Description
Book SynopsisThis 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, Grobner 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.
Trade ReviewIt's all terrific stuff. I hope this book will succeed in bringing many young mathematicians to love cohomology, too, and then to go on from there." —
MAA Reviews"This book is an excellent text on local cohomology and complements well the existing sources. It will surely become a standard reference on this theory." —
Mathematical Reviews Table of Contents
- Basic notions
- Cohomology
- Resolutions and derived functors
- Limits
- Gradings, filtrations, and Grobner bases
- Complexes from a sequence of ring elements
- Local cohomology
- Auslander-Buchsbaum formula and global dimension
- Depth and cohomological dimension
- Cohen-Macaulay rings
- Gorenstein rings
- Connections with sheaf cohomology
- Projective varieties
- The Hartshorne-Lichtenbaum vanishing theorem
- Connectedness
- Polyhedral applications
- $D$-modules
- Local duality revisited
- De Rham cohomology
- Local cohomology over semigroup rings
- The Frobenius endomorphism
- Curious examples
- Algorithmic aspects of local cohomology
- Holonomic rank and hypergeometric systems
- Injective modules and Matlis duality
- Bibliography
- Index
