Serre's Problem on Projective Modules

"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.

List of contents

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.

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 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.

Additional text

From the reviews:

"It is a full-fledged advanced course on themes in higher algebra suited for a specialized graduate seminar, a research seminar, and of course, self-study by an aspiring researcher. … Serre’s Problem on Projective Modules, is very clear and well written … and quickly gets the reader properly air-borne. … the pay-off is huge: this is fantastic stuff. … is a superb book. It’s highly recommended." (Michael Berg, MathDL, March, 2007)
"The book starts with the basics of projective modules and the K₀ and K₁ groups, and then gives the classical, partial results about Serre’s conjecture. … This well-written book is the definitive treatment of ‘Serre’s conjecture’ – its history, solution, and generalizations – and will be of interest to both beginning graduate students and advanced researchers in this field." (David F. Anderson, Zentralblatt MATH, Vol. 1101 (3), 2007)
"Lam has done a magnificent job of organizing the mated al and presenting complete proofs of all the results directly connected with Sen-e's problem. ... The references are complete and make the book a very valuable reference even for experts in the field.... It will be very useful to students wishing to learn about projective modules ... . This is definitely a book that anyone ... interested in projective modules should have on his or her shelf!" (Richard G. Swan, Bulletin of the American Mathematical Society, Vol. 45 (3), July, 2008)

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From the reviews:

"It is a full-fledged advanced course on themes in higher algebra suited for a specialized graduate seminar, a research seminar, and of course, self-study by an aspiring researcher. ... Serre's Problem on Projective Modules, is very clear and well written ... and quickly gets the reader properly air-borne. ... the pay-off is huge: this is fantastic stuff. ... is a superb book. It's highly recommended." (Michael Berg, MathDL, March, 2007)
"The book starts with the basics of projective modules and the K₀ and K₁ groups, and then gives the classical, partial results about Serre's conjecture. ... This well-written book is the definitive treatment of 'Serre's conjecture' - its history, solution, and generalizations - and will be of interest to both beginning graduate students and advanced researchers in this field." (David F. Anderson, Zentralblatt MATH, Vol. 1101 (3), 2007)
"Lam has done a magnificent job of organizing the mated al and presenting complete proofs of all the results directly connected with Sen-e's problem. ... The references are complete and make the book a very valuable reference even for experts in the field.... It will be very useful to students wishing to learn about projective modules ... . This is definitely a book that anyone ... interested in projective modules should have on his or her shelf!" (Richard G. Swan, Bulletin of the American Mathematical Society, Vol. 45 (3), July, 2008)

Product details

• Authors: T Y Lam, T. Y. Lam, T.Y. Lam
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 11.10.2010
• EAN: 9783642062353
• ISBN: 978-3-642-06235-3
• No. of pages: 404
• Dimensions: 181 mm x 21 mm x 253 mm
• Weight: 836 g
• Illustrations: XXII, 404 p.
• Series: Springer Monographs in Mathematics; Springer Monographs in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, C, Mathematics, Theorem, Mathematics and Statistics, Commutative algebra, Commutative rings, Commutative Rings and Algebras, Rings (Algebra), Associative rings, Associative Rings and Algebras, Projective modules, polynomial, polynomial rings