On Artinianness of formal local cohomology, colocalization and coassociated primes (Q2857589)

Suppose \((R,\mathfrak m)\) is a local ring, \(\mathfrak a\) an ideal of \(R\) and \(M\) a finitely generated \(R\)-module. Set \(U=:\mathrm{Spec}(R)\setminus \{\mathfrak m\}\) and \((\widehat{U},\mathcal{O}_{\widehat{u}})\) denote the formal completion of \(U\) along \(V(\mathfrak a)\setminus\{\mathfrak m\}\). Let \(\widehat{\mathcal{F}}\) denote the \(\mathcal{O}_{\widehat{u}}\) sheaf associated to \(\varprojlim_nM/ \mathfrak a^n M\). Then formal cohomology modules \(H^i(\widehat{U},\widehat{\mathcal{F}})\) defined by taking limit inverse of the system \(\{ H^{i+1}_{\mathfrak m}(M/\mathfrak a^n M)\}\) for \(i\geq 1\). The paper under review gives some vanishing and artinian results on formal cohomology modules.