Softcover ISBN:  9781470471590 
Product Code:  GSM/87.S 
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Electronic ISBN:  9781470421175 
Product Code:  GSM/87.E 
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Book DetailsGraduate Studies in MathematicsVolume: 87; 2007; 282 ppMSC: Primary 13; 14; Secondary 55;
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.ReadershipGraduate students and research mathematicians interested in theory and applications of local cohomology.

Table of Contents

Chapters

Lecture 1. Basic notions

Lecture 2. Cohomology

Lecture 3. Resolutions and derived functors

Lecture 4. Limits

Lecture 5. Gradings, filtrations, and Gröbner bases

Lecture 6. Complexes from a sequence of ring elements

Lecture 7. Local cohomology

Lecture 8. AuslanderBuchsbaum formula and global dimension

Lecture 9. Depth and cohomological dimension

Lecture 10. CohenMacaulay rings

Lecture 11. Gorenstein rings

Lecture 12. Connections with sheaf cohomology

Lecture 13. Projective varieties

Lecture 14. The HartshorneLichtenbaum 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


Additional Material

Reviews

It'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


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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.
Graduate students and research mathematicians interested in theory and applications of local cohomology.

Chapters

Lecture 1. Basic notions

Lecture 2. Cohomology

Lecture 3. Resolutions and derived functors

Lecture 4. Limits

Lecture 5. Gradings, filtrations, and Gröbner bases

Lecture 6. Complexes from a sequence of ring elements

Lecture 7. Local cohomology

Lecture 8. AuslanderBuchsbaum formula and global dimension

Lecture 9. Depth and cohomological dimension

Lecture 10. CohenMacaulay rings

Lecture 11. Gorenstein rings

Lecture 12. Connections with sheaf cohomology

Lecture 13. Projective varieties

Lecture 14. The HartshorneLichtenbaum 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

It'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