A primer of algebraic D-modules

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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
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