| Description |
1 online resource (xii, 207 pages) |
| Series |
London Mathematical Society student texts ; 33 |
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London Mathematical Society student texts ; 33.
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| Contents |
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 |
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4. Automorphisms5. 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 filtration; 6. Exercises; Chapter 8. Noetherian rings and modules.; 1. Noetherian modules; 2. Noetherian rings; 3. Good filtration; 4. Exercises |
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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 |
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1. External products of algebras2. External products of modules; 3. Graduations and filtrations; 4. Dimensions and multiplicities; 5. Exercises; Chapter 14. Inverse Image.; 1. Change of rings; 2. Inverse images; 3. Projections; 4. Exercises; Chapter 15. Embeddings.; 1. The standard embedding; 2. Composition; 3. Embeddings 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 |
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2. Direct images3. 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 |
| Summary |
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 over-sophistication. It is aimed at beginning graduate students and the approach taken is algebraic, concentrating on the role of the Weyl algebra. Very few prerequisites are assumed, and the book is virtually self-contained. Exercises are included at the end of each chapter and the reader is given ample references to the more advanced literature. This is an excellent introduction to D-modules for all who are new to this area |
| Bibliography |
Includes bibliographical references (pages 197-202) and index |
| Notes |
Print version record |
| Subject |
D-modules.
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MATHEMATICS -- Algebra -- Intermediate.
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D-modules
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D-Modul
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Weyl-Algebra
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Anneaux (algèbre)
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D-modules, Théorie des.
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| Form |
Electronic book
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| ISBN |
9781107362352 |
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1107362350 |
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9780511623653 |
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0511623658 |
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