Relative Betti Inequalities for Lex-Plus-Powers Ideals

Definitions and mathematical statement

For a standard graded algebra $R$ over a field $k$, with $R_0=k$, and a finitely generated graded $R$-module $M$, the usual graded Betti numbers are \[ b^R_{i,j}(M)=\dim_k\operatorname{Tor}^R_i(M,k)_j, \qquad i\in{\mathbb{Z}}_{\geq 0},\quad j\in{\mathbb{Z}}. \] This definition explains the conventional notation; the supplied record does not explicitly specify the field, grading, or module conventions.

The source asks the following conditional question. With $S,F,P,I,L$ satisfying the hypotheses referred to there as those of the Lex-Plus-Powers Conjecture, does one have \[ b_{i,j}^{S/P}(L)\geq b_{i,j}^{S/F}(I) \qquad\text{for all }i,j? \] Under the graded Betti-number convention above, the index range is $i\in{\mathbb{Z}}_{\geq0}$ and $j\in{\mathbb{Z}}$.

The referenced hypotheses are not reproduced in the supplied record. In particular, it does not define $S,F,P,I,L$, specify the regular-sequence or degree conditions, state the relation between the Hilbert functions, or identify precisely the modules denoted by $I$ and $L$ over the respective quotient rings. Consequently, this is a transcription of the intended question with an unresolved contextual dependency, rather than a fully specified standalone conjecture. The arguments $L$ and $I$ in the displayed inequality are retained exactly; replacing them with quotient modules would require source clarification.

Short English statement

The question asks whether the lex-plus-powers object has at least as many graded Betti numbers over $S/P$ as the corresponding object over $S/F$, in every homological and internal degree. The supplied record omits the hypotheses defining these objects, so the precise scope and module interpretation require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001236$, title "Relative Betti Inequalities for Lex-Plus-Powers Ideals," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives the proposed year as 2009 and supplies no individual proposer.
• Primary source supplied by the registry: Kommutative Algebra (2009), Oberwolfach Reports, report 22, pp. 1157--1224, DOI 10.4171/owr/2009/22. https://doi.org/10.4171/owr/2009/22. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: "Caviglia-Sammartano on lex-plus-powers." https://arxiv.org/abs/1802.03035.
• Supplied literature source: "A study of the lex-plus-powers conjecture." https://doi.org/10.1016/S0022-4049(03)00130-0.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes it as the general Betti-number inequality in the lex-plus-powers setting, states that the report leaves it open outside the pure-power case, and says that later work treats only broad special families. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not a verification performed for this draft.