**footnotetext: E-mail<EMAIL_ADDRESS>(Y.-Z. Liu<EMAIL_ADDRESS>(H. Gao),
<EMAIL_ADDRESS>(Z. Huang).

# Homological Dimensions of Gentle Algebras via Geometric Models ***This
research was partially supported by NSFC (Grant Nos. 11971225, 12171207).

Yu-Zhe Liu1, Hanpeng Gao2, Zhaoyong Huang ,1
1 Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu
Province, China;
2 School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui
Province, China.  Corresponding author.

###### Abstract

Let $A=kQ/I$ be a finite dimensional basic algebra over an algebraically
closed field $k$ which is a gentle algebra with the marked ribbon surface
$(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$. It is known that
$\mathcal{S}_{A}$ can be divided into some elementary polygons
$\\{\Delta_{i}\mid 1\leq i\leq d\\}$ by $\Gamma_{A}$ which has exactly one
side in the boundary of $\mathcal{S}_{A}$. Let $\mathfrak{C}(\Delta_{i})$ be
the number of sides of $\Delta_{i}$ belonging to $\Gamma_{A}$ if the unmarked
boundary component of $\mathcal{S}_{A}$ is not a side of $\Delta_{i}$;
otherwise, $\mathfrak{C}(\Delta_{i})=\infty$, and let
$\mathsf{f}\text{-}\Delta$ be the set of all non-$\infty$-elementary polygons
and $\mathcal{F}_{A}$ (respectively, ${\mathsf{f}\text{-}\mathcal{F}}_{A}$)
the set of all forbidden threads (respectively, of finite length). Then we
have

  1. (1)

The global dimension of $A=\max\limits_{1\leq i\leq
d}{\mathfrak{C}(\Delta_{i})}-1=\max\limits_{\mathit{\Pi}\in\mathcal{F}_{A}}l(\mathit{\Pi})$,
where $l(\mathit{\Pi})$ is the length of $\mathit{\Pi}$.

  2. (2)

The left and right self-injective dimensions of $A=$

$\begin{cases}0,\ \mbox{\text{if {\it Q} is either a point or an oriented
cycle with full relations};}\\\
\max\limits_{\Delta_{i}\in{\mathsf{f}\text{-}\Delta}}\big{\\{}1,{\mathfrak{C}(\Delta_{i})}-1\big{\\}}=\max\limits_{\mathit{\Pi}\in{\mathsf{f}\text{-}\mathcal{F}}_{A}}l(\mathit{\Pi}),\
\mbox{\text{otherwise}.}\end{cases}$

As a consequence, we get that the finiteness of the global dimension of gentle
algebras is invariant under AG-equivalence. In addition, we get that the
number of indecomposable non-projective Gorenstein projective modules over
gentle algebras is also invariant under AG-equivalence.

2020 Mathematics Subject Classification: 16E10; 16G10.

Key words: global dimension; self-injective dimension; gentle algebras; marked
ribbon surfaces; geometric models; AG-equivalence.

##  1 Introduction

Gentle algebras were introduced by Assem and Skowroński [5] as appropriate
context for the study of algebras derived equivalent to hereditary algebras of
type $\widetilde{\mathbb{A}}_{n}$. Note that every gentle algebra is a special
biserial algebra and all indecomposable modules over a special biserial
algebra were described by Butler and Ringel [8, Section 3] and Wald and
Waschbäsch [28, Proposition 2.3]. Thus we can portray all indecomposable
modules over a gentle algebra. To be precise, each indecomposable module over
a gentle algebra is either a string module or a band module. Moreover, every
gentle algebra is also a string algebra and all irreducible morphisms between
string and band modules over a string algebra were studied by Butler and
Ringel [8], Crawley-Boevey [11] and Krause [21]. In addition, by proving that
the left and right self-injective dimensions of a gentle algebra are equal to
the maximal length of certain paths starting with a gentle arrow and bounded
by one if there is no such arrow, Geiß and Reiten [14] obtained that every
gentle algebra is Gorenstein, that is, its left self-injective dimension
$\mathrm{inj.dim}_{A}A$ and right self-injective dimension
$\mathrm{inj.dim}A_{A}$ are finite.

The geometric models of gentle algebras were introduced in [7, 15, 25] and
have been extensively studied in [4, 17, 26] based on the works in [3, 6, 12,
27]. They originated in triangulated surfaces which is used in the study of
cluster algebras and cluster categories, such as [9, 22, 24] and so on. Opper,
Plamondon and Schroll introduced ribbon graphs and marked ribbon surfaces for
gentle algebras [25]. One can calculate all objects, morphisms and AG-
invariants (a derived invariant defined in [2]) in the derived category of a
gentle algebra by marked ribbon surfaces (see [25, Theorems 2.5, 3.3 and
6.1]). Furthermore, Baur and Simoes [7] introduced permissible curves (see
Definition 2.5 below) to describe indecomposable modules, and construct the
geometric model of the category of finitely generated modules over a gentle
algebra; and they also provided the depictions of the Auslander-Reiten
translate $\tau$ and Auslander-Reiten sequences through the rotations of
permissible curves in marked ribbon surfaces. Recently, He, Zhou and Zhu [17]
studied the category of finitely generated modules over skew-gentle algebras
(a generalization of gentle algebras) by punctured marked surfaces (a
generalization of marked ribbon surfaces). They provided a dimension formula
for calculating morphisms between the indecomposable modules $M$ and $\tau M$
by equivalence classes of tagged permissible curves, tagged intersections and
intersection numbers. In addition, by using the Koszul dual, Opper, Plamondon
and Schroll [25, Section 1, 1.7] proved that the global dimension
$\mathrm{gl.dim}A$ of a gentle algebra $A$ is infinite if and only if its
quiver has at least one oriented cycle with full relations.

The aim of this paper is to give some definite formulae for calculating the
global and self-injective dimensions of a given gentle algebra by marked
ribbon surfaces.

Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Then $\mathcal{S}_{A}$ is divided into some elementary
polygons $\\{\Delta_{i}\mid 1\leq i\leq d\\}$ by $\Gamma_{A}$ which has
exactly one side $\subseteq\partial\mathcal{S}_{A}$ (the boundary of
$\mathcal{S}_{A}$). We use $\mathfrak{C}(\Delta_{i})$ to denote the number of
sides of $\Delta_{i}$ belonging to $\Gamma_{A}$ if the unmarked boundary
component of $\mathcal{S}_{A}$ is not a side of $\Delta_{i}$; otherwise,
$\mathfrak{C}(\Delta_{i})=\infty$. We use $\mathsf{f}\text{-}\Delta$ to denote
the set of all non-$\infty$-elementary polygons (see Remark 5.8), and use
$\mathcal{F}_{A}$ (respectively, $\mathsf{f}\text{-}\mathcal{F}_{A}$) to
denote the set of all forbidden threads (respectively, of finite length) (see
Definition 5.9). For any $\mathit{\Pi}\in\mathcal{F}_{A}$, we use
$l(\mathit{\Pi})$ to denote its length. Our main result is as follows.

######  Theorem 1.1.

(Theorems 5.10 and 6.9) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Then we have

  1. (1)

$\mathrm{gl.dim}A=\max\limits_{1\leq i\leq
d}{\mathfrak{C}(\Delta_{i})}-1=\max\limits_{\mathit{\Pi}\in\mathcal{F}_{A}}l(\mathit{\Pi})$.

  2. (2)

$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=$

$\begin{cases}0,\ \mbox{{if $Q$ is either a point or an oriented cycle with
full relations};}\\\
\max\limits_{\Delta_{i}\in\mathsf{f}\text{-}\Delta}\big{\\{}1,{\mathfrak{C}(\Delta_{i})}-1\big{\\}}=\max\limits_{\mathit{\Pi}\in\mathsf{f}\text{-}\mathcal{F}_{A}}l(\mathit{\Pi}),\
\mbox{{otherwise}.}\end{cases}$

The paper is organized as follows. In Section 2, we recall some terminologies
and some preliminary results needed in this paper. In particular, we give the
definition of gentle algebras and some related notions related to their
geometric models. In Section 3, we give the descriptions of some short exact
sequences by geometric models, which will be used frequently in the sequel.

######  Theorem 1.2.

(Theorem 3.12) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface.

  1. (1)

If $c$, $c^{\prime}$ and $c^{\prime\prime}$ are permissible curves such that
the positional relationship of them is given by FIGURE 1 Case I, then there
exists an exact sequence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0.$

  2. (2)

If $c$, $c^{\prime}$, $c_{\mathrm{I}}^{\prime\prime}$ and
$c_{\mathrm{II}}^{\prime\prime}$ are permissible curves such that the
positional relationship of them is given by FIGURE 1 Case II, then there
exists an exact sequence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c_{\mathrm{I}}^{\prime\prime}\oplus
c_{\mathrm{II}}^{\prime\prime})\longrightarrow 0.$

  3. (3)

If $c$, $c_{\mathrm{I}}^{\prime}$, $c_{\mathrm{II}}^{\prime}$ and
$c^{\prime\prime}$ are permissible curves such that the positional
relationship of them is given by FIGURE 1 Case III, then there exists an exact
sequence

$0\longrightarrow M(c_{\mathrm{I}}^{\prime}\oplus
c_{\mathrm{II}}^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0.$

$c$$c^{\prime}$$c^{\prime\prime}$Case I

$c$$c^{\prime}$$c_{\mathrm{I}}^{\prime\prime}$$c_{\mathrm{II}}^{\prime\prime}$Case
II

$c$$c^{\prime\prime}$$c_{\mathrm{I}}^{\prime}$$c_{\mathrm{II}}^{\prime}$Case
III

Figure 1: For each case, the module $M^{\prime}$ corresponding to blue
permissible curve(s) is a submodule of $M$ corresponding to the red
permissible curve, and the module $M^{\prime\prime}$ corresponding to orange
permissible curve(s) is isomorphic to the quotient $M/M^{\prime}$.

In Section 4, we describe the projective covers and injective envelopes of
indecomposable modules over gentle algebras by marked ribbon surfaces
(Theorems 4.6 and 4.9). Then in Section 5, we give the proof of Theorem
1.1(1).

In Section 6, by describing all Gorenstein projective modules in geometric
models, we give a proof of the following equalities

$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\begin{cases}0,\ \mbox{{if $Q$ is
either a point or an oriented cycle with full relations};}\\\
\max\limits_{\Delta_{i}\in\mathsf{f}\text{-}\Delta}\big{\\{}1,{\mathfrak{C}(\Delta_{i})}-1\big{\\}},\
\mbox{{otherwise}.}\end{cases}$

Then by describing minimal projective resolutions of all injective modules, we
obtain a formula for calculating the left and right self-injective dimensions
of $A$ by marked ribbon surfaces (Proposition 6.8). As a consequence, we give
the proof of Theorem 1.1(2) including another proof of the above equalities.
In addition, we prove that the number of indecomposable non-projective
Gorenstein projective modules over gentle algebras is invariant under AG-
equivalence (Proposition 6.12). In Section 7, we give some examples to
illustrate the obtained results; that is, we apply Theorem 1.1 to calculate
the global and self-injective dimensions of some gentle algebras.

##  2 The geometric models of gentle algebras

In this paper, assume that $A=kQ/I$ is a finite dimensional basic algebra over
an algebraically closed field $k$, where $I$ is an admissible ideal of $kQ$
and $Q=(Q_{0},Q_{1})$ is a finite quiver with $Q_{0},Q_{1}$ the sets of all
vertices and arrows, respectively. We use $s,t$ to denote two functions from
$Q_{1}$ to $Q_{0}$ which send each arrow to its source and target,
respectively. The multiplication $\alpha_{1}\alpha_{2}$ of two arrows
$\alpha_{1}$ and $\alpha_{2}$ in $Q_{1}$ is defined by the concatenation if
$t(\alpha_{1})=s(\alpha_{2})$ or zero if $t(\alpha_{1})\neq s(\alpha_{2})$.
For a set $X$, the number of elements of $X$ is denoted by $\sharp X$. We use
$\bmod A$ to denote the category of finitely generated right $A$-modules, and
use $\mathrm{gl.dim}A$ to denote the global dimension of $A$. For a module
$M\in\bmod A$, we use $\mathrm{proj.dim}M$ and $\mathrm{inj.dim}M$ to denote
the projective and injective dimensions of $M$, respectively, and use ${\rm
top}M$, $\mathrm{soc}M$ and $P(M)$ to denote the top, socle and projective
cover of $M$, respectively. We denote by $N\leq M$ (respectively,
$N\leq_{\oplus}M$) if $N$ is a submodule (respectively, direct summand) of
$M$.

###  2.1 Marked ribbon surfaces

In gentle algebras, Opper, Plamondon and Schrollin [25] introduced the notion
of marked ribbon surfaces by defining ribbon graphs first. Marked ribbon
surfaces are often referred to as marked surfaces, such as [4, 7, 17]. In this
paper we still use the terminology from [25] (but we do not need the
definition of ribbon graphs).

######  Definition 2.1.

(Marked ribbon surfaces) [4, 7, 25] A marked ribbon surface is a triple
$(\mathcal{S},\mathcal{M},\Gamma)$, where

  * •

$\mathcal{S}$ is an oriented connected surface with non-empty boundary, and we
use $\partial\mathcal{S}$ to denote its boundary.

  * •

$\mathcal{M}$ is a finite set of points on $\partial\mathcal{S}$, and each
element in $\mathcal{M}$ is called a marked point and denoted by the symbol
$\bullet$.

  * •

$\Gamma$ is called a full formal $\bullet$-arc system of $\mathcal{S}$ such
that the following hold:

    * –

Each element in $\Gamma$, called a $\bullet$-arc (or an arc), is a curve whose
ending points belong to $\mathcal{M}$.

    * –

For any two $\bullet$-arcs $\gamma$ and $\gamma^{\prime}$, there is no
intersection in the inner of $\mathcal{S}$.

    * –

$\mathcal{S}$ is divided into some parts $\\{\Delta_{i}\mid 1\leq i\leq d\\}$
by $\Gamma$ such that each part is a polygon which has exactly a side, say a
single boundary arc, not belonging to $\Gamma$. The set of all polygons
$\Delta=\\{\Delta_{i}\mid 1\leq i\leq d\\}$ is called an original
$\bullet$-dissection (or original dissection) of $\mathcal{S}$ [4], and each
polygon $\Delta_{i}$ above is said to be elementary. We denote by
$\mathfrak{S}(\Delta_{i})$, say the arc set, the set of all sides of
$\Delta_{i}$ belong to $\Gamma$.

We give some supplements to Definition 2.1 as follows.

  1. (1)

Each arc in $\Gamma$ can be seen as a pair $(\gamma,\gamma^{-})$, where
$\gamma$ is a continuous function
$\gamma:(0,1)\to\mathcal{S}\backslash\partial\mathcal{S}$ such that
$\gamma(0^{+})=\lim\limits_{x\to 0^{+}}\gamma(x)$ and
$\gamma(1^{-})=\lim\limits_{x\to 1^{-}}\gamma(x)$ lie in $\mathcal{M}$, and
$\gamma^{-}$ is the formal inverse of $\gamma$, that is,
$\gamma^{-}(x)=\gamma(1-x)$. Based on this, $\gamma=\gamma(x)$ and
$\gamma^{-}=\gamma(1-x)$ correspond to the same arc in $\Gamma$. For the sake
of simplicity, we suppose that $\gamma$ and $\gamma^{-}$ both are elements in
$\Gamma$ and $\gamma\simeq\gamma^{-}\simeq(\gamma,\gamma^{-})$. This
hypothesis can be used to simplify some descriptions, such as Theorem 5.6 and
Example 7.1.

  2. (2)

If there is some $i$ with $1\leq i\leq d$ such that $\Delta_{i}$ is a two-
gons, then $\Delta_{i}$ is one of the two forms (i) and (ii) shown in FIGURE
2. If $\Delta_{i}$ is of the form (i), then adding a new point (will be
denoted by the symbol
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$
and called an extra point) on the side not belonging to $\Gamma$. The set of
all extra points is denoted by $E$.

(i)

(ii)

Figure 2:  (i) and (ii) are 2-gons. In (i), we add an extra point on the side
not belonging to $\Gamma$.

  3. (3)

The positive direction of $\partial\mathcal{S}$ is defined as follows: the
interior of $\mathcal{S}$ is on the left while walking along the boundary.

######  Remark 2.2.

  *   * (1)

In some cases, we do not know whether a point in the surface is a marked point
or an extra point, we use “$\boldsymbol{\times}$” to express it.

  * (2)

We say that arcs $\gamma_{1},\cdots,\gamma_{n}$ with a common endpoint $p$
surround $p$ in the clockwise $($respectively, counterclockwise$)$ order if
$\gamma_{i+1}$ is right (respectively, left) to $\gamma_{i}$ ($1\leq i\leq
n-1$) at the point $p$ (see FIGURE 3).

$p$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\cdots$$\gamma_{n}$

$p$$\gamma_{n}$$\cdots$$\gamma_{3}$$\gamma_{2}$$\gamma_{1}$

Figure 3: $\gamma_{1},\cdots,\gamma_{n}$ with common endpoint $p$ are surround
$p$ in the clockwise $($respectively, counterclockwise$)$ order.

  * (3)

In [4], Amiot, Plamondon and Schroll defined the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-points
and
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-dissection
for marked ribbon surfaces. The
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-point
is either a point belonging to $\partial\mathcal{S}$ which lies between two
adjacent marked points or a boundary component of $\mathcal{S}$ with no marked
point. The
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-dissection
$\Delta_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}}$
of $\mathcal{S}$, whose elements are called
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-arcs,
is dual of $\bullet$-dissection $\Delta=\Delta_{\bullet}$, that is, each
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-arc
is a curve whose endpoints are
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-points,
any two
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-arcs
have no intersection point in $\mathcal{S}\backslash\partial\mathcal{S}$, and
for every $\bullet$-arc (respectively,
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-arc),
there exists a unique
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-arc
(respectively, $\bullet$-arc) such that they have only one intersection point
in $\mathcal{S}\backslash\partial\mathcal{S}$. Indeed,
$\Delta_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}}$
is uniquely determined by $\Delta_{\bullet}$, and it is said to be lamination
of $\mathcal{S}$ by Opper, Plamondon and Schroll in [25], although their
definitions seem different. In this paper, we do not need the definitions of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-point
and
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-dissection,
but we need to define the extra points which are special
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$-points.
They are used in the description of some indecomposable modules.

######  Definition 2.3.

($k$-algebras of marked ribbon surfaces) [4, 25] Let
$\mathcal{S}=(\mathcal{S},\mathcal{M},\Gamma)$ be a marked ribbon surface. We
associate a quiver $Q_{\mathcal{S}}=(Q_{0},Q_{1})$ and relation
$I_{\mathcal{S}}=\langle R\rangle$ to $\mathcal{S}$ as follows:

  1. (1)

The vertexes in $Q_{0}$ correspond to the $\bullet$-arcs in $\Gamma$ and this
corresponding is one-to-one, denoted by $\mathfrak{v}:Q_{0}\to\Gamma$, that
is, each $\bullet$-arc can be seen as a vertex of $Q_{\mathcal{S}}$;

  2. (2)

For every elementary polygon $\Delta_{i}$ with two sides $u,v\in\Gamma$, if
$v$ follows $u$ in the counterclockwise order, then there is an arrow $u\to v$
in $Q_{1}$;

  3. (3)

$R$ is the set of all compositions $\alpha\beta$, where $\alpha:u\to v$ and
$\beta:v\to w$ satisfies that $u$, $v$ and $w$ are sides of the same
elementary polygon.

Then the $k$-algebra $A_{\mathcal{S}}$ of the marked ribbon surfaces
$\mathcal{S}$ is defined by
$A_{\mathcal{S}}:=kQ_{\mathcal{S}}/I_{\mathcal{S}}$.

