2. Preliminaries

In this section, we fix some of the notation and definitions which will be used through this paper. We fix an algebraically closed field K of $\operatorname{char}\neq 2$ .

2.1. Gentle and skew-gentle algebras

In this subsection, we define gentle and skew-gentle algebras.

A quiver Q is a quadruple $(Q_0, Q_1, s, t)$ , where $Q_0$ is the set of vertices, $Q_1$ is the set of arrows and $s, t:Q_1\to Q_0$ are functions indicating the source and target of an arrow. A path w of length $n>0$ in Q is a sequence of arrows $\alpha_1 \dots \alpha_n$ such that $t(\alpha_j)=s(\alpha_{j+1})$ for each $j=1, \dots, n-1$ . For each vertex i, we denote by $e_i$ the trivial path of length 0.

The path algebra KQ is defined as the K-vector space with basis the set of all paths in Q, with multiplication induced by concatenation of paths. A 2-sided ideal I of KQ is admissible if there exists an integer $m\geq 2$ such that $R^m \subset I \subset R^{2}$ , where R is the ideal of KQ generated by the arrows of Q.

Skew-gentle algebras were introduced in [Reference Geiß and de la Peña16]. They are closely linked to the well-studied class of gentle algebras. For instance, they are skew-group algebras of gentle algebras. They are of tame representation type [Reference Geiß and de la Peña16], and their derived categories are also tame [Reference Bekkert, Marcos and Merklen5, Reference Burban and Drozd12]. In [Reference Bekkert, Marcos and Merklen5], a combinatorial description of the indecomposable objects in the bounded derived category of a skew-gentle algebra is given in terms of homotopy strings and bands.

In order to define skew-gentle algebras, we first recall the definition of gentle algebras.

Definition 2.1 A K-algebra $\Lambda$ is gentle if it is Morita equivalent to $KQ/I$ , where

1. (1) Q is a finite quiver such that for every vertex i of Q there are at most two arrows ending at i and at most two arrow starting at i;

2. (2) for every arrow $\alpha$ of Q, there is at most one arrow $\beta$ such that $t(\alpha) = s(\beta)$ and $\alpha\beta\in I$ , and there is at most one arrow $\gamma$ such that $t(\gamma) = s(\alpha)$ and $\gamma\alpha \in I$ ;

3. (3) for every arrow $\alpha$ of Q, there is at most one arrow $\beta'$ such that $\alpha\beta'\notin I$ , and there is at most one arrow $\gamma'$ such that $\gamma'\alpha\not\in I$ ;

4. (4) I is the 2-sided ideal of Q generated by certain paths of length 2;

5. (5) I is an admissible ideal of Q.

If I satisfies (1), (2), (3) and (4), then we say that the quotient $KQ/I$ is locally gentle.

Definition 2.2 A K-algebra A is (locally) skew-gentle if it is Morita equivalent to an algebra $KQ/I$ where

1. (1) $Q_1 = Q^{\prime}_1 \cup S$ , where for $\varepsilon \in S$ , $s(\varepsilon) = t(\varepsilon)$ ,

2. (2) $I = \langle I' \cup \{ \varepsilon^2 - \varepsilon \mid \varepsilon \in S\} \rangle$ ,

3. (3) $KQ' / I'$ is a (locally) gentle algebra where $Q' = (Q^{\prime}_1, Q_0)$ ,

4. (4) if $\varepsilon \in S$ then the vertex $i = s(\varepsilon)$ is the start or the end of exactly one arrow in $Q^{\prime}_1$ and if there is an arrow $\alpha \in Q^{\prime}_1$ with $t(\alpha) = i$ and an arrow $\beta \in Q^{\prime}_1$ with $s(\beta) = i$ then $\alpha \beta \in I'$ . Moreover, there is no other element in S starting at i.

We call a vertex $i \in Q_0$ special if there exists $\varepsilon \in S$ such that $i = s(\varepsilon)$ . We denote the set of special vertices by Sp. If $KQ/I$ is a skew-gentle algebra as above, we call (Q $^{\prime}$ , I $^{\prime}$ , Sp) a skew-gentle triple.

Remark 2.3 If $KQ/I$ is a skew-gentle algebra with non-empty set of special vertices, then the ideal I is not admissible.

An admissible presentation $KQ^{sg} / I^{sg}$ of a skew-gentle algebra A is given as follows. Let $\Lambda=KQ/I$ be the gentle algebra obtained from A by deleting the special loops.

Set

\begin{equation*}Q^{sg}_0(i) = \left\{\begin{array}{l@{\quad}l}\{i^+, i^-\} & \mbox{if $i \in Sp$} \\ \\[-7pt] \{ i \} & \mbox{otherwise} \\\end{array}.\right.\end{equation*}

Define

\begin{equation*}Q^{sg}_0:= \bigcup_{i\in Q_0}Q^{sg}_0(i).\end{equation*}

The arrows of $Q_1^{sg}$ are defined as follows. The set $Q^{sg}_1[i, j]$ of arrows from vertex i to vertex j is given by

\begin{equation*}Q^{sg}_1[i,j]:=\{(i, \alpha,j) \mid \alpha\in Q_1, i \in Q_0^{sg}(s(\alpha)), j \in Q_0^{sg}(t(\alpha))\}.\end{equation*}

The ideal $I^{sg}$ is defined as follows.

\begin{equation*}I^{sg}:=\left\langle \sum_{j\in Q_0^{sg}(s(\beta))}\lambda_j(i,\alpha, j)(j,\beta, k)\mid \alpha\beta\in I, i\in Q_0^{sg}(s(\alpha)), k\in Q_0^{sg}(t(\beta)) \right\rangle,\end{equation*}

where $\lambda_j=-1$ if $j=l^-$ for some $l\in Q_0$ , and $\lambda_j=1$ otherwise.

Note that in general, the relations in $I^{sg}$ are not monomial; instead, the ideal $I^{sg}$ is admissible and quadratic.

Example 2.4 Consider the following quivers

and set $I=\{\varepsilon^2-\varepsilon\}$ and $I'=\{\varepsilon_1^2-\varepsilon_1,\varepsilon_2^2-\varepsilon_2, \alpha_1\alpha_2\}$ . Then, the algebras $A_1=KQ/I$ and $A_2=KQ'/I'$ are skew-gentle and their respective admissible presentations $KQ^{sg}/I^{sg}$ and $KQ'^{sg}/I'^{sg}$ are as follows.

where $I^{sg}$ is the empty set and

\begin{equation*}I'^{sg}=\langle (3^{-}, \alpha_1, 2^+)(2^+, \alpha_2, 1)- (3^{-}, \alpha_1, 2^-)(2^-, \alpha_2, 1), (3^{+}, \alpha_1, 2^+)(2^+, \alpha_2, 1)- (3^{+}, \alpha_1, 2^-)(2^-, \alpha_2, 1) \rangle.\end{equation*}

Note that the algebra $KQ^{sg}/I^{sg} = KQ^{sg}$ corresponds to an orientation of the Dynkin diagram $\mathbb{D}_5$ .

3. Skew-gentle algebras and orbifolds

In this section, we define a graph associated with a skew-gentle algebra generalising the ribbon graph associated with a gentle algebra, and we will show that this graph also has a canonical embedding into a surface.

Proof. Let i be a special vertex of A corresponding the edge i in $G_\Lambda$ and let P and P $^{\prime}$ be the polygons sharing the edge g. By Definition 2.2, the vertex i is either the start or the end of exactly one arrow in Q or there are exactly two arrows $\alpha, \beta$ in Q such that $s(\beta)=t(\alpha)=i$ and $\alpha\beta\in I$ .

Suppose first that we are in the latter case, that is that there are exactly two arrows $\alpha$ and $\beta$ such that $s(\beta)=t(\alpha)=i$ . Assume further that $s(\alpha) $ and $t(\beta)$ are edges in P. Then, the edge j of P $^{\prime}$ preceding i and the edge k following i in the orientation of $S_\Lambda$ are boundary edges. By,[Reference Opper, Plamondon and Schroll35, Proposition 1.12], every polygon in the surface dissection of $S_\Lambda$ has at most one boundary edge which implies that $j=k$ and P $^{\prime}$ is a digon.

In the case that there is exactly one arrow incident with i, the argument is similar.

We now give the definition of a generalised ribbon graph associated with a skew-gentle algebra based on the ribbon graph and surface of the underlying gentle algebra.

Let A be a skew-gentle, $\Lambda=KQ/I$ be the associated gentle algebra obtained from A by deleting all special vertices, and let $(S_\Lambda, M_\Lambda, G_\Lambda)$ its surface dissection. For each edge g of $G_{\Lambda}$ corresponding to a special vertex $i\in Sp$ , let P and P $^{\prime}$ be the polygons sharing the edge g. Suppose further that P $^{\prime}$ is a digon with one boundary edge (which exists by the specialness of v, see Lemma 3.1). We define the local replacement of g in $G_\Lambda$ as the graph embedded graph $G^{\prime}_{\Lambda}$ obtained by contracting the boundary segment of P $^{\prime}$ and identifying the vertices $p_1$ and $p_2$ by collapsing the interior of the polygon P $^{\prime}$ so that g is incident with $p_1 = p_2$ . In the process, we obtain a new marked point in the interior of P, we will denote this point by o and depict by drawing a cross-shaped vertex in the surface. We will refer to these vertices as special vertices. We locally illustrate the local replacement and the resulting new vertex in Figure 1.

