The Stacks project

Proof. Let $N$ be a $B$-module. If $d = 0$, then $N$ is flat as an $A$-module, hence flat as a $B$-module by Lemma 15.104.2. Assume $d > 0$. Choose a resolution $F_\bullet \to N$ by free $B$-modules. Our assumption implies that $K = \mathop{\mathrm{Im}}(F_ d \to F_{d - 1})$ is $A$-flat, see Lemma 15.66.2. Hence it is $B$-flat by Lemma 15.104.2. Thus $0 \to K \to F_{d - 1} \to \ldots \to F_0 \to N \to 0$ is a flat resolution of length $d$ and we see that $N$ has tor dimension at most $d$. $\square$