######  Definition 2.4.

(Gentle algebras) [2, 4, 5, 25] A $k$-algebra $A=kQ/I$ is said to be special
biserial if the following conditions are satisfied.

  1. (1)

For each vertex $v\in Q_{0}$, $\sharp\\{\alpha\in Q_{1}\mid s(\alpha)=v\\}\leq
2$ and $\sharp\\{\alpha\in Q_{1}\mid t(\alpha)=v\\}\leq 2$;

  2. (2)

For each arrow $\beta\in Q_{1}$, $\sharp\\{\alpha\in Q_{1}\mid
s(\beta)=t(\alpha),\alpha\beta\notin I\\}\leq 1$ and $\sharp\\{\alpha\in
Q_{1}\mid s(\alpha)=t(\beta),\beta\alpha\notin I\\}\leq 1$;

  3. (3)

For each arrow $\beta\in Q_{1}$, there exists a bound $n\in\mathbb{N}^{+}$
such that

     * •

For each path $p=\beta_{1}\cdots\beta_{n-1}$ where $t(p)=s(\beta)$, the path
$p\beta$ contains a subpath in $I$;

     * •

For each path $p^{\prime}=\beta_{1}^{\prime}\cdots\beta_{n-1}^{\prime}$ where
$t(\beta)=s(p^{\prime})$, the path $\beta p^{\prime}$ contains a subpath in
$I$.

Moreover, $A=kQ/I$ is said to be gentle if it is a special biserial algebra
such that the following hold:

  1. (4)

$Q$ is a bound quiver;

  2. (5)

All relations in $I$ are paths of length $2$;

  3. (6)

For each arrow $\beta\in Q_{1}$, $\sharp\\{\alpha\in Q_{1}\mid
s(\beta)=t(\alpha),\alpha\beta\in I\\}\leq 1$ and $\sharp\\{\alpha\in
Q_{1}\mid s(\alpha)=t(\beta),\beta\alpha\in I\\}\leq 1$.

Notice that a special biserial algebra is finite dimensional if and only if
$Q$ is finite, thus a gentle algebra is always finite dimensional. Moreover,
Opper, Plamondon and Schroll proved that
$A_{\mathcal{S}}:=kQ_{\mathcal{S}}/I_{\mathcal{S}}$ is gentle, and that there
is a bijection between isomorphism classes of gentle algebras and homotopy
classes of marked ribbon surfaces (see [25, Proposition 1.21]). It should be
pointed out that there are different definitions of gentle algebras. For
example, the definition of gentle algebras in [2, 5] does not require the
fourth condition, and so there are gentle algebras whose quivers are not
bound; and that of gentle algebras in [4, 25] does not require the third
condition, and so there are gentle algebras whose quivers has at least one
oriented cycles with no relations.

###  2.2 Permissible curves and permissible closed curves

Each curve $c$ in a marked ribbon surface
$\mathcal{S}=(\mathcal{S},\mathcal{M},\Gamma)$ can be defined as a function
$c:[0,1]\to\mathcal{S}$ where $c(0)$ and $c(1)$ are its endpoints. We say that
$c(0)$ and $c(1)$ are its starting point and ending point, respectively. In
this paper, we only consider such curves whose points lie in the interior of
$\mathcal{S}$ except endpoints, that is, $\\{c(x)\mid
0<x<1\\}\subseteq\mathcal{S}\backslash\partial\mathcal{S}$, and whose
endpoints are elements belong to $\mathcal{M}\cup E$. Thus each non-closed
curve can be regarded as a function
$c:(0,1)\to\mathcal{S}\backslash\partial\mathcal{S}$ where $c(0^{+})$ and
$c(1^{-})$ are its endpoints and each closed curve can be regarded as a
function $c:[0,1]\to\mathcal{S}\backslash\partial\mathcal{S}$ where
$c(0)=c(1)\in\mathcal{S}\backslash\partial\mathcal{S}$. In this paper, for
each curve $c$ with the endpoints $c(0^{+})$ and $c(1^{-})$ on the marked
ribbon surface, we always suppose that the number of intersections between $c$
and $\Gamma$ is minimal up to homotopy.

######  Definition 2.5.

(Permissible curves and permissible closed curves) [7, Definition 3.1] Let
$\mathcal{S}=(\mathcal{S},\mathcal{M},\Gamma)$ be a marked ribbon surface.

  1. (1)

A curve $c:(0,1)\to\mathcal{S}\backslash\partial\mathcal{S}$ (or a closed
curve $c:[0,1]\to\mathcal{S}\backslash\partial\mathcal{S}$) in $\mathcal{S}$
is said to consecutively cross $u,v\in\Gamma$ if $p_{1}=c\cap u$ and
$p_{2}=c\cap v$ belong to $\mathcal{S}\backslash\partial\mathcal{S}$, and the
segment of $c$ between the points $p_{1}$ and $p_{2}$ does not cross any other
arc in $\Gamma$.

Then each non-closed curve $c$ consecutively crossing
$\gamma_{1},\cdots,\gamma_{r}\in\Gamma$ can be written as $c=c_{1}c_{2}\cdots
c_{r}c_{r+1}$, where each $c_{i}$ ($2\leq i\leq r$) is a segment between
$\gamma_{i-1}$ and $\gamma_{i}$ does not cross any other arc in $\Gamma$, and
$c_{1}$ (respectively, $c_{r+1}$) is a segment between $c(0^{+})$ and
$\gamma_{1}$ (respectively, $\gamma_{r}$ and $c(1^{-})$) does not cross any
other arc in $\Gamma$. The segment $c_{i}$ ($2\leq i\leq r-1$) is called
middle, and the segments $c_{1}$ and $c_{r}$ are called end. Similarly, each
closed curve $c$ consecutively crossing
$\gamma_{1},\cdots,\gamma_{r}\in\Gamma$ can be written as $c=c_{1}c_{2}\cdots
c_{r}$, where each $c_{i}$ $(2\leq i\leq r)$ is a segment between
$\gamma_{i-1}$ and $\gamma_{i}$ does not cross any other arc in $\Gamma$, and
$c_{1}$ is a segment between $\gamma_{r}$ and $\gamma_{1}$.

  2. (2)

Let $B$ be a boundary component with no marked point of $\mathcal{S}$, called
an unmarked boundary component of $\mathcal{S}$, and let
$c:(0,1)\to\mathcal{S}\backslash\partial\mathcal{S}$ be a curve in
$\mathcal{S}$. Write $c=c_{1}c^{\prime}c_{2}$ where $c_{1}$ (respectively,
$c_{2}$) is the end segment between $c(0^{+})$ (respectively, $c(1^{-})$) and
the first (respectively, last) crossing point in $\Gamma$. We define the
winding number of $c_{i}$ $(i=1,2)$ around $B$ as the minimum number of times
travels $\bar{c}$ around $B$ in either direction, where $\bar{c}$ lies in the
homotopy class of $c_{i}$.

  3. (3)

A curve $c$ is called permissible if the following conditions are satisfied.

    1. (a)

The winding number of $c$ around an unmarked boundary component is either 0 or
1;

    2. (b)

If $c$ consecutively crosses two arcs $u$ and $v$ in $\Gamma$, then $u$ and
$v$ have a common endpoint $p$, denoted by $p(u,v,c)$, lying in $\mathcal{M}$,
and locally we have a triangle with $p$ a vertex (as Picture I in FIGURE 4,
the curves $c_{1}$, $c$, $c^{\prime}$ and $c^{\prime\prime}$ are permissible).

$Y_{1}$$Y_{2}$$Y_{3}$$Y_{4}$$Y_{5}$$Y_{6}$$Y_{7}$$Y_{8}$$c_{1}$$c$$c^{\prime}$$c^{\prime\prime}$$Y_{1}$$Y_{2}$$c$$\mathrm{I}$$\mathrm{II}$
Figure 4: Some examples of permissible (closed) curves in marked ribbon
surfaces

  4. (4)

A permissible closed curve is a closed curve $c$ satisfying condition (3)(b)
(as shown Picture II in FIGURE 4, the algebra of the marked ribbon surface is
a 2-Kronecker algebra, and the curve $c$ is a permissible closed curve).

######  Definition 2.6.

(Equivalence class of permissible curves) [7] Let
$\mathcal{S}=(\mathcal{S},\mathcal{M},\Gamma)$ be a marked ribbon surface. Two
permissible curves
$c,c^{\prime}:(0,1)\to\mathcal{S}\backslash\partial\mathcal{S}$ in
$\mathcal{S}$ are called equivalent, denoted by $c\simeq c^{\prime}$, if one
of the following conditions is satisfied.

  1. (1)

$c^{\prime}$ is the reverse curve $c^{-}$ of $c$, that is,
$c(x)=c^{\prime}(1-x)$ for any $x\in(0,1)$.

  2. (2)

There is a sequence of consecutive arcs
$(u_{i}:(0,1)\to\mathcal{S}\backslash\partial\mathcal{S})_{1\leq i\leq n}$ in
the arc system $\Gamma$ such that

    1. (a)

All $u_{i}$ $(1\leq i\leq n)$ are sides of the same elementary polygon
$\Delta_{j}$.

    2. (b)

$c(0^{+})$ (respectively, $c(1^{-})$) is a marked point $\bullet$ or an extra
point
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\circ}}$,
$c(1^{-})=u_{1}(0^{+})$ (respectively, $c(0^{+})=u_{1}(0^{+})$),
$u_{i-1}(1^{-})=u_{i}(0^{+})$ for any $1\leq i\leq n$.

    3. (c)

$c$ is homotopic to the concatenation of $c^{\prime}$ and $(u_{i})_{1\leq
i\leq n}$.

    4. (d)

$c^{\prime}$ starts at an endpoint of $u_{n}$, and the first crossing with
$\Gamma$ of $c$ and $c^{\prime}$ are with the same side of $\Delta_{j}$.

  3. (3)

$c(0^{+})=c^{\prime}(0^{+})$ and $c(1^{-})=c^{\prime}(1^{-})$ (respectively,
$c(0^{+})=c^{\prime}(1^{-})$ and $c(1^{-})=c^{\prime}(0^{+})$) are marked
points or extra points, arcs consecutively crossed by $c$ and $c^{\prime}$
($c^{-}$ and $c^{\prime}$) are same, and each segment of $c$ cut by two arcs
is homotopic to that of $c^{\prime}$.

  4. (4)

$c$ (respectively, $c^{\prime}$) is either a curve satisfying
$c\cap\Gamma=\varnothing$ (respectively, $c^{\prime}\cap\Gamma=\varnothing$)
or a arc; in this case, we call $c,c^{\prime}$ are trivial. Thus each arc can
be regarded as a trivial permissible curve.

######  Example 2.7.

In FIGURE 4 Picture I, the curves $c$, $c^{\prime}$ and $c^{\prime\prime}$ are
permissible and $c\simeq c^{\prime}\simeq c^{\prime\prime}$, but $c\not\simeq
c_{1}$.

##  3 Finitely generated module categories

###  3.1 The geometric models of string and band modules

Let $A=kQ/I$ be a gentle algebra. For each arrow $\alpha\in Q_{1}$, the formal
inverse of $\alpha$, denoted by $\alpha^{-}$, is an arrow with
$s(\alpha^{-})=t(\alpha)$ and $t(\alpha^{-1})=s(\alpha)$. Obviously,
$(\alpha^{-})^{-}=\alpha$. We use $Q_{1}^{-}$ to denote the set of all formal
inverses of arrows in $Q_{1}$. Also we can define that the formal inverse of
any path $p=\alpha_{1}\cdots\alpha_{n}$ is
$p^{-}=\alpha_{n}^{-}\cdots\alpha_{1}^{-}$. In this subsection, we recall the
definitions of string modules and band modules over a gentle algebra $A$ [8],
and review the description of indecomposable modules in marked ribbon surfaces
[7].

A string $s$ is a reduced walk in the quiver $Q$ with no relations, that is,
$s=a_{1}\cdots a_{n}$ with $a_{i}\in Q_{1}\cup Q_{1}^{-}$ has neither subwalks
of the form $aa^{-}$ and $a^{-}a$ nor subwalks of the form $ab$ such that
$ab\in I$ or $b^{-}a^{-}\in I$; particularly, $s=1_{v}$ is a path of length
$0$ corresponding to a vertex $v\in Q_{0}$, it is called a simple string (note
that $1_{v}^{-}=1_{v}$ for each $v\in Q_{0}$). Trivially, we define the
trivial string $s=0$.

Each string $s=a_{1}\cdots a_{n}$ in $A$ corresponds to a module $M(s)\in\bmod
A$ as follows:

  * •

replacing each vertex of $s$ by a copy of the field $k$, that is, for each
$v\in Q_{0}$, $\dim_{k}M(s)1_{v}=1$ if $1_{v}$ is a trivial subwalk of $s$ or
zero if otherwise; and

  * •

the action of an arrow $\alpha\in Q_{1}$ on $M(s)$ is the identity morphism if
$\alpha$ is an arrow of $s$ or zero if otherwise.

Obviously, $s=0$ yields $M(s)=0$; $s$ being simple yields that $M(s)$ is
simple; and $M(s)\cong M(s^{\prime})$ if and only if $s^{\prime}=s^{-}$ or
$s^{\prime}=s$.

A band $b=a_{1}\cdots a_{n}$ is a cyclic string (that is, $t(a_{n})=s(a_{1})$)
such that each power $b^{n}$ is a string but $b$ is not a proper power of any
string.

Each band $b=a_{1}\cdots a_{n}$ in $A$ corresponds to a family of modules
$M(b,\varphi)\in\bmod A$ as follows:

  * •

$\varphi$ is an indecomposable $k$-linear automorphism of $k^{n}$ with $n\geq
0$ (that is, $\varphi\in\mathrm{Aut}(k^{n})$ is a Jordan block);

  * •

replacing each vertex of $s$ by a copy of the vector space $k^{n}$; and

  * •

the action of an arrow $\alpha\in Q_{1}$ on $M(b,\varphi)$ is the identity
morphism if $\alpha=a_{i}$ for $1\leq i\leq n-1$ or $\varphi$ if
$\alpha=a_{n}$ (thus $M(b,\varphi)\alpha=0$ if $\alpha$ is not an arrow of
$b$).

######  Remark 3.1.

By [8], we know that each indecomposable $A$-module is either a string module
or a band module. Baur and Coelho Simões [7] showed that there is a bijection
between the equivalence classes of non-trivial permissible curves in
$\mathcal{S}_{A}$ and non-zero strings of $A$, and there is a bijection
between the homotopy classes of permissible closed curves $c$ in
$\mathcal{S}_{A}$ with $|I_{\Gamma_{A}}(c)|\geq 2$ and the powers of bands of
$A$, where $|I_{\Gamma_{A}}(c)|$ is the number of intersection points of $c$
and $\Gamma_{A}$ [7]. To be precise:

  * •

If $[c]$ is an equivalence class of non-trivial permissible curve, then
$c=c_{1}\cdots c_{r}$, where each $c_{i}$ is a segment cut by two arcs
$\gamma_{i-1}$ and $\gamma_{i}$ (we suppose that $c$ consecutively crosses
$\gamma_{1},\cdots,\gamma_{r-1}$), and $c$ corresponds to such string
$s=s_{1}\cdots s_{r}$ that $s_{i}$ is one of:

    * –

the arrow $\gamma_{i-1}\to\gamma_{i}$, if $c$ counterclockwise crosses
$\gamma_{i-1}$ and $\gamma_{i}$ around their common endpoint
$p(\gamma_{i-1},\gamma_{i},c)$;

    * –

the formal inversion of the arrow $\gamma_{i}\to\gamma_{i-1}$, if $c$
clockwise crosses $\gamma_{i-1}$ and $\gamma_{i}$ around their common endpoint
$p(\gamma_{i-1},\gamma_{i},c)$.

(Note that each arc in $\Gamma$ can be regarded as a vertex of quiver $Q$ by
Definition 2.3).

Thus $c$ can be corresponded to the string module of $s$.

  * •

If $c$ is a homotopy class of permissible closed curves, then $c$ with an
indecomposable invertible linear transformation $\varphi:k^{n}\to k^{n}$
corresponds to a band which induces a band module $M(c,\varphi)$. Band modules
lie in homogeneous tubes of the Auslander-Reiten quiver, thus the Auslander-
Reiten translate $\tau$ acts on them as the identity morphism.

######  Remark 3.2.

[7, Theorem 3.8 and Proposition 3.9] Let $[c]$ (respectively,
$\mathrm{htp}(c)$) be the equivalence class of permissible curve
(respectively, the homotopy class of permissible closed curve) $c$. Define the
following three sets:

  * •

${\mathrm{Eq}\text{-}\mathrm{cls}}(\mathcal{S}_{A})=\\{[c]\mid c\text{ is a
permissible curve}\\}$;

  * •

${\mathrm{Htp}\text{-}\mathrm{cls}}(\mathcal{S}_{A})=\\{(\mathrm{htp}(c),\varphi)\mid
c\text{ is a permissible closed curve with }|I_{\Gamma_{A}}(c)|\geq 2\
\text{and}\ \varphi\ \text{a Jordan block}\\}$;

  * •

$\mathfrak{P}(\mathcal{S}_{A})={\mathrm{Eq}\text{-}\mathrm{cls}}(\mathcal{S}_{A})\cup{\mathrm{Htp}\text{-}\mathrm{cls}}(\mathcal{S}_{A})$.

Then we have a bijection

$M:\mathfrak{P}(\mathcal{S}_{A})\to\\{\text{string
modules}\\}\cup\\{\text{band modules}\\}=\mathrm{ind}(\bmod A),x\mapsto M(x).$

Obviously, each trivial permissible curve corresponds to zero by $M$.

######  Definition 3.3.

(Formal direct sums) Let $\mathcal{S}_{A}$ be the marked ribbon surface of a
gentle algebra $A$. A formal direct sum of $c_{1},\cdots,c_{n}$, denoted by
$\bigoplus_{i=1}^{n}c^{i}$, naturally corresponds to the direct sum of modules
$\bigoplus_{i=1}^{n}M(c^{i})$ by the corresponding $M$ in Remark 3.2.

###  3.2 The geometric models of some short exact sequences

In this subsection, we depict certain short exact sequences by marked ribbon
surfaces for gentle algebras, which are used to describe the minimal
projective and injective resolutions of simple modules in Section 5, and
describe the minimal projective resolutions of injective modules in Section 6.
First at all, we fix some terms and notations.

We always assume that each non-trivial permissible curve
$c:(0,1)\to\mathcal{S}\backslash\partial\mathcal{S}$ in
$\mathcal{S}=(\mathcal{S},\mathcal{M},\Gamma)$ consecutively crossing arcs
$\gamma_{1},\cdots,\gamma_{r-1}$ satisfies that $\gamma_{1}$ is the first arc
crossed by $c$, that is, there is a sequence
$0=x_{0}<x_{1}<x_{2}\cdots<x_{r}=1$ such that $c=c_{1}\cup c_{2}\cup\cdots\cup
c_{r}$ (we write $c=c_{1}c_{2}\cdots c_{r}$ for simplicity), where each
$c_{i}=\\{c(x)\mid x_{i-1}\leq x\leq x_{i}\\}$ is a segment of $c$ satisfying
$c_{i}\cap\gamma_{i}=c(x_{i})=c_{i+1}\cap\gamma_{i}$ if $2\leq i\leq r-1$; and
$c_{1}=\\{c(x)\mid 0<x\leq x_{1}\\}$ and $c_{r}=\\{c(x)\mid x_{r-1}\leq
x<1\\}$ are the end segments of $c$ satisfying
$c_{1}\cap\gamma_{1}=c(x_{1})=c_{2}\cap\gamma_{1}$ and
$c_{r}\cap\gamma_{r-1}=c(x_{r-1})=c_{r-1}\cap\gamma_{r-1}$, respectively.