Figure 1. A local replacement.

The generalised ribbon graph $G_A$ of A is the graph obtained from $G_\Lambda$ by applying a local replacement at each special vertex.

Denote by $M_A$ the marked points on $S_\Lambda$ corresponding to the vertices of the embedded ribbon graph $G_A$ which are not special. Let $O_A=(S, M_A, \mathcal{O})$ be the triple given by the surface $S = S_\Lambda$ , the marked points $M_A$ and the set of special vertices $\mathcal{O}$ .

From now on, consider the special vertices of $G_A$ to be orbifold points of order two and we say that an edge in $G_A$ joining a vertex and an orbifold point is a special edge. Consequently, we refer to $O_A$ as the orbifold of A.

Note that the above construction also works for locally skew-gentle algebras $A = KQ/I$ . In this case, the generalised ribbon graph will have in addition to the special vertices punctures corresponding to either cycles with no relations in the quiver or to cycles with full relations, but with a special loop at each vertex of the cycle.

The set of edges of $G_A$ is in bijection with the vertices of $Q_0$ (note that this includes the special vertices). Then, $G_A$ is a ribbon graph with the cyclic ordering of the edges at each vertex induced by the Sp-maximal paths. Denote by $S_A$ the corresponding oriented surface with boundary such that $G_A$ is a deformation retract of $S_\Lambda$ . Now define a marking map $m{:} M\setminus Sp \to E$ similar to the gentle case. Note that the elements of M corresponding to the idempotents at special vertices in $Q_0$ are not marked. The marking map gives a unique way of gluing the marked vertices of $G_A$ to the boundary of $S_A$ whereas the vertices of $G_A$ corresponding to elements in Sp stay in the interior of the surface where they give rise to the set $\mathcal O$ of orbifold points of order two. This gives rise to a generalised surface dissection which coincides with the construction of the orbifold dissection of $O_A$ in Definition 3.7 below.

Example 3.5 Let $A_1=KQ/I$ and $A_2=KQ'/I'$ be the skew-gentle algebras from Example 2.4. Note that the set of vertices ${M}_{A_1}$ of the generalised ribbon graph $G_{A_1}$ is the set $\{\alpha_1\alpha_2\alpha_3\varepsilon, e_4, e_3, e_2, e_1\}$ where $e_1$ corresponds to the trivial path associated with the special vertex 1 and that the set of vertices $M_{A_2}$ of $A_2$ is the set $\{\varepsilon_1\alpha_1\varepsilon_2\alpha_2, e_3, e_2, e_1\}$ where $e_3$ and $e_2$ correspond to the trivial path associated with the special vertices 3 and 2 respectively. The generalised ribbon graphs $G_{A_1}$ and $G_{A_2}$ can be seen in Figure 2.

Figure 2. Generalised ribbon graphs of $\mathbb D_5$ and $A_2$ from Example 2.4.

Then, the generalised ribbon graphs $G_{A_1}$ and $G_{A_2}$ embedded in their respective orbifolds can be seen in Figure 3.

Figure 3. Generalised ribbon graphs of $\mathbb D_5$ and $KQ'/ I$ from Example 2.4 embedded in their respective orbifolds.

For any skew-gentle algebra A, the edges of $G_A$ cut the orbifold $O_A$ into polygons, some of which contain the points in $\mathcal{O}$ and a special edge connected to them. We call those polygons degenerate polygons. We note that the following two results, Proposition 3.6 and Theorem 3.8, have independently appeared in [Reference Amiot and Brüstle1].

Proof. This directly follows from [Reference Opper, Plamondon and Schroll35, Proposition 1.12] and the construction of $G_A$ by local replacement.

Before stating the next result, we define the following notation. Denote by B one of the skew-gentle algebras with two vertices, one arrow between them and one or two special loops.

Proof. By Proposition 3.6, it is enough to show that given a orbifold dissection $(S,M,\mathcal O, G)$ there exists a skew-gentle algebra A having $(S,M,\mathcal{O})$ as its orbifold and G as its generalised ribbon graph. Given $(S,M, \mathcal{O}, G)$ , define a quiver Q as follows:

1. (1) the vertices of Q are in bijection with the edges of G;

2. (2) if i, j are two edges incident with the same vertex in G then there is an arrow from i to j if j is a direct successor of i in the orientation of the surface, that is there is no other edge of G between i and j. Note that if i is a special edge in a degenerate polygon, then i is its own successor, therefore there is a loop incident to i. Denote by Sp the set of vertices of Q which corresponds to the special edges of G and by S the set of special loops incident to a special vertex $i\in Sp$ .

Observe that by construction any vertex of Q has at most two in going arrows and at most two outgoing arrows, because any edge of G shares at most two (degenerate) polygons. Let I be the ideal of KQ generated by the following relations: if $\alpha{:} i \to j$ and $\beta{:}j\to k$ are two consecutive arrows such that i, j and k correspond to edges of the same (degenerate) polygon and such that neither $\alpha$ nor $\beta$ correspond to a special loop, then $\alpha\beta$ is a relation. Consequently, for any arrow $\alpha$ , there is at most one consecutive arrow $\beta$ such that $\alpha\beta\in I$ and at most one preceding arrow $\gamma$ such that $\gamma\alpha\in I$ . Finally, for each loop $\varepsilon$ , incident to a special vertex, $\varepsilon^2-\varepsilon\in I$ .

We need to show that $A=KQ/I$ is a skew-gentle algebra. By construction, if P is a degenerate polygon, and v is a special edge of P, then the valency of i is at most two, because any special edge belongs to exactly one degenerate polygon. Moreover, if the valency of i is two, by the definition of the generators of I, the composition of the arrow ending at i with the arrow starting at i is a generator.

Finally, it follows from the construction of $KQ/I$ that its orbifold dissection is $(S,M, \mathcal{O}, G)$ .

4. The dual graph and the Koszul dual of a skew-gentle algebra

In this subsection, we show that a skew-gentle algebra is strong Koszul and that its Koszul dual is again skew-gentle. Furthermore, given a skew-gentle algebra A and the corresponding skew orbifold dissection, we construct a dual graph embedded in the orbifold and we show that this dual graph is the orbifold dissection of the Koszul dual of A.

According to [Reference Green21], an algebra $KQ/I$ is strong Koszul if I is quadratic and has a quadratic Gröbner basis. By [Reference Green and Huang19], any strong Koszul algebra is a Koszul algebra, but the converse does not always hold, for example, Sklyanin algebras [Reference Smith40] are Koszul algebras but not strong Koszul.

Since gentle algebras are Koszul and since skew-group constructions preserve the Koszul property [Reference Li33], it is clear that skew-gentle algebras are Koszul. However, it is not known whether the skew-group algebra of a strong Koszul algebra is strong Koszul. The aim of this section is prove that skew-gentle algebras are strong Koszul. As a consequence, we also give a new proof that skew-gentle algebras are Koszul. Following [Reference Green20, Reference Green, Hille and Schroll22], we recall the basic definitions of Gröbner bases. Recall that $\succ$ is an admissible order on B if $\succ$ is a total order on B such that every non-empty subset of B has a minimal element, and $\succ$ is compatible with the multiplicative structure of B, namely the following conditions hold for any $p,q,x,w \in B$ , see [Reference Green20, Section 2.2.1] for details.

1. (1) if $p\succ q$ then $px\succ qx$ when $px\neq 0$ and $qx\neq 0$ .

2. (2) if $p\succ q$ then $wp\succ wq$ when $px\neq 0$ and $qx\neq 0$ .

3. (3) if $p=qx$ , then $p\succ q$ and $p\succ x$ .

To simplify notation, in this section a vertex idempotent $e_i$ associated with a vertex i will be denoted by i. An element $x\in KQ$ is uniform if there are vertices $i, j\in Q_0$ such that $ixj = x$ .

The concepts of simple and complete reductions play an important role in Gröbner basis theory. For example, a useful characterisation of a Gröbner basis is that all its elements completely reduce to zero, see, for example, [Reference Green, Hille and Schroll22, Proposition 2.9]. Furthermore, complete reduction can be used to complete a subset $\mathcal{H}$ of uniform elements of the ideal I to a Gröbner basis.

Definition 4.4 Let $x = \sum_{p\in \mathcal B}\lambda_pp$ and $y=\sum_{q\in\mathcal B}\mu_qq$ in KQ. Suppose that $s = \operatorname{tip}(x), t= \operatorname{tip}(y)$ and $sm = nt$ for some $m, n \in \mathcal B \setminus Q_0$ where the lengths of m and n are strictly less than the length of s. Then the overlap relation, $\mathbf {o}(x, y, m, n)$ , is

\begin{equation*}\mathbf{o}(x, y, m, n) = (\mu_t)xm-(\lambda_s)ny.\end{equation*}

We now show that a skew-gentle algebra is strongly Koszul by showing that it has a quadratic Gröbner basis. For this, we begin by showing that given a skew-gentle algebra A, any admissible order for the underlying gentle algebra $\Lambda$ induces a natural admissible order for A.

