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K. Baur and R. Coelho Simões (2019) “A Geometric Model for the Module Category of a Gentle Algebra,”
International Mathematics Research Notices, Vol. 00, No. 0, pp. 1–36
doi:10.1093/imrn/rnz150
A Geometric Model for the Module Category of a Gentle
Algebra
Karin Baur1,2 and Raquel Coelho Simões3,∗
1Karl-Franzens-Universitat Graz Institut fur Unternehmensrechnung
und Steuerlehre, Institut für Mathematik und Wissenschaftliches
Rechnen, Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria,
2School of Mathematics, University of Leeds, Leeds, LS2 9JT, United
Kingdom, and 3Centro de Análise Funcional, Estruturas Lineares e
Aplicações, Faculdade de Ciências da Universidade de Lisboa, Campo
Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal
∗Correspondence to be sent to: e-mail: rcoelhosimoes@campus.ul.pt
In this article, gentle algebras are realised as tiling algebras, which are associated to
partial triangulations of unpunctured surfaces with marked points on the boundary.
This notion of tiling algebras generalise the notion of Jacobian algebras of triangula-
tions of surfaces and the notion of surface algebras. We use this description to give a
geometric model of the module category of any gentle algebra.
Introduction
Gentle algebras are an important class of finite-dimensional algebras of tame repre-
sentation type. Gentle algebras are Gorenstein [25], closed under tilting and derived
equivalence [39,40], and they are ubiquitous, for instance they occur in contexts such
as Fukaya categories [28], dimer models [8], enveloping algebras of Lie algebras [30], and
cluster theory. As such, there has been widespread interest in this class of algebras; see,
for example, [10,14,16,38] for recent developments in this area.
Received April 17, 2018; Revised April 17, 2019; Accepted June 3, 2019
© The Author(s) 2019. Published by Oxford University Press. All rights reserved. For permissions,
please e-mail: journals.permission@oup.com.
2 K. Baur and R. Coelho Simões
In the context of cluster theory, gentle algebras arise as m-cluster-tilted alge-
bras, m-Calabi–Yau-tilted algebras (see, e.g., [23]) and as Jacobian algebras associated
to triangulations of unpunctured surfaces with marked points on the boundary [2,32].
The combinatorial geometry of the surfaces plays an important role in cluster-
tilting theory and in representation theory in general. It was used to give a geometric
model of categories related to hereditary algebras of several types (see, e.g., [4,6,
13,35,42]) and of generalised cluster categories and module categories of Jacobian
algebras [11]. The combinatorics of the surface is also useful to give a description of
certain m-cluster-tilted algebras in terms of quivers with relations and to study their
derived equivalence classes (see, e.g., [26,27,36]). On the other hand, geometric models
associated to certain categories have also been used to tackle several representation-
theoretic questions on those categories, such as the classification of torsion pairs (see,
e.g., [5,17,29]).
In this article, we realise gentle algebras as tiling algebras, by considering
partial triangulations of the surfaces mentioned above. These algebras were already
considered in [18]and[24] in the case where the surface is a disc, and they are a natural
generalisation of Jacobian algebras and surface algebras [20]. See [21] for other types
of algebras related to partial triangulations. Viewing gentle algebras as tiling algebras
allows us to construct a geometric model for their module categories. The main results
of this article are stated as follows:
Theorem 1. (Theorems 2.8,2.9 and 2.9)LetAbe a finite-dimensional algebra. Then the
following are equivalent.
(1) Ais a gentle algebra.
(2) Ais a tiling algebra.
(3) Ais the endomorphism algebra of a partial cluster-tilting object of a
generalised cluster category associated to some unpunctured surface.
Theorem 2. (Theorems 3.8,3.9 and 3.15)LetAPbe the tiling algebra associated to
a partial triangulation Pof an unpunctured surface Swith marked points Min its
boundary. There are explicit bijections between
(1) the equivalence classes of non-trivial permissible arcs in (S,M)and non-
zero strings of AP,
(2) permissible closed curves and powers of band modules, and
(3) certain moves between permissible arcs and irreducible morphisms of
string modules.
