For a commutative Noetherian ring $R$, let $ \mathcal I$ denote the set of all ideals of $R$. A function $f : \mathcal I \to \mathcal I $ is called a closure operation on $R$ iff for every ideals $I,J$ of $R$, we have $I \subseteq f(I)=f(f(I))$ and $I \subseteq J$ implies $f(I) \subseteq f(J)$.
Now for a fixed $R$-module $M$, if we define $f_M(I):= \mathrm{Ann}_R (M/IM)$ , then it is easy to show that $f_M$ is a closure operation. Now $M/IM \cong M \otimes_R R/I$. Motivated by this, I ask the following:
For which $i>0$ and for which $R$-modules $M$, is it true that $g_{M,i }(I):=\mathrm{Ann}_R (\mathrm{Tor}^R_i (M,R/I))$ defines a closure operation on $R$ ? (If needed, I'm willing to assume $R$ is Cohen-Macaulay or Gorenstein.)