Stable Rank of Skew-Gentle Algebras

• Stable rank is a homological invariant that quantifies the depth of radical layers and non-projective modules in skew-gentle algebras.
• Skew-gentle algebras generalize gentle algebras through finite group actions and orbifold data, preserving radical filtrations and singularity categories.
• Geometric models using orbifold dissections and support τ-tilting modules offer practical tools to compute and interpret the stable rank in these algebras.

The stable rank of skew-gentle algebras is a homological and categorical invariant that reflects, in several parallel senses—module-theoretic, derived, geometric, and K-theoretic—the depth and combinatorial structure of these algebras. Skew-gentle algebras generalize gentle algebras by introducing additional group symmetries, special loops, or orbifold data, and emerge as skew-group algebras over gentle algebras acted upon by a finite group (often $\mathbb{Z}_2$). Their stable rank, as discussed in the literature, is most sharply characterized via the preservation and interpretation of invariants under skew-group constructions, the equivalence of singularity categories, the combinatorics of surface-derived models, and the explicit control of radical and morphism layers in the module category.

1. Singularity Categories, Stable Module Categories, and Decomposition Principles

The foundation for understanding stable rank phenomena in skew-gentle algebras stems from the triangulated singularity category $$\mathcal{D}_{sg}(A) = D^b(\mathrm{mod}\text{-}A)/K^b(\mathrm{proj}\text{-}A)$$, which for gentle algebras is classified as a finite product of orbit categories (cluster categories of type $\mathbb{A}_1$), corresponding to the full repetition-free cycles in the quiver and relations (Kalck, 2012). Explicitly,

$$\mathcal{D}_{sg}(A) \simeq \prod_{c\in C(A)} D^b(k\text{-}\mathrm{mod})/[l(c)],$$

where $C(A)$ is the set of cycles, and $l(c)$ is the length of the cycle.

By rich analogy and category-theoretic equivalences established in the skew-gentle case (Chen et al., 2014), the singularity categories of a skew-gentle algebra $A^{sg}$, its underlying gentle algebra $A$, and the associated gentle algebra $A^g$ are triangle equivalent: $$\mathcal{D}_{sg}(A^{sg}) \simeq \mathcal{D}_{sg}(A) \simeq \mathcal{D}_{sg}(A^g).$$ Therefore, the stable rank—meaning the number of non-projective indecomposable Gorenstein projective modules, or the size and structure of the semisimple additive category underlying $\mathcal{D}_{sg}(A^{sg})$—is controlled by the combinatorics of the cycle data, even after introduction of the skewness. Homological smoothness (finite global dimension) corresponds to vanishing of the singularity category, and thus the “stable rank is zero” if and only if $A^{sg}$ is homologically smooth.

2. Skew Group Algebras and Radical Filtration Invariance

From a module-theoretic and categorical perspective, skew-gentle algebras arise as skew group algebras $\Lambda G$ over a gentle algebra $\Lambda$ with a finite abelian group $G$ (often $G \cong \mathbb{Z}_2$) acting by automorphisms (Sardar et al., 27 Jul 2025, Amiot et al., 2019). The key structural result is that the stable radical length—in the sense of the ordinal $\alpha$ such that $$\mathrm{rad}^{\alpha}(\Lambda) = \mathrm{rad}^{\alpha+1}(\Lambda)$$, which is commonly referred to as the stable rank—is preserved under the skew-group construction: $\mathrm{st}(\Lambda G) = \mathrm{st}(\Lambda).$ For all ordinals $\alpha$, the Galois semi-covering functor $F_\lambda$ induces layerwise isomorphisms

$$\mathrm{rad}^\alpha_{\Lambda G}(F_\lambda M, F_\lambda N) \cong \begin{cases} \bigoplus_{g\in G} \mathrm{rad}^\alpha_\Lambda({}^g M, N) & \text{if } G_M \neq G, \ \bigoplus_{g\in G} \mathrm{rad}^\alpha_\Lambda(M, {}^g N) & \text{if } G_N \neq G, \ \mathrm{rad}_\Lambda^{\alpha |G|}(M, N) & \text{if } G_{MN} = G, \ \end{cases}$$

so that the radical filtration for modules, and thus the maximal length of (non-invertible) morphism factorizations, is unaltered by skewness (Sardar et al., 27 Jul 2025). The “Loewy” or ordinal stable rank is preserved: for gentle and skew-gentle algebras with bands, $\,\omega \leq \mathrm{st}(\Lambda) < \omega^2$ (Sardar et al., 27 Jul 2025).

3. Geometric Models and the Role of Orbifold Dissections

The modern approach to describing module categories and derived invariants for (skew-)gentle algebras is via geometric models: dissected surfaces with (possibly) orbifold points correspond to the algebraic data (Schroll, 2020, Labardini-Fragoso et al., 2020, Amiot, 2021, Qiu et al., 2022). In the skew-gentle case, every algebra is associated with an orbifold dissection, with special loops in the quiver producing orbifold points of order two.

The structure of key invariants—singularity category, Gorenstein dimension, and q-Cartan determinant—can be “read off” from the dissection:

Invariant	Geometric Interpretation	Formula Type
Singularity category	Product over interior polygons (i.e., saturated cycles)	$$\prod_{P\in \mathcal{P}_A^0} D^b(k\text{-}\mathrm{mod})/[\#\text{edges}(P)]$$
Gorenstein dimension	Maximal number $m$ of internal edges in boundary polygons	$\mathrm{Gor.dim}(A) = m + 1$
$q$-Cartan determinant	Product over polygons by edge count (length spectrum)	$\prod_{k\geq 1} (1-(-q)^k)^{c_k}$

All of these invariants—determining the complexity and “stable rank” in the sense of non-projective indecomposables or in terms of derived dimensions—are stable under derived equivalence and thus are constant for gentle and skew-gentle algebras associated to the same surface or orbifold model (Labardini-Fragoso et al., 2020).

