Serre's Problem on Projective Modules

1. Front Matter

   Pages i-xxi

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2. Introduction to Serre’s Conjecture: 1955–1976

   • Tsit Yuen Lam

   Pages 1-7

3. Foundations

   • Tsit Yuen Lam

   Pages 9-69

4. The “Classical” Results on Serre’s Conjecture

   • Tsit Yuen Lam

   Pages 71-97

5. The Basic Calculus of Unimodular Rows

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   Pages 99-138

6. Horrocks’ Theorem

   • Tsit Yuen Lam

   Pages 139-160

7. Quillen’s Methods

   • Tsit Yuen Lam

   Pages 161-202

8. K₁-Analogue of Serre’s Conjecture

   • Tsit Yuen Lam

   Pages 203-232

9. The Quadratic Analogue of Serre’s Conjecture

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   Pages 233-269

10. References for Chapters I–VII

    • Tsit Yuen Lam

    Pages 271-276

11. Appendix: Complete Intersections and Serre’s Conjecture

    • Tsit Yuen Lam

    Pages 277-287

12. New Developments (since 1977)

    • Tsit Yuen Lam

    Pages 289-361

13. References for Chapter VIII

    • Tsit Yuen Lam

    Pages 363-393

14. Back Matter

    Pages 395-401

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“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.

• Area
• Calc
• Factor
• Microsoft Access
• Serre's Conjecture
• Serre's problem
• boundary element method
• calculus
• development
• mathematics
• polynomial
• polynomial rings
• projective modules
• theorem

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)

• Department of Mathematics, University of California, Berkeley, USA

  Tsit Yuen Lam

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