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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].

##### MSC:
 14B15 Local cohomology and algebraic geometry 13D45 Local cohomology and commutative rings
##### Keywords:
$$D$$-module; $$F$$-module; cohomological dimension