Lam Serre's Problem on Projective Modules

1. Auflage 2010
ISBN: 978-3-540-34575-6
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 404 Seiten, eBook

Reihe: Springer Monographs in Mathematics

“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.

Zielgruppe

Research

Autoren/Hrsg.

Lam, T.Y.

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.

Leseproben

"Notes on Chapter III (p. 137-138)

The two elementary proofs of Serre’s Conjecture presented in the beginning sections of this chapter were both discovered shortly after the Quillen-Suslin solution in January 1976. Suslin’s proof was contained in a letter from him to Bass dated May 2, 1976. I ?rst learned about this proof from a 1976 talk of [L. Roberts: 1976], who learned about this proof from a talk of Murthy. Our exposition of Vaserstein’s elementary proof follows the lecture notes of Ferrand’s Bourbaki talk [Ferrand: 1976].

For the original source of this proof, see [Vaserstein: 1976]. In the literature, this proof of Serre’s Conjecture has sometimes been fondly referred to as “Vaserstein’s 8-line proof” (see, e.g. Math Reviews MR 0472826).We must therefore plead guilty to consuming considerably more than eight lines in our exposition! As was observed in our verbose text, Vaserstein’s proof uses a local-global method to reduce the consideration to a local Horrocks-type result (2.6), and is therefore rather close in spirit to Quillen’s proof.

However, the arguments in Vaserstein’s proof are substantially simpler, since one need only deal with type 1 stably free modules (i.e. unimodular rows) in this proof, rather than with general ?nitely generated projective modules. Suslin’s Monic Polynomial Theorem (3.3) was proved by Suslin several years before the solution of Serre’s Conjecture. For coef?cient rings of dimension zero, (3.3) boils down essentially to Noether’s classical Normalization Theorem, so (3.3) may be viewed as a strong generalization of the latter.

Suslin’s result has played a crucial role in his work on cancellation theorems over R[t1, . . . , tn], and has led to the af?rmation of Serre’s Conjecture in some special cases for small values of n, in the period 1973/75. Suslin’s Theorem (3.3), as well as other parts of the work of [Vaserstein-Suslin: 1974], was made widely available to the American and European mathematical communities by the Bourbaki talk of [Bass: 1974], and subsequently by the Queen’s lecture notes of [Swan: 1975]. See also [Geramita: 1974/76]. The Transitivity Theorem (3.6), its Corollary (3.7), and the spectacular Stability Theorem (3.8) all came from [Suslin: 1977a].

The proof of (3.6) offered here is selfcontained, and so is the proof of (3.7) (except when the ground ring has dimension 0). As for the Stability Theorem (3.8), we shall eventually come back to it in the context of the K1-analogue of Serre’s Conjecture. For more details on this, see VI.4. Suslin’s n! Theorem (4.1) is decidedly a highlight in the research work on the completion of unimodular rows, and has important applications to complete intersections; see (VIII.3).

Our exposition in §4 follows [Suslin: 1977b] (which is a part of Sulin’s doctoral dissertation), and in part also [Gupta-Murthy: 1980] and [Mandal: 1997]. For another proof of the n! Theorem, see [Roitman: 1985, Thm. 4] listed in the references on Chapter VIII. The completion proposition (4.13) on linear polynomial unimodular vectors, due to Suslin and Swan, is a natural application of the n! theorem. From an expository point of view, this result serves advance notice for Suslin’s Problem Su(R)n to be introduced and discussed later in IV.3."