On the finiteness properties of local cohomology modules for regular local rings (Q2407612)

Let \(R\) be a Noetherian local ring, \(\mathfrak a\) an ideal and \(M\) a finitely generated module. Grothendieck asked if \[ \text{Ass } \text{Hom}_R (R/\mathfrak a, H_{\mathfrak a}^i(M)) \] is always finite. Although Hartshorne gave a counterexample, the set above or \[ \mathcal F = \text{Ass } \text{Ext}_R^n (R/\mathfrak a, H_{\mathfrak a}^i(M)) \] is often finite. For example [\textit{T. Marley} and \textit{J. C. Vassilev}, J. Algebra 256, No. 1, 180--193 (2002; Zbl 1042.13010)] showed that \(\mathcal F\) is finite if \(R\) is a regular local ring of dimension at most \(4\). In the present paper, the authors prove that \(\mathcal F\) is finite in several cases. One of main theorems is as follows: The set \(\mathcal F\) is finite if \(M = R\), \(R\) is a regular local ring containing a field and \(\dim R = 5\).