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Local Cohomology, Second Edition

Book Description

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.

Table of Contents

  1. Cover
  2. Copyright
  3. Dedication
  4. Contents
  5. Preface to the First Edition
  6. Preface to the Second Edition
  7. Notation and conventions
  8. 1 The local cohomology functors
    1. 1.1 Torsion functors
    2. 1.2 Local cohomology modules
    3. 1.3 Connected sequences of functors
  9. 2 Torsion modules and ideal transforms
    1. 2.1 Torsion modules
    2. 2.2 Ideal transforms and generalized ideal transforms
    3. 2.3 Geometrical significance
  10. 3 The Mayer–Vietoris sequence
    1. 3.1 Comparison of systems of ideals
    2. 3.2 Construction of the sequence
    3. 3.3 Arithmetic rank
    4. 3.4 Direct limits
  11. 4 Change of rings
    1. 4.1 Some acyclic modules
    2. 4.2 The Independence Theorem
    3. 4.3 The Flat Base Change Theorem
  12. 5 Other approaches
    1. 5.1 Use of Čech complexes
    2. 5.2 Use of Koszul complexes
    3. 5.3 Local cohomology in prime characteristic
  13. 6 Fundamental vanishing theorems
    1. 6.1 Grothendieck’s Vanishing Theorem
    2. 6.2 Connections with grade
    3. 6.3 Exactness of ideal transforms
    4. 6.4 An Affineness Criterion due to Serre
    5. 6.5 Applications to local algebra in prime characteristic
  14. 7 Artinian local cohomology modules
    1. 7.1 Artinian modules
    2. 7.2 Secondary representation
    3. 7.3 The Non-vanishing Theorem again
  15. 8 The Lichtenbaum–Hartshorne Theorem
    1. 8.1 Preparatory lemmas
    2. 8.2 The main theorem
  16. 9 The Annihilator and Finiteness Theorems
    1. 9.1 Finiteness dimensions
    2. 9.2 Adjusted depths
    3. 9.3 The first inequality
    4. 9.4 The second inequality
    5. 9.5 The main theorems
    6. 9.6 Extensions
  17. 10 Matlis duality
    1. 10.1 Indecomposable injective modules
    2. 10.2 Matlis duality
  18. 11 Local duality
    1. 11.1 Minimal injective resolutions
    2. 11.2 Local Duality Theorems
  19. 12 Canonical modules
    1. 12.1 Definition and basic properties
    2. 12.2 The endomorphism ring
    3. 12.3 S2-ifications
  20. 13 Foundations in the graded case
    1. 13.1 Basic multi-graded commutative algebra
    2. 13.2 *Injective modules
    3. 13.3 The *restriction property
    4. 13.4 The reconciliation
    5. 13.5 Some examples and applications
  21. 14 Graded versions of basic theorems
    1. 14.1 Fundamental theorems
    2. 14.2 *Indecomposable *injective modules
    3. 14.3 A graded version of the Annihilator Theorem
    4. 14.4 Graded local duality
    5. 14.5 *Canonical modules
  22. 15 Links with projective varieties
    1. 15.1 Affine algebraic cones
    2. 15.2 Projective varieties
  23. 16 Castelnuovo regularity
    1. 16.1 Finitely generated components
    2. 16.2 The basics of Castelnuovo regularity
    3. 16.3 Degrees of generators
  24. 17 Hilbert polynomials
    1. 17.1 The characteristic function
    2. 17.2 The significance of reg2
    3. 17.3 Bounds on reg2 in terms of Hilbert coefficients
    4. 17.4 Bounds on reg1 and reg0
  25. 18 Applications to reductions of ideals
    1. 18.1 Reductions and integral closures
    2. 18.2 The analytic spread
    3. 18.3 Links with Castelnuovo regularity
  26. 19 Connectivity in algebraic varieties
    1. 19.1 The connectedness dimension
    2. 19.2 Complete local rings and connectivity
    3. 19.3 Some local dimensions
    4. 19.4 Connectivity of affine algebraic cones
    5. 19.5 Connectivity of projective varieties
    6. 19.6 Connectivity of intersections
    7. 19.7 The projective spectrum and connectedness
  27. 20 Links with sheaf cohomology
    1. 20.1 The Deligne Isomorphism
    2. 20.2 The Graded Deligne Isomorphism
    3. 20.3 Links with sheaf theory
    4. 20.4 Applications to projective schemes
    5. 20.5 Locally free sheaves
  28. References
  29. Index