A partial survey of local cohomology. (Q2776426)

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 \(H_I^i(M) \neq 0\).NEWLINENEWLINEFor the entire collection see [Zbl 0974.00036].