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.
Srikanth B. Iyengar, University of Nebraska, Lincoln, NE, Graham J. Leuschke, Syracuse University, NY, Anton Leykin, Institute for Mathematics and Its Applications, Syracuse, NY, Claudia Miller, Syracuse University, NY, Ezra Miller, University of Minnesota, Minneapolis, MN, Anurag K. Singh, University of Utah, Salt Lake City, UT, and Uli Walther, Purdue University, West Lafayette, IN
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