2nd Edition. — Cambridge University Press, 2013. — 516 p. — (Cambridge Studies in Advanced Mathematics 136) — ISBN 978-0-521-51363-0.One can take the view that local cohomology is an algebraic child of geometric parents. J.-P. Serre’s fundamental paper ‘Faisceaux alg´ebriques coh´erents’  represents a cornerstone of the development of cohomology as a tool in algebraic geometry: it foreshadowed many crucial ideas of modern sheaf cohomology. Serre’s paper, published in 1955, also has many hints of themes which are central in local cohomology theory, and yet it was not until 1967 that the publication of R. Hartshorne’s ‘Local cohomology’ Lecture Notes  (on A. Grothendieck’s 1961 Harvard University seminar) confirmed the effectiveness of local cohomology as a tool in local algebra. In the fifteen years since we completed the First Edition of this book, we have had opportunity to reflect on how we could change it in order to enhance its usefulness to the graduate students at whom it is aimed. As a result, this Second Edition shows substantial differences from the First. The main ones are described as follows. Contents 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
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