Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge Studies in Advanced Mathematics, Series Number 60)

This Book Provides A Careful And Detailed Algebraic Introduction To Grothendieck's Local Cohomology Theory, And Provides Many Illustrations Of Applications Of The Theory In Commutative Algebra And In The Geometry Of Quasi-affine And Quasi-projective Varieties. Topics Covered Include Castelnuovo–mumford Regularity, The Fulton–hansen Connectedness Theorem For Projective Varieties, And Connections Between Local Cohomology And Both Reductions Of Ideals And Sheaf Cohomology. It Is Designed For Graduate Students Who Have Some Experience Of Basic Commutative Algebra And Homological Algebra, And Also For Experts In Commutative Algebra And Algebraic Geometry. 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. Foundations In The Graded Case -- 13. Graded Versions Of Basic Theorems -- 14. Links With Projective Varieties -- 15. Castelnuovo Regularity -- 16. Bounds Of Diagonal Type -- 17. Hilbert Polynomials -- 18. Applications To Reductions Of Ideals -- 19. Connectivity In Algebraic Varieties -- 20. Links With Sheaf Cohomology. M.p. Brodmann, R.y. Sharp. Includes Bibliographical References And Index. "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."--Publisher's website. This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and illustrates many applications for the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo-Mumford regularity, the Fulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry. Even though the book seems to suggest that it's aimed at a reader without considerable knowledge of algebraic geometry, reality paints a different picture. If one follows just the algebra, one misses the richness and beauty of the geometry that this algebra was called forward to describe. This is the only flaw, but I feel it's a serious one, so it's better to be forewarned. However, if the knowledge of algebraic geometry IS there, the book both enriches and informs. The main objective of this chapter is to introduce the a-torsion functor (throughout the book, a always denotes an ideal in a (non-trivial) commutative Noetherian ring R) and its right derived functors H (i > 0), referred to as the local cohomology functors with respect to a.