Coassociated primes of local homology and local cohomology modules (Q640276)

Let \((R,\mathfrak{m})\) be a commutative noetherian local ring with identity. Let \(\mathfrak{a}\) be an ideal of \(R\) and \(H^{\mathfrak{a}}_i(-)\) denote the \(i\)th local homology functor with respect to \(\mathfrak{a}\). So, by definition, \(H^{\mathfrak{a}}_i(-)\) is the \(i\)th left derived functor of the \(\mathfrak{a}\)-adic completion functor. For an \(R\)-module \(M\), a prime ideal \(\mathfrak{p}\) of \(R\) is said to be a coassociated prime ideal of \(M\) if there exists an Artinian quotient \(L\) of \(M\) such that \(\mathfrak{p}=0:_RL\). The set of all coassociated prime ideals of \(M\) is denoted by \(\mathrm{Coass}_RM\). It is known that if \(M\) is Artinian, then the set \(\mathrm{Coass}_RM\) is finite. Also, \(\mathrm{Cosupp}_RM\) is defined by \[ \mathrm{Cosupp}_RM:=\{\mathfrak{p}\in \mathrm{Spec} R|\mathfrak{p}\supseteq 0:_RL \text{ for some Artinian quotient } L \text{ of } M\}. \] Let \(A\) be an Artinian \(R\)-module. It is known that there is a natural \(R\)-isomorphism \(H^{\mathfrak{a}}_i(A)\cong {\varprojlim}_n \mathrm{Tor}_i^R(R/\mathfrak{a}^n,A)\) for all non-negative integers \(i\). Among other things, for a nonnegative integer \(t\), the authors prove that \(V(\mathfrak{a})\cap \mathrm{Coass}_R(H^{\mathfrak{a}}_t(A))\) is finite provided either: i) \(H^{\mathfrak{a}}_i(A)\) is Artinian for all \(i<t\); or ii) \(\mathrm{Cosupp}_R(H^{\mathfrak{a}}_i(A))\) is finite for all \(i<t\).