Lam Serre's Problem on Projective Modules

1. Auflage 2010
ISBN: 978-3-540-34575-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 404 Seiten

“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.

Autoren/Hrsg.

Lam, T.Y.

Inhaltsverzeichnis

1;Dedication Page;5
2;Preface;6
3;Table of Contents;11
4;Notes to the Reader;14
5;Partial List of Notations;16
6;Introduction to Serre’s Conjecture: 1955–1976;19
7;Chapter I. Foundations;26
7.1;$1. Projective Modules;26
7.2;$2. Flat Modules, Faithfully Flat Modules and Finitely Presented Modules;29
7.3;$3. Local-Global Methods;35
7.4;$4. Stably Free Modules and Hermite Rings;41
7.4.1;Appendix to $4;54
7.5;$5. Elementary Transformations;60
7.6;$6. The Grothendieck Group Ko;66
7.7;$7. The Whitehead Group K1;69
7.8;$8. Examples for SLn (R) . En (R);70
7.9;$9. Suslin’s Normality Theorem;76
7.10;Notes on Chapter I;84
8;Chapter II The “Classical” Results on Serre’s Conjecture;87
8.1;$1. The Case of Rank 1 Projectives;87
8.2;$2. The Case of One Variable;88
8.3;$3. The Case of Noncommutative Base Rings;89
8.4;$4. The Graded Case;93
8.5;$5. The Stable Case;95
8.6;$6. The Case of Two Variables;101
8.7;$7. The Case of Big Rank;107
8.8;Notes on Chapter II;112
9;Chapter III The Basic Calculus of Unimodular Rows;114
9.1;$1. Suslin’s Elementary Proof of Serre’s Conjecture;115
9.2;$2. Vaserstein’s Elementary Proof of Serre’s Conjecture;118
9.2.1;Appendix to $ 2;121
9.3;$3. Suslin’s Monic Polynomial Theorem;122
9.4;$4. Suslin’s n! Theorem;126
9.5;$5. Sectionable Sequences;131
9.6;$6. Self-Duality of Stably Free Modules;140
9.7;$7. A Touch of Suslin Matrices;146
9.8;Notes on Chapter III;152
10;Chapter IV Horrocks’ Theorem;154
10.1;$1. Localization at Monic Polynomials;154
10.2;$2. Statement of Horrocks’ Theorem;158
10.3;$3. Swan’s Proof of Horrocks’ Theorem;160
10.4;$4. Roberts’ Proof of Horrocks’ Theorem;166
10.5;$5. Nashier-Nichols’ Proof of Horrocks’ Theorem;168
10.6;$6. Murthy-Horrocks Theorem;171
10.7;Notes on Chapter IV;175
11;Chapter V Quillen’s Methods;176
11.1;$1. Quillen’s Patching Theorem;176
11.2;$2. Affine Horrocks’ Theorem and Applications;185
11.3;$3. Quillen Induction, and the Bass-Quillen Conjecture;192
11.3.1;Appendix to $3;198
11.4;$4. Laurent Polynomial Rings;201
11.5;$5. Power Series Rings;206
11.6;Notes on Chapter V;215
12;Chapter VI K1–Analogue of Serre’s Conjecture;218
12.1;$1. Patching Theorems for GLn;218
12.2;$2. Patching Theorem for Elementary Group Action;225
12.3;$3. Mennicke Symbols;230
12.4;$4. Suslin’s Stability Theorem;233
12.5;$5. K1–Analogue of Horrocks’ Theorem;235
12.6;$6. Structure Theorem on En (R[t, t-1]) ;239
12.7;Notes on Chapter VI;247
13;Chapter VII The Quadratic Analogue of Serre’s Conjecture;248
13.1;$1. Inner Product Spaces;248
13.2;$2. Karoubi’s Theorem;254
13.3;$3. Harder Theorem: Easier Proof;257
13.4;$4. Parimala’s Counterexamples;262
13.5;$5. Symplectic Spaces and Self-Duality;270
13.6;Notes on Chapter VII;283
14;References for Chapters I–VII;285
15;Appendix: Complete Intersections and Serre’s Conjecture;291
15.1;References;300
16;Chapter VIII New Developments (since 1977);302
16.1;$1. R[t1, . . . , tn] for R Noetherian;303
16.2;$2. Projective Modules over Affine Algebras;309
16.3;$3. Complete Intersections;316
16.4;$4. Monomial Algebras and Discrete Hodge Algebras;327
16.5;$5. Unimodular Rows;331
16.6;$6. The Bass-Quillen Conjecture;341
16.7;$7. R[t1, . . . , tn] for R Non-Noetherian;346
16.8;$8. Noncommutative Polynomial Rings;351
16.9;$9. K1 (and Higher Kn) Analogues;352
16.10;$10. Quadratic Analogues of Serre’s Conjecture;358
16.11;$11. Quantum Versions of Serre’s Conjecture;364
16.12;$12. Algorithmic Methods;366
16.13;$13. Applications of Serre’s Conjecture;368
17;References for Chapter VIII;375
18;Index;406

Leseproben

" (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."