Proof. Let $\succ_{Q}$ be an order on Q inducing an admissible order on $\mathcal{B}$ . Consider $\succ$ an order on $Q^{sg}$ induced by $\succ_{Q}$ as follows.

Let $(i, \alpha, j)$ and $(i', \beta, j')$ be two arrows in $Q^{sg}$ . If $\alpha\neq \beta$ , we say that $(i, \alpha, j)\succ (i', \beta, j')$ if and only if $\alpha \succ_Q \beta$ .

Now, suppose that $\alpha=\beta$ and $s(\alpha)$ or $t(\alpha)$ is a special vertex, then we fix an order on the set of arrows induced by $\alpha{:}i \to j$ as follows.

If i and j are special vertices, then $(i^+, \alpha, j^+) \succ (i^-,\alpha, j^-) \succ (i^+, \alpha, j^-) \succ (i^-, \alpha, j^+)$ . If i is a special vertex and $t(\alpha)$ is not special (resp. i is not special and j is special), then $(i^+, \alpha, j) \succ (i^-,\alpha, j)$ (resp. $(i, \alpha, j^+) \succ (i,\alpha, j^-)$ ).

By construction, the only property we need to check in order for $\succ$ to be admissible, is that every descending chain in $\mathcal{B}^{sg}$ has a minimal element. Let C=( $p_1\succ p_2\succ \dots \succ p_t \succ \dots)$ be a descending chain of paths in $\mathcal{B}^{sg}$ . By construction of $kQ^{sg}$ , each path $p_r$ is a sequence of arrows of the form $(i_{r_1}, \alpha_{r_1},j_{r_1})\dots (i_{r_l}, \alpha_{r_l},j_{r_l})$ such that $\alpha_{r_1}\dots \alpha_{r_l}$ is a path $\widehat{p_r}$ in Q and $C'=(\widehat{p_1}\succ_Q \widehat{p_2}\succ_Q \dots \succ_Q \widehat{p_t} \succ_Q \dots)$ is a descending chain of elements in $\mathcal{B}$ . Since $\succ_Q$ is an admissible order on $\mathcal{B}$ , the chain C $^{\prime}$ has a minimal element $\widehat{p}_r$ , for some $r\in\mathbb N$ , such that $\widehat{p}_r=\widehat{p}_{r+m}$ for all $m\in\mathbb N$ , therefore there exists some $r' \in \mathbb N$ such that $p_{r^{\prime}}=p_{r^{\prime}+m}$ for all $m\in\mathbb N$ .

Thus, the order $\succ$ on the basis of paths $\mathcal{B}^{sg}$ of $KQ^{sg}$ is admissible.

Proof. Let A be a locally skew-gentle algebra, $\Lambda=KQ/I$ be the locally gentle algebra obtained from A by deleting special loops. Consider the admissible presentation of A, namely $A^{sg}=KQ^{sg}/I^{sg}$ . By [Reference Green and Huang19, Theorem 3] it is enough to prove that there exists a quadratic Gröbner basis for the ideal $I^{sg}$ . Let $\succ_Q$ be an admissible order on $\mathcal{B}$ the basis of path of KQ, for example, and more precisely let $\succ_Q$ be a paths length lexicographical order. By Lemma 4.5, there exists an admissible order $\succ$ on $\mathcal{B}^{sg}$ , the basis of paths of $KQ^{sg}$ such that:

1. (1) if $\alpha\neq \beta$ then $(i, \alpha, j)\succ (i', \beta, j')$ if and only if $\alpha \succ_Q \beta$ ;

2. (2) the order on the set of arrows of $Q^{sg}$ associated with an arrow $\alpha$ in Q ending or starting in a special vertex is given as follows: If i and j are special vertices, then $(i^+, \alpha, j^+) \succ (i^-,\alpha, j^-) \succ (i^+, \alpha, j^-) \succ (i^-, \alpha, j^+)$ . If i is a special vertex and $t(\alpha)$ is not special (resp. i not special and j special), then $(i^+, \alpha, j) \succ (i^-,\alpha, j)$ (resp. $(i, \alpha, j^+) \succ (i,\alpha, j^-)$ ).

We claim that the set

\begin{equation*}\mathcal G= \left\{\sum_{j\in Q_0^{sg}(s(\beta))}\lambda_j(i,\alpha, j)(j,\beta, k)\mid \alpha\beta\in I, i\in Q_0^{sg}(s(\alpha)), k\in Q_0^{sg}(t(\beta))\right\},\end{equation*}

where $\lambda_j=-1$ if $j=l^-$ for some $l\in Q_0$ , and $\lambda_j=1$ otherwise, is Gröbner basis for $I^{sg}$ . By [Reference Green, Hille and Schroll22, Theorem 2.13], it is enough to show that every overlap relation of any two elements of $\mathcal{G}$ completely reduces to 0 by $\mathcal{G}$ . It follows from the definition of $A^{sg}$ that any element in $\mathcal{G}$ is a linear combination with at most two summands. Let $x=\lambda_1(i, \alpha, j)(j, \beta, k)+ \lambda_2(i, \alpha, j')(j', \beta,k)$ and $y=\mu_1(j,\beta,k)(k, \gamma, l) +\mu_2(j,\beta, k')(k', \gamma, l)$ be elements in $\mathcal G$ and $n, m\in \mathcal B$ such that $\mathbf o(x, y, m,n)$ is a overlap relation. Observe that in this case, by the definition of the overlap relation, n and m are arrows. It is easy to check that if x or y are monomial relations, then $\mathbf o(x, y, m,n)$ is also a monomial relation in $A^{sg}$ . Suppose that x and y are binomial relations, and suppose $t=\operatorname{tip}(x)=(i, \alpha, j)(j, \beta, k)$ and $t'=\operatorname{tip}(y)=(j,\beta,k)(k, \gamma, l)$ , then the overlap relation is written as follows

\begin{equation*} \begin{aligned} \mathbf{o}(x,y,n,m) &=\mu_1\lambda_2((i, \alpha, j')(j', \beta,k))(k, \gamma, l)-\lambda_1\mu_2 (i, \alpha, j)((j,\beta, k')(k', \gamma, l)) \\[3pt] &= -((i, \alpha, j')(j', \beta,k))(k, \gamma, l)+ (i, \alpha, j)((j,\beta, k')(k', \gamma, l)),\end{aligned}\end{equation*}

where $n=(k, \gamma, l)$ and $m=(i, \alpha, j)$ . Observe that by the definition of the relations in $I^{sg}$ and by definition of the admissible order $\succ$ , we have that $(j,\beta,k)$ is either $(s(\beta)^+, \beta, t(\beta)^+)$ or $(s(\beta)^{-}, \beta, t(\beta)^-)$ . Moreover, there is no element $z\in \mathcal{G}$ such that $\operatorname{tip}(z)$ is starting with $(j', \beta, k)$ or $(j, \beta, k')$ . Therefore, any simple reduction of $\mathbf{o}(x,y,n,m)$ replaces the second element $(i, \alpha, j)(j,\beta, k')(k', \gamma, l)$ with:

\begin{equation*}-((i, \alpha, j')(j', \beta,k))(k, \gamma, l)-(i, \alpha, j')(j',\beta, k')(k', \gamma, l).\end{equation*}

Finally, if there exist an element w in $\mathcal{G}$ such that $\operatorname{tip}(w)$ is starting with $(j', \beta, k')$ then we can reduce $(i, \alpha, j')(j',\beta, k')(k', \gamma, l)$ as follows:

\begin{equation*}-((i, \alpha, j')(j', \beta,k))(k, \gamma, l)-(i, \alpha, j')(j',\beta, k)(k, \gamma, l)=0.\end{equation*}

which implies that the quadratic basis $\mathcal G$ is a Gröbner basis, and as a consequence, A is a Koszul algebra.

Before we state the last result of this section, we recall the definition of Koszul dual, see [Reference Martínez-Villa34] for details. The Koszul dual $A^!$ of a finite dimensional Koszul algebra A is by definition the algebra $\operatorname{Ext}_A(A/ \operatorname{rad}(A), A/ \operatorname{rad}(A))^{op}$ , which is isomorphic to the quadratic dual of A. For the convenience of the reader, we briefly recall the construction of $A^!$ for algebras of the form $KQ/I$ . Let $V=KQ_2$ be the vector space generated by the paths of length two and $\{\gamma_1, \dots, \gamma_r\}$ be a basis of V. Denote by $V^{op}$ the vector space generated by paths of length two in $Q^{op}$ with dual basis $\{\gamma_1^{op}, \dots, \gamma_n^{op}\}$ .