Let $c=c_{1}c_{2}\cdots c_{r}$ be a permissible curve consecutively crossing
$\gamma_{1},\cdots,\gamma_{r-1}$. Then it induces a new permissible curve
$c^{\prime}=c^{\mathfrak{T}}_{i-1}c_{i}\cdots c_{j}c^{\mathfrak{T}}_{j+1}$ for
any $2\leq i\leq j\leq r-1$ such that

  1. (1)

$c^{\mathfrak{T}}_{i-1}$ and $c_{i-1}$ lie in the inner of the same elementary
polygon of $\mathcal{S}$;

  2. (2)

one endpoint of $c^{\mathfrak{T}}_{i-1}$ is $c_{i-1}\cap\gamma_{i-1}$ and the
other is an endpoint of $\gamma_{i-2}$;

  3. (3)

the dual of the condition (1), that is, $c^{\mathfrak{T}}_{j+1}$ and $c_{j+1}$
lie in the inner of the same elementary polygon;

  4. (4)

the dual of the condition (2), that is, one endpoint of
$c^{\mathfrak{T}}_{j+1}$ is $c_{j+1}\cap\gamma_{j}$ and the other is an
endpoint of $\gamma_{j+1}$.

Note that it is a trivial case for $c_{1}^{\mathfrak{T}}c_{2}\cdots
c_{r-1}c_{r}^{\mathfrak{T}}=c_{1}c_{2}\cdots c_{r-1}c_{r}$, that is,
$c_{1}^{\mathfrak{T}}=c_{1}$, and $c_{r}^{\mathfrak{T}}=c_{r}$. We provide an
example in the FIGURE 5, in which $c=\cdots c_{i-1}c_{i}c_{i+1}\cdots
c_{j-1}c_{j}c_{j+1}$ and $c^{\prime}=c_{i}^{\mathfrak{T}}c_{i+1}\cdots
c_{j-1}c_{j}^{\mathfrak{T}}$ are the red and blue permissible curves
respectively. In this example, $c$ corresponds to the string

$\gamma_{1}\longrightarrow\cdots-\\!\\!\\!-\\!\\!\\!-\gamma_{i-1}\longrightarrow\gamma_{i}\longrightarrow\cdots\longrightarrow\gamma_{j}\longrightarrow\gamma_{j+1}-\\!\\!\\!-\\!\\!\\!-\cdots\longleftarrow\gamma_{r-1},$

and $c^{\prime}$ corresponds to the string

$\gamma_{i}\longrightarrow\cdots\longrightarrow\gamma_{j}.$

$p$$p^{\prime}$$q^{\prime}$$q$$\cdots$$\gamma_{m-1}$$\gamma_{2}$$\gamma_{1}$$\gamma_{m}$$\gamma_{i-1}$$\cdots$$\gamma_{i}$$\gamma_{j}$$\cdots$$\gamma_{j+1}$$\gamma_{n-2}$$\cdots$$\gamma_{n-1}$$\cdots$$\gamma_{n}$$\gamma_{r-2}$$\gamma_{r-1}$$\cdots$$c=c_{1}c_{2}\cdots
c_{r}$$c_{1}$$c_{2}$$c_{m}$$c_{i-1}$$c_{j+1}$$c_{n-1}$$c_{n}$$c_{n+1}$$c_{i}^{\mathfrak{T}}$$c_{j}^{\mathfrak{T}}$$c_{i}^{\mathfrak{T}}c_{i+1}\cdots
c_{j-1}c_{j}^{\mathfrak{T}}=c^{\prime}$ Figure 5: Two examples of permissible
curves corresponding to strings.

A string module which corresponds to the permissible curves $c$ (respectively,
its equivalence class $[c]$) is denoted by $M(c)$ (respectively, $M([c])$); a
band module with a Jordan block $\varphi\in\mathrm{Aut}(k^{n})$ which
corresponds to the permissible closed curve $c$ (respectively, its homotopic
class $\mathrm{htp}(c)$) is denoted by $M(c,\varphi)$ (respectively,
$M(\mathrm{htp}(c),\varphi)$).

For the convenience of describing rotations, we suppose that the positive
direction of the boundary $\partial\mathcal{S}$ of the surface $\mathcal{S}$
is the following walking direction: walking along $\partial\mathcal{S}$, the
inner of $\mathcal{S}$ is on the left.

######  Definition 3.4.

(Rotations of Type 1) Let $c^{1}=c_{1}\cdots c_{r}$ be a permissible curve
consecutively crossing $\gamma_{1},\cdots,\gamma_{r-1}$, and let $c^{2}$ be
equivalent to $c^{\prime}c_{i+1}\cdots c_{r}$ (in this case
$c^{1}(1^{-})=c^{2}(1^{-})$), where $c^{\prime}$ is either
$c_{i}^{\mathfrak{T}}$ or a curve with an endpoint being an extra point. We
say that the permissible curve $c^{3}$, denoted by
$\mathrm{prot}_{c^{1}}(c^{2})$, is obtained by the positive rotating $c^{2}$
with respect to $c^{1}$ if it is given as follows:

  1. Step 1.

move the endpoint of $c^{2}$ which is the common endpoint of $c^{2}$ and
$c^{1}$ to the other of $c^{1}$;

  2. Step 2.

following the positive direction of the boundary, move the other endpoint of
$c^{2}$ to the vertex $p$ of $\Delta_{j}$, where

    1. -

$\Delta_{j}$ is the elementary polygon such that $c_{i}$ lies in the inner of
$\Delta_{j}$ (in this case,
$\gamma_{i-1},\gamma_{i}\in\mathfrak{S}(\Delta_{j})$),

    2. -

and $p$ is a vertex such that $c^{3}=\mathrm{prot}_{c^{1}}(c^{2})$ crosses
$\gamma_{i-1}$.

Dually, we can define $\mathrm{nrot}_{c^{1}}(c^{2})$ obtained by the negative
rotating $c^{2}$ with respect to $c^{1}$ (see FIGURE 6).

$\gamma_{i-1}$$\gamma_{i}$$\boldsymbol{\times}$$\boldsymbol{\times}$$c^{1}$$c^{2}$$p$$c^{3}$

$\gamma_{i-1}$$\gamma_{i}$$\vdots$$\boldsymbol{\times}$$\boldsymbol{\times}$$c^{1}$$c^{2}$$p$$c^{3}$

Figure 6: The point “$\boldsymbol{\times}$” is either a marked point or an
extra point.

######  Definition 3.5.

(Rotations of Type 2) Let $c^{1}=c_{1}\cdots c_{r}$ be a permissible curve
consecutively crossing $\gamma_{1},\cdots,\gamma_{r-1}$, and $c^{2}$
equivalent to either $c_{m}^{\mathfrak{T}}c_{m+1}\cdots
c_{n}c_{n+1}^{\mathfrak{T}}$ ($2\leq m\leq n+1\leq r-2$) such that $c^{1}$ and
$c^{2}$ have an intersection, say $x$, in
$\mathcal{S}\backslash\partial\mathcal{S}$. Then $c^{2}$ is divided into two
parts $c^{2}_{\mathrm{I}}$ and $c^{2}_{\mathrm{II}}$ where
$c^{2}_{\mathrm{I}}(1^{-})=x=c^{2}_{\mathrm{II}}(0^{+})$. We say that the
permissible curve $c^{3}_{\mathrm{I}}$ (respectively, $c^{3}_{\mathrm{II}}$),
denoted by $\mathrm{prot}_{c^{1}}(c^{2}_{\mathrm{I}})$ (respectively,
$\mathrm{prot}_{c^{1}}(c^{2}_{\mathrm{II}})$), is obtained by the positive
rotating $c^{2}_{\mathrm{I}}$ (respectively, $c^{2}_{\mathrm{II}}$) with
respect to $c^{1}$ if it is given as follows:

  1. Step 1.

move the endpoint $x$ of $c^{2}_{\mathrm{I}}$ (respectively,
$c^{2}_{\mathrm{II}}$) to the end point $c^{1}(0^{+})$ (respectively,
$c^{1}(1^{-})$);

  2. Step 2.

following the positive direction of boundary, move the endpoint
$c^{2}_{\mathrm{I}}(0^{+})$ of $c^{2}_{\mathrm{I}}$ (respectively, the
endpoint $c^{2}_{\mathrm{II}}(1^{-})$ of $c^{2}_{\mathrm{II}}$) to the next
vertex of $\Delta_{j}$, where $\Delta_{j}$ is an elementary polygon such that
$c_{m}$ (respectively, $c_{n+1}$) lies in the inner of $\Delta_{j}$ (in this
case, we have $\gamma_{m-1},\gamma_{m}\in\mathfrak{S}(\Delta_{j})$ (resp.,
$\gamma_{n},\gamma_{n+1}\in\mathfrak{S}(\Delta_{j})$)).

Dually, we can define $\mathrm{nrot}_{c^{1}}(c^{2}_{\mathrm{I}})$ and
$\mathrm{nrot}_{c^{1}}(c^{2}_{\mathrm{II}})$ obtained by the negative rotating
$c^{2}_{\mathrm{I}}$ and $c^{2}_{\mathrm{II}}$ with respect to $c^{1}$,
respectively (see FIGURE 7).

$\boldsymbol{\times}$$\boldsymbol{\times}$$c^{1}$$x$$c^{2}_{\mathrm{I}}$$c^{2}_{\mathrm{II}}$$c^{3}_{\mathrm{I}}$$c^{3}_{\mathrm{II}}$$\gamma_{m}$$\gamma_{m-1}$$\gamma_{n}$$\gamma_{n+1}$
Figure 7: The point “$\boldsymbol{\times}$” is either a marked point or an
extra point.

######  Proposition 3.6.

Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. If there are three permissible curves $c$, $c^{\prime}$,
$c^{\prime\prime}$ such that the following conditions are satisfied (see
FIGURE 8):

  1. (1)

$c^{\prime}$ and $c$ have a common endpoint $m_{s}=c(0^{+})$,
$c^{\prime\prime}$ and $c$ have another common endpoint $m_{t}=c(1^{-})$.

  2. (2)

$c$ consecutively crosses arcs $\gamma^{\prime}$ and $\gamma^{\prime\prime}$
such that $p(\gamma^{\prime},\gamma^{\prime\prime},c)$ is on the right of $c$;
$c^{\prime}$ crosses $\gamma^{\prime}$ such that $c^{\prime}$ and
$\gamma^{\prime\prime}$ have a common endpoint which is not $m_{t}$; and
$c^{\prime\prime}$ crosses $\gamma^{\prime\prime}$ such that
$c^{\prime\prime}$ and $\gamma^{\prime}$ have a common endpoint which is not
$m_{s}$.

  3. (3)

$c^{\prime\prime}\simeq\mathrm{prot}_{c}(c^{\prime})$ $($equivalently,
$c^{\prime}\simeq\mathrm{nrot}_{c}(c^{\prime\prime}))$ (Definition 3.4);.

Then we have a short exact sequence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0.$

$c$$c^{\prime}$$c^{\prime\prime}$$\gamma^{\prime}$$\gamma^{\prime\prime}$$m_{s}$$m_{t}$

$c$$c^{\prime\prime}$$c^{\prime}$$\gamma^{\prime\prime}$$\gamma^{\prime}$$m_{t}$$m_{s}$

Figure 8: The short exact sequence is described in marked ribbon surface.

###### Proof.

Notice that $c$ corresponds to such a string which is of the form $s_{c}:=\ \
\cdots-\\!\\!\\!-\gamma^{\prime}\longleftarrow\gamma^{\prime\prime}-\\!\\!\\!-\cdots.$
Then $c^{\prime}$ and $c^{\prime\prime}$ correspond to $s_{c^{\prime}}:=\ \
\cdots-\\!\\!\\!-\gamma^{\prime}\ \text{ and }\ \ s_{c^{\prime\prime}}:=\ \
\gamma^{\prime\prime}-\\!\\!\\!-\cdots,\ \text{ respectively. }$ Furthermore,
$M(c)$, $M(c^{\prime})$ and $M(c^{\prime\prime})$ are string modules
corresponding to $s_{c}$, $s_{c^{\prime}}$ and $s_{c^{\prime\prime}}$,
respectively, and hence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0$

is exact. ∎

######  Lemma 3.7.

Let $A$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Let $c=c_{1}\cdots
c_{r}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ be a
permissible curve consecutively crossing $\gamma_{1},\cdots\gamma_{r-1}$ such
that the endpoints $p$ and $q$ of $\gamma_{i}$ $(1<i<r-1)$ are on the left and
right of $c$, respectively $($see FIGURE 9$)$. Then for any permissible curve
$c^{\prime}$ consecutively crossing $\gamma_{m},\cdots,\gamma_{n}$ $(1\leq
m<i<n\leq r-1)$ such that

  * •

$\gamma_{m},\cdots,\gamma_{i}$ have a common endpoint
$p=p(\gamma_{m},\gamma_{m+1},c)=\cdots=p(\gamma_{i-1},\gamma_{i},c)$, and

  * •

$\gamma_{i},\cdots,\gamma_{n}$ have a common endpoint
$q=p(\gamma_{i},\gamma_{i+1},c)=\cdots=p(\gamma_{n-1},\gamma_{n},c)$,

we have $M(c^{\prime})\leq M(c)$.

$p$$q$$\gamma_{i}$$\gamma_{i-1}$$\gamma_{i+1}$$\cdots$$\cdots$$\gamma_{m+1}$$\gamma_{n-1}$$\gamma_{m}$$\gamma_{n}$$c$$c^{\prime}$$c_{\mathrm{I}}^{\prime\prime}$$c_{\mathrm{II}}^{\prime\prime}$
Figure 9: $c$ and $c^{\prime}$ given in Lemma 3.7 corresponding to two module
$M(c)$ and $M(c^{\prime})$, we have $M(c^{\prime})\leq M(c)$. The formal
direct sum $c_{\mathrm{I}}^{\prime\prime}\oplus
c_{\mathrm{II}}^{\prime\prime}$ corresponds to the module
$M(c_{\mathrm{I}}^{\prime\prime}\oplus c_{\mathrm{II}}^{\prime\prime})$ which
isomorphic to the quotient $M(c)/M(c^{\prime})$.

###### Proof.

The strings $s_{c}$ and $s_{c^{\prime}}$ corresponding to $c$ and $c^{\prime}$
are

$\displaystyle\cdots\longrightarrow\gamma_{m}\longrightarrow\gamma_{m+1}\longrightarrow\cdots\longrightarrow\gamma_{i}\longleftarrow\cdots\longleftarrow\gamma_{n-1}\longleftarrow\gamma_{n}\longleftarrow\cdots\
\text{ and }$
$\displaystyle\gamma_{m}\longrightarrow\gamma_{m+1}\longrightarrow\cdots\longrightarrow\gamma_{i}\longleftarrow\cdots\longleftarrow\gamma_{n-1}\longleftarrow\gamma_{n}\text{,
respectively. }$

Then the string module corresponding to $s_{c^{\prime}}$ is a submodule of the
module corresponding to $s_{c}$, and thus $M(c^{\prime})\leq M(c)$. ∎

######  Proposition 3.8.

Keeping the notations in Lemma 3.7, we have that
$c^{\prime}=c_{m}^{\mathfrak{T}}c_{m+1}\cdots
c_{n}c_{n+1}^{\mathfrak{T}}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$
crosses $\gamma_{i}$ and that $c$ and $c^{\prime}$ have an intersection.
Assume that

  * •

$c_{\mathrm{I}}^{\prime\prime}$ is a permissible curve which is given as
follows:

    * –

$c^{\prime}$ is divided into two parts by $c$, where denote the part with
endpoints $c^{\prime}(0^{+})$ and $c^{\prime}\cap\gamma_{i}$ by
$c_{\mathrm{I}}^{\prime}$;

    * –

$c_{\mathrm{I}}^{\prime\prime}\simeq\mathrm{prot}_{c}(c^{\prime}_{\mathrm{I}})$
$($Definition 3.5$)$;

  * •

and $c_{\mathrm{II}}^{\prime\prime}$ is a permissible curve which is given as
follows:

    * –

$c^{\prime}$ is divided into two parts by $c$, where denote the part with
endpoints $c^{\prime}\cap\gamma_{i}$ and $c^{\prime}(1^{-})$ by
$c_{\mathrm{II}}^{\prime}$;

    * –

$c_{\mathrm{II}}^{\prime\prime}\simeq\mathrm{prot}_{c}(c^{\prime}_{\mathrm{II}})$
$($Definition 3.5$)$.

Then the quotient module $M(c)/M(c^{\prime})$ is isomorphic to
$M(c_{\mathrm{I}}^{\prime\prime}\oplus c_{\mathrm{II}}^{\prime\prime})$ (see
FIGURE 10 $(1)$).

$c$$c^{\prime}$$c_{\mathrm{I}}^{\prime\prime}$$c_{\mathrm{II}}^{\prime\prime}$(1)

$c$$c^{\prime\prime}$$c_{\mathrm{I}}^{\prime}$$c_{\mathrm{II}}^{\prime}$(2)

Figure 10: The descriptions of short exact sequences given in Propositions 3.8
and 3.11.

###### Proof.

Similar to the proof of Lemma 3.7, we have that $M(c)/M(c^{\prime})$
corresponds to two permissible curves

$\tilde{c}_{\mathrm{I}}^{\prime\prime}=c_{1}\cdots
c_{m-1}c_{m}^{\mathfrak{T}}\ {\rm and}\
\tilde{c}_{\mathrm{II}}^{\prime\prime}=c_{n+1}^{\mathfrak{T}}c_{n+2}\cdots
c_{r}.$

We can see that $\tilde{c}_{\mathrm{I}}^{\prime\prime}$ consecutively crosses
arcs $\gamma_{1},\cdots,\gamma_{m-1}$ and
$\tilde{c}_{\mathrm{II}}^{\prime\prime}$ consecutively crosses arcs
$\gamma_{n+1},\cdots,\gamma_{r-1}$ ($1\leq m\leq i\leq n\leq r-1$) (see FIGURE
9). Thus $\tilde{c}_{\mathrm{I}}^{\prime\prime}$ and
$\tilde{c}_{\mathrm{II}}^{\prime\prime}$ lie in the equivalence class
$[c_{\mathrm{I}}^{\prime\prime}]$ and $[c_{\mathrm{II}}^{\prime\prime}]$,
respectively, and therefore $M(c)/M(c^{\prime})\cong
M(\tilde{c}_{\mathrm{I}}^{\prime\prime}\oplus\tilde{c}_{\mathrm{II}}^{\prime\prime})\cong
M(c_{\mathrm{I}}^{\prime\prime}\oplus c_{\mathrm{II}}^{\prime\prime})$. ∎

######  Remark 3.9.

If $\gamma_{1}=\gamma_{m}=\gamma_{i}$ (respectively,
$\gamma_{i}=\gamma_{n}=\gamma_{r}$), then
$c^{\prime}=c_{1}^{\mathfrak{T}}c_{2}\cdots c_{n}c_{n+1}^{\mathfrak{T}}\simeq
c_{1}c_{2}\cdots c_{n}c_{n+1}^{\mathfrak{T}}$ (respectively,
$c^{\prime}=c_{m}^{\mathfrak{T}}c_{m+1}\cdots
c_{r-1}c_{r}^{\mathfrak{T}}\simeq c_{m}^{\mathfrak{T}}c_{m+1}\cdots
c_{r-1}c_{r}$), and thus $c_{\mathrm{I}}^{\prime\prime}$ (respectively,
$c_{\mathrm{II}}^{\prime\prime}$) is zero.

The following two results are dual to Lemma 3.7 and Proposition 3.8,
respectively.

