Colocalization of formal local cohomology modules (Q1982559)

Let \(\mathfrak{a}\) denote an ideal of a local ring \((R,\mathfrak{m})\). Let \(H^i_{\mathfrak{m}}(\cdot)\) denote the \(i\)-th local cohomology functor with respect to \(\mathfrak{m}\). For a finitely generated \(R\)-module \(M\) the reviewer introduced \(\mathfrak{F}^i_{\mathfrak{a}}(M) := \varprojlim H^i_{\mathfrak{m}}(M/\mathfrak{a}^nM)\) as the \(i\)-th formal local cohomology of \(M\) with respect to \(\mathfrak{a}\) [\textit{P. Schenzel}, J. Algebra 315, No. 2, 894--923 (2007; Zbl 1131.13018)]. Here the author contributes with structural results about the formal local cohomology modules: (1) The following conditions are equivalent: (i) \(\mathfrak{F}^i_{\mathfrak{a}}(M)\) is an Artinian \(R\)-module for all \(i < n\). (ii) \(\mathfrak{F}^i_{\mathfrak{a}}(M)\) is a representable \(R\)-module for all \(i < n\). (iii) The co-localization \({}_{\mathfrak{p}}(\mathfrak{F}^i_{\mathfrak{a}}(M))\) (in the sense of \textit{L. Melkersson} and \textit{P. Schenzel} [Proc. Edinb. Math. Soc., II. Ser. 38, No. 1, 121--131 (1995; Zbl 0824.13011)]) is a representable \(R_{\mathfrak{p}}\)-module for all \(i <n\) and all \(\mathfrak{p} \in \operatorname{Spec} R\). -- (2) \(\mathfrak{F}^i_{\mathfrak{a}}(M)\) is a minimax \(R\)-module for all \(i < n\) if and only if \(\operatorname{Cos}_R (\mathfrak{F}^i_{\mathfrak{a}}(M)) \subseteq V(\mathfrak{a}) \) for all \(i < n\). Here \(\operatorname{Cos}_R\) is the co-support defined by Melkersson and the reviewer (see [loc. cit.).