On functors that detect $S_n$

Let $A$ be a Noetherian ring. For each $k$ where $0 \leq k \leq \dim A$ we construct left exact functors $D_k$ on $Mod(A)$. Let $D^i_k$ be the $i^{th}$-right derived functor of $D_k$. Let $M$ be a finitely generated $A$-module. Under mild conditions on $A$ and $M$ we prove that vanishing of some finitely many $D^i_k(M)$ is equivalent to $M$ satisfying $S_n$.