######  Lemma 3.10.

Let $A$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Let $c=c_{1}\cdots
c_{r}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ be a
permissible curve consecutively crossing $\gamma_{1},\cdots\gamma_{r-1}$ such
that the endpoints $p$ and $q$ of $\gamma_{i}$ $(1\leq i\leq r-1)$ are on the
right and left of $c$, respectively. Then for any permissible curve
$c^{\prime\prime}$ consecutively crossing $\gamma_{m},\cdots,\gamma_{n}$
$(1\leq m<i<n\leq r-1)$ such that

  * •

$\gamma_{m},\cdots,\gamma_{i}$ have a common endpoint
$p=p(\gamma_{m},\gamma_{m+1},c)=\cdots=p(\gamma_{i-1},\gamma_{i},c)$, and

  * •

$\gamma_{i},\cdots,\gamma_{n}$ have a common endpoint
$q=p(\gamma_{i},\gamma_{i+1},c)=\cdots=p(\gamma_{n-1},\gamma_{n},c)$,

we have that $M(c^{\prime\prime})$ is a quotient module of $M(c)$.

######  Proposition 3.11.

Keeping the notations in Lemma 3.10, we have that
$c^{\prime\prime}=c_{m}^{\mathfrak{T}}c_{m+1}\cdots
c_{n}c_{n+1}^{\mathfrak{T}}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$
crosses $\gamma_{i}$ and that $c$ and $c^{\prime\prime}$ have an intersection.
Assume that

  * •

$c_{\mathrm{I}}^{\prime}$ is a permissible curve which is given as follows:

    * –

$c^{\prime\prime}$ is divided into two parts, where denote the part with
endpoints $c^{\prime\prime}(0^{+})$ and $c^{\prime\prime}\cap\gamma_{i}$ by
$c_{\mathrm{I}}^{\prime\prime}$;

    * –

$c_{\mathrm{I}}^{\prime}\simeq\mathrm{nrot}_{c}(c_{\mathrm{I}}^{\prime\prime})$
$($Definition 3.5$)$;

  * •

and $c_{\mathrm{II}}^{\prime}$ is a permissible curve which is given as
follows:

    * –

$c^{\prime\prime}$ is divided into two parts, where denote the part with
endpoints $c^{\prime\prime}\cap\gamma_{i}$ and $c^{\prime\prime}(1^{-})$ by
$c_{\mathrm{II}}^{\prime\prime}$;

    * –

$c_{\mathrm{II}}^{\prime}\simeq\mathrm{nrot}_{c}(c_{\mathrm{II}}^{\prime\prime})$
$($Definition 3.5$)$.

Then $M(c^{\prime\prime})$ is isomorphic to
$M(c)/M(c_{\mathrm{I}}^{\prime}\oplus c_{\mathrm{II}}^{\prime})$ (see FIGURE
10 $(2)$).

By Propositions 3.6, 3.8 and 3.11, we get the following theorem.

######  Theorem 3.12.

Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface.

  1. (1)

If $c$, $c^{\prime}$ and $c^{\prime\prime}$ are permissible curves satisfying
the conditions in Proposition 3.6, then there is a short exact sequence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0.$

  2. (2)

If $c$ and $c^{\prime}$ are permissible curves satisfy the conditions in Lemma
3.7, and if $c_{\mathrm{I}}^{\prime\prime}$ and
$c_{\mathrm{II}}^{\prime\prime}$ are permissible curves satisfying the
conditions in Proposition 3.8, then there is a short exact sequence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c_{\mathrm{I}}^{\prime\prime}\oplus
c_{\mathrm{II}}^{\prime\prime})\longrightarrow 0.$

  3. (3)

(The dual of (2)) If $c$ and $c^{\prime\prime}$ are permissible curves
satisfying the conditions in Lemma 3.10, and if $c_{\mathrm{I}}^{\prime}$ and
$c_{\mathrm{II}}^{\prime}$ are permissible curves satisfying the conditions in
Proposition 3.11, then there is a short exact sequence

$0\longrightarrow M(c_{\mathrm{I}}^{\prime}\oplus
c_{\mathrm{II}}^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0.$

##  4 Special modules

###  4.1 Simple modules and top (respectively, socle) of indecomposable
modules

A permissible curve in $\mathcal{S}_{A}$ corresponding to a simple $A$-module
is a permissible curve crossing a unique arc in $\Gamma_{A}$, we call it
simple. The top (respectively, socle) of an $A$-module is semi-simple, thus we
can use a family of simple permissible curves to describe the top
(respectively, socle) of any $A$-module. Moreover, for a permissible curve $c$
consecutively crossing arcs $\gamma$ and $\gamma^{\prime}$, recall that
$p(\gamma,\gamma^{\prime},c)$ is the marked point lying in
$\gamma\cap\gamma^{\prime}$ such that it is the vertex of the triangle decided
by $\gamma$, $\gamma^{\prime}$ and the segment given by $\gamma$ and
$\gamma^{\prime}$ cutting $c$. We will use this notation frequently in the
sequel.

######  Lemma 4.1.

Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Let $c=c_{1}c_{2}\cdots c_{r}$ be a non-trivial permissible
$($non-closed$)$ curve in $\mathcal{S}_{A}$ consecutively crossing arcs
$\gamma_{1},\gamma_{2},\cdots,\gamma_{r-1}$ $($note: $c_{i}$ is the segment
between $\gamma_{i-1}$ and $\gamma_{i}$, $c_{1}$ is the segment between
$\partial\mathcal{S}_{A}$ and $\gamma_{1}$, and $c_{r}$ is the segment between
$\gamma_{r-1}$ and $\partial\mathcal{S}_{A}$$)$. If there exists some $i$ with
$1\leq i\leq r$ such that one of the following conditions is satisfied:

  * •

Case 1 (respectively, Case 1′): $p(\gamma_{i-1},\gamma_{i},c)$ is on the right
(respectively, left) of $c$, and $p(\gamma_{i},\gamma_{i+1},c)$ is on the left
(respectively, right) of $c$ $(2\leq i\leq r-2)$;

  * •

Case 2 (respectively, Case 2′): $c_{i}=c_{1}$ is an end segment such that
$p(\gamma_{1},\gamma_{2},c)$ is on the left (respectively, right) of $c$;

  * •

Case 3 (respectively, Case 3′): $c_{i}=c_{r}$ is an end segment such that
$p(\gamma_{r-2},\gamma_{r-1},c)$ is on the right (respectively, left) of $c$,

then $M(c_{\mathrm{simp}}^{\gamma_{i}})\leq_{\oplus}{\rm top}M(c)$
$($respectively,
$M(c_{\mathrm{simp}}^{\gamma_{i}})\leq_{\oplus}\mathrm{soc}M(c))$, where
$c_{\mathrm{simp}}^{\gamma_{i}}$ is the simple permissible curve crossing
$\gamma_{i}$ and $M:\mathfrak{P}(\mathcal{S}_{A})\to\mathrm{ind}\bmod A$ is
the bijection given in Remark 3.2.

$c$$\bullet$$p(\gamma_{i},\gamma_{i+1},c)$$\bullet$$p(\gamma_{i-1},\gamma_{i},c)$$\gamma_{i-1}$$\gamma_{i+1}$$\gamma_{i}$Case
1Sentences

$c$$\bullet$$p(\gamma_{1},\gamma_{2},c)$$\bullet$$\boldsymbol{\times}$$\gamma_{2}$$b$$\gamma_{1}$Case
2: $b$ is either an arcor a side on $\partial\mathcal{S}_{A}$;

$c$$\bullet$$\bullet$$p(\gamma_{r-2},\gamma_{r-1},c)$$\gamma_{r-2}$$\boldsymbol{\times}$$b$$\gamma_{r-1}$Case
3: $b$ is either an arcor a side on $\partial\mathcal{S}_{A}$;

Figure 11: $M(c_{\mathrm{simp}}^{\gamma_{i}})$ is a direct summand of
$\mathrm{top}M(c)$ (the point “$\boldsymbol{\times}$” is either a marked point
or an extra point), where $i=1$ in Case 2 and $i=r-1$ in Case 3. Cases
$1^{\prime}$, $2^{\prime}$ and $3^{\prime}$ are dual.

###### Proof.

We only prove the case for top, and the case for socle is dual. For any string
module $M$, the source, denoted by $i$, of the string $s$ corresponding to $M$
is one of the following three types (we denote by $\gamma_{i}$ the arc
corresponding to $i$):

$\text{Type 1:}\ \ \cdots\longleftarrow i\longrightarrow\cdots\ ;\text{Type
2:}\ \ i\longrightarrow\cdots\ ;\ \text{and}\ \text{Type 3:}\ \
\cdots\longleftarrow i\ .$

Then the module $S(i)$, the simple module corresponding to the vertex $i$, is
a direct summand of ${\rm top}M$, that is, $S(i)\leq_{\oplus}{\rm top}M$. On
the other hand, the permissible curve $c=c_{1}\cdots c_{r}$ corresponding to
$s$ satisfies one of the conditions in Cases 1–3 accordingly (in Cases 2 and
3, we have $i=1$ and $i=r-1$, respectively). Thus $S(i)$ corresponds to the
permissible curve $c_{\mathrm{simp}}^{\gamma_{i}}$ and
$M(c_{\mathrm{simp}}^{\gamma_{i}})\leq_{\oplus}{\rm top}M\cong{\rm top}M(c)$.
∎

######  Lemma 4.2.

Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Let $c=c_{1}c_{2}\cdots c_{r}$ be a non-trivial permissible
closed curve in $\mathcal{S}_{A}$ consecutively crossing arcs
$\gamma_{1},\gamma_{2},\cdots,\gamma_{r-1},\gamma_{r}=\gamma_{0}$ ($c_{i}$ is
the segment between $\gamma_{\overline{i-1}}$ and $\gamma_{\overline{i}}$,
where $\overline{i}$ is $i$ modulo $r$), and let
$\varphi\in\mathrm{Aut}(k^{m})$ with $m\geq 1$ be a Jordan block. If there
exists some $i$ with $1\leq i\leq r$ such that the following condition is
satisfied:

  * •

Case RL (respectively, Case LR):
$p(\gamma_{\overline{i-1}},\gamma_{\overline{i}},c)$ is on the right
(respectively, left) of $c$ and
$p(\gamma_{\overline{i}},\gamma_{\overline{i+1}},c)$ is on the left
(respectively, right) of $c$,

then $M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus
m}\leq_{\oplus}\mathrm{top}(M(c,\varphi))$ $($respectively,
$M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus
m}\leq_{\oplus}\mathrm{soc}(M(c,\varphi)))$.

###### Proof.

For any band module $M$, a source, denoted by $i$, of the band corresponding
to $M$ must be Type 1 in the proof of Lemma 4.1. Note that
$M(c,\varphi)e_{i}\cong k^{m}$ is $k$-linearly isomorphic to
$M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus m}$, where $e_{i}$ is the trivial
path corresponding to $i\in Q_{0}$. Now, similar to the proof of Lemma 4.1, we
have $M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus
m}\leq_{\oplus}\mathrm{top}M(c,\varphi)$. The case for socle is dual. ∎

As a consequence of Lemmas 4.1 and 4.2, we get the following proposition.

######  Proposition 4.3.

  1.   2. (1)

Under the assumptions in Lemma 4.1, let

$\displaystyle\mathfrak{I}$ $\displaystyle=\\{1\leq i\leq r-1\mid i\text{
satisfies one of Cases 1, 2 and 3 in Lemma \ref{lemm. tops of string mods.}
}\\},$ $\displaystyle(respectively,\mathfrak{I}$ $\displaystyle=\\{1\leq i\leq
r-1\mid i\text{ satisfies one of Cases 1${}^{\prime}$, 2${}^{\prime}$ and
3${}^{\prime}$ in Lemma \ref{lemm. tops of string mods.} }\\}).$

Then
$\mathrm{top}M(c)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}})$
(respectively,
$\mathrm{soc}M(c)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}}))$.

  3. (2)

Under the assumptions in Lemma 4.2, let

$\displaystyle\mathfrak{I}$ $\displaystyle=\\{1\leq i\leq r-1\mid i\text{
satisfies Case RL in Lemma \ref{lemm. tops of band mods.} }\\},$
$\displaystyle(respectively,\mathfrak{I}$ $\displaystyle=\\{1\leq i\leq
r-1\mid i\text{ satisfies Case LR in Lemma \ref{lemm. tops of band mods.}
}\\}).$

Then
$\mathrm{top}M(c,\varphi)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus
m}$ (respectively,
$\mathrm{soc}M(c,\varphi)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus
m}$).

###### Proof.

(1) Let $M\in\bmod A$ be string and

$\mathfrak{I}^{\prime}=\\{1\leq i\leq r-1\mid i\text{ is a source which is one
of Types 1, 2 and 3 in the proof of Lemma \ref{lemm. tops of string
mods.}}\\}.$

Then
$\mathrm{top}M\cong\mathrm{top}M(c)\cong\bigoplus\limits_{i\in\mathfrak{I}^{\prime}}S(i)\cong
M(\bigoplus\limits_{i\in\mathfrak{I}}c_{\mathrm{simp}}^{\gamma_{i}})\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}})$
($\bigoplus\limits_{i\in\mathfrak{I}}c_{\mathrm{simp}}^{\gamma_{i}}$ is the
formal direct sum of $c_{\mathrm{simp}}^{\gamma_{i}}$, see Definition 3.3).

(2) It is similar to (1). ∎

###  4.2 Projective modules and projective covers

Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface of $A$. If $s=\alpha_{1}\alpha_{2}\cdots\alpha_{n}$
($\alpha_{i}\in Q_{0}\cup Q_{0}^{-}$) is a string corresponding to a
projective module, then $s$ is one of the following cases:

  * •

$s$ is left-maximal inverse, that is, $s$ is inverse and $\alpha^{-}s\in I$
for any $\alpha\in Q_{0}$ with $t(\alpha_{1}^{-})=s(\alpha)$;

  * •

$s$ is right-maximal direct, that is, $s$ is direct and $s\alpha\in I$ for any
$\alpha\in Q_{0}$ with $t(\alpha_{n})=s(\alpha)$;

  * •

there exists a unique $i$ with $1\leq i\leq n-1$ such that
$\alpha_{1}\cdots\alpha_{i}$ is left-maximal inverse and
$\alpha_{i+1}\cdots\alpha_{n}$ is right-maximal direct.

Therefore, a permissible curve
$c:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ (which
consecutively crosses arcs $\gamma_{1},\cdots,\gamma_{r-1}$) corresponds to an
indecomposable projective module by
$M:\mathfrak{P}(\mathcal{S}_{A})\to\mathrm{ind}(\bmod A)$ defined in Remark
3.2 if and only if $c$ is a permissible curve such that the following
conditions are satisfied:

  * •

(PQC) (Projective Quotient Condition) There exits $i$ with $1\leq i\leq r-1$
such that

    * –

(PQC 1) $m:=p(\gamma_{1},\gamma_{2},c)=\cdots=p(\gamma_{i-1},\gamma_{i},c)$
which is on the right of $c$; and

    * –

(PQC 2)
$m^{\prime}:=p(\gamma_{i},\gamma_{i+1},c)=\cdots=p(\gamma_{r-2},\gamma_{r-1},c)$
which is on the left of $c$;

  * •

(MC) (Maximal Conditions) There are no arcs $\gamma$ and $\gamma^{\prime}$
such that

    * –

(LMC) $\gamma,\gamma_{1},\gamma_{2},\cdots,\gamma_{i}$ have a common endpoint
$m$ and they surround $m$ in the clockwise order; and

    * –

(RMC) $\gamma_{i},\gamma_{i+1},\cdots,\gamma_{n},\gamma^{\prime}$ have a
common endpoint $m^{\prime}$ and they surround $m^{\prime}$ in the
counterclockwise order.

FIGURE 12 provides an example of a permissible curve corresponding to a
projective module.

$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{c}$$\cdots$$\cdots$
Figure 12: Each point $\bullet$ is either a marked point or an extra point.
The red curve $c$ is a permissible curve satisfying (PQC) and (MC), thus the
string module corresponding to $c$ is a projective module.

######  Definition 4.4.

(Projective permissible curves) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. For an arc $\gamma\in\Gamma_{A}$, a permissible curve $c$
consecutively crossing arcs $\gamma_{1},\cdots,\gamma_{r-1}$ is called a
projective permissible curve of $\gamma$, denoted by
$c_{\mathrm{proj}}^{\gamma}$, if $c$ is decided by the pair $(c_{1},c_{2})$ of
two curves $c_{1}:(0,1]\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$
and $c_{2}:[0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$, where

  * •

there is a unique $i$ with $1\leq i\leq r-1$ such that $\gamma=\gamma_{i}$;

  * •

$c(0^{+})=c_{1}(0^{+})$, $c(1^{-})=c_{2}(1^{-})$,
$c_{1}(1)=c_{2}(0)=\gamma\cap c$;

  * •

$c$ is obtained by connecting $c_{1}$ and $c_{2}$, that is, $c=c_{1}\cup
c_{2}$; and

  * •

$c_{1}$ and $c_{2}$ consecutively cross arcs $\gamma_{1},\cdots,\gamma_{i}$
and $\gamma_{i},\cdots,\gamma_{r-1}$ such that (PQC 1, LMC) and (PQC 2, RMC)
are satisfied, respectively.

We use $\mathfrak{P}_{\mathrm{proj}}(\mathcal{S}_{A})$ to denote the set of
all equivalence classes of projective permissible curves. Obviously, we have
$\mathfrak{P}_{\mathrm{proj}}(\mathcal{S}_{A})\subseteq{\mathrm{Eq}\text{-}\mathrm{cls}}(\mathcal{S}_{A})$.

######  Lemma 4.5.

(Projective modules) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Then there exists a one-to-one mapping between the equivalence
classes of projective permissible curves and the isomorphic classes of
indecomposable projective modules which is given by

$\displaystyle
M|_{\mathfrak{P}_{\mathrm{proj}}(\mathcal{S}_{A})}:\mathfrak{P}_{\mathrm{proj}}(\mathcal{S}_{A})\to\mathrm{ind}(\mathrm{proj}A),c\mapsto
M(c).$

###### Proof.

A permissible $c$ lies in $\mathfrak{P}_{\mathrm{proj}}(\mathcal{S}_{A})$ if
and only if it corresponds to a string $s$ which is the connection of a left-
maximal (trivial) inverse string $s_{1}$ and a right-maximal (trivial) direct
string $s_{2}$, that is, $s=\alpha_{1}\alpha_{2}\cdots\alpha_{n}$ is of the
form

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Thus the string module corresponding to $s$ is projective. (If
$\alpha_{1}\cdots\alpha_{i}$ (respectively, $\alpha_{i+1}\cdots\alpha_{n}$) is
trivial, then the module corresponding to $s$ is $P(v)\cong e_{v}A$, where
$v\in Q_{0}$ is the source of $\alpha_{i+1}$ (respectively, $\alpha_{i}$) and
$e_{v}$ is the primitive idempotent corresponding to the vertex $v$). ∎

Now we consider the projective cover of a module by marked ribbon surfaces of
gentle algebras.

######  Theorem 4.6.

(Projective covers of indecomposable modules) Let $A=kQ/I$ be a gentle algebra
and $\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Let $M=M(c)$ (respectively, $M=M(c,\varphi)$) be a string
module (respectively, band module) which corresponds to the permissible curve
$c$ (respectively, the permissible closed curve $c$ with a Jordan block over a
$k$-linear space $V$). Then

$P(M)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{proj}}^{\gamma_{i}})\
\ \ \
(respectively,P(M)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{proj}}^{\gamma_{i}})^{\oplus\dim_{k}V}),$

where $\\{\gamma_{i}\mid i\in\mathfrak{I}\\}$ and $\mathfrak{I}$ are the same
as that in Lemma 4.1 and Proposition 4.3$(2)$ (respectively, Lemma 4.2 and
Proposition 4.3$(2)$), respectively.

