On the support of local cohomology
Finiteness properties of local cohomology modules are studied in several research papers. For example we recall the following two results which are less known. \textit{M. Katzman} [in: Commutative algebra: geometric, homological, combinatorial and computational aspects. Papers presented during the conference, Sevilla, Spain, June 18--21, 2003 and the conference on commutative algebra, Lisbon, Portugal, June 23--27, 2003. Boca Raton, FL: Chapman \& Hall/CRC. Lecture Notes in Pure and Applied Mathematics 244, 165--174 (2006; Zbl 1097.13022)] (based on a work of Lyubeznik) proved that if \(R\) is a noetherian ring of positive prime characteristic and \(I\) is an ideal generated by n elements, then the support of \(H^n_I (R)\) is closed. \textit{C. Rotthaus} and \textit{L. M. Sega} [J. Algebra 283, No. 1, 232--247 (2005; Zbl 1105.13021)] proved that the support of the top local cohomology of the irrelevant ideal is always closed in a standard graded noetherian ring. Having these in mind, the paper under review deals with the following question: Let \(R\) be a noetherian ring, \(M\) a finitely generated \(R\)-module, and \(I\) an ideal generated by \(n\) elements. Is the support of \(H^n_I (M)\) is closed? The paper under review gives a positive answer to the question when the cohomological dimension of \(I\) is two or the the base ring is local of dimension at most four. There are several reductions and samples in the paper concerning the above question.
scientific article