Softcover ISBN:  9780821847602 
Product Code:  ULECT/47 
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AMS Member Price:  $36.00 
Electronic ISBN:  9781470416423 
Product Code:  ULECT/47.E 
List Price:  $42.00 
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Book DetailsUniversity Lecture SeriesVolume: 47; 2008; 158 ppMSC: Primary 14; Secondary 32;
This book, which grew out of lectures by E. Kunz for students with a background in algebra and algebraic geometry, develops local and global duality theory in the special case of (possibly singular) algebraic varieties over algebraically closed base fields. It describes duality and residue theorems in terms of Kähler differential forms and their residues. The properties of residues are introduced via local cohomology. Special emphasis is given to the relation between residues to classical results of algebraic geometry and their generalizations. The contribution by A. Dickenstein gives applications of residues and duality to polynomial solutions of constant coefficient partial differential equations and to problems in interpolation and ideal membership. D. A. Cox explains toric residues and relates them to the earlier text.
The book is intended as an introduction to more advanced treatments and further applications of the subject, to which numerous bibliographical hints are given.ReadershipGraduate students and research mathematicians interested in algebra, algebraic geometry, complex analyis, and computer algebra.

Table of Contents

Articles

Chapter 1. Local cohomology functors

Chapter 2. Local cohomology of noetherian affine schemes

Chapter 3. Čech cohomology

Chapter 4. Koszul complexes and local cohomology

Chapter 5. Residues and local cohomology for power series rings

Chapter 6. The cohomology of projective schemes

Chapter 7. Duality and residue theorems for projective space

Chapter 8. Traces, complementary modules, and differents

Chapter 9. The sheaf of regular differential forms on an algebraic variety

Chapter 10. Residues for algebraic varieties. Local duality

Chapter 11. Duality and residue theorems for projective varieties

Chapter 12. Complete duality

Alicia Dickenstein  Chapter 13. Applications of residues and duality

David A. Cox  Chapter 14. Toric residues


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This book, which grew out of lectures by E. Kunz for students with a background in algebra and algebraic geometry, develops local and global duality theory in the special case of (possibly singular) algebraic varieties over algebraically closed base fields. It describes duality and residue theorems in terms of Kähler differential forms and their residues. The properties of residues are introduced via local cohomology. Special emphasis is given to the relation between residues to classical results of algebraic geometry and their generalizations. The contribution by A. Dickenstein gives applications of residues and duality to polynomial solutions of constant coefficient partial differential equations and to problems in interpolation and ideal membership. D. A. Cox explains toric residues and relates them to the earlier text.
The book is intended as an introduction to more advanced treatments and further applications of the subject, to which numerous bibliographical hints are given.
Graduate students and research mathematicians interested in algebra, algebraic geometry, complex analyis, and computer algebra.

Articles

Chapter 1. Local cohomology functors

Chapter 2. Local cohomology of noetherian affine schemes

Chapter 3. Čech cohomology

Chapter 4. Koszul complexes and local cohomology

Chapter 5. Residues and local cohomology for power series rings

Chapter 6. The cohomology of projective schemes

Chapter 7. Duality and residue theorems for projective space

Chapter 8. Traces, complementary modules, and differents

Chapter 9. The sheaf of regular differential forms on an algebraic variety

Chapter 10. Residues for algebraic varieties. Local duality

Chapter 11. Duality and residue theorems for projective varieties

Chapter 12. Complete duality

Alicia Dickenstein  Chapter 13. Applications of residues and duality

David A. Cox  Chapter 14. Toric residues