###### Proof.

For each indecomposable module $M$, its projective cover $P(M)$ is isomorphic
to the projective cover $P(\mathrm{top}M)$ of its top. Thus we only need to
consider the projective cover of each simple module. Note that, for each
simple permissible curve $c_{\mathrm{simp}}^{\gamma}$, we have
$P(M(c_{\mathrm{simp}}^{\gamma}))=M(c_{\mathrm{proj}}^{\gamma})$. Now suppose
$\mathrm{top}M\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}})$
(when $M$ is string, each $\gamma_{i}$ correspond the vertex of $Q$ such that
one of three cases in Lemma 4.1 holds; when $M$ is band, each $\gamma_{i}$
corresponds to the vertex of $Q$ such that the condition in Lemma 4.2 holds).
If $M$ is a string module, then

$\displaystyle P(M)$ $\displaystyle\cong P(\mathrm{top}M)\cong
P(\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}}))\cong\bigoplus\limits_{i\in\mathfrak{I}}P(M(c_{\mathrm{simp}}^{\gamma_{i}}))\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{proj}}^{\gamma_{i}}).$

If $M$ is a band module, then

$\displaystyle P(M)$ $\displaystyle\cong P(\mathrm{top}M)\cong
P(\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus\dim_{k}V})\cong\bigoplus\limits_{i\in\mathfrak{I}}P(M(c_{\mathrm{simp}}^{\gamma_{i}})^{\oplus\dim_{k}V})\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{proj}}^{\gamma_{i}})^{\oplus\dim_{k}V}.$

∎

###  4.3 Injective modules and injective envelopes

If a permissible curve
$c:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ consecutively
crosses arcs $\gamma_{1},\cdots,\gamma_{r-1}$ and corresponds to an injective
module by
$M:{\mathrm{Eq}\text{-}\mathrm{cls}}(\mathcal{S}_{A})\cup{\mathrm{Htp}\text{-}\mathrm{cls}}(\mathcal{S}_{A})\to\mathrm{ind}(\bmod
A)$, then there is a unique $i$ with $1\leq i\leq n-1$ such that the following
conditions are satisfied:

  * •

(ISC) (Injective Submodule Condition) There exits $i$ with $1\leq i\leq r-1$
such that

    * –

(ISC 1) $m:=p(\gamma_{1},\gamma_{2},c)=\cdots=p(\gamma_{i-1},\gamma_{i},c)$
which is on the left of $c$; and

    * –

(ISC 2)
$m^{\prime}:=p(\gamma_{i},\gamma_{i+1},c)=\cdots=p(\gamma_{r-2},\gamma_{r-1},c)$
which is on the right of $c$;

  * •

(MC′) There are no arcs $\gamma$ and $\gamma^{\prime}$ such that

    * –

(LMC′) $\gamma,\gamma_{1},\gamma_{2},\cdots,\gamma_{i}$ have a common endpoint
$m$ and they surround $m$ in the counterclockwise order; and

    * –

(RMC′) $\gamma_{i},\gamma_{i+1},\cdots,\gamma_{n},\gamma^{\prime}$ have a
common endpoint $m^{\prime}$ and they surround $m^{\prime}$ in the clockwise
order.

We define injective permissible curves and
$\mathfrak{P}_{\mathrm{inj}}(\mathcal{S}_{A})$ as follows.

######  Definition 4.7.

(Injective permissible curves) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. For an arc $\gamma\in\Gamma_{A}$, a permissible curve $c$
consecutively crossing arcs $\gamma_{1},\cdots,\gamma_{r-1}$ is called an
injective permissible curve of $\gamma$, denoted by
$c_{\mathrm{inj}}^{\gamma}$, if $c$ is decided by the pair $(c_{1},c_{2})$ of
two curves $c_{1}:(0,1]\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$
and $c_{2}:[0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$, where

  * •

there is a unique $i$ with $1\leq i\leq r-1$ such that $\gamma=\gamma_{i}$;

  * •

$c(0^{+})=c_{1}(0^{+})$, $c(1^{-})=c_{2}(1^{-})$,
$c_{1}(1)=c_{2}(0)=\gamma\cap c$;

  * •

$c$ is obtained by connecting $c_{1}$ and $c_{2}$, i.e., $c=c_{1}\cup c_{2}$;
and

  * •

$c_{1}$ and $c_{2}$ consecutively crosses arcs $\gamma_{1},\cdots,\gamma_{i}$
and $\gamma_{i},\cdots,\gamma_{r-1}$ such that (ISC 1, LMC′) and (ISC 2, RMC′)
are satisfied, respectively.

We use $\mathfrak{P}_{\mathrm{inj}}(\mathcal{S}_{A})$ to denote the set of all
equivalence classes of injective permissible curves. Obviously, we have
$\mathfrak{P}_{\mathrm{inj}}(\mathcal{S}_{A})\subseteq{\mathrm{Eq}\text{-}\mathrm{cls}}(\mathcal{S}_{A})$.
The following results are dual to Lemma 4.5 and Theorem 4.6.

######  Lemma 4.8.

(Injective modules) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Then there exists a one-to-one mapping between the equivalence
classes of injective permissible curves and the isomorphic classes of
indecomposable injective modules which is given by

$\displaystyle\mathfrak{P}|_{\mathrm{inj}}(\mathcal{S}_{A}):\mathfrak{P}_{\mathrm{inj}}(\mathcal{S}_{A})\to\mathrm{ind}(\mathrm{inj}A),c\mapsto
M(c).$

######  Theorem 4.9.

(Injective envelopes of indecomposable modules) Let $A=kQ/I$ be a gentle
algebra and $\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its
marked ribbon surface. Let $M=M(c)$ (respectively, $M=M(c,\varphi)$) be a
string module (respectively, band module) which corresponds to the permissible
curve $c$ (respectively, the permissible closed curve $c$ with a Jordan block
over a $k$-linear space $V$). Then

$E(M)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{inj}}^{\gamma_{i}})\
\ \ \
\big{(}\text{respectively},E(M)\cong\bigoplus\limits_{i\in\mathfrak{I}}M(c_{\mathrm{inj}}^{\gamma_{i}})^{\oplus\dim_{k}V}\big{)},$

where $\\{\gamma_{i}\mid i\in\mathfrak{I}\\}$ and $\mathfrak{I}$ are the same
as that in Lemma 4.1 and Proposition 4.3$(2)$ (respectively, Lemma 4.2 and
Proposition 4.3$(2)$), respectively.

##  5 Global dimension

###  5.1 The descriptions of projective and injective resolutions of simple
modules in geometric models

In this subsection, we consider the minimal projective and injective
resolutions of simple modules through marked ribbon surfaces.

######  Definition 5.1.

(P-condition and I-condition) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Let $\gamma$ be an arc in $\Gamma_{A}$ with two endpoints $p$,
$q$. We say that $\gamma$ satisfies the P-condition at $p$ (respectively,
I-condition at $p$) if the following conditions are satisfied:

  * •

there are arcs $\gamma_{1},\cdots,\gamma_{r}$ such that $p$ is the common
endpoint of $\gamma$ and $\gamma_{i}$ (for any $1\leq i\leq r$), and

  * •

$\gamma,\gamma_{1},\cdots,\gamma_{r}$ (respectively,
$\gamma_{1},\cdots,\gamma_{r},\gamma$) surround $p$ in the counterclockwise
order.

Furthermore, we say that $\gamma$ satisfies the P-condition (respectively,
I-condition) if the P-conditions (respectively, I-conditions) at $p$ and $q$
are satisfied.

######  Remark 5.2.

If an arc $\gamma$ with endpoints $p$ and $q$ satisfies the P-condition at $p$
or $q$, then $M(c_{\mathrm{proj}}^{\gamma})$ is not simple.

We provide an example in FIGURE 13. In this figure, $\gamma$ satisfies the
P-condition.

$\gamma$$p$$q$$\gamma_{1}$$\gamma_{2}$$\cdots$$\gamma_{r}$$\gamma_{1}^{\prime}$$\gamma_{2}^{\prime}$$\cdots$$\gamma_{s}^{\prime}$
Figure 13: The arc $\gamma$ satisfying the P-condition.

######  Lemma 5.3.

Under the conditions in Definition 5.1, let $c=c_{\mathrm{simp}}^{\gamma}$ be
a simple permissible curve, where $\gamma$ is an arc in $\Gamma_{A}$ with
endpoints $p$ and $q$ such that the P-condition is satisfied at $p$
(respectively, $q$) but not at $q$ (respectively, $p$). Then the kernel of the
projective cover
$p(M(c_{\mathrm{simp}}^{\gamma})):M(c_{\mathrm{proj}}^{\gamma})\twoheadrightarrow
M(c_{\mathrm{simp}}^{\gamma})$ is the module $M(c^{\prime})$, where
$c^{\prime}$ is the permissible curve consecutively crossing
$\gamma_{1},\cdots,\gamma_{r}$ $($respectively,
$\gamma_{1}^{\prime},\cdots,\gamma_{s}^{\prime})$.

###### Proof.

We only prove the case for $\gamma$ satisfying the P-condition at $p$ but not
at $q$, and the argument for the other case is similar. As shown in FIGURE 14,
let $\gamma=\tilde{\gamma}_{1}$, and $\gamma_{j}=\tilde{\gamma}_{j+1}$ ($1\leq
j\leq r$). Then we may suppose
$c=c_{\mathrm{proj}}^{\gamma}=c_{\mathrm{proj}}^{\tilde{\gamma}_{1}}\simeq
c_{1}c_{2}\cdots c_{r+2}$, where each $c_{t}$ ($2\leq t\leq r+1$) is a segment
between $\tilde{\gamma}_{t-1}$ and $\tilde{\gamma}_{t}$, and $c_{1}$,
$c_{r+2}$ are end segments of $c$.

$c_{\mathrm{simp}}^{\gamma}$$c_{\mathrm{proj}}^{\gamma}$$\boldsymbol{\times}$$c^{1}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\boldsymbol{\times}$$\bullet$$\boldsymbol{\times}$$\boldsymbol{\times}$$\gamma$$p$$q$$\gamma_{1}=\gamma^{1}_{1}$$\gamma_{2}=\gamma^{1}_{2}$$\cdots$$\gamma_{r}=\gamma^{1}_{r}$
Figure 14:  The projective cover
$p(M(c_{\mathrm{simp}}^{\gamma})):M(c_{\mathrm{proj}}^{\gamma})\to
M(c_{\mathrm{simp}}^{\gamma})$ of $M(c_{\mathrm{simp}}^{\gamma})$ and its
kernel $\mathrm{Ker}\ p(M(c_{\mathrm{simp}}^{\gamma}))$ $\cong$ $M(c^{1})$,
where
$c^{1}\simeq\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma}}(c_{\mathrm{simp}}^{\gamma})$
by Proposition 3.11.

Let
$c^{\prime\prime}=c_{\mathrm{simp}}^{\gamma}=c_{\mathrm{simp}}^{\tilde{\gamma}_{1}}$
and let $c^{\prime}$ be the permissible curve consecutively crossing
$\gamma_{1},\cdots,\gamma_{r}$. Then $c^{\prime\prime}\simeq
c_{1}c_{2}^{\mathfrak{T}}$ and
$c^{\prime}\simeq\mathrm{nrot}_{c}(c^{\prime\prime})$ (by Definition 3.4) and
the positional relationship of $c$, $c^{\prime}$ and $c^{\prime\prime}$
satisfies the conditions in Proposition 3.6 (that is, $c$ consecutively
crosses $\tilde{\gamma}_{2}$ and $\tilde{\gamma}_{1}$, $c^{\prime}$ crosses
only $\tilde{\gamma}_{2}$ and $c^{\prime\prime}$ crosses $\tilde{\gamma}_{1}$,
where $\tilde{\gamma}_{2}$ and $\tilde{\gamma}_{1}$ are viewed as arcs
$\gamma^{\prime}$ and $\gamma^{\prime\prime}$ in Proposition 3.6,
respectively). Thus we get the following exact sequence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0,$

by Theorem 3.12(1). ∎

######  Lemma 5.4.

Under the conditions in Definition 5.1, let $c=c_{\mathrm{simp}}^{\gamma}$ be
a simple permissible curve, where $\gamma$ is an arc in $\Gamma_{A}$ with
endpoints $p$ and $q$ such that the P-condition is satisfied. Then the kernel
of the projective cover
$p(M(c_{\mathrm{simp}}^{\gamma})):M(c_{\mathrm{proj}}^{\gamma})\twoheadrightarrow
M(c_{\mathrm{simp}}^{\gamma})$ is the module $M(c_{\mathrm{I}}^{\prime}\oplus
c_{\mathrm{II}}^{\prime})$, where $c_{\mathrm{I}}^{\prime}$ consecutively
crosses $\gamma_{1}^{\prime},\cdots,\gamma_{s}^{\prime}$ and $c_{2}^{\prime}$
consecutively crosses $\gamma_{1},\cdots,\gamma_{r}$.

###### Proof.

Let $\gamma_{i}^{\prime}=\tilde{\gamma}_{s-i+1}$ ($1\leq i\leq s$),
$\gamma=\tilde{\gamma}_{s+1}$, and $\gamma_{j}=\tilde{\gamma}_{s+j+1}$ ($1\leq
j\leq r$). Then we may suppose
$c=c_{\mathrm{proj}}^{\gamma}=c_{\mathrm{proj}}^{\tilde{\gamma}_{s+1}}\simeq
c_{1}c_{2}\cdots c_{s+r+2}$ where each $c_{t}$ ($2\leq t\leq s+r+1$) is a
segment between $\tilde{\gamma}_{t-1}$ and $\tilde{\gamma}_{t}$, and $c_{1}$,
$c_{s+r+2}$ are end segments of $c$. Let
$c^{\prime\prime}=c_{\mathrm{simp}}^{\gamma}$, and let $x$ be the intersection
of $c$ and $c^{\prime\prime}$ (note that $\sharp(c\cap c^{\prime\prime})=1$ in
this case) such that
$c_{\mathrm{I}}^{\prime\prime},c_{\mathrm{II}}^{\prime\prime}$ are the curves
obtained by $x$ dividing $c^{\prime\prime}$. Then the positional relationship
of
$c,\mathrm{prot}_{c}(c_{\mathrm{I}}^{\prime\prime}),\mathrm{prot}_{c}(c_{\mathrm{II}}^{\prime\prime})$
and $c^{\prime\prime}$ satisfies the conditions in Proposition 3.11. We get
the following exact sequence

$0\longrightarrow
M(\mathrm{prot}_{c}(c_{\mathrm{I}}^{\prime\prime})\oplus\mathrm{prot}_{c}(c_{\mathrm{II}}^{\prime\prime}))\longrightarrow
M(c)\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{p(M(c_{\mathrm{simp}}^{\gamma}))}M(c^{\prime\prime})\longrightarrow
0$

by Theorem 3.12(3). Notice that
$\mathrm{prot}_{c}(c_{\mathrm{I}}^{\prime\prime})\simeq
c_{\mathrm{I}}^{\prime}$ and
$\mathrm{prot}_{c}(c_{\mathrm{II}}^{\prime\prime})\simeq
c_{\mathrm{II}}^{\prime}$, thus

$\mathrm{Ker}p(M(c_{\mathrm{simp}}^{\gamma}))=M(\mathrm{prot}_{c}(c_{\mathrm{I}}^{\prime\prime})\oplus\mathrm{prot}_{c}(c_{\mathrm{II}}^{\prime\prime}))\cong
M(c_{\mathrm{I}}^{\prime}\oplus c_{\mathrm{II}}^{\prime}).$

∎

######  Lemma 5.5.

Under the conditions Definition 5.1 and Lemma 5.3, let $c^{1}$ be the
permissible curve $c^{\prime}$ corresponding to
$\mathrm{Ker}p(M(c_{\mathrm{simp}}^{\gamma}))$, where
$p(M(c_{\mathrm{simp}}^{\gamma}))$ is given by Lemma 5.3. Then
$P(M(c^{1}))=P(M(c_{\mathrm{proj}}^{\gamma_{1}}))$ is indecomposable, and the
kernel of $P(M(c^{1}))\twoheadrightarrow M(c^{1})$ is
$M(\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma_{1}}}(c^{1}))$. To be precise, if
$\gamma^{2}_{1},\gamma^{2}_{2},\cdots,\gamma^{2}_{s}$ are arcs with the
endpoint of $\gamma_{1}$ which is not $p$ such that
$\gamma,\gamma_{1},\gamma^{2}_{1}$ are sides of the same elementary polygon of
$\mathcal{S}_{A}$, then
$\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma_{1}}}(c^{1})\simeq c^{2}$, where
$c^{2}$ is the permissible curve consecutively crosses
$\gamma^{2}_{1},\gamma^{2}_{2},\cdots,\gamma^{2}_{s}$.

###### Proof.

Let $\gamma_{r-i}=\tilde{\gamma}_{i+1}$ $(0\leq i\leq r-1)$,
$\gamma^{2}_{j}=\tilde{\gamma}_{r+j}$ $(1\leq j\leq s)$. Then
$c^{\prime\prime}:=c^{1}=c_{1}^{\prime\prime}c_{2}^{\prime\prime}\cdots
c_{r+1}^{\prime\prime}$, where each $c_{i}^{\prime\prime}$ ($2\leq i\leq r$)
is a segment between $\tilde{\gamma}_{i-1}$ and $\tilde{\gamma}_{i}$, and
$c_{1}^{\prime\prime}$, $c_{r+1}^{\prime\prime}$ are its end segments; and
$c:=c_{\mathrm{proj}}^{\gamma_{1}}=c_{\mathrm{proj}}^{\tilde{\gamma}_{r}}=c_{1}c_{2}\cdots
c_{r+s+1}$ where each $c_{j}$ ($2\leq j\leq r+s$) is a segment between
$\tilde{\gamma}_{j-1}$ and $\tilde{\gamma}_{j}$, and $c_{1}$, $c_{j}$ are its
end segments. Thus
$c^{\prime\prime}=c_{1}^{\prime\prime}c_{2}^{\prime\prime}\cdots
c_{r+1}^{\prime\prime}\simeq c_{1}c_{2}\cdots c_{r+1}^{\mathfrak{T}}$. We have
that the permissible curve $c^{\prime}=c_{r+1}^{\mathfrak{T}}c_{r+2}\cdots
c_{r+s+1}$ which consecutively crosses
$\tilde{\gamma}_{r+1}=\gamma^{2}_{1},\tilde{\gamma}_{r+2}=\gamma^{2}_{2},\cdots,\tilde{\gamma}_{r+s}=\gamma^{2}_{s}$,
and the positional relationship of $c$, $c^{\prime}$ and $c^{\prime\prime}$
satisfies the conditions in Proposition 3.6. So, by Theorem 3.12(1), we get
the following short exact sequence

$0\longrightarrow M(c^{\prime})\longrightarrow M(c)\longrightarrow
M(c^{\prime\prime})\longrightarrow 0,$

where $M(c^{\prime\prime})=M(c^{1})$,
$M(c)=M(c_{\mathrm{proj}}^{\gamma_{1}})$, and
$M(c^{\prime})=M(c_{r+1}^{\mathfrak{T}}c_{r+2}\cdots c_{r+s+1})\cong
M(\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma_{1}}}(c^{1}))$ (Note that
$c^{\prime}\simeq\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma_{1}}}(c^{1})$ since
$\gamma$ and $\gamma_{1}$ are sides of the same elementary polygon). ∎

$c_{\mathrm{simp}}^{\gamma}$$c_{\mathrm{proj}}^{\gamma}$$\boldsymbol{\times}$$c^{1}$$c_{\mathrm{proj}}^{\gamma^{1}_{1}}$$\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma^{1}_{1}}}(c^{1})$$\boldsymbol{\times}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\boldsymbol{\times}$$\boldsymbol{\times}$$\boldsymbol{\times}$$\gamma$$p$$q$$\cdots$$\gamma^{2}_{1}$$\gamma^{2}_{2}$$\cdots$$\gamma^{2}_{s}$
Figure 15:  The projective cover
$p(M(c_{\mathrm{simp}}^{\gamma})):M(c_{\mathrm{proj}}^{\gamma_{1}^{1}})\to
M(c_{\mathrm{simp}}^{\gamma})$ of $M(c_{\mathrm{simp}}^{\gamma})$ and its
kernel $\mathrm{Ker}\ p(M(c_{\mathrm{simp}}^{\gamma}))\cong M(c^{2})$, where
$c^{2}\simeq\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma_{1}}}(c^{1})$ by
Proposition 3.11.