Module Categories of Gentle Algebras 3
In the cases when the partial triangulation is either a triangulation or a cut of
a triangulation, our model in Theorem 2 recovers the one given for surface algebras
[20] (see also [2,11]). We note that, recently, a geometric description for the bounded
derived category of a gentle algebra (up to shift) was introduced in [37] (see also [34]for
the case of homologically smooth graded gentle algebras). The special case of discrete
derived categories was previously studied in [9]. There are differences in the surfaces
considered in these models for the derived category and our model for the module
category. Namely, we can have more marked points and fewer boundary components
with no marked points (more details can be found in the end of Section 2). On the
other hand, an indecomposable module, seen as an object of the derived category, has
a different geometric representation than the representation in the model presented
in this paper. For instance, a string module with infinite projective resolution is
represented by an infinite arc circling around an unmarked boundary component in
the geometric model of the derived category in [37], while in our model it is represented
by a finite arc. Moreover, unlike in the model in [37], the same module can be represented
by non-isotopic arcs in our model, and not all arcs in our model give rise to an
indecomposable module. For more details, we refer the reader to Remark 3.12 and
Example 3.13.
We also give a simple description, in terms of the partial triangulations of
surfaces, of all the representation-finite gentle algebras. Theorem 2(1) and (3) give a
complete geometric description of this class of gentle algebras, and examples include
gentle algebras with no double arrows nor loops, and whose cycles are oriented and
with radical square zero (these arise from discs) and derived-discrete algebras (these
arise from annuli).
This paper is organised as follows. In Section 1, we give the necessary back-
ground on gentle algebras. In Section 2we introduce the notion of tiling algebra and
give the proof of Theorem 1. Section 3is devoted to the geometric model for the module
category of a tiling algebra, and it includes a geometric description of morphisms
between indecomposable modules.
1 Gentle algebras: basic notions and properties
In this section, we will recall the definition of gentle algebras and basic properties of
their module categories. The main reference is [12].
Let kdenote an algebraically closed field, Qa quiver with set of vertices Q0
and set of arrows Q1,Ian admissible ideal of kQ,andAthe bound quiver algebra
4 K. Baur and R. Coelho Simões
kQ/I.Givena∈Q1, we denote its source by s(a)and its target by t(a). We will
read paths in Qfrom left to right, and mod(A)will denote the category of right
Amodules.
Definition 1.1. [3] A finite-dimensional algebra Ais gentle if it admits a presentation
A=kQ/Isatisfying the following conditions:
(G1) Each vertex of Qis the source of at most two arrows and the target of at
most two arrows.
(G2) For each arrow αin Q, there is at most one arrow βin Qsuch that αβ ∈ Iand
there is at most one arrow γsuch that γα ∈ I.
(G3) For each arrow αin Q, there is at most one arrow δin Qsuch that αδ ∈Iand
there is at most one arrow μsuch that μα ∈I.
(G4) Iis generated by paths of length 2.
Throughout this section, A=kQ/Idenotes a gentle algebra. To each arrow a∈
Q1, we associate a formal inverse a−1,withs(a−1)=t(a)and t(a−1)=s(a). The set
of formal inverses of arrows in Q1is denoted by Q−1. We can extend this to any path
p=a1···anin Q, by setting p−1=a−1
n···a−1
1.
Astring w is a reduced walk in the quiver that avoids relations, that is, w=
a1···an,withai∈Q∪Q−1, for which there are no subwalks of the form aa−1or a−1a,
for some a∈Q1, nor subwalks ab such that ab ∈Ior (ab)−1∈I.Thelength of a string
w=a1···anis n. For each vertex v∈Q0, there are two trivial strings 1+
vand 1−
v.
