Top formal local cohomology module (Q2322530)

Let \((R,\mathfrak{m})\) be a noetherian local ring, \(I\subseteq R\) be an ideal of \(R\) and \(M\) a finitely generated \(R\)-module. Put \(\mathcal{F}_I^i(M):=\varprojlim H^i_{\mathfrak{m}}(M/I^tM).\) The main result of the paper is that if \(M\neq IM\) and if \(l:=\dim(M/IM),\) then \(\mathcal{F}_I^l(M)\) is Artinian if and only if \(l>\dim(\tilde{M}/I\tilde{M}),\) where \(\tilde{M}:=M/H^0_I(M).\) Several consequences are proven and nice examples are given.