Serre's Problem on Projective Modules
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.
Avis des internautes - Rédiger un commentaire
Aucun commentaire n'a été trouvé aux emplacements habituels.
Chapter I Foundations
Chapter II The Classical Results on Serres Conjecture
Chapter III The Basic Calculus of Unimodular Rows
Chapter IV Horrocks Theorem
Chapter V Quillens Methods
Chapter VI K1Analogue of Serres Conjecture
Autres éditions - Tout afficher
affine algebras algebraic K-theory algebraically closed analogue assume Bass Bass-Quillen Conjecture Bhatwadekar cancellation Chapter commutative ring complete intersections Corollary defined denotes division ring elementary exists extended f.g. projective module fact field finitely presented free modules GLn(R Hermite rings homomorphism Horrocks implies induction integer invertible isomorphism K-Hermite Krull dimension Laurent polynomial Laurent polynomial ring Lemma Math matrix maximal ideal Mohan Kumar monic polynomial monoid monomial Murthy noetherian ring noncommutative notations Note paper Parimala polynomial rings prime ideal projective modules proof Proposition proved quadratic spaces Quillen Quillen-Suslin Theorem R-module rank regular local ring regular ring Remark result sequence Serre Serre’s Conjecture Serre’s Problem set-theoretic Spec Sridharan stably free modules surjection Suslin Swan symplectic structure trivial unimodular element unimodular rows Vaserstein vector bundles