The vertex vis both the source and target of each corresponding trivial string, and the
formal inverse acts by swapping the sign, that is, (1±
v)−1=1∓
v. For technical reasons, we
also consider the empty string, often called the zero string. We will denote it by w=0.
Astringw=a1···anis direct (inverse, resp.) if ai∈Q1(ai∈Q−1
1, resp.), for all
i=1, ...,n.
Astringiscyclic if its source and target coincide. A band is a cyclic string bfor
which each power bnisastring,butbitself is not a proper power of any string.
Each string win Adefines a string module M(w)∈mod(A). The underlying
vector space of M(w)is obtained by replacing each vertex of wby a copy of the field k,
and the action of an arrow aon M(w)is the identity morphism if ais an arrow of w,
and is zero otherwise. If w=0, then M(w)=0. Note that M(w)M(v)if and only if
v=wor v=w−1.
Each band b=a1···an(up to cyclic permutation and inversion) defines a family
of band modules M(b,n,ϕ), where n∈Nand ϕis an irreducible automorphism of kn.
Module Categories of Gentle Algebras 5
Here, each vertex of bis replaced by a copy of the vector space knand the action of an
arrow aon M(b,n,ϕ) is induced by identity morphisms if a=ai,fori=1, ...,n−1
and by ϕif a=an. All string and band modules are indecomposable, and every
indecomposable Amodule is either a string module or a band module (cf. [12]).
Given two strings v=a1···an,w=b1···bmof length at least one, the
composition vw is defined if vw =a1···anb1···bmisastring.
In order to define composition of strings with trivial strings and avoid
ambiguity in the description of irreducible morphisms between string modules, we
need to consider two sign functions σ,ε:Q1→{−1, 1}satisfying the following
conditions:
•If b1= b2∈Q1are such that s(b1)=s(b2),thenσ(b1)=−σ(b2).
•If a1= a2∈Q1are such that t(a1)=t(a2),thenε(a1)=−ε(a2).
•If ab is not in a relation, then σ(b)=−ε(a).
We refer the reader to [12, p. 158], for an algorithm for choosing these functions. We
can extend the maps σand εto all strings as follows. Given an arrow b∈Q1,wehave
σ(b−1)=ε(b)and ε(b−1)=σ(b).Ifw=a1···anis a string of length n1, σ(w)=σ(a1)
and ε(w)=ε(an).Finally,σ(1±
v)=∓1andε(1±
v)=±1.
If vis a string and w=1±
x,thenvw is defined if t(v)=xand ε(v)=±1.
Analogously, if v=1±
xand wis a string, then vw is defined if s(w)=xand
σ(w)=∓1. In a nutshell, composition of strings vw of a gentle algebra is defined
if σ(w)=−ε(v).
Now we recall the description of irreducible morphisms between string A
modules, given in [12]. Let wbeastringandM(w)be the corresponding string module.
The irreducible morphisms starting at M(w)can be described in terms of modifying the
string won the left or on the right.
Case 1: There is an arrow a∈Q1such that aw is a string. Then let vbe the
maximal direct string starting at s(a), and define wto be the string v−1aw.
Case 2: There is no arrow a∈Q1such that aw is a string. Then, either wis
a direct string or we can write w=va−1w, where a∈Q1,wis a string, and vis a
maximal direct substring starting at s(w). In the former case, we define wto be the
zero string, and in the latter case w:=w.
In the literature, wis said to be obtained from wby either adding a hook on
s(w)(in Case 1) or removing a cohook from s(w)(in Case 2).
The definition of wris dual. One can check that (w)r=(wr), and we shall
denote this string by wr,. Figures 1and 2illustrates these concepts.
6 K. Baur and R. Coelho Simões
Fig. 1. Left: adding a hook on s(w). Right: removing a cohook from s(w).
Fig. 2. Left: adding a hook on t(w). Right: removing a cohook from t(w).
