On the Annihilation of local homology modules (Q2891117)

Let \(R\neq 0\) be a commutative Noetherian ring, \(I\) an ideal of \(R\), and \(A\) an Artinian \(R\)-module. The \(i\)-th local homology module of \(A\) with respect to \(I\) is defined by NEWLINE\[NEWLINEH^I_i(A)=\underset{\underset{t}\longleftarrow}\lim {\text{Tor}}^R_i (R/I^t,A).NEWLINE\]NEWLINE The authors main theorem says that for ideals \(I\subseteq J\) in \(R\) and every non-negative integer \(n\) the product \(\prod_{p+q=n}{\text{Ann}}({\text{Tor}}^R_p(R/J,H^I_q(A))\) annihilates \({\text{Tor}}^R_n(R/J,A)\). If, in addition, \(H^I_i(A)\) is Artinian for all \(i<n\), then the theorem implies that \(I\subseteq{\text{Rad}}({\text{Ann}}(H^J_i(A))\) for all \(i<n\).NEWLINENEWLINE Finally the author proves: Let \(n\) be as above, \(I\) generated by \(n\) elements of \(R\), and \(\tilde I=\bigcap_{t\geq 1}\bigcap_{i=0}^n{\text{Ann}}({\text{Tor}}_i^R(R/I^t,A))\). Then NEWLINE\[NEWLINE\tilde I^{\binom n{[n/2]}}=\bigcap_{i=0}^{n-1}{\text{Ann}}(H^I_i(A)).NEWLINE\]