media, 17 mai 2010 - 404 pages
“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 algebro-geometric analogue of 1 n the af?ne n-space over k. In topology, the n-space is contractible, so there are only trivial bundles over it. Would the analogue of the latter also hold for the n-space 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 mid-1950s. 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 K-theory.
 

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Table des matières

19551976
1
Chapter I Foundations
8
Chapter II The Classical Results on Serres Conjecture
71
Chapter III The Basic Calculus of Unimodular Rows
98
Chapter IV Horrocks Theorem
139
Chapter V Quillens Methods
161
Chapter VI K1Analogue of Serres Conjecture
203
Chapter VII The Quadratic Analogue of Serres Conjecture
233
References for Chapters IVII
271
Complete Intersections and Serres Conjecture
277
Chapter VIII New Developments since 1977
288
References for Chapter VIII
363
Index
394
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