Proposition 1.2. [12]Letwbe a string of a gentle algebra Aand M(w)be the
corresponding string module.
(1) There are at most two irreducible morphisms starting at M(w), namely
M(w)→M(w)and M(w)→M(wr). These maps are either the natural
inclusion (in Case 1) or the natural projection (in Case 2).
(2) If M(w)is not injective, then the Auslander–Reiten sequence starting at
M(w)is given by 0 →M(w)→M(w)⊕M(wr)→M(wr,)→0. In particular,
τ−1(M(w)) =M(wr,).
This proposition actually holds more generally for string algebras. However, we
only state it for gentle algebras, since this is the case we are interested in. The remaining
irreducible morphisms are between band modules of the form M(b,n,ϕ) and M(b,n+
1, ϕ). Band modules lie in homogeneous tubes, and so the Auslander–Reiten translate
acts on them as the identity morphism.
2 Gentle algebras as tiling algebras
Throughout this paper, Sdenotes an unpunctured, connected oriented Riemann surface
with non-empty boundary and with a finite set Mof marked points on the boundary.
We assume that, if Sis a disc, then Mhas at least four marked points. The pair (S,M)is
simply called a marked surface. We allow unmarked boundary components of S,that
is, boundary components with no marked points in M.
An arc in (S,M)isacurveγin Ssatisfying the following properties:
•The endpoints of γare in M.
•γintersects the boundary of Sonly in its endpoints.
•γdoes not cut out a monogon or a digon.
Module Categories of Gentle Algebras 7
We always consider arcs up to isotopy relative to their endpoints. We call a
collection of arcs Pthat does not intersect themselves or each other in the interior of S
apartial triangulation. A triple (S,M,P)satisfying the conditions above is a tiling.
Before giving the definition of tiling algebra associated to (S,M,P), we need to
introduce some terminology and notation. We follow the nice exposition given in [20].
An arc vin (S,M,P)has two end segments, defined by trisecting the arc and
deleting the middle piece. We denote these two end segments by vand v.
Given a marked point pin M,letmand m be two points in the same boundary
component of psuch that m,m ∈ Mand pis the only marked point lying in the
boundary segment δbetween mand m. Draw a curve cisotopic to δbut lying in
the interior of Sexcept for its endpoints mand m.Thecomplete fan at p is the
sequence v1,...,vkof arcs in P,whichccrosses in the clockwise order. Any subsequence
vi,vi+1,...,vjis called a fan at p. The arc vi(vj, resp.) is said to be the start (end, resp.)
of this fan. If the marked point pis not the endpoint of any arc in P, then the (complete)
fan at pis called an empty fan.
Every arc vin Plies in either one non-empty complete fan, in case visaloop,
that is, both endpoints coincide, or two distinct non-empty complete fans.
Definition 2.1. The tiling algebra APassociated to the partial triangulation Pof (S,M)
is the bound quiver algebra AP=kQP/IP, where (QP,IP)are described as follows:
(1) The vertices in (QP)0are in one-to-one correspondence with the arcs in P.
(2) There is an arrow a:as→atin (QP)1if the arcs asand atshare an endpoint
pa∈Mand atis the immediate successor of asin the complete fan at pa.
(3) IPis generated by paths ab of length 2 that satisfy one of the following
conditions:
•b=a, that is, the arcs corresponding to the sources and targets
of aand bcoincide and this arc is a loop;
•b= aand either pa= pbor we are in the situation presented in
Figure 3,withaa loop.
In other words, ab ∈IPif and only if either pa= pbor pa=pband t(a)=s(b)
corresponds to a loop arc.
Remark 2.2. If a,b∈(QP)1with t(b)=s(a)are such that ab ∈IP, then the arcs
corresponding to s(a),t(a)=s(b),andt(b)bound the same region in (S,M,P).However,
the converse is not true in general.
