Lemma 15.65.6. Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \geq 0$. The following are equivalent

$M$ has tor dimension $\leq d$, and

there exists a resolution

\[ 0 \to F_ d \to \ldots \to F_1 \to F_0 \to M \to 0 \]

with $F_ i$ a flat $R$-module.

In particular an $R$-module has tor dimension $0$ if and only if it is a flat $R$-module.

**Proof.**
Assume (2). Then the complex $E^\bullet $ with $E^{-i} = F_ i$ is quasi-isomorphic to $M$. Hence the Tor dimension of $M$ is at most $d$ by Lemma 15.65.3. Conversely, assume (1). Let $P^\bullet \to M$ be a projective resolution of $M$. By Lemma 15.65.2 we see that $\tau _{\geq -d}P^\bullet $ is a flat resolution of $M$ of length $d$, i.e., (2) holds.
$\square$

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