Following [Reference Martínez-Villa34], the orthogonal ideal $I_2^{\bot}$ is generated by

\begin{equation*} B=\{v\in V^{op}\mid \langle u,v\rangle=0 \text{\ for every \ } u\in I \},\end{equation*}

where $\langle , \rangle{:}V \times V^{op}\to k $ is a bilinear form defined on bases elements as follows:

\begin{equation*}\langle \gamma_i, \gamma_j^{op}\rangle=\begin{cases}0 & \text{\ if \ } \gamma_i\neq \gamma_j,\\[3pt] 1 & \text{\ otherwise. }\end{cases}\end{equation*}

Then, the Koszul dual $A^!$ of A is the path algebra $KQ^{op}/ I_{2}^{\bot}$ .

Proof. Let W be the set of generators of the ideal $I^{sg}$ , namely

\begin{equation*}W= \left\{\sum_{j\in Q_0^{sg}(s(\beta))}\lambda_j(i,\alpha, j)(j,\beta, k)\mid \alpha\beta\in I, i\in Q_0^{sg}(s(\alpha)), k\in Q_0^{sg}(t(\beta))\right\},\end{equation*}

where $\lambda_j=-1$ if $j=l^{-}$ for some $l\in Q_0$ , and $\lambda_j=1$ otherwise.

Then, the orthogonal ideal $I_2^{\bot}$ is generated by

\begin{equation*} B=\{v\in V^{op}\mid \langle u,v\rangle=0 \text{\ for every \ } u\in W \},\end{equation*}

where $\langle , \rangle:V \times V^{op}\to k $ is a bilinear form defined on bases elements as follows:

\begin{equation*}\langle \gamma_i, \gamma_j^{op}\rangle=\begin{cases}0 & \text{\ if \ } \gamma_i\neq \gamma_j,\\ \\[-7pt]1 & \text{\ otherwise. }\end{cases}\end{equation*}

By definition of B, any path of length two which is not a summand of a minimal generator of $I^{sg}$ is an element of B. Let x be uniform element in B which is by definition of V a linear combination of at most two paths. To fix notation, let $x=\lambda_1\rho_1+\lambda_2\rho_2$ be a linear combination of two paths $\rho_1$ and $\rho_2$ with the same source and target with $\lambda_1, \lambda_2 \in K^*$ .

Suppose for contradiction that $\rho_1$ is a monomial relation. Then $\langle \rho_1, x\rangle=0$ implies that $\lambda_1=0$ which is a contradiction. Therefore without loss of generality, we have $\rho_1 - \rho_2 \in V$ and $\langle \rho_1-\rho_2, x\rangle=0$ implies that $\lambda_1=\lambda_2$ that is $x = \rho_1+\rho_2$ .

Using the isomorphism $\varphi{:}KQ^{sg} \to KQ^{sg}$ defined by

\begin{equation*}\varphi((i, \alpha, j))=\begin{cases}-(i, \alpha, j) & \text{ if } i=s(\alpha)^+ \text{ and } j=t(\alpha)^+ \text{ or } i=s(\alpha)^- \text{ and } j=t(\alpha)^-;\\ \\[-7pt](i, \alpha, j) & \text{ otherwise. }\end{cases}\end{equation*}

the result follows.

The following Theorem shows how to compute the generalised ribbon graph of the Koszul dual of a skew-gentle algebra.

Theorem-Definition 4.8 Let A be a skew-gentle algebra, and let $(S,M,\mathcal{O}, G_A)$ be the orbifold dissection of A. Denote by $G_A^*$ the graph embedded in a surface obtained from $(S, M, \mathcal{O})$ as follows.

• • In each boundary edge of the dissection $G_A$ , there is exactly one vertex of $G_A^*$ . In addition, any unmarked boundary in $(S,M,\mathcal{O})$ is replaced by a vertex of $G_A^*$ .

• • Any orbifold point in $G_A$ is also an orbifold point in $G_A^*$ .

Then for every non-special edge v of $G_A,$ there is a unique edge in $G_A^*$ crossing v exactly once. Every special edge of $G_A$ corresponds to an edge of $G_A^*$ connecting the orbifold point with the unique vertex of $G_A^*$ such that the resulting edge does not cross any edge of $G_A$ . We call these edges of $G_A^*$ special edges.

Then $G_A^*$ is the graph of the Koszul dual $A^{!}$ .

Proof. This follows directly from the construction of $G_A^*$ and Proposition 4.7.

In Figure 4, we can see the dual graphs of the generalised ribbon graphs from Example 3.5.

Figure 4. Dissections for the skew-gentle algebra and its Koszul dual associated with $\mathbb{D}_5$ on the left and $A_2$ on the right from Example 2.4

and $\mathcal{R}_1=\{\alpha_1\alpha_2,\alpha_2\alpha_3,\alpha_3\alpha_1,\varepsilon^2-\varepsilon\}$ and $\mathcal{R}_2=\{\alpha_1\alpha_2,\alpha_2\alpha_3, \varepsilon^2-\varepsilon\}$ . The algebras $A_1=KQ/\langle\mathcal{R}_1\rangle$ and $A_2=KQ/\langle \mathcal{R}_2\rangle$ are skew-gentle. Orbifold dissections and their duals are depicted in Figure 5.

Figure 5. Orbifold dissection and dual graph for $A_1$ and $A_2$

5. Graded curves in an orbifold dissection

In this section, we define graded curves in an orbifold with marked points. We begin by recalling from [Reference Chas and Gadgil13] the notion of homotopy in an orbifold based on what is called skein relations in that paper.

Let A be a skew-gentle algebra and O be the associated orbifold. We recall the notion of O-free homotopy from [Reference Chas and Gadgil13].

As a consequence of Definition 5.1 the curves in Figure 7 are O-homotopic in O.

Our main result requires a notion of grading on arcs and closed curves. This grading depends on the dual graph $G^*_A$ of a skew-gentle algebra A. Therefore before giving the definition, we will need some results on the geometry of the graph $G^*_A$ . Moreover, since every unmarked boundary in a orbifold dissection $(S,M, \mathcal{O}, G)$ of A is replaced by a vertex of $G^*$ , we will view unmarked boundary components as marked points in the interior. In particular, it will be useful to think of infinite arcs wrapping around a boundary component as infinite arcs wrapping around a marked points in the interior. We note that in our model, we only consider infinite arcs as in Definition 5.2(3), that is only those infinite arcs that wrap around an unmarked boundary component (i.e. a marked point in the interior of the surface) in a clockwise way.

Figure 6. Moves in a disk containing exactly one orbifold point.

Figure 7. O-homotopic curves in a disk containing one orbifold point.

Proof. Let $\Lambda$ be the gentle algebra obtained from A by deleting all special loops in Q and let $(S, M_\Lambda, G_\Lambda)$ be surface dissection associated with $\Lambda$ . By [Reference Opper, Plamondon and Schroll35, Lemma 2.6] and Remark 3.2, the dual graph $G^*_{\Lambda}$ of $\Lambda$ subdivides $S_{\Lambda}$ into polygons and degenerate polygons, where the edges of each such polygon are edges of $G^*_\Lambda$ and exactly one boundary segment containing exactly one marked point of $M_\Lambda$ .

It is enough to observe that after a local replacement of a special edge $\gamma$ in $G_{\Lambda}$ , the unique edge $\gamma^*$ of $G^*_\Lambda$ crossing $\gamma$ corresponds to a special edge in $G^*$ . By definition $\gamma^*$ is connected to an orbifold point $o_x$ . Let P and P $^{\prime}$ be the polygons in of $G^*_\Lambda$ in S sharing the common edge $\gamma^*$ . As a consequence of the local replacement, P and P $^{\prime}$ correspond to a single degenerate polygon in S containing $o_x$ , see Figure 8. The result follows.

Figure 8. Degenerate polygon after the local replacement of a special edge in $G_\Lambda$ and $G^*_{\Lambda}$ .

Figure 9. O-homotopic curves.

By the Lemma 5.3, $G^*_A$ induces a dissection of $O_A$ . Furthermore, the generalised polygons of $G^*_A$ are in bijection with vertices of $G_A$ .

We note that if $(S, M,\mathcal{O}, G^*)$ is an orbifold dissection and if $\gamma$ is a possibly infinite arc or a closed curve in O, then $\gamma$ is completely determined by the possibly infinite sequence of edges of $G^*$ which it crosses. If $\gamma$ is a closed curve, then this sequence is determined up to cyclic permutation.

Let $\left[x_{i}, x_{i+1}\right]$ be the oriented segment going from $x_i$ to $x_{i+1}$ . Then, both $x_i$ and $x_{i+1}$ are edges of the same (degenerate) polygon P which contains exactly one marked point $m \in M$ . A grading on $\gamma$ is a function $f:\left(x_i\right) \to \mathbb Z$ satisfying the following conditions.

\begin{equation*}f(x_{i+1})=\begin{cases}f(x_i)+1 & \textrm{if m is to the left of $\left[x_{i}, x_{i+1}\right]$ in P;} \\ \\[-7pt] f(x_i)-1 & \textrm{if m is to the right of $\left[x_{i}, x_{i+1}\right]$ in P.}\end{cases}\end{equation*}

If $\gamma$ is an (infinite) arc, we say that $(\gamma, f)$ is an (infinite) graded arc on $S_{A}$ . If $\gamma$ is a closed curve successively crossing the edges $x_1, \dots, x_n$ of $G^*$ , we say that $(\gamma, f)$ is a graded closed curve if the grading f is such that $f(x_1) = f(x_n)$ .

