Exercise 16, § 3

Let $A$ be a Macaulay ring and $M$ a finitely generated $A$-module of finite projective dimension. Prove that for $M$ to be flat, it is necessary and sufficient that for every ideal $J$ of $A$ generated by a completely secant sequence the canonical homomorphism $J \otimes_A M \to JM$ be bijective (prove by induction on the length of the sequence that this condition implies $\operatorname{Tor}^A_i(A/J, M) = 0$ for all $i > 0$, then, by descending induction on $n$, $\operatorname{Tor}^A_n(N, M) = 0$ for every finitely generated $A$-module $N$ and every integer $n > 0$). Example: a torsion-free module over a Dedekind ring is flat (cf. VII, § 2, exer. 14).