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.

- Cover
- Copyright
- Dedication
- 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
- References
- Index