6. Indecomposable objects in the derived category of a skew-gentle algebra

In this section, given a skew-gentle algebra A, we show that the geometric model constructed in Section 3 is a model for the bounded derived category $D^b(A)$ . More precisely, using the equivalence $D^b(A)$ and $K^{-,b}(proj-A)$ , we establish a one to one correspondence between the homotopy strings and bands encoding the indecomposable objects in $K^{-,b}(proj-A)$ and graded arcs and curves in the orbifold associated with A.

6.1 Homotopy strings and bands

We begin by briefly recalling the definition of homotopy strings and bands from [Reference Bekkert, Marcos and Merklen5]. Throughout this section, let A be a skew-gentle algebra and let $\Lambda=KQ/I$ be the associated gentle algebra obtained from A by deleting all special loops.

The definition of homotopy strings and bands for skew-gentle algebras is based on another gentle algebra $A_+$ underlying A in the following way. Let B a minimal set of relations of $\Lambda=KQ/I$ and set $J = \langle B\setminus \{\alpha\beta\mid \alpha\beta\in I, t(\alpha)\in Sp\} \rangle$ . Then by [Reference Bekkert, Marcos and Merklen5] $A_+= KQ/ J$ is a gentle algebra.

For every arrow $\alpha \in Q_1$ , we define a formal inverse arrow $\overline{\alpha}$ where $s(\overline{\alpha}) = t(\alpha)$ and $t(\overline{\alpha}) = s(\alpha)$ . For each path $w=\alpha_1\dots\alpha_k$ we define $\overline{(\alpha_1\dots \alpha_k)}=\overline{\alpha_k}\dots \overline{\alpha_1}$ , $s(\overline{w})=e(w)$ and $t(\overline{w})=s(w)$ . A walk w is sequence $w_1 \dots w_n$ where $w_i$ is either an arrow or an inverse arrow such that $s(w_{k+1})=e(w_k)$ .

A string is a walk $w=w_1\dots w_n$ such that $w_{k+1}\neq \overline{w_k}$ for $1\leq k <n$ and such that no substring w $^{\prime}$ of w or its inverse $\overline{w'}$ is in J. For every $i\in Q_0$ , we denote by $e_i$ the string corresponding to the trivial walk at i.

A string $w=w_1 \dots w_n$ is a direct (resp. inverse) homotopy letter if $w_k$ is a direct (resp. inverse) arrow, for all $1\leq k \leq n$ . A homotopy walk $\sigma$ is a sequence $\sigma_1\dots \sigma_r$ , where $\sigma_k$ is a direct or inverse homotopy letter such that $s(\sigma_{k+1}) = t(\sigma_k)$ . We say that $\sigma=\sigma_1\dots \sigma_r$ is a direct (resp. inverse) homotopy walk if $\sigma_k$ is direct (resp. inverse), for all $1\leq k \leq r$ . A walk $\sigma$ (resp. a homotopy walk) is closed if $s(\sigma)=t(\sigma)$ . Given a (homotopy) walk $\sigma=\sigma_1 \dots \sigma_r$ we denote by $\sigma[m]=\sigma_{m+1} \dots \sigma_r\sigma_1 \dots \sigma_j$ its rotations where $m=1, \dots, r-1.$

Definition 6.1 Let A be a skew-gentle algebra with $\Lambda=KQ/I$ be the associated gentle algebra obtained from A be deleting all special loops, and $A_+=KQ/J$ the associated gentle algebra as defined above.

• (1) A homotopy string is a homotopy walk $\sigma=\sigma_1\dots \sigma_r$ such that

  • • if both $\sigma_i, \sigma_{i+1}$ are direct (resp. inverse) homotopy letters such that $\sigma_i$ (resp. $\overline{\sigma_{i+1}}$ ) is not ending at a special vertex, then $\sigma_i \sigma_{i+1}\in J $ (resp. $\overline{\sigma_i \sigma_{i+1}} \in J$ ),

  • • if $\sigma_i, \overline{\sigma_{i+1}}$ (resp. $\overline{\sigma_{i}}, \sigma_{i+1}$ ) are direct homotopy letters and $\sigma_i$ (resp. $\overline{\sigma_{i}}$ ) is not ending at a special vertex, then $\sigma_i \sigma_{i+1}$ is a string. A non-trivial homotopy string $\sigma$ is ${symmetric}$ if $\sigma=\overline{\sigma}$ and ${asymmetric}$ otherwise.

• (2) A ${homotopy\ band}$ is a closed homotopy string $\sigma=\sigma_1 \dots \sigma_{n}$ with an equal number of direct and inverse homotopy letters such that $\sigma$ is not a proper power of some homotopy string $\sigma '$ and such that every power of $\sigma$ is a homotopy string.

A non-trivial homotopy band $\sigma$ is symmetric if $\sigma= \overline{\sigma}[m]$ for some m and asymmetric otherwise.

Remark 6.2 By definition, any symmetric band $\sigma$ is a word such that

\begin{equation*}\sigma=\sigma=\sigma_1 \dots \sigma_k=a_1 \dots a_r\overline{a_r}\dots \overline{a_1}b_1 \dots b_s \overline{b_s} \dots \overline{b_1},\end{equation*}

where $t(a_r)$ and $t(b_s)$ are special vertex, having the following picture:

Definition 6.3 A right (resp. left) infinite homotopy string is a sequence $\sigma = \sigma_0 \sigma_1 \sigma_2 \ldots $ (resp. $\sigma= \dots \sigma_{-2}\sigma_{-1} \sigma_0 $ ) such that for some k, all $\sigma_i$ (resp. $\sigma_{-i}$ ) are direct (resp. inverse) arrows, for $i >k$ and such that every finite subword of $\sigma$ is a homotopy string.

An infinite homotopy string is a sequence $\sigma= \dots \sigma_{-2}\sigma_{-1}\sigma_0\sigma_1 \sigma_2 \dots$ such that $\dots \sigma_{-2}\sigma_{-1}$ is a left infinite homotopy string and $\sigma_0\sigma_1\sigma_{2}$ is a right infinite homotopy string.

6.2 String and Band Complexes

In order to define the complexes induced by homotopy strings and bands as introduced in [Reference Bekkert, Marcos and Merklen5], we need to introduce a grading on homotopy strings and bands. Our definition closely follows [Reference Opper, Plamondon and Schroll35].

Definition 6.4 Let $\sigma=\sigma_1 \dots \sigma_r$ be a finite homotopy string.

A grading on $\sigma$ is a sequence of integers $\mu=(\mu_0, \dots, \mu_r)$ such that

\begin{equation*}\mu_{i+1}=\begin{cases} \mu_i+1 & \textrm{if $\sigma_{i+1}$ is a direct homotopy letter;}\\ \\[-7pt] \mu_{i}-1 & \textrm{otherwise,}\end{cases}\end{equation*}

for each $i\in \{1,\dots, r-1\}$ . The pair $(\sigma, \mu)$ is a graded homotopy string.

Moreover, if $\sigma$ is a homotopy band, the pair $(\sigma, \mu)$ is a graded homotopy band if $(\sigma, \mu)$ is a graded homotopy string and i is considered modulo r.

In a similar way, we define a grading on (left, right) infinite homotopy strings.

Let A be a skew-gentle algebra with admissible presentation $A^{sg} = KQ^{sg} / I^{sg}$ . For a vertex $i \in Q_0^{sg}$ , we write $P_i$ for the projective indecomposable $A^{sg}$ -module at vertex i.

Following [Reference Bekkert, Marcos and Merklen5], to each graded homotopy string or band $(\sigma, \mu)$ we associate a complex of projective A-modules $P^\bullet_{(\sigma, \mu)}$ which is not necessarily indecomposable, but is a sum of at most two indecomposable complexes. For this, we freely view A-modules as modules over $A^{sg}$ and we define the following projective A-module for every $i \in Q_0$ .

\begin{equation*}P(i)=\begin{cases}P_{i^+}\oplus P_{i^-} & \textrm{if {i} is a special vertex;}\\ \\[-7pt] P_i & \textrm{otherwise.}\end{cases}\end{equation*}

The definition of a complex associated with a symmetric homotopy band $\sigma$ relies on a set of matrices $\mathcal{M}$ with coefficients in K where $M \in \mathcal{M}$ is such that $M=\begin{pmatrix} A &\quad C \\ B &\quad D\end{pmatrix}\in \textrm{Mat}(l+l', m+m')$ for some strictly positive integers l, l $^{\prime}$ ,m,m $^{\prime}$ . We refer the reader to [Reference Bekkert, Marcos and Merklen5, Section 3.1] for the precise definition of $\mathcal{M}$ .