Keep the notations in Lemma 5.5 and denote by $\gamma^{1}_{i}$ the permissible
curve $\gamma_{i}$ for any $1\leq i\leq r$. In FIGURE 15,
$p(M(c_{\mathrm{simp}}^{\gamma})):M(c_{\mathrm{proj}}^{\gamma})\twoheadrightarrow
M(c_{\mathrm{simp}}^{\gamma})$ is the projective cover of
$M(c_{\mathrm{simp}}^{\gamma})$ which is simple, and the module $M(c^{1})$
corresponding to the orange permissible curve $c^{1}$ is the kernel of
$p(M(c_{\mathrm{simp}}^{\gamma}))$. In addition,
$c_{\mathrm{proj}}^{\gamma^{1}_{1}}$, the red permissible curve in this
Figure, provides the projective cover of $M(c^{1})$, that is,
$p(M(c^{1})):M(c_{\mathrm{proj}}^{\gamma^{1}_{1}})\twoheadrightarrow
M(c^{1})$. Then $\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma^{1}_{1}}}(c^{1})$ is
the kernel of $p(M(c^{1}))$. Furthermore, let
$\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma^{1}_{1}}}(c^{1})=c^{2}$. If there
are arcs $\gamma^{3}_{1},\cdots,\gamma^{3}_{t}$ having a common endpoint which
is the endpoint of $\gamma^{2}_{1}$ but not the common endpoint of
$\gamma^{2}_{1},\cdots,\gamma^{2}_{s}$ such that
$\gamma,\gamma^{1}_{1},\gamma^{2}_{1},\gamma^{3}_{1}$ are sides of the same
elementary polygon of $\mathcal{S}_{A}$, then the kernel of
$M(c_{\mathrm{proj}}^{\gamma^{2}_{1}})\twoheadrightarrow M(c^{2})$ is
isomorphic to $M(c^{3})$, where
$c^{3}\simeq\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma^{2}_{1}}}(c^{2})$ is a
permissible curve consecutively crossing
$\gamma^{3}_{1},\cdots,\gamma^{3}_{t}$. Repeating this process, we get the
minimal projective resolution of $M(c_{\mathrm{simp}}^{\gamma})$ as follows:

$\cdots\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{2}_{1}})\longrightarrow
M(c_{\mathrm{proj}}^{\gamma^{1}_{1}})\longrightarrow
M(c_{\mathrm{proj}}^{\gamma})\longrightarrow
M(c_{\mathrm{simp}}^{\gamma})\longrightarrow 0.$

Thus we obtain a one-one mapping
$M(c_{\mathrm{proj}}^{\gamma^{t}_{1}})\mapsto\gamma^{t}_{1}$, where
$\gamma^{0}_{1}=\gamma$ and $\gamma^{0}_{1},\gamma^{1}_{1},\cdots$ are sides
of the same elementary polygon of $\mathcal{S}_{A}$, see FIGURE 16.

$c_{\mathrm{simp}}^{\gamma}$$c_{\mathrm{proj}}^{\gamma}$$\boldsymbol{\times}$$c^{1}$$c_{\mathrm{proj}}^{\gamma^{1}_{1}}$$c^{2}=\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma^{1}_{1}}}(c^{1})$$c_{\mathrm{proj}}^{\gamma^{2}_{1}}$$c^{3}=\mathrm{nrot}_{c_{\mathrm{proj}}^{\gamma^{2}_{1}}}(c^{2})$$\boldsymbol{\times}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\boldsymbol{\times}$$\boldsymbol{\times}$$\boldsymbol{\times}$$\boldsymbol{\times}$$\gamma$$p$$q$$\cdots$$\cdots$$\gamma^{3}_{1}$$\gamma^{3}_{2}$$\cdots$$\gamma^{3}_{t}$
Figure 16:  Keep the notations in Figure 15. Then the positional relationship
of $c^{2}$, $c_{\mathrm{proj}}^{\gamma^{2}_{1}}$ and $c^{3}$ satisfies the
conditions in Proposition 3.11.

In the case that $\gamma$ satisfies the P-condition at $q$ but not at $p$, we
get dually the minimal projective resolution of
$M(c_{\mathrm{simp}}^{\gamma})$ as follows:

$\cdots\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{\prime
2}_{1}})\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{\prime
1}_{1}})\longrightarrow M(c_{\mathrm{proj}}^{\gamma})\longrightarrow
M(c_{\mathrm{simp}}^{\gamma})\longrightarrow 0.$

Moreover, if $\gamma$ satisfies the P-condition, then the minimal projective
resolution of $M(c_{\mathrm{simp}}^{\gamma})$ is as follows:

$\cdots\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{2}_{1}})\oplus
M(c_{\mathrm{proj}}^{\gamma^{\prime 2}_{1}})\longrightarrow
M(c_{\mathrm{proj}}^{\gamma^{1}_{1}})\oplus
M(c_{\mathrm{proj}}^{\gamma^{\prime 1}_{1}})\longrightarrow
M(c_{\mathrm{proj}}^{\gamma})\longrightarrow
M(c_{\mathrm{simp}}^{\gamma})\longrightarrow 0.$

######  Proposition 5.6.

Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Let $\gamma^{0}$ be an arc
$\gamma^{0}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ whose
endpoints are $p:=\gamma^{0}(0^{+})$ and $q:=\gamma^{0}(1^{-})$. As shown in
FIGURE 17, assume that

  * •

(PD) $($respectively, (ID)) there are arcs
$\gamma^{1},\cdots,\gamma^{n}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$
such that following hold:

    * –

(PD1)$=$(ID1) $\gamma^{0},\gamma^{1},\cdots,\gamma^{n}$ are sides of the same
elementary polygon of $\mathcal{S}_{A}$;

    * –

(PD2)$=$(ID2) $\gamma^{1}(0^{+})=q$ and
$\gamma^{i-1}(1^{-})=\gamma^{i}(0^{+})$ for any $2\leq i\leq n$;

    * –

(PD3) (respectively, (ID3)) in the clockwise $($respectively,
counterclockwise$)$ order at the common endpoint of $\gamma^{i-1}$ and
$\gamma^{i}$, $\gamma^{i}$ appears after $\gamma^{i-1}$ for any $1\leq i\leq
n$;

    * –

(PD4)$=$(ID4) for any $\gamma\in\Gamma_{A}$, none of the arcs
$\gamma^{0},\gamma^{1},\cdots,\gamma^{n},\gamma$ satisfies one of above three
conditions;

and assume that

  * •

(PD)′ $($respectively, (ID)′) there are arcs $\gamma^{\prime
1},\cdots,\gamma^{\prime
m}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ such that
following hold.

    * –

(PD1)′$=$(ID1)′ $\gamma^{\prime 0}:=\gamma^{0},\gamma^{\prime
1},\cdots,\gamma^{\prime m}$ are sides of the same elementary polygon of
$\mathcal{S}_{A}$;

    * –

(PD2)′$=$(ID2)′ $\gamma^{\prime 1}(0^{+})=p$ and $\gamma^{\prime
j-1}(1^{-})=\gamma^{\prime j}(0^{+})$ for any $2\leq j\leq m$;

    * –

(PD3)′ (respectively, (ID3)′) in the counterclockwise $($respectively,
clockwise$)$ order at the common endpoint of $\gamma^{\prime j-1}$ and
$\gamma^{\prime j}$, $\gamma^{\prime j}$ appears after $\gamma^{j-1}$ for any
$1\leq j\leq m$;

    * –

(PD4)′$=$(ID4)′ for any $\gamma^{\prime}\in\Gamma_{A}$, none of the arcs
$\gamma^{\prime 0},\gamma^{\prime 1},\cdots,\gamma^{\prime m},\gamma^{\prime}$
satisfies one of above three conditions.

Then the projective dimension $($respectively, injective dimension$)$ of
$M(c_{\mathrm{simp}}^{\gamma^{0}})$ is $\max\\{n,m\\}$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\gamma^{0}$$\gamma^{1}$$\gamma^{n}$$\partial\mathcal{S}_{A}$$p=\gamma^{0}(0^{+})$$q=\gamma^{0}(1^{-})$(1):
$(\mathbf{PD})$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\gamma^{\prime
0}=\gamma^{0}$$\gamma^{\prime 1}$$\gamma^{\prime
m}$$\partial\mathcal{S}_{A}$$p=\gamma^{0}(0^{+})$$q=\gamma^{0}(1^{-})$(2):
$(\mathbf{PD})^{\prime}$

Figure 17: $\gamma^{0},\gamma^{1},\cdots,\gamma^{n}$ are arcs satisfying
$(\mathbf{PD})$, and $\gamma^{\prime 0},\gamma^{\prime
1},\cdots,\gamma^{\prime m}$ are arcs satisfying $(\mathbf{PD})^{\prime}$. The
figures of $(\mathbf{ID})$ and $(\mathbf{ID})^{\prime}$ are dual.

###### Proof.

Notice that $M(c_{\mathrm{proj}}^{\gamma})\mapsto\gamma$ for any
$\gamma\in\\{\gamma^{i}\mid 1\leq i\leq n\\}\cup\\{\gamma^{j}\mid 1\leq j\leq
m\\}\cup\\{\gamma^{0}\\}$ by Lemmas 5.3, 5.4 and 5.5, thus the minimal
projective resolution of $M(c_{\mathrm{simp}}^{\gamma^{0}})$ is as follows:

$\displaystyle 0$ $\displaystyle\longrightarrow
M(c_{\mathrm{proj}}^{\gamma^{\max\\{n,m\\}}})\oplus
M(c_{\mathrm{proj}}^{\gamma^{\prime\max\\{n,m\\}}})\longrightarrow\cdots$
$\displaystyle\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{2}})\oplus
M(c_{\mathrm{proj}}^{\gamma^{\prime 2}})\longrightarrow
M(c_{\mathrm{proj}}^{\gamma^{1}})\oplus M(c_{\mathrm{proj}}^{\gamma^{\prime
1}})\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{0}})\longrightarrow
M(c_{\mathrm{simp}}^{\gamma^{0}})\longrightarrow 0.$ (5.1)

If $m\geq n$, then $M(c_{\mathrm{proj}}^{\gamma^{i}})=0$ for any $i>n$, and so
$\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma^{0}})=m$. If $n\geq m$, then
$\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma^{0}})=n$ similarly. Dually, we
can prove the case for injective dimension. ∎

The exact sequence (5.1) is a description of the minimal projective resolution
of a simple module in marked ribbon surfaces. We will give two examples in
Section 7 to illustrate it, see Examples 7.1 and 7.2.

###  5.2 The descriptions of global dimension in geometric models

######  Definition 5.7.

(Consecutive arcs numbers) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Recall from Definition 2.1 that $\Gamma_{A}$ divides
$\mathcal{S}_{A}$ to some elementary polygons $\\{\Delta_{i}\mid 1\leq i\leq
d\\}$ whose sides are arcs in $\Gamma_{A}$ except one side which is a boundary
arc. We define $\mathfrak{C}(\Delta_{i})$, the consecutive arcs numbers of
$\Delta_{i}$, as the number of sides belonging to $\Gamma_{A}$, if the
unmarked boundary component of $\mathcal{S}_{A}$ is not a side of
$\Delta_{i}$; otherwise, $\mathfrak{C}(\Delta_{i})=\infty$.

######  Remark 5.8.

An elementary polygon $\Delta_{i}$ is called an $\infty$-elementary polygon if
$\Delta_{i}$ has a side which is an unmarked boundary component $b$ of
$\mathcal{S}_{A}$. Obviously, $\mathfrak{C}(\Delta_{i})=\infty$ if and only if
$\Delta_{i}$ is $\infty$-elementary. In this case, for any side belonging to
$\Gamma_{A}$ of $\Delta_{i}$, say $\gamma^{0}$, there are arcs
$\gamma^{0},\gamma^{1},\cdots,\gamma^{n}$ satisfying the conditions (PD1),
(PD2) and (PD3) in Theorem 5.6. Let $\gamma^{t}=\gamma^{\bar{t}}$ for all
$t>n$ where $\bar{t}$ equals to $t$ modulo $n+1$. Then for any $t\geq 0$,
$\gamma^{0},\gamma^{1},\cdots,\gamma^{t}$ are arcs such that (PD1), (PD2) and
(PD3) hold. Thus the minimal projective resolution of
$M(c_{\mathrm{simp}}^{\gamma^{0}})$ is as follows:

$\cdots\longrightarrow P^{i}\longrightarrow\cdots\longrightarrow
P^{2}\longrightarrow P^{1}\longrightarrow
M(c_{\mathrm{proj}}^{\gamma^{0}})\longrightarrow
M(c_{\mathrm{simp}}^{\gamma^{0}})\longrightarrow 0,$

where each $M(c_{\mathrm{proj}}^{\gamma^{\overline{i}}})$ is a quotient of
$P^{i}$. Therefore
$\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma^{0}})=\infty$.

We recall some notions from [2, Section 2].

######  Definition 5.9.

Let $A=kQ/I$ be a gentle algebra. A non-trivial permitted (respectively,
forbidden) path is either a path of length $1$, or a path
$\mathit{\Pi}=\alpha_{1}\cdots\alpha_{m}$ ($m\geq 2$) such that
$\alpha_{i}\alpha_{i+1}\notin I$ (respectively, $\alpha_{i}\alpha_{i+1}\in I$)
for any $1\leq i\leq m-1$. A non-trivial permitted (respectively, forbidden)
thread is a maximal non-trivial permitted (respectively, forbidden) path, that
is, for each $\beta\in Q_{1}$ satisfying $t(\beta)=s(\alpha_{1})$ we have
$\beta\alpha_{1}\in I$ (respectively, $\beta\alpha_{1}\notin I$), and for each
$\beta^{\prime}\in Q_{1}$ satisfying $t(\alpha_{m})=s(\beta^{\prime})$ we have
$\alpha_{m}\beta^{\prime}\in I$ (respectively, $\alpha_{m}\beta^{\prime}\notin
I$). A trivial permitted (respectively, forbidden) thread is a trivial path
$v\in Q_{0}$ such that one of following holds: (i) $v$ is a source point and
$\sharp\\{\beta\in Q_{1}\mid t(\beta)=v\\}=0$; (ii) $v$ is a sink point and
$\sharp\\{\beta^{\prime}\in Q_{1}\mid s(\beta^{\prime})=v\\}=0$; (iii)
$\sharp\\{\beta\in Q_{1}\mid t(\beta)=v\\}=1=\sharp\\{\beta^{\prime}\in
Q_{1}\mid s(\beta^{\prime})=v\\}$ and $\beta\beta^{\prime}\notin I$
(respectively, $\beta\beta^{\prime}\in I$).

We use $\mathcal{P}_{A}$ (respectively, $\mathcal{F}_{A}$) to denote the set
of all permitted (respectively, forbidden) threads. By [25, Section 1], we
have a bijection $\mathfrak{p}:\mathcal{P}_{A}\to\mathcal{M}_{A}$
(respectively, $\mathfrak{f}:\mathcal{F}_{A}\to\\{\Delta_{i}\mid 1\leq i\leq
d\\}$), where $\mathfrak{p}(H)$ is the marked point $p$ such that every arc
with endpoint $p\in\mathcal{M}_{A}$ corresponds to a vertex of
$H\in\mathcal{P}_{A}$ (respectively, $\mathfrak{f}(\mathit{\Pi})$ is the
elementary polygons whose sides belonging to $\Gamma_{A}$ correspond to the
vertices of $\mathit{\Pi}\in\mathcal{F}_{A}$).

For any path $\wp$, we use $l(\wp)$ to denote the length of $\wp$. In
particular, for an oriented cycle with full relation of the quiver $(Q,I)$ of
a gentle algebra $A$, that is, an oriented cycle
$\alpha_{1}\alpha_{2}\cdots\alpha_{\ell}$ ($t(\alpha_{\ell})=s(\alpha_{1})$)
such that $\alpha_{\overline{i}}\alpha_{\overline{i+1}}\in I$ where
$\overline{i}$ is equal to $i$ modulo $\ell$, it deduces a forbidden threads
of infinite length
$\cdots\alpha_{\ell}\alpha_{1}\alpha_{2}\cdots\alpha_{\ell}\alpha_{1}\alpha_{2}\cdots$.

######  Theorem 5.10.

Let $A=kQ/I$ be gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface, and let $\\{\Delta_{i}\mid 1\leq i\leq d\\}$ be the set of all
elementary polygons of $\mathcal{S}_{A}$. Then

$\mathrm{gl.dim}A=\max\limits_{1\leq i\leq
d}\mathfrak{C}(\Delta_{i})-1=\max\limits_{\mathit{\Pi}\in\mathcal{F}_{A}}l(\mathit{\Pi}).$

###### Proof.

If there exists an elementary polygons $\Delta_{i}$ which is of the form given
in Remark 5.8, then $\mathfrak{C}(\Delta_{i})=\infty$. In this case, for each
side $\gamma$ of $\Delta_{i}$, we have
$\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma})=\infty$ by Remark 5.8, and the
assertion follows.

Otherwise, for each elementary polygons $\Delta_{i}$ whose sides are denoted
by $\gamma^{0},\gamma^{1},\cdots,\gamma^{m_{i}}$ such that (PD) holds, then
$m_{i}+1=\mathfrak{C}(\Delta_{i})$. For the arc $\gamma^{t}$ ($0\leq t\leq
m_{i}-1$), there is an elementary polygon $\Delta^{\prime}_{i}$ such that
$\gamma^{t}$ is the unique common side of $\Delta_{i}$ and
$\Delta_{i}^{\prime}$. We can find a sequence of arcs, say $\gamma^{\prime
0}(=\gamma^{t}),\gamma^{\prime 1},\cdots,\gamma^{\prime n_{it}}$ such that
(PD) holds, and thus the minimal projective resolution of
$M(c_{\mathrm{simp}}^{\gamma^{t}})$ is as follows:

$\cdots\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{t+2}})\oplus
M(c_{\mathrm{proj}}^{\gamma^{\prime 2}})\longrightarrow
M(c_{\mathrm{proj}}^{\gamma^{t+1}})\oplus M(c_{\mathrm{proj}}^{\gamma^{\prime
1}})\longrightarrow M(c_{\mathrm{proj}}^{\gamma^{t}})\longrightarrow
M(c_{\mathrm{simp}}^{\gamma^{t}})\longrightarrow 0.$

It stops from the $\max\\{m_{i}-t,n_{it}\\}$-th projective module. Notice that
$n_{it}\leq\mathfrak{C}(\Delta_{i}^{\prime})-1$, so
$\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma^{t}})=\max\\{m_{i},n_{it}\\}\leq\max\\{\mathfrak{C}(\Delta_{i})-1,\mathfrak{C}(\Delta_{i}^{\prime})-1\\}$,
and hence

$\mathrm{gl.dim}A\leq\max\limits_{1\leq i\leq
d}\\{\max\\{\mathfrak{C}(\Delta_{i})-1,\mathfrak{C}(\Delta_{i}^{\prime})-1\\}\\}=\max\limits_{1\leq
i\leq d}\mathfrak{C}(\Delta_{i})-1.$

Let $\Delta_{\ell}$ be the elementary polygon such that $\max\limits_{1\leq
i\leq
d}\mathfrak{C}(\Delta_{i})=\mathfrak{C}(\Delta_{\ell})\mathop{=\\!=\\!=\\!=\\!=\\!=}\limits^{\text{denoted
by}}m_{\ell}$, and let
$\gamma^{0},\gamma^{1},\cdots,\gamma^{m_{\ell}}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$
be its sides such that $\gamma_{i}(1^{-})=p_{i}=\gamma_{i+1}(0^{+})$ and
$\gamma_{i+1}$ follows $\gamma_{i}$ around $p_{i}$ in the counterclockwise
order ($0\leq i\leq m_{\ell}-1$). Then
$\gamma^{0},\gamma^{1},\cdots,\gamma^{m_{\ell}}$ satisfy the condition (PD),
and thus

$\mathrm{gl.dim}A\geq\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma^{0}})=m_{\ell}=\mathfrak{C}(\Delta_{\ell})-1=\max\limits_{1\leq
j\leq d}\mathfrak{C}(\Delta_{j})-1$

by Proposition 5.6. It follows that $\mathrm{gl.dim}A=\max\limits_{1\leq i\leq
d}\mathfrak{C}(\Delta_{i})-1$.

