Strong Lefschetz Property of the Zeroth Jordan Component

Definitions and mathematical statement

The supplied record asks whether there exists an Artinian Gorenstein algebra $A$ such that \[ \operatorname{SL}(A)\quad\text{and}\quad\neg\operatorname{SL}(Q(0)), \] where $\operatorname{SL}$ denotes the strong Lefschetz property and $Q(0)$ is the component designated by the source. This is an existence question, not an asserted example or theorem.

The record does not specify the ground field, its characteristic, whether $A$ is local or graded, or the construction of $Q(0)$. Its edited statement calls the decomposition a "Jordan decomposition," whereas its original statement says only "decomposition." No definition identifying $Q(0)$ is supplied. Consequently, the displayed question preserves the intended logical form but is not yet a fully specified mathematical proposition.

For clarity, one conventional graded definition is as follows; its hypotheses are conditional and are not supplied hypotheses of this problem. Let $K$ be a field and let $B=\bigoplus_{i\geq 0}B_i$ be a finite-dimensional commutative graded $K$-algebra with $B_0=K$, generated by $B_1$. Then $B$ has the strong Lefschetz property if there exists $\ell\in B_1$ such that, for every pair of integers $i,d\geq 0$, multiplication induces a map \[ \times\ell^d:B_i\longrightarrow B_{i+d} \] with \[ \operatorname{rank}_K(\times\ell^d)=\min\{\dim_K B_i,\dim_K B_{i+d}\}. \] Under these same hypotheses, failure of this property means that for every $\ell\in B_1$ there exist $i,d\geq 0$ for which the rank is strictly smaller than this minimum. In this connected graded setting, the Gorenstein condition can be expressed by \[ \dim_K\operatorname{Soc}(B)=1,\qquad \operatorname{Soc}(B)=\{b\in B:B_+b=0\},\qquad B_+=\bigoplus_{i>0}B_i. \] These definitions do not determine whether this graded setting is intended, whether $Q(0)$ is an algebra or a module, or which grading and multiplication operators should be used for it. Those choices and the defining decomposition must be recovered from the cited source before a precise specialization of the existence question can be made.

Short English statement

The question asks whether an Artinian Gorenstein algebra can have the strong Lefschetz property while its designated component $Q(0)$ fails that property. The supplied record leaves the decomposition, the ground field, and the local or graded setting unspecified, so the statement requires clarification and human mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004526$, source problem code catalogue entry, title "Strong Lefschetz Property of the Zeroth Jordan Component." No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied report citation: Lefschetz Properties in Algebra, Geometry and Combinatorics (2021), Oberwolfach Reports, pp. 1539--1600, DOI 10.4171/owr/2020/31. https://doi.org/10.4171/owr/2020/31. The supplied literature list also identifies this as "Oberwolfach Reports 2020/31."
• Supplied literature reference: Harima--Watanabe, "Central Simple Modules." https://doi.org/10.1016/j.jpaa.2006.10.016.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not an independently verified literature assessment. The record labels the problem open and states that general Jordan-type and central-simple-module criteria do not settle whether $A$ can be strong Lefschetz while $Q(0)$ is not. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired. This is a supplied editorial claim, not independent mathematical verification of this draft.