Definition 6.5

1. 1. Let $(\sigma,\mu)$ be a graded homotopy string with $\sigma=\sigma_1 \dots \sigma_r$ . Then let

   \begin{equation*}P^\bullet_{(\sigma, \mu)}= \dots \longrightarrow P^{-1} \longrightarrow P^0 \longrightarrow P^1 \longrightarrow \dots\end{equation*}
   be the complex such that $P^j=\bigoplus_{\substack{0\leq i \leq r \\ \mu_i=j}} P(i)$ , for all $j \in \mathbb Z$ and where the differentials are induced by the homotopy letters.

   If $\sigma$ is an asymmetric homotopy string, we say that a complex P is an asymmetric string complex if it is isomorphic to $P_{(\sigma, \mu)}$ in $K^{b,-}(proj - A)$ .

   If $\sigma$ is a symmetric homotopy string, then $P_{(\sigma, \mu)}$ decomposes into the direct sum of two indecomposables complexes of projective A-modules. We will refer to the indecomposable summands as $P_{(\sigma, \mu,0)}$ and $P_{(\sigma, \mu,1)}$ . We call a complex P isomorphic to either $P_{(\sigma, \mu, 0)}^{\bullet}$ or $P_{(\sigma, \mu, 1)}^{\bullet}$ , a dimidiate string complex.

2. 2. Let $(\sigma, \mu)$ be a graded homotopy band where $\sigma=\sigma_1 \dots \sigma_r$ is an asymmetric homotopy band. Let $\operatorname{ind}K\left[x\right]$ be the set of non-trivial powers of irreducible polynomials over K with leading coefficient equal to 1 and different from x and $x-1$ . Then, for each $p(x)\in \operatorname{ind}k\left[x\right]$ , let

   \begin{equation*}P^\bullet_{(\sigma, \mu), p(x)}= \dots \longrightarrow P^{-1} \longrightarrow P^0 \longrightarrow P^1 \longrightarrow \dots\end{equation*}
   be the complex such that $P^j=\bigoplus_{\substack{0\leq i \leq r \\ \mu_i=j}} P(i) \otimes_K K^{\operatorname{deg} p(x)}$ in degree j. We call a complex P an asymmetric band complex if it is isomorphic in $K^{b,-}(proj - A)$ to $P_{(\sigma, \mu), p(x)}$ where $\sigma$ is an asymmetric band.

3. 3. Let $(\sigma, \mu)$ be a graded homotopy band with $\sigma=a_1 \dots a_r\overline{a_r}\dots \overline{a_1}b_1 \dots b_s \overline{b_s} \dots \overline{b_1}$ symmetric and such that $e(a_r)$ and $e(b_s)$ are special vertices. Let $M=\begin{pmatrix} A & \quad C \\ B &\quad D\end{pmatrix}\in \mathcal M$ . Then the complex

   \begin{equation*}P^\bullet_{(\sigma, \mu)}= \dots \longrightarrow P^{-1} \longrightarrow P^0 \longrightarrow P^1 \longrightarrow \dots\end{equation*}
   is given by
   \begin{equation*}P^j=\bigoplus_{\substack{0\leq i \leq r \\ \mu_i=j}} Q(i)\oplus \bigoplus_{\substack{2r+1\leq i \leq 2r+s \\ \mu_i=j}} Q(i)\end{equation*}

   for all $j\in \mathbb Z$ , where Q(i) is a projective A-module which depends on the size of the matrices A,B,C,D in M as follows:

   \begin{equation*}Q(i)=\begin{cases} P(i)\otimes_K K^{l} \oplus P(i)\otimes_K K^{l'} & \textrm{if $i=\mu_0, \dots, \mu_{r-1}$}\\ \\[-7pt] P(i)\otimes_K K^{m} \oplus P(i)\otimes_K K^{m'} & \textrm{if $i=\mu_{2r+1}, \dots, \mu_{2r+s-1}$}\\ \\[-7pt] P_{i^+}\otimes_K K^{l} \oplus P_{i^-}\otimes_K K^{l'} & \textrm{if $i=\mu_{r}$}\\ \\[-7pt] P_{i^+}\otimes_K K^{m} \oplus P_{i^-}\otimes_K K^{m'} & \textrm{if $i=2r+s$}\end{cases}.\end{equation*}

   We say that a complex P is a dimidiate band complex if it is isomorphic in $K^{b,-}(proj - A)$ to $P_{(\sigma, \mu), M}$ where $\sigma$ is a symmetric band.

We will not give the definitions of the differentials of the above complexes, since we do not need those in the geometric description of the indecomposable objects of the bounded derived category of a skew-gentle algebra.

Remark 6.6 In [Reference Schroll and Valdivieso39], we give a correspondence of intersections of graded curves and homomorphisms in the bounded derived category of a skew-gentle algebra.

For a grading $\mu= (\mu_1, \ldots, \mu_r)$ on a homotopy string or band, define a grading shift [m] as $\mu[m] = (\mu_1 +m, \ldots, \mu_r +m)$ for $m \in \mathbb{Z}$ . Observe that the complex induced by $(\sigma, \mu[m])$ is $P^\bullet_{(\sigma, \mu[m])}=P^\bullet_{(\sigma, \mu)}[m]$ .

6.3 Main result on indecomposable objects of the derived category of a skew-gentle algebra

Before stating one of the main theorems of this paper, we recall that we identify unmarked boundary components and punctures in a surface S. We also recall that given an orbifold dissection $O=(S,M,\mathcal O, G)$ we define graded arcs and graded closed curves up to O-homotopy.

Theorem 6.7 Let A be a skew-gentle algebra with orbifold dissection $O=(S, M, \mathcal{O}, G^*)$ . Then the homotopy strings and bands parametrising the indecomposable objects in $D^b(A)$ are in bijection with graded arcs and curves in O. More precisely,

1. (1) the set of homotopy strings are in bijection with graded arcs $(\gamma, f)$ , where $\gamma$ is a finite arc in O or an infinite arc whose infinite rays wrap around unmarked boundary components in the anti-clockwise orientation;

2. (2) the set of homotopy bands are in bijection with graded primitive closed curves $(\gamma, f)$ in O.

The proof is very similar to that of the corresponding result for gentle algebras in [Reference Opper, Plamondon and Schroll35] with suitable adjustments to be made for special loops in the algebra on the one side and polygons containing orbifold points in the surface on the other. For the convenience of the reader, we give the whole proof in detail.

The following definition gives the construction of a homotopy word induced by the intersection of an arc or curve in the surface with the edges of the dual graph.

Definition 6.8 Let A be a skew-gentle algebra with quiver $Q_A=Q$ and $\gamma$ be an arc or a closed curve in its orbifold dissection $(S,M, \mathcal{O}, G^*)$ . We set $\left(x_i\right)$ to be the ordered multiset of edges of $G^*$ crossed by $\gamma$ following its trajectory. Furthermore, if $\gamma$ crosses $G^*$ at least twice. Let $\left[x_{i}, x_{i+1}\right]$ be the oriented segment going from $x_i$ to $x_{i+1}$ , recall that both edges are in a (degenerate) polygon P which contains exactly one marked point $m \in M$ .

We define the homotopy letter $\sigma(x_i)$ associated with the oriented segment $\left[x_i, x_{i+1}\right]$ as follows:

1. 1. If the marked point m is on the left of $\left[x_i, x_{i+1}\right]$ , then let $w_1, \dots, w_r$ be the edges between $x_i=w_1$ and $x_{i+1}=w_r$ in the clockwise order. By Theorem 4.8, these correspond to vertices of the quiver Q of A which are joined by arrows $\alpha_1, \dots, \alpha_{r-1}$ . Then define $\sigma(x_i):=(\alpha_1 \dots \alpha_{r-1})$ .

2. 2. If the marked point m is on the right of $\left[x_i, x_{i+1}\right]$ , then let $w_1, \dots, w_r$ be the edges between $x_{i+1}=w_1$ and $x_i=w_r$ in the clockwise order. By Theorem 4.8, these correspond to vertices of the quiver Q of A which are joined by arrows $\alpha_1, \dots, \alpha_{r-1}$ . Then define $\sigma(x_i):=\overline{(\alpha_1 \dots \alpha_{r-1})}$ .

If a graded arc $(\gamma, f)$ crosses $G^*$ exactly once, namely at the edge x, then we set $\sigma(\gamma)$ to be the trivial string $e_x$ , and $\mu(f)=(f(x))$ the grading on $\sigma(\gamma)$ .

Observe that by Theorem 4.8, $\sigma(x_i)$ is indeed a homotopy letter.

Lemma 6.9 Let $(S, M, \mathcal{O}, G^*)$ be the orbifold dissection of a skew-gentle algebra A given by the dual $G^*$ of the generalised ribbon graph of A and let $(\gamma, f)$ be an arc or a graded primitive closed curve on $(S,M,\mathcal{O})$ with $\left(x_i \right)$ the ordered multiset of edges of the dual graph $G^*$ crossed by $\gamma$ following its trajectory.

