Orthogonality (wrt. Ext, Tor) in commutative noetherian rings

Hi,

it is a folklore, that:

let $p$, $q$ be two primes of a commutative Gorenstein ring $R$. $$\operatorname{Tor}^k(E(R/p), E(R/q)) \neq 0 \iff p = q\mbox{ and }k = \operatorname{height} p.$$ where $E(R/p)$ is an injective hull of the cyclic module $R/p$ and $\operatorname{Tor}^k(-,-)$ means $k$-th Tor (derived functor to tensor product).

My question is: are there some other modules satisfying this orthogonality? (over some commutative noetherian rings)

So what I'm looking for:

for any $p \in \operatorname{Spec}(R)$ a module $M(p)$ such that $\operatorname{Ext}^k$ or $\operatorname{Tor}^k$ of $M(p)$ and $M(q)$ is not zero iff $p=q$ and $k = \operatorname{height} p$. (so the same as for $M(p) = E(R/p)$ and Tor in the Gorenstein case above)

Thank you,

• I think, that again for Gorenstein commutative rings, k(p) (residue field), p \in Spec(R) and Tor should work as well. – Zdenek Dec 20 '12 at 10:20