Local Cohomology
An Algebraic Introduction with Geometric Applications

This popular graduate text has been thoroughly revised and updated to incorporate recent developments in the field.

M. P. Brodmann (Author), R. Y. Sharp (Author)

9780521513630, Cambridge University Press

Hardback, published 15 November 2012

505 pages, 330 exercises
23.6 x 15.7 x 3 cm, 0.86 kg

'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

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.

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.

Subject Areas: Algebraic topology [PBPD], Topology [PBP], Algebraic geometry [PBMW], Geometry [PBM], Algebra [PBF], Mathematics [PB]