**Graduate Studies in Mathematics**

Volume: 87;
2007;
282 pp;
Hardcover

MSC: Primary 13; 14;
Secondary 55

Print ISBN: 978-0-8218-4126-6

Product Code: GSM/87

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

**Electronic ISBN: 978-1-4704-2117-5
Product Code: GSM/87.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

#### Supplemental Materials

# Twenty-Four Hours of Local Cohomology

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*Srikanth B. Iyengar; Graham J. Leuschke; Anton Leykin; Claudia Miller; Ezra Miller; Anurag K. Singh; Uli Walther*

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.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Twenty-Four Hours of Local Cohomology

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xiii14 free
- Introduction xv16 free
- Lecture 1. Basic Notions 120 free
- Lecture 2. Cohomology 1534
- Lecture 3. Resolutions and Derived Functors 2948
- Lecture 4. Limits 4160
- Lecture 5. Gradings, Filtrations, and Grobner Bases 5574
- Lecture 6. Complexes from a Sequence of Ring Elements 6786
- Lecture 7. Local Cohomology 7796
- Lecture 8. Auslander-Buchsbaum Formula and Global Dimension 87106
- Lecture 9. Depth and Cohomological Dimension 97116
- Lecture 10. Cohen-Macaulay Rings 105124
- Lecture 11. Gorenstein Rings 117136
- Lecture 12. Connections with Sheaf Cohomology 131150
- Lecture 13. Projective Varieties 141160
- Lecture 14. The Hartshorne-Lichtenbaum Vanishing Theorem 147166
- Lecture 15. Connectedness 153172
- Lecture 16. Polyhedral Applications 159178
- Lecture 17. D-modules 171190
- Lecture 18. Local Duality Revisited 179198
- Lecture 19. De Rham Cohomology 191210
- Lecture 20. Local Cohomology over Semigroup Rings 203222
- Lecture 21. The Frobenius Endomorphism 217236
- Lecture 22. Curious Examples 229248
- Lecture 23. Algorithmic Aspects of Local Cohomology 239258
- Lecture 24. Holonomic Rank and Hyper geometric Systems 247266
- Appendix. Injective Modules and Matlis Duality 257276
- Bibliography 269288
- Index 277296
- Back Cover Back Cover1304