Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?

I mean by *regular* that the localization at every prime ideal is a regular local ring.

The question arose from my interest in the desingularization problem. To have a desingularization algorithm of arithmetic schemes, one first needs to know the regularity of a given scheme.

The definition of the regularity is point-wise. Naively one has to check the regularity point by point of $\mathrm{Spec}\, R$ for a given ring $R$. It is not an algorithm in the sense that it never halts if $R$ is regular.

If $R$ is defined over a prime filed, $\mathbb{Q}$ or $\mathbb{F}_p$, then one can use the Jacobian criterion: first compute the Jacobian ideal and then check if it is trivial by computing its reduced Gröbner basis. In positive characteristic, one may also use Kunz's criterion in terms of Frobenius maps. As far as I know, there is no such a global criterion for rings over $\mathbb{Z}$. Serre's criterion by the finiteness of global dimension looks global at the first glance. But one needs to know the projective dimensions of *infinitely many* modules.

So, my guess is that the answer to the question would be NO. Does someone know the answer or related works?