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All rings are commutative with unit. A ring $R$ is called regular if it satisfies

(Reg) Every finitely generated ideal of $R$ has finite projective dimension.

Clearly this gives the usual definition if $R$ is noetherian. In the non-noetherian world, valuation rings are regular (this is the reason why I got interested), as are polynomial rings in arbitrary sets of variables over a field.
I am wondering about the reason for this definition, as opposed in particular to:

(Reg') Every finitely presented $R$-module has finite projective dimension.

Clearly (Reg') implies (Reg), and they are equivalent if $R$ is coherent (every finitely generated ideal is finitely presented).
To me, (Reg') looks much more natural than (Reg), and from my limited knowledge of the literature it seems that the association "coherent+regular" is quite popular. So, my question is

What are examples of regular rings not satisfying (Reg')?

Since my post is really about motivating the definition, I am of course more interested in "natural" examples (not just constructed for the purpose).
On the other hand, all examples are welcome since I don't know any.

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  • $\begingroup$ Are these notions local on $R$? Perhaps it would be better to impose these conditions on the local rings of $R$? Anyway, I just wanted to mention that for a valuation ring $V$, not only $V$ itself is regular (in the stronger sense) but also so is every finite type, flat $V$-algebra $R$ with regular $R$-fibers (e.g., $R$ could be $V$-smooth), at least if one takes the local definition I have alluded to. This is a result of Gabber--Ramero, Prop. 11.4.1 in arxiv.org/abs/math/0409584 $\endgroup$
    – Kestutis Cesnavicius
    Commented Apr 3, 2019 at 20:03
  • $\begingroup$ @KestutisCesnavicius: Thanks for the reference. Both properties are stable under any localization, and are Zariski-local on the spectrum. I doubt that they are local in the stronger sense of localizing at primes, but this should be true if the ring is assumed coherent, which is the case for valuation rings and finitely generated flat algebras over them. See Knaf, J. of Alg. Appl. 7 (2008) #5, 575-591. $\endgroup$
    – Laurent Moret-Bailly
    Commented Apr 3, 2019 at 21:41
  • $\begingroup$ If I understand correctly, to show that regular Noetherian is regular in this sense, you have to use the fact that for Noetherian rings being proj dimension <n at a point implies it in a neighborhood? $\endgroup$
    – Andy Jiang
    Commented Jul 19 at 17:40

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