Lemma 10.74.8. Let $R$ be a ring. Let $M$ be an $R$-module. The following are equivalent:

The module $M$ is flat over $R$.

For all $i>0$ the functor $\text{Tor}_ i^ R(M, -)$ is zero.

The functor $\text{Tor}_1^ R(M, -)$ is zero.

For all ideals $I \subset R$ we have $\text{Tor}_1^ R(M, R/I) = 0$.

For all finitely generated ideals $I \subset R$ we have $\text{Tor}_1^ R(M, R/I) = 0$.

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