Hochschild Cohomology for Graded Skew-Gentle Algebras

• The paper introduces a compact projective bimodule (graded CS-) resolution to effectively compute HH*(A) for graded skew-gentle algebras.
• It derives an explicit bigrading and K-basis in HH*(A) by classifying eight distinct cocycle types linked to the underlying quiver combinatorics.
• It establishes the algebra structure through well-defined cup products and Gerstenhaber brackets, connecting algebraic invariants with orbifold surface geometry.

A graded skew-gentle algebra is defined for a fixed base field $K$, a finite quiver $Q = (Q_0, Q_1, s, t)$, a set of quadratic monomial relations $R \subset KQ$, and a distinguished subset $\mathrm{Sp} \subset Q_1$ of loops. The data $(Q, R, \mathrm{Sp})$ form a graded skew-gentle triple if (a) the pair $$(Q, R \cup \{\varepsilon^2 \mid \varepsilon \in \mathrm{Sp}\})$$ is a graded gentle pair, and (b) each $\varepsilon \in \mathrm{Sp}$ has degree zero. The associated algebra is

$$A = KQ \big/ \langle R \cup \{\varepsilon^2 - \varepsilon \mid \varepsilon \in \mathrm{Sp}\} \rangle.$$

Setting $\mathrm{Sp} = \varnothing$ recovers the graded gentle case. These algebras are closely related to the partially wrapped Fukaya categories of orbifold surfaces with stops, with geometric information encoded in the quiver and its relations (Bian et al., 8 Jan 2026).

1. Projective Resolution and Computation of Hochschild Cohomology

The Hochschild cohomology $HH^*(A)$ of a graded skew-gentle algebra is computed using a compact projective bimodule resolution, the graded CS-resolution, which is much smaller than the standard bar resolution. This resolution is given by

$$\cdots \to A \otimes \Gamma_n \otimes A \xrightarrow{d_n} A \otimes \Gamma_{n-1} \otimes A \to \cdots \to A \otimes \Gamma_0 \otimes A \to A \to 0,$$

where $\Gamma_n$ consists of paths of length $n$ with no subpaths in $R \cup \{\varepsilon^2\}$. Applying $\operatorname{Hom}_{A^e}(-,A)$ yields a cochain complex whose cohomology is $HH^*(A)$.

2. Cohomology Basis and Grading Structure

The cohomology $HH^*(A)$ acquires a bigrading: $HH^N(A) = \bigoplus_{n + j = N} HH^{n, j}(A),$ with an explicit $K$-basis in each bidegree, described as eight types of cocycles. These basis classes reflect geometric and combinatorial data from the quiver:

• Maximal paths $\alpha$ yield cohomology in degree $(0,|\alpha|)$.
• Primitive cocomplete cycles, arrows outside a spanning tree and special loops, $\Gamma$-maximal paths, and certain complete cycles contribute further classes.
• The explicit formula for the dimension in degree $N$ is a sum of the counts of maximal paths and other structures of total length $N$, according to the eight cocycle types.

Each generator's cohomological and internal degrees follow from the path combinatorics and the quiver's grading scheme (Bian et al., 8 Jan 2026).

3. Algebra Structure: Cup Product

The associative cup product on $HH^*(A)$ is defined by transporting the usual cup product to the CS-resolution. For basis cocycles, only four types of nontrivial products occur, as codified in a classification table:

Product Type	Nonzero Product Condition	Formula
$\alpha_s \smile \alpha_s$	Always	$\alpha^2_s$
$C_{gr} \smile C_{gr}$	Always	$(-1)^{\ell(C)} C_{gr}^2$
$(c,c)\smile \alpha_s$	$c$ divides $\alpha$	$(c,c\alpha)$
$(c,c)\smile C_{gr}$	$c \in C$	$(-1)^{|c|\cdot|C|} (cC,c)$

All other products among basis elements vanish. The algebra $HH^*(A)$ is generated by five classes: $$\{\alpha,\,\alpha_s,\, (c,c),\, (\gamma,\alpha),\,C_{gr} \}$$, with relations enforcing vanishing of unwanted products and gluing relations such as

$$(c,c) \smile C_{gr} = (d,d) \smile C_{gr} \text{ for } c,d \in C,$$

and similarly for the $\alpha_s$ type.

4. Gerstenhaber Bracket and Lie Structure

A graded Lie algebra structure (the Gerstenhaber bracket) is induced on $sHH^*(A)$ via the transferred insertion operation: $$[f,g] = f \circ g - (-1)^{(|f| - 1)(|g| - 1)} g \circ f,$$ where the only nontrivial bracket among the generators is

$[(c,c), v] = (\deg_c(v))\, v,$

with $\deg_c(v)$ counting the occurrences of $c$ within the combinatorial data of $v$. All other brackets among the eight generator types vanish. The degree-one part, $HH^1(A)$, is an abelian Lie algebra except in the case of a one-vertex/one-loop quiver, where exceptional nonzero brackets appear.

5. Geometric Interpretation via Orbifold Surfaces

To each graded skew-gentle algebra is associated a graded marked ribbon graph $G$, and hence a graded marked orbifold surface $(S, M, \mathcal{O}, \eta)$, where $M$ is the collection of un-orbifolded marked points, $\mathcal{O}$ is the set of order-2 orbifold points (one for each special loop), and $\eta$ is a line field determined by the grading.

There is a bijective correspondence between basis generators of $HH^*(A)$ and classes of curves on $S$:

Class in $HH^*(A)$	Geometric Curve Correspondence
$(\alpha, \alpha)$	Boundary component with one marked point
$\alpha_s$	$G$-puncture (unstopped)
$C_{gr}$	$G^*$-puncture (fully stopped)
$(c, c)$	Generator of $\pi_1(S_{\rm smooth})$ ($\cong HH^1(A)$)

Total degree is given by $\deg_{\rm tot} = w_\eta(\mathsf{C}) + 1$, with $w_\eta(\mathsf{C})$ the winding number. The cup products correspond to concatenation of curves, and the only nontrivial bracket is the winding-derivative bracket

$[(c,c), \mathsf{C}] = w_c(\mathsf{C}) \mathsf{C},$

where $w_c(\mathsf{C})$ counts parallel traversals of $\mathsf{C}$ along the generator loop represented by $(c,c)$.

6. Explicit Illustration: Three-Point Orbifold Example

Consider the quiver

$$2 \bigcirc^{\varepsilon_2} \longrightarrow 1 \longleftarrow 3 \bigcirc^{\varepsilon_3},$$

with relations $ba = 0$, $cb = 0$ and special loops $\mathrm{Sp} = \{\varepsilon_2, \varepsilon_3\}$. In this case:

• There is a unique $\Gamma$-maximal path of length 3, $\alpha = c\,\varepsilon_3\,b\,\varepsilon_2\,a$.
• The eight basis types reduce to three generators: $\alpha_s \in HH^{0,|\alpha|}(A)$, $(c,c) \in HH^{1,0}(A)$, and $(cba,e_1) \in HH^{3, -|\alpha|}(A)$.
• The only nonzero products are

$$\alpha_s \smile \alpha_s = \alpha^2_s, \qquad (c,c) \smile (c,c) = 0, \qquad (c,c) \smile \alpha_s = (c,c\alpha).$$

• The only nonzero bracket is

$[(c,c), \alpha_s] = 1 \cdot \alpha_s.$

Geometrically, the surface $S$ is a disk with one boundary-marked point and two orbifold punctures, with the three cohomology generators corresponding to (i) a boundary loop, (ii) a loop at the interior marked point, and (iii) small loops around the orbifold points (Bian et al., 8 Jan 2026).