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Twenty-Four Hours of Local Cohomology
 
Srikanth B. Iyengar University of Nebraska, Lincoln, NE
Graham J. Leuschke Syracuse University, Syracuse, NY
Anton Leykin Institute for Mathematics and Its Applications, Syracuse, NY
Claudia Miller Syracuse University, Syracuse, NY
Ezra Miller University of Minnesota, Minneapolis, MN
Anurag K. Singh University of Utah, Salt Lake City, UT
Uli Walther Purdue University, West Lafayette, IN
Front Cover for Twenty-Four Hours of Local Cohomology
Available Formats:
Softcover ISBN: 978-1-4704-7159-0
Product Code: GSM/87.S
List Price: $64.00
MAA Member Price: $57.60
AMS Member Price: $51.20
Electronic ISBN: 978-1-4704-2117-5
Product Code: GSM/87.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $48.00
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $96.00
MAA Member Price: $86.40
AMS Member Price: $76.80
Front Cover for Twenty-Four Hours of Local Cohomology
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  • Front Cover for Twenty-Four Hours of Local Cohomology
  • Back Cover for Twenty-Four Hours of Local Cohomology
Twenty-Four Hours of Local Cohomology
Srikanth B. Iyengar University of Nebraska, Lincoln, NE
Graham J. Leuschke Syracuse University, Syracuse, NY
Anton Leykin Institute for Mathematics and Its Applications, Syracuse, NY
Claudia Miller Syracuse University, Syracuse, NY
Ezra Miller University of Minnesota, Minneapolis, MN
Anurag K. Singh University of Utah, Salt Lake City, UT
Uli Walther Purdue University, West Lafayette, IN
Available Formats:
Softcover ISBN:  978-1-4704-7159-0
Product Code:  GSM/87.S
List Price: $64.00
MAA Member Price: $57.60
AMS Member Price: $51.20
Electronic ISBN:  978-1-4704-2117-5
Product Code:  GSM/87.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $48.00
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $96.00
MAA Member Price: $86.40
AMS Member Price: $76.80
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 872007; 282 pp
    MSC: 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.

    Readership

    Graduate 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. 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
  • 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
  • Request Review Copy
  • Get Permissions
Volume: 872007; 282 pp
MSC: 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.

Readership

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. 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
  • 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
Please select which format for which you are requesting permissions.