Local Cohomology

Local Cohomology : An Algebraic Introduction with Geometric Applications

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This second edition of a successful graduate text provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, including in multi-graded situations, and provides many illustrations of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Serre's Affineness Criterion, the Lichtenbaum-Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and Faltings' Annihilator Theorem, local duality and canonical modules, the Fulton-Hansen Connectedness Theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. The book is designed for graduate students who have some experience of basic commutative algebra and homological algebra and also experts in commutative algebra and algebraic geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine to challenging, and hints are provided for some of the more difficult ones.
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Product details

  • Electronic book text
  • Cambridge University Press (Virtual Publishing)
  • Cambridge, United Kingdom
  • Revised
  • 2nd Revised edition
  • 1139785648
  • 9781139785648

About M. P. Brodmann

M. P. Brodmann is Emeritus Professor in the Institute of Mathematics at the University of Zurich. R. Y. Sharp is Emeritus Professor of Pure Mathematics at the University of Sheffield.
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Review quote

Review of the first edition: '... Brodmann and Sharp have produced an excellent book: it is clearly, carefully and enthusiastically written; it covers all important aspects and main uses of the subject; and it gives a thorough and well-rounded appreciation of the topic's geometric and algebraic interrelationships ... I am sure that this will be a standard text and reference book for years to come.' Liam O'Carroll, Bulletin of the London Mathematical Society Review of the first edition: 'The book is well organised, very nicely written, and reads very well ... a very good overview of local cohomology theory.' Newsletter of the European Mathematical Society Review of the first edition: '... a careful and detailed algebraic introduction to Grothendieck's local cohomology theory.' L'Enseignement Mathematique '... the book opens the view towards the beauty of local cohomology, not as an isolated subject but as a tool helpful in commutative algebra and algebraic geometry.' Zentralblatt MATH 'From the point of view of the reviewer (who learned all his basic knowledge about local cohomology reading the first edition of this book and doing some of its exercises), the changes previously described (the new Chapter 12 concerning canonical modules, the treatment of multigraded local cohomology, and the final new section of Chapter 20 about locally free sheaves) definitely make this second edition an even better graduate textbook than the first. Indeed, it is well written and, overall, almost self-contained, which is very important in a book addressed to graduate students.' Alberto F. Boix, Mathematical Reviews
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Table of contents

Preface to the First Edition; Preface to the Second Edition; Notation and conventions; 1. The local cohomology functors; 2. Torsion modules and ideal transforms; 3. The Mayer-Vietoris sequence; 4. Change of rings; 5. Other approaches; 6. Fundamental vanishing theorems; 7. Artinian local cohomology modules; 8. The Lichtenbaum-Hartshorne Theorem; 9. The Annihilator and Finiteness Theorems; 10. Matlis duality; 11. Local duality; 12. Canonical modules; 13. Foundations in the graded case; 14. Graded versions of basic theorems; 15. Links with projective varieties; 16. Castelnuovo regularity; 17. Hilbert polynomials; 18. Applications to reductions of ideals; 19. Connectivity in algebraic varieties; 20. Links with sheaf cohomology; Bibliography; Index.
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