Serre's Problem On Projective Modules

by Lam, T. Y.

• ISBN13: 9783540233176
• ISBN10: 3540233172
• Format: Hardcover
• Copyright: 2006-02-13
• Publisher: Springer Nature

Summary

This book covers the area of "Serre's Problem on Projective Modules" thoroughly, and from very many different angles. It presents the broadest and most comprehensive view available in the literature on the mathematics of "Serre's Conjecture". The author's original treatment of Serre's Conjecture in "Lecture Notes in Mathematics" (Vol. 635) has been completely worked over and expanded into a multifaceted exposition suitable for experts and yet easily accessible to beginners. Segments of new material in the book include an enriched coverage of stably free modules and the basic calculus of unimodular rows, Suslin's Normality and n -Factorial Theorems, as well as his fundamental work on the K 1 analogue of Serre's Conjecture for polynomial rings. Another new feature of the book is a long survey chapter on the further developments in many acts and ramifications on Serre's Problem that took place in the "post-Quillen-Suslin era", from 1977 to the present.

Table of Contents

Dedication	v
Preface	vii
Table of Contents	xiii
Notes to the Reader	xvii
Partial List of Notations	xix
Introduction to Serre's Conjecture: 1955-1976	1	(8)
Chapter I. Foundations	9	(62)
1. Projective Modules	9	(3)
2. Flat Modules, Faithfully Flat Modules and Finitely Presented Modules	12	(6)
3. Local-Global Methods	18	(6)
4. Stably Free Modules and Hermite Rings	24	(19)
Appendix to §4	37	(6)
5. Elementary Transformations	43	(6)
6. The Grothendieck Group K0	49	(3)
7. The Whitehead Group K1	52	(1)
8. Examples for SLn (R)not equal to En (R)	53	(6)
9. Suslin's Normality Theorem	59	(8)
Notes on Chapter I	67	(4)
Chapter II. The "Classical" Results on Serre's Conjecture	71	(28)
1. The Case of Rank 1 Projectives	71	(1)
2. The Case of One Variable	72	(1)
3. The Case of Noncommutative Base Rings	73	(4)
4. The Graded Case	77	(2)
5. The Stable Case	79	(6)
6. The Case of Two Variables	85	(6)
7. The Case of Big Rank	91	(5)
Notes on Chapter II	96	(3)
Chapter III. The Basic Calculus of Unimodular Rows	99	(40)
1. Suslin's Elementary Proof of Serre's Conjecture	100	(3)
2. Vaserstein's Elementary Proof of Serre's Conjecture	103	(4)
Appendix to §2	106	(1)
3. Suslin's Monic Polynomial Theorem	107	(4)
4. Suslin's n! Theorem	111	(5)
5. Sectionable Sequences	116	(9)
6. Self-Duality of Stably Free Modules	125	(6)
7. A Touch of Suslin Matrices	131	(6)
Notes on Chapter III	137	(2)
Chapter IV. Horrocks' Theorem	139	(22)
1. Localization at Monic Polynomials	139	(4)
2. Statement of Horrocks' Theorem	143	(2)
3. Swan's Proof of Horrocks' Theorem	145	(6)
4. Roberts' Proof of Horrocks' Theorem	151	(2)
5. Nashier-Nichols' Proof of Horrocks' Theorem	153	(3)
6. Murthy-Horrocks Theorem	156	(4)
Notes on Chapter IV	160	(1)
Chapter V. Quillen's Methods	161	(42)
1. Quillen's Patching Theorem	161	(9)
2. Affine Horrocks' Theorem and Applications	170	(7)
3. Quillen Induction, and the Bass-Quillen Conjecture	177	(9)
Appendix to §3	183	(3)
4. Laurent Polynomial Rings	186	(5)
5. Power Series Rings	191	(9)
Notes on Chapter V	200	(3)
Chapter VI. K1–Analogue of Serre's Conjecture	203	(30)
1. Patching Theorems for GL,	203	(7)
2. Patching Theorem for Elementary Group Action	210	(5)
3. Mennicke Symbols	215	(3)
4. Suslin's Stability Theorem	218	(2)
5. K1–Analogue of Horrocks' Theorem	220	(4)
6. Structure Theorem on En7 (R[t t-¹])	224	(8)
Notes on Chapter VI	232	(1)
Chapter VII. The Quadratic Analogue of Serre's Conjecture	233	(38)
1. Inner Product Spaces	233	(6)
2. Karoubi's Theorem	239	(3)
3. Harder Theorem: Easier Proof	242	(5)
4. Parimala's Counter-Examples	247	(8)
5. Symplectic Spaces and Self-Duality	255	(13)
Notes on Chapter VII	268	(3)
References for Chapters I—VII	271	(6)
Appendix: Complete Intersections and Serre's Conjecture	277	(12)
Chapter VIII. New Developments (since 1977)	289	(74)
1. R [t1,..., tn] for R Noetherian	290	(6)
2. Projective Modules over Affine Algebras	296	(7)
3. Complete Intersections	303	(11)
4. Monomial Algebras and Discrete Hodge Algebras	314	(4)
5. Unimodular Rows	318	(10)
6. The Bass-Quillen Conjecture	328	(5)
7. R[t1,...,tn] for R Non-Noetherian	333	(5)
8. Noncommutative Polynomial Rings	338	(1)
9. K1 (and Higher Kn) Analogues	339	(6)
10. Quadratic Analogues of Serre's Conjecture	345	(6)
11. Quantum Versions of Serre's Conjecture	351	(2)
12. Algorithmic Methods	353	(2)
13. Applications of Serre's Conjecture	355	(8)
References for Chapter VIII	363	(32)
Index	395