Regularity of Cycles for Linearly Resolved Ideals

Definitions and mathematical statement

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ have its standard grading, with homogeneous maximal ideal $\mathfrak m=(x_1,\ldots,x_n)$. The source considers a homogeneous ideal $I\subset S$ generated in one degree $d$, of linear type on the punctured spectrum, and having a linear free resolution.

In the usual interpretation, linear type off the origin means that for every $\mathfrak p\ne\mathfrak m$ the canonical map \[ \operatorname{Sym}_{S_{\mathfrak p}}(I_{\mathfrak p}) \longrightarrow\mathcal R_{S_{\mathfrak p}}(I_{\mathfrak p}) =\bigoplus_{q\geq0}I_{\mathfrak p}^{q}t^q \] is an isomorphism. For nonzero $I$, a $d$-linear resolution has minimal free modules $F_i=S(-d-i)^{\beta_i}$. For a finitely generated graded $S$-module $M$, write \[ \operatorname{reg}_S M=\max_i\left\{\sup\{a:\operatorname{Tor}_i^S(M,K)_a\ne0\}-i\right\}, \] with a separate convention needed if $M=0$.

The requested conjectural equality is \[ \operatorname{reg}_S(B_j)=jd \] for the intended indices $j$. However, the supplied record does not define the family $B_j$, its construction from $I$, its grading shifts, or its index range. Consequently the hypotheses on $I$ are meaningful but the conclusion is not yet a fully determined mathematical assertion. The title mentioning cycles does not supply those definitions, and the draft does not choose a module construction from that title. The exact punctured-spectrum convention and any characteristic or nonzero-module restrictions also require checking against the source.

The supplied literature triage dated 21 August 2026 reports that the conjecture in its extracted form is unresolved in the cited primary paper, with special cases and asymptotic consequences proved. Those results and their hypotheses are not reproduced, and that assessment does not resolve the missing definition of $B_j$. The registry's self-contained/corrected verification label is retained as metadata rather than taken to supply absent mathematical data.

Short English statement

The ideal is generated in one degree, locally of linear type away from the homogeneous maximal ideal, and linearly resolved. The source asks for $\operatorname{reg}(B_j)=jd$, but never defines $B_j$, its grading, or its index range. That central ambiguity remains explicit.

Sources

• Local UnsolvedMath registry, permanent ID $30000079$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; supplied verification reports a self-contained edit against the official report, although the provided record still omits $B_j$'s definition.
• "Combinatorial Commutative Algebra (2005)," Oberwolfach Reports, pp. 1703--1746, DOI 10.4171/owr/2004/32: https://doi.org/10.4171/owr/2004/32. Proposed year and DOI refer to 2004, while the citation label says 2005.
• Supplied "Eisenbud--Huneke--Ulrich, The Regularity of Tor": https://arxiv.org/abs/math/0405373. The other supplied literature reference is the original report DOI above; the excerpt does not provide the module definition or precise special-case statements.
• Literature checked 21 August 2026; supplied status open. Registry created 25 August 2026 and updated 16 September 2026. This dated curation triage requires independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.