4. Derived and Geometric Invariance, Support $\tau$-Tilting, and Cluster Combinatorics

The stable rank has a precise geometric and combinatorial realization via support $\tau$-tilting modules, maximal rigid modules, and their corresponding collections of non-crossing (tagged) curves or arcs on a surface (He et al., 2020, Qiu et al., 2022, Labardini-Fragoso et al., 2023, Chang, 10 Mar 2025). In this setup:

• The number of tagged permissible curves in a maximal collection (i.e., a generalized dissection) equals the number of basic support $\tau$-tilting modules, which provides a geometric measure of the stable rank (He et al., 2020).
• The intersection-dimension formula $$\mathrm{Int}(\gamma_1,\gamma_2) = \dim_k \mathrm{Hom}(M(\gamma_1, K_1), T M(\gamma_2, K_2)) + \dim_k \mathrm{Hom}(M(\gamma_2, K_2), T M(\gamma_1, K_1))$$ allows for the computation of ranks of morphisms and, ultimately, the calculation of Grothendieck-group ranks or “stable size”.

Furthermore, Caldero–Chapoton functions and $g$-vectors encode the stable (free) rank of support $\tau$-tilting modules and are robust under mutation—meaning that in categorified cluster algebra settings, the stable rank (free rank at vertices) is a derived invariant (Labardini-Fragoso et al., 2023).

5. Classification Results, Symmetry, and Reflection Invariance

Trivial extensions of skew-gentle algebras yield symmetric algebras isomorphic to skew-Brauer graph algebras, with the associated graph combinatorics dictating the stable structure (Elsener et al., 2 Oct 2024). For a skew-gentle algebra with “small” combinatorial data (Brauer tree with one special vertex), the stable invariants are as low as possible (often interpreted as stable rank one). Derived equivalences via reflection or admissible cuts do not change the stable rank, since the construction is controlled entirely by the graph and surface data. All selfinjective skew-gentle algebras are gentle; thus the maximally “stable” case coincides with known easy invariants (Nakayama, gentle, or their trivial extensions) (Chen, 2022, Elsener et al., 2 Oct 2024).

6. Matrix Problem Approach and Homological Dimensions

A matrix problem reduction of the representation theory for gentle and skew-gentle algebras (Burban et al., 2017) shows that:

• Indecomposable derived objects correspond to string and band diagonals (strings/bands) encoded as decorated matrices (tame matrix problem).
• The existence of special cycles (related to the cycles behind the stable rank computation) determines when the global dimension is infinite, and consequently when the singularity category is nontrivial—again, matching the surface or orbifold combinatorics.

This precise control provides a complete classification of radical layers, morphisms, and extension groups—essential to the analysis of stable rank (Sardar et al., 27 Jul 2025, Burban et al., 2017).

7. Explicit Calculation, Derived Equivalence, and Preservation Across Frameworks

Under the push-down functor along a Galois semi-covering, all radical layers and irreducible morphisms are preserved (Sardar et al., 27 Jul 2025). The stable rank of a skew-gentle algebra (the minimal ordinal at which the radical chain stabilizes) equals that of the original gentle algebra. This invariance holds universally for all skew group algebra constructions with finite group actions and is confirmed for skew-gentle algebras obtained from gentle algebras with bands: $\omega \leq \mathrm{st}(\Lambda) < \omega^2$ where $\Lambda$ is skew-gentle with at least one band (Sardar et al., 27 Jul 2025).

Summary Table: Key Invariants and Methods

Perspective	Description and Result	Reference
Singularity / stable module category	$$\mathcal{D}_{sg}(A^{sg})\cong \mathcal{D}_{sg}(A)\cong$$ product of cluster categories (cycles)	(Kalck, 2012, Chen et al., 2014)
Radical filtration (module category)	Stable rank (radical length) preserved under skew group algebra: $\mathrm{st}(A^{sg}) = \mathrm{st}(A)$	(Sardar et al., 27 Jul 2025)
Geometric (orbifold dissection)	Stable rank, Gorenstein dim., and Cartan determinant read off from polygons/intersections/orbifold points	(Labardini-Fragoso et al., 2020, Qiu et al., 2022)
Support $\tau$-tilting, cluster	Number of basic support $\tau$-tilting modules (or maximal dissection) measures stable rank	(He et al., 2020, Labardini-Fragoso et al., 2023)
Symmetry and trivial extension	Stable rank minimal when trivial extension is a skew-Brauer tree algebra of multiplicity 1 and unique special vertex	(Elsener et al., 2 Oct 2024)

Concluding Remarks

The stable rank of skew-gentle algebras—interpreted as the depth of the radical, the size of the stable module category, or the number of indecomposable factors in the singularity category—is precisely controlled by the combinatorial-topological data underlying the algebra (cycles in the quiver, surface or orbifold dissections). All advanced frameworks—coverings, geometric models, cluster and tilting theory, K-theory, and combinatorial matrix problems—agree: stable rank is invariant under skew group algebra constructions, derived equivalence, and surface symmetries. Thus, analysis of the gentle “part” suffices to determine the stable ranks and their numerical values, with all skewness effects effectively absorbed by derived and geometric invariance (Chen et al., 2014, Chen, 2022, Sardar et al., 27 Jul 2025, Elsener et al., 2 Oct 2024).