1. (1) If $(\gamma, f)$ is a finite graded arc which crosses the edges of $G^*$ exactly r times and at least twice, then $\sigma(\gamma)= \prod_{i=1}^{r-1}\sigma(x_i)$ is a homotopy string and $\mu(f) = (f(x_1),\dots,f(x_r))$ is a grading on $\sigma(\gamma)$ .

2. (2) If $(\gamma, f)$ is an infinite graded arc, then $\sigma(\gamma)=\prod\sigma(x_i)$ is a infinite homotopy string and $\mu(f) = (f(x_i))$ is a grading on $\sigma(\gamma)$ .

3. (3) If $(\gamma, f)$ is a primitive graded closed curve and if $x_1, x_2, \dots, x_r$ are the distinct edges of $G^*$ crossed by $\gamma$ (in that order), then $\sigma(\gamma)=\prod_{i=0}^r\sigma(x_i)$ is a homotopy band and $\mu(f) = (f(x_1),\dots,f(x_{r-1}))$ is a grading on $\sigma(\gamma)$ .

Proof. By construction, $\sigma(\gamma_i)$ is a homotopy letter, for all i. As before, let $\Lambda= KQ/I$ be the associated gentle algebra obtained from A be deleting all special loops, B be a minimal set of relations of $\Lambda$ and $A+ = KQ/J$ , where $J = \langle B \setminus \{\alpha \beta \mid t(\alpha) \in Sp \}\rangle$ . By Theorem 4.8, either the composition or of the last arrow of $\sigma(\gamma_i)$ and the first arrow of $\sigma(\gamma_{i+1})$ (or the composition of their inverses) are in J or the end of $\sigma(\gamma_i)$ is a special vertex. Then (2) directly follows from the definition of homotopy strings in Section 6.1.

Now let $(\gamma, f)$ be an infinite graded arc. By Lemma 5.3, $\gamma$ wraps around a puncture p with at least one incident edge and by the above, every finite homotopy sub-walk of $\sigma(\gamma)$ is a homotopy string. Denote by $x_k, x_{k+1}, \dots, x_r$ the set of edges of $G^*$ that $\gamma$ crosses when $\gamma$ does a complete turn around p. By Theorem 4.8, their associated homotopy letters $\sigma(x_k), \dots, \sigma(x_r)$ are homotopy letters of length one, and $\sigma(x_k) \dots \sigma(x_r)$ or its inverse is an oriented cycle. Furthermore, we have that $\sigma(x_r) \sigma(x_{r+1}) \in J$ with $\sigma(x_{r+1}) = \sigma(x_{k})$ and more generally, $\sigma(x_{r+i}) = \sigma(x_{k+i-1})$ for all $i \geq 1$ . Thus, $\sigma(\gamma)$ is eventually periodic and an infinite homotopy string.

In cases (1) and (2), directly follows from the definition of the grading f on $\gamma$ that $\mu(f)$ is a grading on the associated homotopy string.

To prove (3), assume that $(\gamma, f)$ is a graded primitive closed curve. Let $\sigma(\gamma)=\prod_{i=0}^r\sigma(\gamma_i)$ . It is enough to observe that $s(\sigma(\gamma_1))=t(\sigma_{\gamma_r})$ to ensure that any rotation of $\sigma(\gamma)$ is a homotopy string. By definition, the existence of the grading f implies that there is the same number of inverse and direct homotopy letters in $\sigma(\gamma)$ , which implies that $\sigma(\gamma)$ is a homotopy band.

Lemma 6.10 Let $O=(S,M, \mathcal{O}, G^*)$ be the orbifold dissection of a skew-gentle algebra A given by the dual graph generalised ribbon graph $G^*$ of A.

1. (1) Let $(\sigma, \mu)$ be a finite (resp. infinite) graded homotopy string. Then there exists a unique finite (resp. infinite) graded arc $(\gamma, f)$ on O such that $(\sigma(\gamma), \mu(f))=(\sigma, \mu)$ .

2. (2) Let $(\sigma,\mu)$ be a graded homotopy band. Then there exists a unique graded closed curve $(\gamma, f)$ on O (up to O-homotopy) such that $(\sigma(\gamma), \mu(f))=(\sigma, \mu)$ .

Proof. Let $\Lambda= KQ/I$ be the associated gentle algebra obtained from A be deleting all special loops, B be a minimal set of relations of $\Lambda$ and $A+ = KQ/J$ , where $J = \langle B \setminus \{\alpha \beta \mid t(\alpha) \in Sp \}\rangle$ .

Let $(\sigma, \mu)$ be a finite graded homotopy string. By definition $\sigma$ is a sequence $\sigma_1 \dots \sigma_n$ of homotopy letters $\sigma_i$ where $s(\sigma_{i+1})=t(\sigma_i)$ and such that either $\sigma_i\sigma_{i+1}\in J$ or $\overline{\sigma_i\sigma_{i+1}}\in J$ or $s(\sigma_{i+1})=t(\sigma_i)$ is a special vertex.

It is enough to prove that for each homotopy letter $\sigma_i$ there is a (unique) oriented segment $\gamma_i$ such that the topological concatenation of those segments induces a graded arc $\gamma$ on O and that two distinct homotopy strings give rise to two graded arcs which are not O-homotopic.

Denote by $x_i$ and $x_{i+1}$ the edges of $G^*$ corresponding to $s(\sigma_i)$ and $t(\sigma_i)$ respectively. By construction of $G^*$ , there is exactly one (degenerate) polygon $P_i$ in O such that $x_i$ and $x_{i+1}$ are edges of $P_i$ .

If $P_i$ is not degenerate or $x_i$ and $x_{i+1}$ are not special edges, then up to homotopy there is a unique oriented segment $\gamma_i$ in the interior of $P_i$ starting at the mid-point of $x_i$ and ending at the mid-point of $x_{i+1}$ such that $\gamma_i$ has no self-crossing and does not cross any other edge of $G^*$ .

If $x_i$ is not a special edge and $x_{i+1}$ is a special edge (or $x_i$ is a special edge and $x_{i+1}$ is not a special edge), then up to homotopy there is a unique oriented segment $\gamma_i$ in the interior of $P_i$ starting at the mid-point of (resp. the orbifold point of the edge) $x_i$ and ending at the orbifold point of the edge (resp. the mid-point of) $x_{i+1}$ such that $\gamma_i$ has no self-crossing and does not cross any other edge of $G^*$ , see Figure 10.

Figure 10. The arc $\gamma$ from $x_i$ to $x_{i+1}$ .

It is clear that the ending point of the oriented segment $\gamma_i$ is the starting point of $\gamma_{i+1}$ and that the concatenation $\gamma_{1}*\dots *\gamma_{n}$ is an oriented segment from $x_1=s(\sigma_1\dots \sigma_n)$ to $x_{n+1}=t(\sigma_1 \dots \sigma_n)$ .

Now, let $P_0$ (resp. $P_{n+1}$ ) be the (degenerate) polygon which shares $x_1$ (resp. $x_{n+1}$ ) with $P_1$ (resp. $P_n$ ) and let $m_0$ (resp. $m_{n+1}$ ) be the marked point in the boundary segment of $P_0$ (resp. $P_{x_{n+1}}$ ). Observe that if $x_1$ (resp. $x_n$ ) is a special edge, then $P_0$ and $P_1$ (resp. $P_n$ and $P_{n+1}$ ) coincide.

Suppose that $x_1$ (resp. $x_n$ ) is not a special edge, then there exists a unique oriented segment from the marked point $m_0$ (resp. the point $x_{n+1}$ ) to middle point of $x_1$ (resp. marked point $m_{n+1}$ ), which lies in $P_0$ (resp. $P_{n+1}$ ), without self intersection and it is not crossing other edge of $G^*$ .

If $x_1$ (resp. $x_n$ ) is a special edge, then there is a unique oriented segment $\gamma_0$ (resp. $\gamma_{n+1}$ ) from $m_0$ (resp. the middle point of $x_{n+1}$ ) to the orbifold point of $x_1$ (resp. $m_{n+1}$ ) such that $\gamma_0*\gamma_1$ (resp. $\gamma_{n}*\gamma_{n+1}$ )lies in $P_1=P_0$ (resp. $P_{n}=P_{n+1}$ ) and is not crossing any other edge of $G^*$ , see Figure 11.

Figure 11. Oriented segment $\gamma_0*\gamma_1.$

Then, $\gamma=\gamma_0*\gamma_1* \dots * \gamma_{n}*\gamma_{n+1}$ is a finite arc and by construction $\sigma(\gamma)=\sigma$ . Moreover, the grading $\mu$ on $\sigma$ induces a natural grading f on $\gamma$ .

To finish the proof, suppose that $\gamma'$ is a finite arcs such that $\sigma(\gamma')=\sigma=\sigma(\gamma)$ . Without loss of generality, suppose that $\gamma'$ is in minimal position, then $\sigma(\gamma')$ is already a reduced homotopy string. We claim that $\gamma$ and $\gamma'$ are O-homotopic. Any curve is completely determined by the ordered multiset of edges of $G^*$ that it crosses and $\sigma(\gamma')=\sigma=\sigma(\gamma)$ , thus $\gamma$ and $\gamma'$ have the same multiset of edges of crossings with $G^*$ and $\gamma$ and $\gamma'$ are O-homotopic.

