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Serre's Problem on Projective Modules

Springer Verlag
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Here’s the first paragraph of Lam’s Preface:

’Serre’s Conjecture,’ for the most part of the second half of the 20th century, referred to the famous statement made by J.-P. Serre in 1955 [made en passant in one of the most beautiful and important papers ever written, Faisceaux Algébriques Cohérents] to the effect that one did not know if finitely generated projective modules were free over a polynomial ring k[x1, … , xn], where k is a field. This statement was motivated by the fact that the affine scheme defined by k[x1, … , xn] is the algebro-geometric analogue of the affine n-space over k. In topology, the 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 Spec k[x1, … , xn] correspond to finitely generated projective modules over k[x1, … , xn], the question was tantamount to whether such projective modules were free, for any base field k.

This beautifully lucid account of the raison d’être for T. Y. Lam’s book, Serre’s Problem on Projective Modules, provides an example of the clarity and concision of Lam’s style of exposition, replete with solid motivations for everything, making him remarkably easy to follow even as he leads the reader into particularly dense parts of the mathematical forest under consideration. To wit, the subject at hand possesses intrinsic connections to algebraic geometry, but it’s equally true, if not more so, that it engenders pure algebra, perhaps even “algebra with a vengeance,” as Weil said of Chevalley’s book on algebraic function fields in a famous Bulletin review.

So it is that Lam presents a sophisticated discourse on a wealth of topics arranged around the central theme of Serre’s Conjecture, including, to give but two illustrations, very nice discussions of chunks of algebraic K-theory, i.e. the Grothendieck and Whitehead groups, and some scrumptious matrix theory, such as Suslin’s “n! Theorem,” whose statement is this: Working over a base ring, R, say that a matrix row, (b1, b2, …, bn) is right unimodular if b1R + b2R + … bnR = R; then the following are equivalent:

  1. any finitely generated (f. g.) stably free right R-module is free;
  2. any f. g. stably free right R-module of type 1 is free;
  3. any right unimodular row over R can be completed to an invertible matrix.

(Recall that a right R-module, P, is stably free if there is some m such that the direct sum of P with Rm is free.) Manifestly, as Lam points out in his Notes to Chapter I, this beautiful result should be fitted in the historical context of Hermite’s Theorem which asserts that any row of integers can be completed to a matrix in GLn(Z) whose determinant is their GCD, which is, in turn, a generalization of the Euclidean Algorithm, of course. (For the connection with n! see p. 111 ff. of Lam’s book.) What gems!

But this book in fact constitutes far more than an excursion into an algebraist’s jewelry-shop’s gallery: 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. As is the case with all of Lam’s books, Serre’s Problem on Projective Modules, is very clear and well-written, as already pointed out, and quickly gets the reader properly air-borne. Given its sophistication, and the fact that working though the book will require serious commitment and proper effort, the pay-off is huge: this is fantastic stuff.

By the way, Lam use of the phrase “Serre’s Problem,” as opposed to the more traditional “Serre’s Conjecture” (which is in fact the title of one of his earlier books), is meant to convey that the above question indeed has an affirmative answer. As Lam indicates in the Preface, the problem was solved independently by Quillen and Suslin in January of 1976, with a flurry of interesting subsequent activity (all described very evocatively): in May of 1976, in a letter to Bass, Suslin went on to present an “elementary” proof of Serre’s assertion, and around the same time, Vaserstein gave (yes!) an 8-line proof. These fascinating historical details, with all the indicated mathematics covered thoroughly and impeccably, add contextual orchestration to Lam’s exposition. They occur throughout the book, in chapter end-notes like the one mentioned above, and significantly enhance the presentation.

Finally I want to take note of Lam’s very moving dedication of the book to his child, Jumei, who died during the writing of his earlier book, Serre’s Conjecture. Says Lam: “may Jumei’s little spirit be with Mom, Dad and her siblings again and always, until we rejoin her.” God rest her soul.

Serre’s Problem on Projective Modules is a superb book. It’s highly recommended.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

Date Received: 
Thursday, June 22, 2006
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T. Y. Lam
Springer Monographs in Mathematics
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Michael Berg

 Introduction.- The "Classical" Results on Serre's Conjecture.- Two Elementary Proofs of Serre's Conjecture.- Horrocks' Theorem.- Quillen's Method.- K1 Analogue of Serre's Conjecture.- The Quadratic Analogue of Serre's Conjecture.- New Developments (since 1977).- References.

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Friday, March 2, 2007