Brodmann M.P., Sharp R.Y. Local Cohomology: An Algebraic Introduction with Geometric Applications

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’ [77] 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 [25] (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.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