Quotient condition for bounded-codimension complete intersections

Determine whether the property of being a complete intersection of codimension at most c satisfies the quotient condition for every integer c≥1.

Background

The paper studies the property $\mathsf{CI}_{\leq c}$ of Noetherian local rings that are complete intersections whose codimension is at most $c$. The quotient condition (QC) requires that, whenever a Noetherian ring satisfies the property at all of its localizations, the corresponding property-locus in every prime quotient contains a nonempty open subset.

The paper proves that $\mathsf{CI}_{\leq c}$ does not satisfy the Nagata criterion (NC) for any $c\geq 1$, while it establishes stability under localization and suitable regular-sequence reductions and proves openness of the $\mathsf{CI}_{\leq c}$-locus for rings satisfying $\mathsf{Reg}$-Q0. The status of the quotient condition for these bounded-codimension complete-intersection classes is left unresolved.