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Hi,

it is a folklore, that:

let p, q $p$, $q$ be two primes of a commutative Gorenstein ring R.

Tor^k(E(R/p), $R$. $$\operatorname{Tor}^k(E(R/p), E(R/q)) \not= neq 0 \iff p = q q\mbox{ and }k = height \operatorname{height} p.$$ where E(R/p) $E(R/p)$ is an injective hull of the cyclic module R/p $R/p$ and Tor^k(-,-) $\operatorname{Tor}^k(-,-)$ means k-th $k$-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 $p \in Spec(R) \operatorname{Spec}(R)$ a module M(p) $M(p)$ such that Ext^k $\operatorname{Ext}^k$ or Tor^k $\operatorname{Tor}^k$ of M(p) $M(p)$ and M(q) $M(q)$ is not zero iff p=q $p=q$ and $k = height p\operatorname{height} p$. (so the same as for M(p) $M(p) = E(R/p) E(R/p)$ and Tor in the Gorenstein case above)

Thank you,

3 added 202 characters in body

Hi,

it is a folklore, that:

let p, q be two primes of a commutative Gorenstein ring R.

Tor^k(E(R/p), E(R/q)) \not= 0 iff p = q and k = height p.

where E(R/p) is an injective hull of the cyclic module R/p and 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 Spec(R) a module M(p) such that Ext^k or Tor^k of M(p) and M(q) is not zero iff p=q and k = height p. (so the same as for M(p) = E(R/p) and Tor in the Gorenstein case above)

Thank you,

2 edited title; added 11 characters in body
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