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AG (Algebraic geometry)?
Cat
arXiv:1001.0862 Abstract local cohomology functors from arXiv Front: math.CT by Yuji Yoshino, Takeshi Yoshizawa We propose to define the notion of abstract local cohomology functors. The derived functors of the ordinary local cohomology functor with support in the closed subset defined by an ideal and the generalized local cohomology functor associated with a given pair of ideals are characterized as elements of the set of all the abstract local cohomology functors.
Twenty-Four Hours of Local Cohomology. AMS.
Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge Studies in Advanced Mathematics) fra Mathematics by: M. P. Brodmann, R. Y. Sharp
Brodmann and Sharp: Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge Studies in Advanced Mathematics)
Section 4.6 in Weibel: An introduction to homological algebra.
I think there are some seminar notes by Hartshorne (1961) (in the Moore library)
Hartshorne: LNM0041 Local cohomology. A heavier and more general treatment might possibly be found in Residues and duality, by the same author.
nLab page on Local cohomology