Serre's Problem on Projective Modules (eBook)

Dedication Page 5
Preface 6
Table of Contents 11
Notes to the Reader 14
Partial List of Notations 16
Introduction to Serre’s Conjecture: 1955–1976 19
Chapter I. Foundations 26
$1. Projective Modules 26
$2. Flat Modules, Faithfully Flat Modules and Finitely Presented Modules 29
$3. Local-Global Methods 35
$4. Stably Free Modules and Hermite Rings 41
Appendix to $4 54
$5. Elementary Transformations 60
$6. The Grothendieck Group Ko 66
$7. The Whitehead Group K1 69
$8. Examples for SLn (R) . En (R) 70
$9. Suslin’s Normality Theorem 76
Notes on Chapter I 84
Chapter II The “Classical” Results on Serre’s Conjecture 87
$1. The Case of Rank 1 Projectives 87
$2. The Case of One Variable 88
$3. The Case of Noncommutative Base Rings 89
$4. The Graded Case 93
$5. The Stable Case 95
$6. The Case of Two Variables 101
$7. The Case of Big Rank 107
Notes on Chapter II 112
Chapter III The Basic Calculus of Unimodular Rows 114
$1. Suslin’s Elementary Proof of Serre’s Conjecture 115
$2. Vaserstein’s Elementary Proof of Serre’s Conjecture 118
Appendix to $ 2 121
$3. Suslin’s Monic Polynomial Theorem 122
$4. Suslin’s n! Theorem 126
$5. Sectionable Sequences 131
$6. Self-Duality of Stably Free Modules 140
$7. A Touch of Suslin Matrices 146
Notes on Chapter III 152
Chapter IV Horrocks’ Theorem 154
$1. Localization at Monic Polynomials 154
$2. Statement of Horrocks’ Theorem 158
$3. Swan’s Proof of Horrocks’ Theorem 160
$4. Roberts’ Proof of Horrocks’ Theorem 166
$5. Nashier-Nichols’ Proof of Horrocks’ Theorem 168
$6. Murthy-Horrocks Theorem 171
Notes on Chapter IV 175
Chapter V Quillen’s Methods 176
$1. Quillen’s Patching Theorem 176
$2. Affine Horrocks’ Theorem and Applications 185
$3. Quillen Induction, and the Bass-Quillen Conjecture 192
Appendix to $3 198
$4. Laurent Polynomial Rings 201
$5. Power Series Rings 206
Notes on Chapter V 215
Chapter VI K1–Analogue of Serre’s Conjecture 218
$1. Patching Theorems for GLn 218
$2. Patching Theorem for Elementary Group Action 225
$3. Mennicke Symbols 230
$4. Suslin’s Stability Theorem 233
$5. K1–Analogue of Horrocks’ Theorem 235
$6. Structure Theorem on En (R[t, t-1]) 239
Notes on Chapter VI 247
Chapter VII The Quadratic Analogue of Serre’s Conjecture 248
$1. Inner Product Spaces 248
$2. Karoubi’s Theorem 254
$3. Harder Theorem: Easier Proof 257
$4. Parimala’s Counterexamples 262
$5. Symplectic Spaces and Self-Duality 270
Notes on Chapter VII 283
References for Chapters I–VII 285
Appendix: Complete Intersections and Serre’s Conjecture 291
References 300
Chapter VIII New Developments (since 1977) 302
$1. R[t1, . . . , tn] for R Noetherian 303
$2. Projective Modules over Affine Algebras 309
$3. Complete Intersections 316
$4. Monomial Algebras and Discrete Hodge Algebras 327
$5. Unimodular Rows 331
$6. The Bass-Quillen Conjecture 341
$7. R[t1, . . . , tn] for R Non-Noetherian 346
$8. Noncommutative Polynomial Rings 351
$9. K1 (and Higher Kn) Analogues 352
$10. Quadratic Analogues of Serre’s Conjecture 358
$11. Quantum Versions of Serre’s Conjecture 364
$12. Algorithmic Methods 366
$13. Applications of Serre’s Conjecture 368
References for Chapter VIII 375
Index 406

"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."