Large Regularity Quadratic Ideals with Linear First Syzygies

Definitions and mathematical statement

Let $K$ be a field and let $S_m=K[x_1,\ldots,x_m]$ have its standard grading, with $\deg x_i=1$. For a nonzero finitely generated graded $S_m$-module $M$, put \[ \beta^{S_m}_{i,j}(M)=\dim_K\operatorname{Tor}^{S_m}_i(M,K)_j, \qquad \operatorname{reg}(M)=\max\{j-i:\beta^{S_m}_{i,j}(M)\ne0\}. \] A quadratic ideal means a homogeneous ideal generated by forms of degree two. If its minimal homogeneous generators are $q_1,\ldots,q_r$, its first syzygy module is the kernel of \[ S_m(-2)^r\longrightarrow I,\qquad e_a\longmapsto q_a. \] Having linear first syzygies means that this kernel is generated by homogeneous relations whose coefficients are linear forms, equivalently by elements of total degree three. In terms of the quotient resolution, this is the condition \[ \beta^{S_m}_{2,j}(S_m/I)=0\quad\text{for all }j\ne3. \]

The first question is whether there exists a family $(I_m)_{m\ge1}$, with $I_m\subseteq S_m$ quadratic and having linear first syzygies, such that \[ \lim_{m\to\infty}\frac{\operatorname{reg}(S_m/I_m)}{m}>0. \] Here $S_m$ makes explicit the ambient polynomial ring denoted by $S$ in the source. The displayed condition retains an actual limit; replacing it by a positive limit superior would change the question. The source does not specify whether an infinite positive limit is allowed, or whether $K$ is fixed arbitrarily, existentially chosen, or subject to restrictions on characteristic.

For the second question, under the usual Stanley--Reisner interpretation, let $\Delta$ be a finite simplicial complex on $[n]=\{1,\ldots,n\}$, and set \[ T=K[x_1,\ldots,x_n],\qquad I_\Delta=\left(x_{i_1}\cdots x_{i_r}:\{i_1,\ldots,i_r\}\notin\Delta\right),\qquad K[\Delta]=T/I_\Delta. \] The ring $K[\Delta]$ is Cohen--Macaulay when its depth at the homogeneous maximal ideal equals its Krull dimension. In the usual graded-resolution convention used here, property $N_2$ means \[ \beta^T_{1,j}(K[\Delta])=0\quad(j\ne2),\qquad \beta^T_{2,j}(K[\Delta])=0\quad(j\ne3), \] so its defining ideal is quadratic and has linear first syzygies.

The second question asks whether there exists a constant $\lambda$ such that, for every such $\Delta$, \[ \left(K[\Delta]\text{ is Cohen--Macaulay and satisfies }N_2\right) \quad\Longrightarrow\quad \operatorname{reg}K[\Delta]\le\lambda. \] Thus the bound is to be independent of $\Delta$, including its number of vertices and dimension, under this unrestricted reading. The supplied record states no narrower class of complexes. It does not specify whether $\lambda$ may depend on $K$, or whether the assertion is uniform over fields. The Stanley--Reisner and $N_2$ conventions above explicate standard meanings; their precise intended scope requires source review.

The supplied curation labels the record partially solved and describes a monomial-ideal analogue as answered negatively by examples of arbitrarily large regularity with linear syzygies, while stating that these examples do not settle the first family question. The record does not specify precisely which assertion its phrase "monomial-ideal analogue" denotes, or whether the cited examples satisfy the Cohen--Macaulay hypothesis of the second question. Consequently, that assessment is retained as a supplied qualification and is not used here to declare either displayed question resolved.

Short English statement

The first question asks for quadratic ideals in increasingly many variables, with linear first syzygies, whose quotient regularity divided by the number of variables has a positive limit. The second asks for a uniform regularity bound for Cohen--Macaulay Stanley--Reisner rings satisfying $N_2$. Field quantifiers, the permitted dependence of the bound, and the scope of the supplied partial-resolution assessment remain unresolved. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003311$, source problem code catalogue entry, titled "Large Regularity Quadratic Ideals with Linear First Syzygies," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The proposed year is recorded as 2016.
• Supplied report citation: Asymptotic Phenomena in Local Algebra and Singularity Theory (2017), Oberwolfach Reports, pp. 3199--3259, DOI 10.4171/owr/2016/57. https://doi.org/10.4171/owr/2016/57. The background attributes the problem collection to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source labeled "Constantinescu--Kahle--Varbaro": https://doi.org/10.1112/S0010437X18007728.
• Supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2016/57.
• Supplied literature source labeled "linear syzygies and regularity": https://doi.org/10.1007/s13348-015-0141-3.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the status partially solved, describes a monomial-ideal analogue as negatively resolved by arbitrarily large regularity with linear syzygies, and says that this does not settle the stated family question for arbitrary quadratic ideals. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, with the supplied explanation that notation, typography, and sentence structure were repaired without changing mathematical content. This is a source-provided label, not independent mathematical verification of this draft.