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$\DeclareMathOperator\Sh{Sh}$Informally, a quasi-excellent ring is a Noetherian ring with a few technical extra regularity conditions that make it turns out to be the largest class of rings that allow for all singularities to be resolved. It is excellent if it is also catenary.

Is there a constructive definition of this property? This is a fairly natural question to ask where one would be tempted to conjecture that there is one, because resolution of singularity is a geometric concept that you can expand to relative schemes, and in that case you would be led to the conjecture that a relative affine scheme over $S$ has no singularities if it can be viewed generated by an excellent ring in the internal language of $\Sh(S)$.

In general however, the internal language of $\Sh(S)$ does not satisfy the law of the excluded middle, which is slightly annoying because unlike more natural notions like coherence, Noetherian rings are notably less well behaved constructively and split into several non-equivalent definitions that are classically equivalent. Is there a definition of quasi-excellent rings that constructively captures the geometric intuition?

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    $\begingroup$ I do not understand your first sentence. Are you assuming resolution of singularities in positive characteristic? $\endgroup$
    – Jason Starr
    Commented Oct 25, 2022 at 22:13
  • $\begingroup$ @JasonStarr The property I was looking for is that Resolution of singularities for excellent rings should follow from resolution of singularities in complete integral local Noetherian rings, where modifications of the latter to be constructively well behaved is allowed. The assumption does not have to be proven, just the implication for a suitably well formulated assumption. The assumption happens to be difficult even classically, and an open problem for positive characteristic, but AFAIK the implication was shown classically by Hironaka and Grothendieck in the 60s. $\endgroup$
    – saolof
    Commented Oct 26, 2022 at 11:47
  • $\begingroup$ Actually, the thing that may be more important is the proof of the converse, that a ring is quasi-excellent if all singularities of its integral algebras can be resolved. It may be interesting to see what parts of that proof survive constructively and if a good definition of a ring property can be extracted from that. $\endgroup$
    – saolof
    Commented Oct 30, 2022 at 17:52

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