Description
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.
Chapter
§2. Krull dimension of a ring
§3. Dimension of an algebraic set
§5. Tangent spaces and regular rings
§6. Dimension of a module
§3. Calculus versus topology
§4. Cech cohomology and derived functors
Lecture 3. Resolutions and Derived Functors
§1. Free, projective, and flat modules
§1. An example from topology
§3. The category of diagrams
§5. Diagrams over diagrams
§7. Diagrams over the pushout poset
Lecture 5. Gradings, Filtrations, and Grobner Bases
§1. Filtrations and associated graded rings
§3. Monomial orders and initial forms
§4. Weight vectors and flat families
§5. Buchberger's algorithm
§6. Grobner bases and syzygies
Lecture 6. Complexes from a Sequence of Ring Elements
§2. Regular sequences and depth: a first look
§3. Back to the Koszul complex
Lecture 7. Local Cohomology
§2. Direct limit of Ext modules
§3. Direct limit of Koszul cohomology
§4. Return of the Cech complex
Lecture 8. Auslander-Buchsbaum Formula and Global Dimension
§1. Regular sequences and depth redux
§3. Auslander-Buchsbaum formula
Lecture 9. Depth and Cohomological Dimension
§2. Cohomological dimension
Lecture 10. Cohen-Macaulay Rings
§1. Noether normalization
§2. Intersection multiplicities
Lecture 11. Gorenstein Rings
§2. Recognizing Gorenstein rings
§3. Injective resolutions of Gorenstein rings
Lecture 12. Connections with Sheaf Cohomology
§3. Local cohomology and sheaf cohomology
Lecture 13. Projective Varieties
§1. Graded local cohomology
§2. Sheaves on projective varieties
§3. Global sections and cohomology
Lecture 14. The Hartshorne-Lichtenbaum Vanishing Theorem
Lecture 15. Connectedness
§1. Mayer-Vietoris sequence
Lecture 16. Polyhedral Applications
§3. The h-vector of a simplicial complex
§4. Stanley-Reisner rings
§5. Local cohomology of Stanley-Reisner rings
§6. Proof of the upper bound theorem
§1. Rings of differential operators
Lecture 18. Local Duality Revisited
§4. Global canonical modules
Lecture 19. De Rham Cohomology
§1. The real case: de Rham's theorem
§4. Local and de Rham cohomology
Lecture 20. Local Cohomology over Semigroup Rings
§2. Cones from semigroups
§3. Maximal support: the Ishida complex
§4. Monomial support: Zd-graded injectives
Lecture 21. The Frobenius Endomorphism
§1. Homological properties
§2. Frobenius action on local cohomology modules
Lecture 22. Curious Examples
§1. Dependence on characteristic
§2. Associated primes of local cohomology modules
Lecture 23. Algorithmic Aspects of Local Cohomology
§1. Holonomicity of localization
§2. Local cohomology as a D-module
§3. Bernstein-Sato polynomials
§4. Computing with the Frobenius morphism
Lecture 24. Holonomic Rank and Hyper geometric Systems
§1. GKZ A-hypergeometric systems
§3. Euler-Koszul homology
Appendix. Injective Modules and Matlis Duality
Twenty-Four Hours of Local Cohomology
( Graduate Studies in Mathematics )
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