9780521513630

Local Cohomology: An Algebraic Introduction with Geometric Applications

by
  • ISBN13:

    9780521513630

  • ISBN10:

    0521513634

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 1/7/2013
  • Publisher: Cambridge University Press
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Summary

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

Preface to the First Edition
Preface to the Second Edition
Notation and conventions
The local cohomology functors
Torsion modules and ideal transforms
The Mayer-Vietoris sequence
Change of rings
Other approaches
Fundamental vanishing theorems
Artinian local cohomology modules
The Lichtenbaum-Hartshorne Theorem
The Annihilator and Finiteness Theorems
Matlis duality
Local duality
Canonical modules
Foundations in the graded case
Graded versions of basic theorems
Links with projective varieties
Castelnuovo regularity
Hilbert polynomials
Applications to reductions of ideals
Connectivity in algebraic varieties
Links with sheaf cohomology
Bibliography
Index
Table of Contents provided by Publisher. All Rights Reserved.

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