Skip to main content
  • Book
  • © 2006

Serre's Problem on Projective Modules

Authors:

(view affiliations)
  • An invaluable summary of research work done in the period from 1978 to the present

Part of the book series: Springer Monographs in Mathematics (SMM)

Buying options

eBook
USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-34575-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 159.99
Price excludes VAT (USA)
Hardcover Book
USD 159.99
Price excludes VAT (USA)

This is a preview of subscription content, access via your institution.

Table of contents (12 chapters)

  1. Front Matter

    Pages i-xxi
  2. Foundations

    • Tsit Yuen Lam
    Pages 9-69
  3. The Basic Calculus of Unimodular Rows

    • Tsit Yuen Lam
    Pages 99-138
  4. Horrocks’ Theorem

    • Tsit Yuen Lam
    Pages 139-160
  5. Quillen’s Methods

    • Tsit Yuen Lam
    Pages 161-202
  6. K1-Analogue of Serre’s Conjecture

    • Tsit Yuen Lam
    Pages 203-232
  7. References for Chapters I–VII

    • Tsit Yuen Lam
    Pages 271-276
  8. New Developments (since 1977)

    • Tsit Yuen Lam
    Pages 289-361
  9. References for Chapter VIII

    • Tsit Yuen Lam
    Pages 363-393
  10. Back Matter

    Pages 395-401

About this book

“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.

Keywords

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

Reviews

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)

Authors and Affiliations

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

    Tsit Yuen Lam

Bibliographic Information

Buying options

eBook
USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-34575-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 159.99
Price excludes VAT (USA)
Hardcover Book
USD 159.99
Price excludes VAT (USA)