8 K. Baur and R. Coelho Simões
Fig. 3. Case ab =0whenpa=pb.
Lemma 2.3. Every oriented cycle in QPof length 2 has a zero relation.
Proof. Let be an oriented cycle of
length k2. Let pibe the marked point associated to arrow ai,fori=1, ...,k.If
ai=ai+1or pi= pi+1for some i=1, ...k+1 (where k+1=1), then aiai+1∈IP,by
definition. So assume ai= ai+1and p:=pi=pi+1, for all i=1, ...k+1. Then the cycle c
corresponds to a fan at pwith v1as both the start and end. In other words, v1isaloop,
and by definition of IP, it follows that aka1∈IP, which finishes the proof.
Corollary 2.4. Every tiling algebra is finite dimensional.
Atriangulation of (S,M)is a partial triangulation cutting the surface into
triangles. Note that not every surface admits a triangulation.
Let (S,M)be a new marked surface obtained from (S,M,P)by adding a marked
point in each unmarked boundary component. Note that these new marked points do
not play a role in the definition of a tiling algebra, that is, the tiling algebras associated
to (S,M,P)and (S,M,P)are isomorphic. On the other hand, the partial triangulation
Pin (S,M)can be completed to a triangulation T(cf. [22, Lemma 2.13]).
Let AT=kQT/ITbe the Jacobian algebra associated to T(see [2] for more details).
In particular, each vertex vof QTcorresponds to an arc vof T, and the relations in AT
are precisely the compositions of any two arrows in an oriented 3-cycle in QT,which
correspond to internal triangles in T, that is, triangles whose three sides are arcs in T.
Definition 2.5. Let AP,T=kQP,T/IP,Tbe the algebra obtained from (QT,IT)as follows:
Module Categories of Gentle Algebras 9
Vertices: (QP,T)0:=(QP)0, that is, the vertices are in one-to-one correspondence
with the arcs in P.
Arrows: For each direct string v→u1→ ··· → uk→vin QT, where v,v∈P
and u1,...,uk∈T\P, we have an arrow v→vin QP,T. All the arrows in QP,Tare
obtained in this way.
Relations: Let be a path of length 2 in QP,T. This means we
have two direct strings and
in QT,withui,u
j∈T\Pfor all i,j,andv,v,v ∈P.Ifαβ ∈IT,thenab ∈IP,T.
Lemma 2.6. The tiling algebra APis isomorphic to AP,T, for any triangulation T
completing Pin (S,M).
Proof. It is clear that the set of vertices (QP)0coincides with (QP,T)0, which corre-
sponds to the arcs in P.
An arrow a:v→vbelongs to (QP,T)1if and only if there is a direct string
v→u1→ ··· → ur→vin QT, where each uicorresponds to an arc in T\P.Thisis
equivalent to saying that the sequence v,u1,...,ur,vis a fan at a marked point pof
(S,M,T), which in turn is equivalent to saying that the sequence v,visafanatthe
marked point pof (S,M,P),thatis,thata:v→v∈(QP)1. Hence, QP=QP,T.
Finally, we have ab ∈IP,Tif and only if the following is a subquiver of QT:
where the oriented 3-cycle has radical square zero. We can deduce that this is equivalent
to saying that ab ∈IP,bynotingthatt(α) =t(a)=s(b)=s(β),pa=pα,andpb=pβ,
and by having in mind the possible internal triangles in Tthat give rise to the oriented
3-cycle in the subquiver above.
Remark 2.7. If Pis a triangulation, then our notion of tiling algebra recovers the
concept of Jacobian algebra. Furthermore, our notion generalises the notion of surface
algebra, which is defined for cuts of triangulations (cf. [20]).
