This question is about zcn's comment on the answer to this question. It's a good point. So I ask it for use of everybody:
if $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD.
2026-03-25 15:43:48.1774453428
If $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD
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The result mentioned is first due to MacRae. It was then reproved (and generalized) by Buchsbaum and Eisenbud. Rather than try and reproduce the proof, let me just refer to their paper, which contains many other well-known results such as the Buchsbaum-Eisenbud acyclicity criterion:
Some structure theorems for finite free resolutions
The result in question is Corollary 5.3. Notice that although local rings have a notion of minimal generators for an ideal, $2$-generated here does not refer to a minimal generating set (in particular, it follows from the linked question that such a ring is a domain). Finally, note that the local condition is crucial here: any ideal in a Dedekind domain is $2$-generated and projective, but a Dedekind domain is a UFD iff it is a PID.