Problem and significance assessment

Paper: P36, Dominance and common deformations of short local rings, Henry Zweiman.

Published question and known progress

Kobayashi–Takahashi, arXiv:2603.10810v1, Question 7.1, asks whether a Gorenstein local ring that is not a complete intersection must be dominant if its maximal ideal has cube zero or if its multiplicity is at most embedding codimension plus two. Section 7 of the author-hosted manuscript was checked, including the exact question and supporting partial results.

Kimura, arXiv:2609.05781v1, posted September 5, 2026, proves the Artinian case for rings containing a field, and the higher-dimensional equicharacteristic case with infinite residue field (Theorem 7.3). The proof chain through Sections 2–4, especially Theorem 4.5, and the Section 7 application were read. This is the essential prior theorem, not a result of the present paper. The first Gorenstein dominant non-complete-intersection examples are already Kimura's.

Proposed contribution

The common deformation applies to every Artinian local ring with maximal-ideal cube zero, without a Gorenstein hypothesis or a residue-field restriction. It presents the ring and its associated graded ring as quotients of a common complete Cohen–Macaulay ring by regular elements outside the maximal-ideal square. Dominance and uniform dominance therefore transfer in both directions, with dominant-index inequalities in the finite case.

Applied to the Gorenstein class, it removes the coefficient-field restriction in Kimura's theorem. The standard Artinian reduction, followed by Liu's flat-descent theorem for finite residue fields, gives the full statement of Question 7.1. These are combined into one paper.

Reusable mathematical content

The deformation transfers any property known to be equivalent across regular linear-parameter quotients. The paper proves the dominance and uniform-dominance applications explicitly. Its broader scope includes non-Gorenstein short rings, but their dominance is reduced to a graded problem, not automatically proved. A future dominant-index estimate for the graded Gorenstein class would immediately give one in unequal characteristic. This is a plausible route to further use; no citation-count prediction is justified.

Priority audit and limits

Searches combined short and cube-zero local rings, Gorenstein rings, associated graded rings, common deformations, Cohen rings, mixed or unequal characteristic, and dominance. No matching characteristic-free dominance transfer or full Question 7.1 solution was located in the checked sources as of September 9, 2026. The recent Kimura source showed only version 1 on arXiv at the time of review.

Earlier work by Avramov–Iyengar–Şega on short rings and Rossi–Şega on compressed Gorenstein rings was screened for structural antecedents. Their known free-resolution and Poincaré-series results are not claimed here. This was not an exhaustive historical audit of all lifting constructions. A search without a matching result is evidence about the search, not a certificate of novelty.

The paper is an internally checked proposed preprint. Independent expert validation, historical priority, and significance remain provisional. It is not a proof that all short Gorenstein rings are uniformly dominant, nor a solution for rings of arbitrary Loewy length.