Since the map $\mathfrak{f}$ above is bijective, the length $l(\mathit{\Pi})$
of the forbidden thread $\mathit{\Pi}$ is equal to
$\mathfrak{C}(\mathfrak{f}(\mathit{\Pi}))-1$ and
$\mathrm{gl.dim}A=\max\limits_{\mathit{\Pi}\in\mathcal{F}_{A}}l(\mathit{\Pi})$.
∎

###  5.3 AG-invariants and AG-equivalence

The AG-invariant of a gentle algebra $A=kQ/I$ introduced by Avella-Alaminos
and Geiß in [2] is a function $\phi_{A}:\mathbb{N}^{2}\to\mathbb{N}$ with
$\mathbb{N}$ the set of natural numbers mapping $\phi[(m_{t},n_{t})]$ to the
number of pairs $(m_{t},n_{t})$ in the sequence $(m_{t},n_{t})_{1\leq t\leq
r}$, where $(m_{i},n_{i})_{1\leq i\leq r}$ is obtained as follows:

  * Step 1.

Let $H_{0}=\alpha^{0}_{1}\cdots\alpha^{0}_{u_{0}}$ ($u_{0}\in\mathbb{N}$) be a
(trivial) permitted thread of $A$.

  * Step 2.

If $H_{0}$ is non-trivial and there exists a non-trivial forbidden thread
$F=\beta^{0}_{1}\cdots\beta^{0}_{v_{0}}$ ($v_{0}\in\mathbb{N}$) such that
$t(\beta^{0}_{v_{0}})=t(\alpha^{0}_{u_{0}})$ and
$\alpha^{0}_{u_{0}}\neq\beta^{0}_{v_{0}}$, consider $\mathit{\Pi}_{0}=F$;
otherwise, consider the (trivial) forbidden thread $\mathit{\Pi}_{0}$
satisfying $t(\mathit{\Pi}_{0})=t(H_{0})$.

  * Step 3.

If $\mathit{\Pi}_{0}$ is non-trivial and there exists a non-trivial
permissible thread $P=\alpha^{1}_{1}\cdots\alpha^{1}_{u_{1}}$
($u_{1}\in\mathbb{N}$) such that $s(\alpha^{1}_{1})=s(\beta^{0}_{1})$ and
$\alpha^{1}_{1}\neq\beta^{0}_{1}$, consider $H_{1}=P$; otherwise, consider the
(trivial) permitted thread $H_{1}$ satisfying $s(H_{1})=s(\mathit{\Pi}_{0})$.

  * Step 4.

If we obtain a permitted thread $H_{i}$ (respectively, forbidden thread
$\mathit{\Pi}_{i}$), then just as Step 2 (respectively, Step 3), determine the
forbidden thread $\mathit{\Pi}_{i}$ (respectively, permitted thread
$H_{i+1}$). Repeat this step until $H_{m_{1}}=H_{0}$ first appears, we induce
a pair $(m_{1},n_{1})$ where $n_{1}=\sum_{i=0}^{m_{1}-1}l(\mathit{\Pi}_{i})$.

  * Step 5.

Until all permitted threads are considered in Steps 1–4, we obtain a sequence
of pairs $(m_{1},n_{1})$, $\cdots$, $(m_{r^{\prime}},n_{r^{\prime}})$ with
$1\leq r^{\prime}\leq r$.

  * Step 6.

For every oriented cycle of length $\ell$ having full relations, we add a pair
$(0,\ell)$ in the sequence obtained in Step 5, and obtain the sequence
$(m_{t},n_{t})_{1\leq t\leq r}$ where $m_{t}=0$ if and only if $t>r^{\prime}$.

The bijection $\mathfrak{f}:\mathcal{F}_{A}\to\\{\Delta_{i}\mid{1\leq i\leq
d}\\}$ provides a method for calculating AG-invariants by marked ribbon
surfaces. Indeed, each elementary polygon, say
$\Delta^{t}_{j}\in\\{\Delta_{i}\mid 1\leq i\leq d\\}$, with a unique single
boundary arc on the boundary component $b^{t}$ of $\mathcal{S}_{A}$
corresponds to a forbidden thread $\mathfrak{f}^{-1}(\Delta^{t}_{j})$, and
$\sharp\mathfrak{S}(\Delta^{t}_{j})$ is equal to:

  * •

either $l(\mathfrak{f}^{-1}(\Delta^{t}_{j}))+1$, if $\Delta^{t}_{j}$ is not
$\infty$-elementary;

  * •

or the number of vertices of $\Delta^{t}_{j}$, otherwise.

Thus $(m_{t},n_{t})_{1\leq t\leq r}$ can be obtained by

$\displaystyle m_{t}$ $\displaystyle=\text{ the number of marked points on the
boundary component }b^{t},$ $\displaystyle n_{t}$ $\displaystyle=\sum_{j\in
J_{t}}(\sharp\mathfrak{S}(\Delta^{t}_{j})-1)=\sum_{j\in
J_{t}}\sharp\mathfrak{S}(\Delta^{t}_{j})-m_{t},$

where $J_{t}\subseteq\\{1,2,\cdots,d\\}$ is a set of all elementary polygons
which have a side is on the boundary component $b^{t}$ (in this case, we have
$\bigcup_{1\leq t\leq r}=\\{1,2,\cdots,d\\}$ and $J_{t}\cap
J_{t^{\prime}}=\varnothing$ holds for all $1\leq t\neq t^{\prime}\leq r$). See
[23, Theorem 3.3].

######  Remark 5.11.

The AG-invariant of a gentle algebras is an invariant up to derived
equivalence; that is, if two gentle algebras $A$ and $B$ are derived
equivalent, then the number of cycles of $A$ and that of $B$ are identical and
$\phi_{A}=\phi_{B}$. Furthermore, if $A$ and $B$ are gentle one-cycle, that
is, the quivers of $A$ and $B$ both have at most one cycle, then
$\phi_{A}=\phi_{B}$ yields that $A$ is derived equivalent to $B$ ([2, Theorem
A, Proposition B, Theorem C]).

We say that two gentle algebras $A$ and $A^{\prime}$ are AG-equivalent if
$\phi_{A}=\phi_{A^{\prime}}$. By [2, Proposition B], derived equivalence
yields AG-equivalence. We know that the finiteness of the global dimension is
invariant under derived equivalence. In the following, we show that it is also
invariant under AG-equivalence. We need the following observation.

######  Proposition 5.12.

For a gentle algebra $A=kQ/I$, we have that $\mathrm{gl.dim}A=\infty$ if and
only if there is $\ell\geq 1$ such that $\phi_{A}(0,\ell)\neq 0$.

###### Proof.

For a gentle algebra $A$, if there is $\ell\geq 1$ such that
$\phi_{A}(0,\ell)=x\geq 1$, then its marked ribbon surface
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ has $x$
unmarked boundary component(s), say $b^{1},b^{2},\cdots,b^{x}$. So
$\mathcal{S}_{A}$ has $x$ $\infty$-elementary polygon(s)
$\Delta^{1},\Delta^{2},\cdots,\Delta^{x}$. Thus $\mathrm{gl.dim}A=\infty$ by
Theorem 5.10.

Conversely, if $\phi_{A}(0,\ell)=0$ for any $\ell\geq 1$, then each boundary
component $b^{t}$ of $\mathcal{S}_{A}$ has at least one marked point and each
elementary polygon $\Delta^{t}_{j}$ ($j\in J$ with $J$ some finite set) whose
single boundary arc included in $b^{t}$ is not a unmarked boundary component
of $\mathcal{S}_{A}$. Then $\mathfrak{C}(\Delta^{t}_{j})<\infty$, and so
$\mathrm{gl.dim}A=\max\limits_{1\leq t\leq r,j\in
J}\mathfrak{C}(\Delta^{t}_{j})-1<\infty$ by Theorem 5.10. ∎

As a consequence, we get the following corollary, which was proved by Opper,
Plamondon and Schroll by using the Koszul dual of gentle algebras.

######  Corollary 5.13.

[25, Section 1, 1.7] Let $A=kQ/I$ be a gentle algebra. Then
$\mathrm{gl.dim}A=\infty$ if and only if its quiver $(Q,I)$ has at least one
oriented cycle with full relations.

###### Proof.

If $\mathrm{gl.dim}A=\infty$, then its marked ribbon surface has at least one
$\infty$-elementary polygon $\Delta_{j}$ by Theorem 5.10. Suppose
$\mathfrak{S}(\Delta_{j})=\\{\gamma_{0},\cdots,\gamma_{\ell-1}\\}$, where
$\gamma_{\overline{i}}(1^{-})=\gamma_{\overline{i+1}}(0^{+})$ for any $i\geq
0$ and $\overline{i}$ is equals to $i$ modulo $\ell$. Then $(Q,I)$ has an
oriented cycle with full relations of length $\ell$ whose vertices are
$\\{\mathfrak{v}^{-1}(\gamma_{i})\mid 0\leq i\leq\ell-1\\}$.

Conversely, if $(Q,I)$ has at least one oriented cycle of length $\ell(\geq
1)$ with full relations, then $\phi_{A}(0,\ell)>0$, and thus
$\mathrm{gl.dim}A=\infty$ by Proposition 5.12. ∎

The following result shows that the finiteness of the global dimension of
gentle algebras is invariant under AG-equivalence.

######  Corollary 5.14.

If two gentle algebras $A$ and $A^{\prime}$ are AG-equivalent, then
$\mathrm{gl.dim}A<\infty$ if and only if $\mathrm{gl.dim}A^{\prime}<\infty$.

###### Proof.

By Proposition 5.12, we have that $\mathrm{gl.dim}A<\infty$ if and only if
$\phi_{A}(0,\ell)=0$ for any $\ell\geq 1$. Suppose that $A$ and $A^{\prime}$
are AG-equivalent and $\mathrm{gl.dim}A<\infty$. Then
$\phi_{A}=\phi_{A^{\prime}}$ and
$\phi_{A^{\prime}}(0,\ell)=\phi_{A}(0,\ell)=0$ for any $\ell\geq 1$. Thus
$\mathrm{gl.dim}A^{\prime}<\infty$ by Proposition 5.12. ∎

##  6 Self-injective dimension

A module $G\in\bmod A$ is called Gorenstein projective if there exists an
exact sequence of projective modules

$\cdots\longrightarrow
P^{-2}\mathop{\longrightarrow}\limits^{d^{-2}}P^{-1}\mathop{\longrightarrow}\limits^{d^{-1}}P^{0}\mathop{\longrightarrow}\limits^{d^{0}}P^{1}\mathop{\longrightarrow}\limits^{d^{1}}P^{2}\longrightarrow\cdots$

which remains still exact after applying the functor $\mathrm{Hom}_{A}(-,A)$,
such that $G\cong\mathrm{Im}d^{-1}$ [1, 13]. Obviously, every projective
module is Gorenstein projective. For each module $M$, its Gorenstein
projective dimension $\mathrm{G}\text{-}\mathrm{proj.dim}M$, is defined as

$\displaystyle\inf\\{n\mid$ $\displaystyle\text{ there exists an exact
sequence }0\to G_{n}\to\cdots\to G_{1}\to G_{0}\to M\to 0$
$\displaystyle\text{in}\ \bmod A\ \text{ with all }G_{i}\text{ Gorenstein
projective}\\}.$

###  6.1 The descriptions of Gorenstein projective modules

For a gentle algebra $A=kQ/I$, the following theorem shows that all
indecomposable Gorenstein projective modules can be determined by its quiver
$(Q,I)$.

######  Theorem 6.1.

[20] Let $A=kQ/I$ be a gentle algebra. An $A$-module $G$ is Gorenstein
projective if and only if $G$ is isomorphic to a projective module or an
$\alpha A$ where $\alpha\in Q_{1}$ is an arrow on some oriented cycle with
full relations of $A$ $($note that $\alpha A$ is a direct summand of
$\mathrm{rad}P(s(\alpha))\cong\mathrm{rad}(e_{s(\alpha)}A)$$)$.

In this subsection, we depict all indecomposable Gorenstein projective modules
in marked ribbon surfaces.

######  Lemma 6.2.

Let $A=kQ/I$ be a gentle algebra having an oriented cycle with full relation,
and let $\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ be its
marked ribbon surface and $\gamma\in\Gamma_{A}$ an arc.

  1. (1)

If $\gamma$ is a side of an $\infty$-elementary polygon $\Delta_{i}$ of
$\mathcal{S}_{A}$, then there is an arc $\gamma^{\prime}$ such that

     * •

$\gamma^{\prime}$ and $\gamma$ are two adjacent sides of $\Delta_{i}$, the
common endpoint of them is denoted by $p$; and

     * •

$\gamma^{\prime}$ and $\gamma$ surround $p$ in the counterclockwise order.

  2. (2)

Moreover, suppose that $c_{\mathrm{proj}}^{\gamma^{\prime}}=c_{1}\cdots c_{r}$
is the projective permissible curve of $\gamma^{\prime}$, where $c_{j}$
$(1\leq j\leq r)$ is the segment from $\gamma^{\prime}$ to $\gamma$, then the
module $M(c_{j}^{\mathfrak{T}}c_{j+1}\cdots c_{r})$ is a non-projective
Gorenstein projective module (see FIGURE 18).

$\bullet$$\bullet$$\bullet$$\Delta_{i}$$\gamma^{\prime}$$\gamma$$c_{\mathrm{proj}}^{\gamma^{\prime}}$$c^{\prime}$$\boldsymbol{\times}$$\boldsymbol{\times}$
Figure 18: $c^{\prime}\simeq c_{j}^{\mathfrak{T}}c_{j+1}\cdots c_{r}$ is the
permissible curve corresponded by non-projective Gorenstein-projective module
$\alpha A$, where $\alpha:\gamma^{\prime}\to\gamma$ is the arrow on the
oriented cycle with full relations corresponded by $\Delta_{i}$.

###### Proof.

(1) Since the quiver $(Q,I)$ of $A$ has an oriented cycle with full relation,
we have $\mathrm{gl.dim}A=\infty$ by Corollary 5.13, and moreover each
oriented cycle of length $\ell$ with full relation corresponds to a pair
$(0,\ell)$ of the AG-invariant $\phi_{A}$ and an $\infty$-elementary
$\ell$-polygon of marked ribbon surface $\mathcal{S}_{A}$. Let $q$ be another
endpoint of $\gamma$. Obviously, in the case for $p=q$, we have
$\gamma=\gamma^{\prime}$. If $p\neq q$, then $\Delta_{i}$ corresponds to some
oriented cycle with full relation
$0\mathop{\to}\limits^{\alpha_{1}}1\mathop{\to}\limits^{\alpha_{2}}2\to\cdots\to\ell-1\mathop{\to}\limits^{\alpha_{\ell}}0$
such that the side $\gamma=\mathfrak{v}^{-1}(x)$ of $\Delta_{i}$ corresponds
to some vertex $x\in\\{0,1,\cdots,\ell-1\\}$ through the map
$\mathfrak{v}:Q_{0}\to\Gamma_{A}$ which is defined in Definition 2.3. Then
$\gamma^{\prime}=\mathfrak{v}(\overline{x-1})$, where $\overline{x-1}$ is
equal to $x-1$ modulo $\ell$.

(2) Following Definition 2.3(1), for simplicity, we use $\gamma$ to denote the
vertex $\mathfrak{v}^{-1}(\gamma)\in Q_{0}$. We have that
$c_{\mathrm{proj}}^{\gamma^{\prime}}$ corresponds to the string

$\cdots\mathop{\longleftarrow}\limits^{\alpha(c_{j-1})}\gamma^{\prime}\mathop{\longrightarrow}\limits^{\alpha(c_{j})}\gamma\mathop{\longrightarrow}\limits^{\alpha(c_{j+1})}\gamma^{\prime\prime}\mathop{\longrightarrow}\limits^{\alpha(c_{j+2})}\cdots,$

where each arrow $\alpha(c_{j})\in Q_{1}$ is induced by the segment $c_{j}$
for any $1\leq j\leq r-1$ (see Definition 2.3(2)). Then
$c_{j}^{\mathfrak{T}}c_{j+1}\cdots c_{r}$, denoted by $c$, corresponds to the
string

$\gamma\mathop{\longrightarrow}\limits^{\alpha(c_{j+1})}\gamma^{\prime\prime}\mathop{\longrightarrow}\limits^{\alpha(c_{j+2})}\cdots.$

Thus
$M(c)\cong\alpha(c_{j})A\leq_{\oplus}\mathrm{rad}M(c_{\mathrm{proj}}^{\gamma^{\prime}})$
is Gorenstein projective by Theorem 6.1.

On the other hand, the side $\mathfrak{v}(\overline{x+1})$ of $\Delta_{i}$
satisfies that

  * •

$\gamma$ and $\mathfrak{v}(\overline{x+1})$ are two adjacent sides of
$\Delta_{i}$ (the common endpoint of them is denoted by $q$); and

  * •

$\gamma$ and $\mathfrak{v}(\overline{x+1})$ surround $q$ in the
counterclockwise order.

Then $c\not\simeq c_{\mathrm{proj}}^{\gamma}$ and thus $M(c)$ is not
projective by Lemma 4.5. ∎

######  Proposition 6.3.

(Gorenstein projective modules) Let $A=kQ/I$ be a gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. For a permissible curve $c$, the indecomposable module $M(c)$
is Gorenstein projective if and only if one of the following holds:

  * •

(GP1) $c$ is a projective permissible curve;

  * •

(GP2) $c\simeq c_{1}\cdots c_{r}$ consecutively crosses arcs
$\gamma_{1},\cdots,\gamma_{r-1}$ which have a common endpoint

$p=p(\gamma_{1},\gamma_{2},c)=\cdots=p(\gamma_{r-2},\gamma_{r-1},c)$

such that:

    * –

$\gamma_{i+1}$ appears after $\gamma_{i}$ $(1\leq i\leq r-2)$ in the
counterclockwise order surrounding $p$;

    * –

$\gamma_{1}$ is a side of some $\infty$-elementary polygon $\Delta_{j}$ with
$1\leq j\leq d$;

    * –

the end segment of $c$ between $c(0^{+})$ and $\gamma_{1}$ lies in the inner
of $\Delta_{j}$.

###### Proof.

We first prove the sufficiency. By Lemma 4.5, if $c$ is a projective
permissible curve, then it is trivial that $M(c)$ is a Gorenstein projective
module.