The proof for infinite homotopy strings and bands is similar to the above.

Proof of Theorem 6.7. By [Reference Bekkert, Marcos and Merklen5, Theorem 3], the indecomposable objects of the derived category $\mathcal D^b(A)$ are completely described by the graded homotopy strings and bands. The result then follows from Lemma 6.9 and Lemma 6.10.

7. Applications

7.1 Singularity category of a skew-gentle algebra

The stable derived category or singularity category $\mathcal D_{sg}(A)$ of an algebra A is defined as the Verdier quotient of the bounded derived category with respect to the perfect derived category.

In [Reference Kalck27, Theorem 2.5], the singularity category of a gentle algebra was explicitly described as a finite product of triangulated orbit categories which turn out to be n-cluster categories of type $\mathbb A_1$ , as follows. Let $\Lambda=KQ/I$ be a gentle algebra. A cycle $\alpha_1 \dots \alpha_n$ of positive length on Q is saturated if each of the length-2 paths $\alpha_{1}\alpha_{2},\ldots,\alpha_{n-1}\alpha_n,\alpha_n\alpha_1$ belongs to I. Let $\mathcal C(\Lambda)$ be the set of cyclical permutation equivalence classes of saturated cycles without repeated arrows. For $n>2$ denote by $\mathcal D^b(K-mod)/\left[n\right]$ the triangulated orbit category as defined in [Reference Keller28]. Then [Reference Kalck27, Theorem 2.5] shows that $\mathcal D_{sg}(\Lambda)$ and $\prod_{c\in \mathcal C(\Lambda)} \mathcal D^b(K-mod)/\left[n_c\right]$ are equivalent as triangulated categories, where $n_c$ denotes the length of any cycle in the equivalence class $c\in \mathcal C(\Lambda)$ .

By construction of the dual ribbon graph $G_\Lambda$ of a gentle algebra $\Lambda$ , it is easy to see that there is a bijection between $\mathcal C(\Lambda)$ and the polygons in $G_\Lambda$ with no boundary edges. We will call such polygons interior polygons and we denote by $\mathcal{P}^\circ_\Lambda$ the set of interior polygons. Furthermore, for $P \in \mathcal{P}^\circ_\Lambda$ denote by $n_P$ the number of edges of P. A concrete description of $\mathcal D_{sg}(\Lambda)$ is then given by

\begin{equation*}\mathcal D_{sg}(\Lambda) \simeq \prod_{P \in \mathcal{P}^\circ_\Lambda } \mathcal D^b(K-mod)/\left[n_P\right].\end{equation*}

The geometric description of the singularity category of a skew-gentle algebra follows from the fact that, by [Reference Chen and Lu14, Theorem 3.5], a skew-gentle algebra A and its underlying gentle algebra $\Lambda$ have equivalent singularity categories and that the generalised ribbon graph of a skew-gentle algebra and that of a gentle algebra have the same number of interior polygons and corresponding interior polygons have the same number of edges. More precisely, we have the following.

Theorem 7.1 Let A be a skew-gentle algebra and $(S, M, \mathcal{O}, G)$ be the orbifold dissection induced by the generalised ribbon graph G of A. Then

\begin{equation*}\mathcal D_{sg}(A) \simeq \prod_{P \in \mathcal{P}^\circ_A } \mathcal D^b(K-mod)/\left[n_P\right],\end{equation*}

where $\mathcal{P}^\circ_A$ is the set of interior polygons of G and $n_P$ is the number of edges of P, for $P \in \mathcal{\mathcal{P}^\circ_A }$ .

7.3 q-Cartan matrices

A classical invariant for graded algebras is the so-called q-Cartan matrix which generalises the classical Cartan matrix of a graded finite dimensional algebra, see for example [Reference Fuller15]. For this, recall that $A=KQ/I$ has a grading induced by paths length if I is generated by homogeneous relations. The q-Cartan matrix $C_A(q)=(c_{ij}(q))$ of A, for an indeterminate q, is the matrix with entries

\begin{equation*}c_{ij}(q)= \sum_n \operatorname{dim}_K (e_iAe_j)_nq^n \in \mathbb Z[q]\end{equation*}

for vertices i, j in Q and where $(e_iAe_j)_n$ is the component of degree n of $e_iAe_j$ . The q-Cartan matrix is invariant under graded derived equivalence and specialises to the classical Cartan matrix by setting $q=1$ .

In [Reference Bessenrodt and Holm7, Theorem 4.2], the determinant of the q-Cartan matrix $C_A(q)$ of a (skew-)gentle algebra A is computed in terms of saturated cycles. In the following we show that this description can be read-off the orbifold dissection of A.

Theorem 7.3 Let A be a skew-gentle algebra and $(S, M, \mathcal{O}, G)$ be the orbifold dissection induced by the generalised ribbon graph $G_A$ of A. Denote by $c_k$ the number of (degenerate) interior polygons of G with k edges and let $\Lambda$ be the gentle algebra obtained from A by deleting the special loops. Then the q-Cartan matrix $C_A(q)$ has determinant

\begin{equation*} \operatorname{det}C_A(q) = \operatorname{det}C_\Lambda(q)= \prod_{k\geq 1} (1-(-q)^k)^{c_k}.\end{equation*}

8. Example

In this section, we will illustrate the geometric model and some of the results in the previous sections on an example. Let $A=KQ/I$ be the skew-gentle algebra with admissible presentation $A^{sg}=KQ^{sg}/I^{sg}$ , where Q and $Q^{sg}$ are as follows

and where $I=\langle \alpha_1\alpha_2, \alpha_4\alpha_3, \varepsilon^2-\varepsilon\rangle$ and $I^{sg}=\langle (3,\alpha_1, 1)(1,\alpha_2,2), (3,\alpha_4, 2)(2,\alpha_3 4^+), (3,\alpha_4, 2)(2,\alpha_3 4^-)\rangle$ .

Following Remark 3.4, the set of vertices $M_A$ of the ribbon graph is the set $\{\alpha_2\alpha_3\epsilon, \alpha_1,\alpha_4, e_4\}$ and its generalised ribbon graph $G_A$ can be seen in Figure 12. Note that the only difference between a ribbon graph and a generalised ribbon graph is that in a generalised ribbon graph some of the (leaf) vertices, namely those giving rise to orbifold points, are designated to be special.

Figure 12. Generalised ribbon graphs of algebra A.

Since A is a skew-gentle algebra, the generalised ribbon graph $G_A$ is embedded in an orbifold with one orbifold point of order 2 which corresponds to the special vertex $e_4$ . The corresponding orbifold dissection of A and its dual graph are depicted in Figure 13.

Figure 13. Orbifold dissection induced by $G_A$ and its dual graph.

Let $(\gamma, f)$ be the graded curve where $\gamma$ is as in Figure 14 and where the grading is given by $f=(1,2,1,0)$ . The homotopy string $\sigma(\gamma)$ associated with $\gamma$ is $(\alpha_2\alpha_3)(\overline{\alpha_3})(\overline{\alpha_4})$ and the grading $\mu(f)$ induced by f. This induces the following asymmetric string complex $P^\bullet_{(\sigma(\gamma), \mu(f))}$ in $K^{b,-}(proj-A^{sg})$ .

Figure 14. The curves $\gamma$ and $\delta$ in the orbifold dissection $(S,M, \mathcal{O}, G_A).$

Let $(\delta, g)$ be the graded closed curve where $\delta$ is depicted in Figure 14 and $g=(0,1,0,-1,0)$ . The asymmetric homotopy band $\sigma(\delta)$ associated with $\delta$ is $(\alpha_2\alpha_3)(\overline{\alpha_3})(\overline{\alpha_4})(\alpha_1)$ and as before $\mu(g)$ is induced by g. Let $q(x)\in \operatorname{ind} K[x]$ be a non-trivial power of an irreducible polynomial over K with leading coefficient equal to 1 and different from x and $x-1$ . The asymmetric band complex $P^\bullet_{(\sigma(\delta), \mu(g), p(x))}$ in $K^{b,-}(proj A^{sg})$ induced by $(\sigma(\delta), \mu(g))$ is

where $P(i)=P_i$ for $i\neq 4$ and $P(4)=P_{4^+}\oplus P_{4^-}$ .

Observe that the set of interior polygons of G is empty. Thus by Theorem 7.1, the singularity category $\mathcal D_{sg}(A)$ is equivalent to the category with one element, and by Theorem 7.3, the determinant $\operatorname{det}C_A(q)$ of the q-Cartan matrix $C_A(q)$ is zero.

By 7.2, to compute the Gorenstein dimension of A, we need to count the maximal number d of internal edges of boundary polygons of the dissection, in this case, the dissection has one digon and two 4-gon, then $d=3$ , and as a consequence, the Gorenstein dimension of A is $3-1=2$ .