Coincidence of Degeneration Orders for Representation-Directed Algebras

Definitions and mathematical statement

For an algebra $A$, let $\mathcal C_A$ denote the class of objects on which the two degeneration orders in the supplied source are intended to be defined. Write $\preccurlyeq_{\mathrm{geom},A}$ and $\preccurlyeq_{\mathrm{homol},A}$ for those geometric and homological degeneration relations, respectively. These symbols are placeholders for definitions missing from the supplied record; in particular, $\mathcal C_A$ is not being identified here with a category of modules or complexes.

Conjectural target (with unresolved definitions). For every representation-directed algebra $A$, the two relations coincide. Equivalently, the intended statement has the logical form \[ \forall A\text{ representation-directed},\qquad \forall X,Y\in\mathcal C_A,\qquad X\preccurlyeq_{\mathrm{geom},A}Y \ \Longleftrightarrow\ X\preccurlyeq_{\mathrm{homol},A}Y. \] Equality of these relations means that they contain precisely the same ordered pairs of admissible objects.

Unresolved qualifications. The supplied record does not define representation-directed algebras, specify the base field or finiteness assumptions, identify the objects being compared, or define either degeneration relation. It also does not state any common dimension, dimension-vector, or other admissibility restriction on a pair $(X,Y)$. Consequently, the display records the conjecture's quantifier structure but is not a fully specified mathematical formulation. In particular, no identification of the homological relation with inequalities involving dimensions of $\operatorname{Hom}$ or $\operatorname{Ext}$ spaces is justified by the supplied text. The original extracted sentence contains an unresolved pronoun and corrupted diagram text, so it does not recover these missing definitions.

Short English statement

The supplied conjecture asks whether geometric degeneration and homological degeneration give exactly the same comparisons for every representation-directed algebra. The record omits the definitions and underlying objects needed to make this precise. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005126$, title Coincidence of Degeneration Orders for Representation-Directed Algebras; exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2022.
• Supplied report citation: Interactions between Algebraic Geometry and Noncommutative Algebra (2023), Oberwolfach Reports, pp. 1239--1312, DOI 10.4171/owr/2022/23. https://doi.org/10.4171/owr/2022/23. This is the authoritative report reference supplied by the record; its contents have not been independently consulted for this draft. The original extract reads "We conjecture that for representation-directed algebras they do," followed by corrupted text.
• Supplied literature reference labeled "Bongartz, degeneration paper": https://doi.org/10.1016/0001-8708(96)00035-1. The record supplies no full bibliographic citation or specific result from this paper.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, says that the report states the coincidence as a conjecture, and reports that no proof in the stated generality was found. It lists the report and Bongartz reference above as checked sources. This is supplied dated curation derived from the OWR record, not independently verified literature research.
• The registry labels the statement corrected_verified and reports checking it against the official report. That verification claim is supplied provenance and has not been independently confirmed here.