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Cover: 9783642062353 | Serre's Problem on Projective Modules | T. Y. Lam | Taschenbuch | 2010
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Serre's Problem on Projective Modules
Taschenbuch von T. Y. Lam
Sprache: Englisch

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Beschreibung
¿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.
¿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.
Zusammenfassung
This book thoroughly covers the area of "Serre's Problem on Projective Modules", as it presents a broad and comprehensive view on the mathematics of "Serre's Conjecture", its history, its different solutions, and the important subsequent developments. This book is an invaluable summary of research work done in the period from 1978 to the present.
Inhaltsverzeichnis
to Serre¿s Conjecture: 1955¿1976.- Foundations.- The ¿Classical¿ Results on Serre¿s Conjecture.- The Basic Calculus of Unimodular Rows.- Horrocks¿ Theorem.- Quillen¿s Methods.- K1-Analogue of Serre¿s Conjecture.- The Quadratic Analogue of Serre¿s Conjecture.- References for Chapters I¿VII.- Appendix: Complete Intersections and Serre¿s Conjecture.- New Developments (since 1977).- References for Chapter VIII.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 428
Reihe: Springer Monographs in Mathematics
Inhalt: xxii
404 S.
ISBN-13: 9783642062353
ISBN-10: 3642062350
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Lam, T. Y.
Auflage: Softcover reprint of hardcover 1st ed. 2006
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Springer Monographs in Mathematics
Maße: 235 x 155 x 24 mm
Von/Mit: T. Y. Lam
Erscheinungsdatum: 21.10.2010
Gewicht: 0,645 kg
preigu-id: 107117020
Zusammenfassung
This book thoroughly covers the area of "Serre's Problem on Projective Modules", as it presents a broad and comprehensive view on the mathematics of "Serre's Conjecture", its history, its different solutions, and the important subsequent developments. This book is an invaluable summary of research work done in the period from 1978 to the present.
Inhaltsverzeichnis
to Serre¿s Conjecture: 1955¿1976.- Foundations.- The ¿Classical¿ Results on Serre¿s Conjecture.- The Basic Calculus of Unimodular Rows.- Horrocks¿ Theorem.- Quillen¿s Methods.- K1-Analogue of Serre¿s Conjecture.- The Quadratic Analogue of Serre¿s Conjecture.- References for Chapters I¿VII.- Appendix: Complete Intersections and Serre¿s Conjecture.- New Developments (since 1977).- References for Chapter VIII.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 428
Reihe: Springer Monographs in Mathematics
Inhalt: xxii
404 S.
ISBN-13: 9783642062353
ISBN-10: 3642062350
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Lam, T. Y.
Auflage: Softcover reprint of hardcover 1st ed. 2006
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Springer Monographs in Mathematics
Maße: 235 x 155 x 24 mm
Von/Mit: T. Y. Lam
Erscheinungsdatum: 21.10.2010
Gewicht: 0,645 kg
preigu-id: 107117020
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