Uniqueness of Surface Models for Locally Gentle Algebras

Definitions and mathematical statement

Let $k$ denote the ground field, which is not specified in the supplied record. For a $k$-algebra $B$, finite dimensionality means $\dim_k B<\infty$. The record does not specify the definition of a locally gentle algebra, the data constituting a tiled-surface model, or the equivalence relation on such models.

The original excerpt places the conjecture beside the following reported characterization, attributed there to reference [2]: \[ \begin{aligned} &A\text{ is a string algebra};\\ &\Longleftrightarrow\quad A=B/R,\quad B\text{ locally gentle},\quad R\text{ an ideal generated by paths of length at least }2,\\ &\hspace{43mm}\text{with every oriented cycle in }A\text{ having a relation};\\ &\Longleftrightarrow\quad A\text{ is a labelled tiling algebra}. \end{aligned} \] This is supplied background, not an independently verified theorem statement. In particular, the quiver presentation and the precise meaning of an oriented cycle having a relation are not specified.

The original conjecture asserts that choosing $B$ finite dimensional makes the associated surface unique up to rotating certain collections of tiles. Its uniqueness content can only be expressed schematically from the supplied evidence: for a fixed algebra being modelled, if $M_1$ and $M_2$ are admissible surface models arising from choices with finite-dimensional $B$, then \[ M_1\sim_{\mathrm{rot}}M_2, \] where $\sim_{\mathrm{rot}}$ stands for the equivalence intended by "rotating certain collections of tiles." Neither the admissible models nor these permitted rotations are defined in the excerpt. Moreover, the preceding context needed to identify the fixed algebra and the allowed choices of $B$ is absent. Thus this display records the apparent uniqueness assertion without supplying a complete formal conjecture.

The registry's clean question instead asks whether every finite-dimensional locally gentle algebra admits a tiled-surface model unique up to rotations of the tiles. This includes an existence assertion and treats the locally gentle algebra itself as the object being modelled. The supplied excerpt does not establish that this formulation is equivalent to the original assertion involving a choice of $B$ in $A=B/R$. Both formulations are retained here as unresolved alternatives.

The subsequent passage concerns a separate discussion of laminations, attributed to [3]. There each tile is a polygon with at least three sides and at most one boundary edge. A lamination is described as a collection of non-crossing curves such that whenever a curve successively crosses two edges of a tile, those edges are adjacent. The passage lists tiling arcs $e_1,\ldots,e_m$, with $m>n$: the endpoints of $e_1,\ldots,e_n$ are punctures, whereas each of $e_{n+1},\ldots,e_m$ has one puncture endpoint and one endpoint at a boundary vertex. A curve crossing the tiling is said to give a vector in ${\mathbb{Z}}^m$, with positive and negative crossings described by S-shapes and Z-shapes, and trivial crossings described by a fan or an unfinished alternative. The excerpt ends before completing that prescription. These restrictions and descriptions belong to the lamination discussion; the record does not establish whether they also constrain the conjectured surface models.

Short English statement

The supplied conjecture says that imposing finite dimensionality on the locally gentle algebra $B$ used in a surface construction should make the resulting surface unique up to certain rotations of collections of tiles. The registry paraphrases this as existence and uniqueness of a surface model for every finite-dimensional locally gentle algebra. The excerpt does not define the models or rotations, or resolve the difference between these targets, so a complete precise formulation requires human review of the surrounding source.

Sources

• Local UnsolvedMath registry: permanent ID $30005425$, title "Uniqueness of Surface Models for Locally Gentle Algebras," exact source problem code catalogue entry; supplied complete source record, including its original excerpt, clean statement, background, and verification fields.
• Supplied source citation: Representation Theory of Quivers and Finite-Dimensional Algebras (2023), Oberwolfach Reports, pp. 397--486, DOI 10.4171/owr/2023/7. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2023/7. No individual proposer is supplied; the proposed year is 2023. The record identifies the report as documentation of workshop discussions and open problems.
• The excerpt attributes its string-algebra characterization to [2], its lamination discussion to [3], and the triangulated-surface comparison to [4], with [5] also cited. Bibliographic identities for [2]--[5] are not supplied. It describes [3] as aiming to determine maximal-lamination size and define mutation of curves in a maximal lamination.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, says the finite-dimensional-surface uniqueness conjecture for string-algebra tilings is stated without proof, and reports that no later primary source located in that curation establishes it in the stated generality. Its listed checked source is the original report linked above. This is supplied dated curation, not independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry supplies status open and statement status corrected_verified, together with a claim that the clean statement was checked against the official report and preserves its hypotheses, notation, and target. These are supplied registry assertions, not verification performed for this draft. The report itself has not been consulted for this response.