Variant of [Corollary 1, Vasconcelos]. See also [Iyengar] and .

Proposition 23.11.3. Let $R$ be a Noetherian ring. Let $I \subset R$ be an ideal which has finite projective dimension and such that $I/I^2$ is finite locally free over $R/I$. Then $I$ is a regular ideal (More on Algebra, Definition 15.32.1).

Proof. By Algebra, Lemma 10.68.6 it suffices to show that $I_\mathfrak p \subset R_\mathfrak p$ is generated by a regular sequence for every $\mathfrak p \supset I$. Thus we may assume $R$ is local. If $I/I^2$ has rank $r$, then by Lemma 23.11.2 we find a regular sequence $x_1, \ldots , x_ r \in I$ generating $I/I^2$. By Nakayama (Algebra, Lemma 10.20.1) we conclude that $I$ is generated by $x_1, \ldots , x_ r$. $\square$

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