Lemma 15.61.6. Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. The following are equivalent

$A$ and $B$ are Tor independent over $R$,

for every pair of primes $\mathfrak p \subset A$ and $\mathfrak q \subset B$ lying over the same prime $\mathfrak r \subset R$ the rings $A_\mathfrak p$ and $B_\mathfrak q$ are Tor independent over $R_\mathfrak r$, and

For every prime $\mathfrak s$ of $A \otimes _ R B$ the module

\[ \text{Tor}_ i^ R(A, B)_\mathfrak s = \text{Tor}_ i^{R_\mathfrak r}(A_\mathfrak p, B_\mathfrak q)_\mathfrak s \](where $\mathfrak p = A \cap \mathfrak s$, $\mathfrak q = B \cap \mathfrak s$ and $\mathfrak r = R \cap \mathfrak s$) is zero.

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