A Primer of Algebraic D-Modules

✍ Scribed by S. C. Coutinho

• Publisher: Cambridge University Press
• Year: 1995
• Tongue: English
• Leaves: 223
• Series: London Mathematical Society Student Texts
• Category: Library

✦ Synopsis

The theory of D-modules is a rich area of study combining ideas from algebra and differential equations, and it has significant applications to diverse areas such as singularity theory and representation theory. This book introduces D-modules and their applications, avoiding all unnecessary technicalities. The author takes an algebraic approach, concentrating on the role of the Weyl algebra. The author assumes very few prerequisites, and the book is virtually self-contained. The author includes exercises at the end of each chapter and gives the reader ample references to the more advanced literature. This is an excellent introduction to D-modules for all who are new to this area.

✦ Table of Contents

Front Cover
TITLE
COPYRIGHT
DEDICATION
CONTENTS
PREFACE
INTRODUCTION
1. THE WEYL ALGEBRA.
2. ALGEBRAIC D-MODULES.
3. THE BOOK: AN OVERVIEW.
4. PRE-REQUISITES .
CHAPTER 1 - THE WEYL ALGEBRA
1. DEFINITION
2. CANONICAL FORM
3. GENERATORS AND RELATIONS
4. EXERCISES
CHAPTER 2 - IDEAL STRUCTURE OF THE WEYL ALGEBRA
1. THE DEGREE OF AN OPERATOR
2. IDEAL STRUCTURE.
3. POSITIVE CHARACTERISTIC.
4. EXERCISES.
CHAPTER 3 - RINGS OF DIFFERENTIAL OPERATORS
1. DEFINITIONS.
2. THE WEYL ALGEBRA.
3. EXERCISES.
CHAPTER 4 - JACOBIAN CONJECTURE
1. POLYNOMIAL MAPS.
2. JACOBIAN CONJECTURE
3. DERIVATIONS
4. AUTOMORPHISMS
5. EXERCISES.
CHAPTER 5 - MODULES OVER THE WEYL ALGEBRA
1. THE POLYNOMIAL RING.
2. TWISTING.
3. HOLOMORPHIC FUNCTIONS.
4. EXERCISES.
CHAPTER 6 - DIFFERENTIAL EQUATIONS
1. THE D-MODULE OF AN EQUATION.
2. DIRECT LIMIT OF MODULES.
3. MICROFUNCTIONS.
4. EXERCISES.
CHAPTER 7 - GRADED AND FILTERED MODULES
1. GRADED RINGS
2. FILTERED RINGS.
3. ASSOCIATED GRADED ALGEBRA.
4. FILTERED MODULES.
5. INDUCED FILTRATIONS.
6. EXERCISES
CHAPTER 8 - NOETHERIAN RINGS AND MODULES
1. NOETHERIAN MODULES.
2. NOETHERIAN RINGS.
3. GOOD FILTRATIONS.
CHAPTER 9 - DIMENSION AND MULTIPLICITY
1. THE HILBERT POLYNOMIAL.
2. DIMENSION AND MULTIPLICITY.
3. BASIC PROPERTIES.
4. BERNSTEIN'S INEQUALITY.
5. EXERCISES
CHAPTER 10 - HOLONOMIC MODULES
1. DEFINITION AND EXAMPLES.
2. BASIC PROPERTIES.
3. FURTHER EXAMPLES.
4. EXERCISES.
CHAPTER 11 - CHARACTERISTIC VARIETIES
1. THE CHARACTERISTIC VARIETY.
2. SYMPLECTIC GEOMETRY.
3. NON-HOLONOMIC IRREDUCIBLE MODULES
4. EXERCISES
CHAPTER 12 - TENSOR PRODUCTS
1. BIMODULES.
2. TENSOR PRODUCTS.
3. THE UNIVERSAL PROPERTY
4. BASIC PROPERTIES.
5. LOCALIZATION.
6. EXERCISES
CHAPTER 13 - EXTERNAL PRODUCTS
1. EXTERNAL PRODUCTS OF ALGEBRAS.
2. EXTERNAL PRODUCTS OF MODULES.
3. GRADUATIONS AND FILTRATIONS.
4. DIMENSIONS AND MULTIPLICITIES
5. EXERCISES
CHAPTER 14 - INVERSE IMAGES
1. CHANGE OF RINGS
2. INVERSE IMAGES.
3. PROJECTIONS.
4. EXERCISES
CHAPTER 15 - EMBED DINGS
1. THE STANDARD EMBEDDING
2. COMPOSITION.
3. EMBED DINGS REVISITED.
4. EXERCISES
CHAPTER 16 - DIRECT IMAGES
1. RIGHT MODULES
2. TRANSPOSITION
3. LEFT MODULES
4. EXERCISES
CHAPTER 17 - KASHIWARA'S THEOREM
1. EMBEDDINGS
2. KASHIWARA'S THEOREM
3. EXERCISES
CHAPTER 18 - PRESERVATION OF HOLONOMY
1. INVERSE IMAGES
2. DIRECT IMAGES.
3. CATEGORIES AND FUNCTORS.
4. EXERCISES
CHAPTER 19 - STABILITY OF DIFFERENTIAL EQUATIONS
1. ASYMPTOTIC STABILITY
2. GLOBAL UPPER BOUND
3. GLOBAL STABILITY ON THE PLANE
4. EXERCISES
CHAPTER 20 - AUTOMATIC PROOF OF IDENTITIES
1. HOLONOMIC FUNCTIONS.
2. HYPEREXPONENTIAL FUNCTIONS.
3. THE METHOD.
4. EXERCISES
CODA
APPENDIX 1 - DEFINING THE ACTION OF A MODULE USING GENERATORS
APPENDIX 2 - LOCAL INVERSION THEOREM
REFERENCES
INDEX
Back cover