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A partial survey of local cohomology. (English) Zbl 1061.14005
Lyubeznik, Gennady (ed.), Local cohomology and its applications. Selected papers of the international workshop, Guanajuato, Mexico. New York, NY: Marcel Dekker (ISBN 0-8247-0741-9/pbk). Lect. Notes Pure Appl. Math. 226, 121-154 (2002).
This article is a survey of an unusual use of local cohomology. If \(A\) is a Noetherian local ring with maximal ideal \(\mathfrak m\) and \(M\) a finitely generated module of dimension \(d\), then the local cohomology modules \(H_{\mathfrak m}^i(M)\) of \(M\) with respect to \(\mathfrak m\) is Artinian, \(H_{\mathfrak m}^i(M) = 0\) for \(i > d\) and \(H_{\mathfrak m}^d(M) \neq 0\). However, if \(A\) is not local and if \(I\) is not a maximal ideal, then \(H_I^i(M)\) is not Artinian and it is possible for \(H_I^d(M)\) to be zero. The author introduces many results on such local cohomology. He mainly considers the \(D\)-module structure (or \(F\)-module structure) of \(H_I^i(R)\) and the cohomological dimension, that is, the largest integer \(i\) such that \(H_I^i(M) \neq 0\).
For the entire collection see [Zbl 0974.00036].

14B15 Local cohomology and algebraic geometry
13D45 Local cohomology and commutative rings