Intersection vectors over skew-tilings

TL;DR

This work extends Fu-Geng’s intersection-vector paradigm from gentle to skew-gentle algebras by developing the skew-tiling and unfolded-tiling framework. It proves that, under a mild condition, a multiset of tagged permissible arcs is uniquely determined by its intersection vector, and establishes a precise correspondence between intersection data on skew-tilings and their unfoldings. The paper then characterizes when different tau-rigid modules over skew-gentle algebras are distinguished by their dimension vectors, tying this to the absence of minimal oriented cycles of even length with full zero relations and to Cartan-determinant criteria. The results are supported by a Morita-equivalence picture across skew-gentle, skewed-gentle, and unfolded tiling algebras, along with two explicit examples illustrating the theory and its computational consequences.

Abstract

We prove that under a mild condition, a multiset of tagged permissible arcs over a skew-tiling is uniquely determined by its intersection vector. As an application, it is proved that -- up to isomorphism -- different $τ$-rigid modules over a skew-gentle algebra $A$ arising from a skew-triple $(Q,Sp,I)$ have different dimension vectors if and only if $(Q,I)$ has no minimal oriented cycle of even-length with full zero relations. This generalizes a recent work of Fu-Geng for gentle algebras.

Figures (14)

• Figure 1: A once-punctured monogon
• Figure 2: Basic tiles of type (I)-(III)
• Figure 3: Basic tiles of type (IV)-(VI)
• Figure 5: $\mathbf{l}_P, b_P, \{\mathbf{a}_P^{-},\mathbf{a}_P^{+}\}$
• Figure 6: From $\mathbf{T}$ to $\mathbf{T}^{\bowtie}$

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Definition 2.6
• Definition 2.7
• Proposition 2.8