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This book is aimed to provide an introduction to local cohomology which takes cognizance of the breadth of its interactions with other areas of mathematics. It covers topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to sheaf cohomology and to de Rham cohomology, Gröbner bases in the commutative setting as well as for D-modules, the Frobenius morphism and characteristic p methods, finiteness properties of local cohomology modules, semigroup rings and polyhedral geometry, and hypergeometric systems arising from semigroups. The book begins with basic notions in geometry, sheaf theory, and homological algebra leading to the definition and basic properties of local cohomology. Then it develops the theory in a number of different directions, and draws connections with topology, geometry, combinatorics, and algorithmic aspects of the subject.
CONTENTS
Preface | Introduction | Lecture 1. Basic Notions | Lecture 2. Cohomology | Lecture 3. Resolutions and Derived Functors | Lecture 4. Limits | Lecture 5. Gradings, Filtrations, and Grobner Bases | Lecture 6. Complexes from a Sequence of Ring Elements | Lecture 7. Local Cohomology | Lecture 8. Auslander-Buchsbaum Formula and Global Dimension | Lecture 9. Depth and Cohomological Dimension | Lecture 10. Cohen-Macaulay Rings | Lecture 11. Gorenstein Rings | Lecture 12. Connections with Sheaf Cohomology | Lecture 13. Projective Varieties | Lecture 14. The Hartshorne-Lichtenbaum Vanishing Theorem | Lecture 15. Connectedness | Lecture 16. Polyhedral Applications | Lecture 17. D-modules | Lecture 18. Local Duality Revisited | Lecture 19. De Rham Cohomology | Lecture 20. Local Cohomology over Semigroup Rings | Lecture 21. The Frobenius Endomorphism | Lecture 22. Curious Examples | Lecture 23. Algorithmic Aspects of Local Cohomology | Lecture 24. Holonomic Rank and Hypergeometric Systems | Appendix. Injective Modules and Matlis Duality | Bibliography | Index
ISBN - 9780821868836
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Pages : 304
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