Serre's Problem on Projective ModulesSpringer 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. |
Table des matières
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 |
363 | |
394 | |
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Expressions et termes fréquents
affine algebraic already answer apply argument assume assumption Bass bundles cancellation Chapter clearly closed commutative ring complete intersections construction contains Corollary course Dedekind domain defined definition denotes desired determinant dimension direct domain elementary elements example exists extended f.g. projective fact field finite given gives Hermite holds Horrocks ideal implies induction instance integer invertible isomorphism Krull later Lemma localization Math matrix maximal ideal monic natural noetherian ring normal Note observation obtained particular polynomial rings presented prime principle Problem projective modules proof Proposition proved quadratic question Quillen R-module rank references regular Remark result Reviewer satisfies sequence Serre’s Conjecture space Spec statement structure suffices suitable Suslin Swan symplectic Theorem theory unimodular rows unit vector write