If $c$ satisfies (GP2), then there is a unique arc $\gamma^{\prime}$ such that
$\gamma^{\prime}$ and $\gamma:=\gamma_{1}$ satisfy the conditions given in
Lemma 6.2 (1). Suppose that $c_{\mathrm{proj}}^{\gamma^{\prime}}=c_{1}\cdots
c_{s}$ consecutively crosses $\tilde{\gamma}_{1}\cdots\tilde{\gamma}_{s-1}$
($s\geq r+1$). Then there is a unique $h$ with $1\leq h\leq s-1$ such that
$\tilde{\gamma}_{h-1}=\gamma^{\prime}$, $\tilde{\gamma}_{h}=\gamma_{1}$,
$\tilde{\gamma}_{h+1}=\gamma_{2}$, $\cdots$,
$\tilde{\gamma}_{s-1}=\gamma_{r-1}$. To be more precise,
$c_{\mathrm{proj}}^{\gamma^{\prime}}$ corresponds to the following string:

$\displaystyle\tilde{\gamma}_{1}\longleftarrow\cdots\longleftarrow\tilde{\gamma}_{h-2}\longleftarrow\tilde{\gamma}_{h-1}\mathop{\longrightarrow}\limits^{\alpha}\tilde{\gamma}_{h}\longrightarrow\tilde{\gamma}_{h+1}\longrightarrow\cdots\longrightarrow\tilde{\gamma}_{s-1}$
$\displaystyle\text{($\tilde{\gamma}_{h-1}=\gamma^{\prime}$ and
$\tilde{\gamma}_{h}=\gamma$ are adjacent sides of $\Delta_{j}$)}.$

We have that $c$ consecutively crosses $\tilde{\gamma}_{h}$,
$\tilde{\gamma}_{h+1}$, $\cdots$, $\tilde{\gamma}_{s-1}$, where
$\tilde{\gamma}_{h}\in\mathfrak{S}(\Delta_{j})$ and
$\tilde{\gamma}_{m}\notin\mathfrak{S}(\Delta_{j})$ for any $h+1\leq m\leq
s-1$. Thus $M(c)\cong\alpha A$ is a Gorenstein projective module satisfying
$M(c)\leq_{\oplus}\mathrm{rad}M(c_{\mathrm{proj}}^{\tilde{\gamma}_{h-1}})$. In
this case, $M(c)$ is non-projective.

Next we prove the necessity. By Theorem 6.1, if $M(c)$ is Gorenstein
projective, then $M(c)$ is isomorphic to either a projective module or an
$\alpha A$ where $\alpha$ is an arrow on some oriented cycle with full
relations.

If $M(c)$ is projective, then $c$ is a projective permissible curve. Now
suppose that $M(c)$ is isomorphic to $\alpha A$. Let $v:=s(\alpha)$, It
follows from Lemma 4.5 that $\alpha
A\cong\mathrm{rad}P(v)\cong\mathrm{rad}M(c_{\mathrm{proj}}^{\tilde{\gamma}(v)})$
for some arc $\tilde{\gamma}(v)\in\Gamma_{A}$, where
$c_{\mathrm{proj}}^{\tilde{\gamma}(v)}=c_{1}\cdots c_{s}$ is a projective
permissible curve consecutively crossing arcs
$\tilde{\gamma}_{1},\cdots,\tilde{\gamma}_{s-1}$. Then $v$ corresponds to some
$\tilde{\gamma}_{h}$ with $2\leq h\leq s-2$ through
$\mathfrak{v}:Q_{0}\to\Gamma_{A}$, that is,
$\tilde{\gamma}_{h}=\tilde{\gamma}(v)$ (note that $h\neq 1$ and $h\neq s-1$
because $v$ is a vertex on the oriented cycle with full relations). Then
$c_{\mathrm{proj}}^{\tilde{\gamma}(v)}=c_{\mathrm{proj}}^{\tilde{\gamma}_{h}}$
corresponds to the string

$\tilde{\gamma}_{1}\longleftarrow\cdots\longleftarrow\tilde{\gamma}_{h-1}\mathop{\longleftarrow}\limits^{\alpha_{1}}\tilde{\gamma}_{h}\mathop{\longrightarrow}\limits^{\alpha_{2}}\tilde{\gamma}_{h+1}\longrightarrow\cdots\longrightarrow\tilde{\gamma}_{s-1},$

and the arrow $\alpha$ is either $\alpha_{1}$ or $\alpha_{2}$. Without loss of
generality, suppose $\alpha=\alpha_{1}$. Since $M(c)\cong\alpha_{1}A$ is
Gorenstein projective, the arcs $\tilde{\gamma}_{h-1}$ and
$\tilde{\gamma}_{h}$ are adjacent sides of some $\infty$-elementary polygon
$\Delta_{j}$ of $\mathcal{S}_{A}$. Let $\tilde{\gamma}_{h-1}=\gamma_{1}$,
$\tilde{\gamma}_{h-2}=\gamma_{2}$, $\cdots$,
$\tilde{\gamma}_{1}=\gamma_{h-1}$. Then the permissible curve $c$
consecutively crossing $\gamma_{1},\cdots,\gamma_{h-1}$ satisfies the
condition (GP2). ∎

######  Example 6.4.

Let $A$ be a gentle algebra whose marked ribbon surface $\mathcal{S}_{A}$ is
given in FIGURE 19. We can find 12 indecomposable Gorenstein projective
modules including

  * •

9 indecomposable projective modules (see green permissible curves):

${{}_{{}^{4}_{5}}}{{}^{1}}{{}_{{}^{2}_{{}^{6}_{7}}}},\
{{}_{{}^{6}_{7}}}{{}^{2}}{{}_{{}^{3}_{{}^{8}_{9}}}},\
{{}_{{}^{8}_{9}}}{{}^{3}}{{}_{{}^{1}_{{}^{4}_{5}}}},\ {{}^{4}_{5}},\
{{}^{6}_{7}},\ {{}^{8}_{9}},\ {5},\ {7},\ {9};$

  * •

and 3 indecomposable non-projective Gorenstein projective modules (see pink
permissible curves):

${{}^{{}_{2}}_{{}^{6}_{7}}}\big{(}\leq_{\oplus}\mathrm{rad}\big{(}{{}_{{}^{4}_{5}}}{{}^{1}}{{}_{{}^{2}_{{}^{6}_{7}}}}\big{)}\big{)},\
{{}^{{}_{3}}_{{}^{8}_{9}}}\big{(}\leq_{\oplus}\mathrm{rad}\big{(}{{}_{{}^{6}_{7}}}{{}^{2}}{{}_{{}^{3}_{{}^{8}_{9}}}}\big{)}\big{)},\
{{}^{{}_{1}}_{{}^{4}_{5}}}\big{(}\leq_{\oplus}\mathrm{rad}\big{(}{{}_{{}^{8}_{9}}}{{}^{3}}{{}_{{}^{1}_{{}^{4}_{5}}}}\big{)}\big{)}.$

$1$$2$$3$$4$$5$$6$$7$$8$$9$ Figure 19: An example for calculating all
indecomposable Gorenstein projective modules

###  6.2 The descriptions of self-injective dimension

Geiß and Reiten showed that gentle algebras are Gorenstein [14, Theorem 3.4].
In this subsection we provide a geometric interpretation of this result by
marked ribbon surfaces. We use $\mathsf{f}\text{-}\Delta$ to denote the set of
all non-$\infty$-elementary polygons, and use
$\mathsf{f}\text{-}\mathcal{F}_{A}$ to denote the set of all forbidden threads
of finite length. We first prove the following proposition.

######  Proposition 6.5.

Let $A=kQ/I$ be a non-simple gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. Then

$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\begin{cases}0,\ \mbox{\text{if
$Q$ is an oriented cycle with full relations};}\\\
\max\limits_{\Delta_{i}\in\mathsf{f}\text{-}\Delta}\big{\\{}1,{\mathfrak{C}(\Delta_{i})}-1\big{\\}},\
\mbox{\text{otherwise}.}\end{cases}$

###### Proof.

If $Q$ is an oriented cycle with full relations, then $A$ is a self-injective
Nakayama algebra.

Now suppose that $Q$ is not an oriented cycle with full relations. Since $A$
is a gentle algebra, it follows from [14, Theorem 3.4] that $A$ is Gorenstein.
Then by [18, Theorem] and [19, Theorem 1.4], we have
$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\sup\\{\mathrm{G}\text{-}\mathrm{proj.dim}S\mid
S$ is simple$\\}$.

The proof is divided to the following two cases: (i) $(Q,I)$ has at least one
oriented cycle with full relations; (ii) $(Q,I)$ has no oriented cycle with
full relations.

(i) Note that if $v\in Q_{0}$ is a vertex on the oriented cycle with full
relations, then $\mathrm{proj.dim}S(v)=\infty$. Consider the arrow $\alpha$ on
this oriented cycle with full relations such that its sink is $v$. We have
that $\alpha A$ is an indecomposable non-projective Gorenstein projective
module, and moreover it is the Gorenstein projective cover of $S(v)$. In the
marked ribbon surface $\mathcal{S}_{A}$, $v$ corresponds to a side, denoted by
$\gamma_{1}$, of an $\infty$-elementary polygon $\Delta_{j}$ and
$M^{-1}(\alpha A)$ is such a permissible curve $c$ consecutively crossing
$\gamma_{1},\cdots,\gamma_{r}$ ($r\geq 1$), where
$\gamma_{1},\cdots,\gamma_{r}$ satisfy the condition (GP2) given in
Proposition 6.3 and $\gamma_{1}=\mathfrak{v}(v)$.

In the case for $r=1$, it is obvious that $S(v)$ is a Gorenstein projective
module and $\mathrm{G}\text{-}\mathrm{proj.dim}S(v)=0$. Suppose $r>1$. Then
there exists an elementary polygon $\Delta_{j^{\prime}}$ such that
$\gamma_{1}$ and $\gamma_{2}$ are two sides of $\Delta_{j^{\prime}}$. If
$\Delta_{j^{\prime}}$ is not $\infty$-elementary then we get
$\mathrm{G}\text{-}\mathrm{proj.dim}S(v)=\mathfrak{C}(\Delta_{j^{\prime}})-1$
by using an argument similar to that of Proposition 5.6 (the arcs $\gamma_{1}$
and $\gamma_{2}$ can be viewed as $\gamma^{0}$ and $\gamma^{1}$ shown in
Proposition 5.6, respectively). If $\Delta_{j^{\prime}}$ is
$\infty$-elementary, then consider the permissible curve $c^{\prime}$
consecutively crossing $\gamma_{2},\cdots,\gamma_{r}$ (in this case
$\gamma_{2}\in\mathfrak{S}(\Delta_{j^{\prime}})$) where
$\gamma_{2},\cdots,\gamma_{r}$ satisfy (GP2). We have that $M(c^{\prime})$ is
the kernel of $M(c)\twoheadrightarrow S(v)\cong
M(c_{\mathrm{simp}}^{\gamma_{1}})$ by Theorem 3.12(1) because the positional
relationship of $c^{\prime}$, $c$ and $c_{\mathrm{simp}}^{\gamma_{1}}$
satisfies the conditions given in Proposition 3.6. Then
$\mathrm{G}\text{-}\mathrm{proj.dim}S(v)=1$. Consequently we have

$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\max\big{\\{}1,\max\limits_{\Delta_{i}\in\mathsf{f}\text{-}\Delta}{\mathfrak{C}(\Delta_{i})}-1\big{\\}}.$

(ii) If $(Q,I)$ has no oriented cycle with relation, then $A$ has finite
global dimension by Theorem 5.10. In this case we have

$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\mathrm{gl.dim}A=\max\limits_{1\leq
i\leq d}\mathfrak{C}(\Delta_{i})-1,$

where $\\{\Delta_{i}\mid 1\leq i\leq d\\}=\mathsf{f}\text{-}\Delta$ is the set
of all elementary polygons of $\mathcal{S}_{A}$. ∎

The proof of Proposition 6.5 depends on Theorem 6.1. In the following, we will
give its another proof which does not need to use the properties of Gorenstein
projective modules. We assume that the quiver of $A$ is neither an oriented
cycle with full relations nor a point.

######  Lemma 6.6.

Let $\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ be a marked
ribbon surface of a gentle algebra $A$, and let $\gamma$ be an arc on
$\mathcal{S}_{A}$ with endpoints $p$ and $q$. As shown in FIGURE 20, suppose
that $c_{\mathrm{inj}}^{\gamma}$ consecutively crosses arcs
$\gamma^{0}_{1},\cdots,\gamma^{0}_{i-1},\gamma$,
$\tilde{\gamma}^{0}_{j-1},\cdots,\tilde{\gamma}^{0}_{1}$, and
$c_{\mathrm{proj}}^{\gamma}$ consecutively crosses arcs
$\tilde{\gamma}^{0}_{n}$, $\cdots$, $\tilde{\gamma}^{0}_{j+1}$, $\gamma$,
$\gamma^{0}_{i+1}$, $\cdots$, $\gamma^{0}_{m}$, where $p$ is the common
endpoint of $\gamma^{0}_{1},\cdots,\gamma^{0}_{m}$ and $q$ is the common
endpoint of $\tilde{\gamma}^{0}_{1},\cdots,\tilde{\gamma}^{0}_{n}$ ($1\leq
i\leq m$, $1\leq j\leq n$ and $\gamma=\gamma^{0}_{i}=\tilde{\gamma}^{0}_{j}$).
Then:

  1. (1)

the projective cover of $M(c_{\mathrm{inj}}^{\gamma})$ is
$p(M(c_{\mathrm{inj}}^{\gamma})):M(c_{\mathrm{proj}}^{\gamma^{0}_{1}}\oplus
c_{\mathrm{proj}}^{\tilde{\gamma}^{0}_{1}})\twoheadrightarrow
M(c_{\mathrm{inj}}^{\gamma})$;

  2. (2)

$\mathrm{Ker}p(M(c_{\mathrm{inj}}^{\gamma}))\cong M(c^{\prime}\oplus
c_{\mathrm{proj}}^{\gamma}\oplus\tilde{c}^{\prime})$, where $c^{\prime}$ and
$c^{\prime\prime}$ are two permissible curves shown in FIGURE 20.

$c_{\mathrm{inj}}^{\gamma}$$c_{\mathrm{proj}}^{\gamma}$$c_{\mathrm{proj}}^{\gamma^{1}_{1}}$$c_{\mathrm{proj}}^{\tilde{\gamma}^{1}_{1}}$$c^{\prime}$$\tilde{c}^{\prime}$$\boldsymbol{\times}$$\boldsymbol{\times}$$\boldsymbol{\times}$$\boldsymbol{\times}$$p$$q$$x$$y$$\gamma$$\gamma^{0}_{1}$$\gamma^{0}_{2}$$\vdots$$\gamma^{0}_{i-1}$$\tilde{\gamma}^{0}_{j+1}$$\tilde{\gamma}^{0}_{j+2}$$\vdots$$\tilde{\gamma}^{0}_{n}$$\gamma^{0}_{m}$$\vdots$$\gamma^{0}_{i+2}$$\gamma^{0}_{i+1}$$\tilde{\gamma}^{0}_{j-1}$$\vdots$$\tilde{\gamma}^{0}_{2}$$\tilde{\gamma}^{0}_{1}$$\gamma^{1}_{r}$$\vdots$$\gamma^{1}_{1}$$\tilde{\gamma}^{1}_{s}$$\vdots$$\tilde{\gamma}^{1}_{1}$$\gamma^{2}_{1}$$\vdots$
Figure 20: The formal direct sum $c_{\mathrm{proj}}^{\gamma_{1}^{1}}\oplus
c_{\mathrm{proj}}^{\tilde{\gamma}_{1}^{1}}$ of permissible curves provides the
projective cover of the injective module $M(c_{\mathrm{inj}}^{\gamma})$.

###### Proof.

(1) This follows from Proposition 4.3 and Theorem 4.6.

(2) Indeed,
$\mathrm{top}M(c_{\mathrm{inj}}^{\gamma})=M(c_{\mathrm{simp}}^{\gamma^{0}_{1}})\oplus
M(c_{\mathrm{simp}}^{\tilde{\gamma}^{0}_{1}})$, and there exists no arc
$\gamma^{0}$ (respectively, $\tilde{\gamma}^{0}$) with endpoint $p$
(respectively, $q$) such that
$\gamma^{0},\gamma^{0}_{1},\cdots,\gamma^{0}_{i-1},\gamma$ (respectively,
$\gamma$, $\tilde{\gamma}^{0}_{j-1}$, $\cdots$, $\tilde{\gamma}^{0}_{1}$,
$\tilde{\gamma}$) surround $p$ (respectively, $q$) in the counterclockwise
order. Thus $M(c_{\mathrm{inj}}^{\gamma})$ corresponds to the string

$\gamma^{0}_{1}\longrightarrow\gamma^{0}_{2}\longrightarrow\cdots\longrightarrow\gamma^{0}_{i-1}\longrightarrow\gamma\longleftarrow\tilde{\gamma}^{0}_{j-1}\longleftarrow\cdots\longleftarrow\tilde{\gamma}^{0}_{2}\longleftarrow\tilde{\gamma}^{0}_{1},$

and $M(c_{\mathrm{simp}}^{\gamma^{0}_{1}})$ and
$M(c_{\mathrm{simp}}^{\tilde{\gamma}^{0}_{1}})$ correspond to the strings

$\displaystyle\gamma^{1}_{r}\longleftarrow\cdots\longleftarrow\gamma^{1}_{1}\longleftarrow\gamma^{0}_{1}\longrightarrow\gamma^{0}_{2}\longrightarrow\cdots\longrightarrow\gamma^{0}_{i-1}\longrightarrow\gamma\longrightarrow\gamma^{0}_{i+1}\longrightarrow\cdots\longrightarrow\gamma^{0}_{m}\
\text{ and }$
$\displaystyle\tilde{\gamma}^{0}_{n}\longleftarrow\cdots\longleftarrow\tilde{\gamma}^{0}_{j+1}\longleftarrow\gamma\longleftarrow\tilde{\gamma}^{0}_{j-1}\longleftarrow\cdots\longleftarrow\tilde{\gamma}^{0}_{2}\longleftarrow\tilde{\gamma}^{0}_{1}\longrightarrow\tilde{\gamma}^{1}_{1}\longrightarrow\cdots\longrightarrow\tilde{\gamma}^{1}_{s}\text{,
respectively.}$

Then the kernel of $p(M(c_{\mathrm{inj}}^{\gamma}))$ is isomorphic to
$M_{1}\oplus M_{2}\oplus M_{3}$, where $M_{1}$, $M_{2}$ and $M_{3}$ correspond
to strings

$\displaystyle\gamma^{1}_{r}\longleftarrow\cdots\longleftarrow\gamma^{1}_{1},\
\tilde{\gamma}^{0}_{n}\longleftarrow\cdots\longleftarrow\tilde{\gamma}^{0}_{j+1}\longleftarrow\gamma\longrightarrow\gamma^{0}_{i+1}\longrightarrow\cdots\longrightarrow\gamma^{0}_{m}\text{
and
}\tilde{\gamma}^{1}_{1}\longrightarrow\cdots\longrightarrow\tilde{\gamma}^{1}_{s},\
\text{ respectively. }$

Since $c_{\mathrm{proj}}^{\gamma^{0}_{1}}$ (respectively,
$c_{\mathrm{proj}}^{\tilde{\gamma}^{0}_{1}}$) is projective, then both
conditions (LMC) and (RMC) are satisfied. The condition (RMC) (respectively,
(LMC)) shows that there is no arc $\gamma^{0}_{m+1}$ (respectively,
$\tilde{\gamma}^{0}_{n+1}$) such that
$\gamma^{0}_{1},\cdots,\gamma^{0}_{m},\gamma^{0}_{m+1}$ (respectively,
$\tilde{\gamma}^{0}_{1},\cdots,\tilde{\gamma}^{0}_{n},\tilde{\gamma}^{0}_{n+1}$)
have the same endpoint $p$ (respectively, $q$) and surround $p$ (respectively,