Inverse limit of local homology (Q2826259)

Let \(R\) be a commutative ring with identity, \(I\) and \(\mathfrak a\) ideals of \(R\) and \(M\) an \(R\)-module. A systematic investigation on the structure of the formal local cohomology modules \(\mathfrak{F}_{\mathfrak a,I}^i(M):=\underset{n}{\varprojlim}H_{I}^i(M/\mathfrak a^n M)\) was initiated by \textit{P. Schenzel} [J. Algebra 315, No. 2, 894--923 (2007; Zbl 1131.13018)].NEWLINENEWLINEIn this paper, the authors introduce and study the dual of formal local cohomology modules. The define NEWLINE\[NEWLINE\mathfrak{F}^{\mathfrak a,I}_i(M):=\underset{n}{\varprojlim}H^{I}_i(M/\mathfrak a^n M).NEWLINE\]NEWLINE They establish several vanishing, non-vanishing and Artinianness results for the modules \(\mathfrak{F}^{\mathfrak a,I}_i(M)\).NEWLINENEWLINERecall that by the Cuong and Nam definition, for each non-negative integer \(i\), the \(i\)-th local homology module of \(M\) with respect to \(I\) is defined by \(H^I_i(M):=\underset{n}{\varprojlim}\mathrm{Tor}_i^R(R/I^n,M).\)