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.

### Contents

1. The Weyl algebra; 2. Ideal structure of the Weyl algebra; 3. Rings of differential operators; 4. Jacobian conjectures; 5. Modules over the Weyl algebra; 6. Differential equations; 7. Graded and filtered modules; 8. Noetherian rings and modules; 9. Dimension and multiplicity; 10. Holonomic modules; 11. Characteristic varieties; 12. Tensor products; 13. External products; 14. Inverse image; 15. Embeddings; 16. Direct images; 17. Kashiwara's theorem; 18. Preservation of holonomy; 19. Stability of differential equations; 20. Automatic proof of identities.

### Review

"...an excellent textbook for a first encounter with D-module theory." Arno van den Essen, Mathematical Reviews