Set-Theoretic Cohen–Macaulay Ideals in Characteristic Zero

Definitions and mathematical statement

Let $K$ be a field of characteristic zero and let $s,t$ be algebraically independent over $K$. Put \[ R=K[w,x,y,z],\qquad A=K[s^4,s^3t,st^3,t^4]\subset K[s,t], \] and define the surjective $K$-algebra homomorphism \[ \varphi:R\longrightarrow A,\qquad w\longmapsto s^4,\quad x\longmapsto s^3t,\quad y\longmapsto st^3,\quad z\longmapsto t^4. \] Let $\mathfrak a=\ker\varphi$. The question is whether, for every such field $K$, there exists an ideal $\mathfrak b\subset R$ such that \[ \sqrt{\mathfrak b}=\mathfrak a \qquad\text{and}\qquad R/\mathfrak b\text{ is Cohen--Macaulay}. \] Here $\sqrt{\mathfrak b}=\{r\in R:r^m\in\mathfrak b\text{ for some }m\geq1\}$, and a Noetherian ring is Cohen--Macaulay if its localizations at prime ideals have depth equal to their Krull dimension. The radical equality requires the same underlying closed support, while permitting a different, nonreduced scheme structure. No homogeneity, complete-intersection presentation, specified generators, or bound on their degrees is imposed on $\mathfrak b$.

The supplied literature triage checked 2026-08-21 identifies this as a characteristic-zero set-theoretic Cohen--Macaulay question posed in the primary source. It reports positive-characteristic and some characteristic-zero results in related settings, but no ideal meeting the requirement in the general formulation of this question. These contextual results are not a solution for the displayed characteristic-zero semigroup ideal, nor is the question enlarged to all ideals.

Short English statement

Can the explicitly given semigroup ideal in four variables over any characteristic-zero field be the radical of an ideal with Cohen--Macaulay quotient? The replacement ideal may define a nonreduced thickening and is not required to be homogeneous.

Sources

• Local UnsolvedMath registry, permanent ID 30000224, catalogue entry, Set-Theoretic Cohen--Macaulay Ideals in Characteristic Zero.
• Kommutative Algebra (2006), Oberwolfach Reports, pp. 1073--1126, DOI https://doi.org/10.4171/owr/2005/19.
• Singh--Walther, supplied label "local cohomology and pure morphisms": https://arxiv.org/abs/math/0701524.
• Supplied reference "local cohomology of Du Bois singularities": https://arxiv.org/abs/1605.02755.
• Supplied statement verification reports recovery of the ring, homomorphism, and kernel from the primary report; supplied literature triage checked 2026-08-21.