We will now state two neat consequences of Lemma 2.6. Assume (S,M)is such
that each boundary component of Scontains a marked point in M. Then one can
associate a category, called generalised cluster category CS,M,to(S,M). Moreover, every
indecomposable object in CS,Mcan be described in terms of curves in the surface. In
10 K. Baur and R. Coelho Simões
particular, there is a bijection between (partial) triangulations Pof (S,M)and (partial)
cluster-tilting objects Pin CS,M. Given a triangulation Tof (S,M), the endomorphism
algebra BTof the corresponding cluster-tilting object Tis known to be isomorphic to the
Jacobian algebra AT. More details can be found in [1], [2], [11], and [32].
Theorem 2.8. Let Abe a finite-dimensional algebra. The following conditions are
equivalent:
(1) Ais a tiling algebra.
(2) Ais isomorphic to the endomorphism algebra of a partial cluster-tilting
object in a generalised cluster category associated to some unpunctured
surface.
Proof. By Lemma 2.6, the algebra Ais a tiling algebra if and only if A=AP,T,for
some partial triangulation Pof a marked surface (S,M), which admits a triangulation,
and some triangulation Tcompleting P. By the above, partial triangulations Pof
marked surfaces (S,M)admitting triangulations are in bijection with partial cluster-
tilting objects Pin the generalised cluster category CS,M. Denote the corresponding
endomorphism algebra by BP. It is then enough to show that AP,TBP.
Since the Jacobian algebra ATis isomorphic to the endomorphism algebra BTof
the cluster-tilting object T, we know that the vertices of the quiver ATare in one-to-one
correspondence with the indecomposable summands of Tand each non-zero path in the
quiver of ATcorrespond to a non-zero morphism in CS,Mbetween the indecomposable
summands associated to the endpoints of the path. We use the same notation for vertices
in ATand indecomposable objects in CS,Mand for arrows in ATand corresponding
morphisms in CS,M.
Given two vertices vand vof ATcorresponding to two indecomposable
summands of P, each arrow v→vin BPcorresponds a non-zero morphism from vto v,
which does not factor through any other summand of P. By the above, this corresponds
to a direct string v→u1→ ··· → ur→vin QT, where each uilies in T\P. Thus, the
set of arrows of (BP)coincides with (QP,T)1.
Similarly, a path v1→v2→ ··· → vkin BPcorresponds to a zero morphism in
CS,Mif and only if there are direct strings vi→ui1→ui2→ ··· → uiir→vi+1, for each
i=1, ...,k−1, such that each uij ∈T\P, and the composition of all these strings yields
a relation. Given the notion of the Jacobian algebra AT, this is equivalent to saying that
there is a relation in ITof the form uiir→vi+1→ui+1,1. Thus, the ideal of BPcoincides
with that of AP,T, which finishes the proof.
Module Categories of Gentle Algebras 11
Theorem 2.8 generalises [20, Theorem 3.7]. The following proposition is a
generalisation of [2, Lemma 2.5] and [20, Theorem 3.8].
Theorem 2.9. Every tiling algebra APis gentle.
Proof. Let Pbe a partial triangulation of (S,M),Tbe a triangulation completing P,
and ATbe the corresponding Jacobian algebra, which is known to be gentle
[2, Lemma 2.5].
By Lemma 2.6,APAP,T.Letvbe a vertex in AP,T, and suppose, for a
contradiction, that there are three distinct arrows starting at v. This means there are
three distinct direct strings in ATstarting at v. These direct strings are subpaths of
three distinct non-trivial maximal direct strings in ATwhich contain the vertex v.
However, since ATis gentle, every vertex lies in at most two non-trivial maximal direct
strings, so we reached a contradiction. Similarly, we can prove that every vertex in AP,T
is the target of at most two arrows.
Consider the following subquiver of QP,T:
Then, we have the following subquiver of QT:
Since ATis gentle, either ab =0orac =0, which implies that either αβ =0orαγ =0.
A similar argument proves the dual. Finally, (G4) holds by definition.
Theorem 2.10. Every gentle algebra is a tiling algebra.
Proof. We can associate to a gentle algebra Aa certain graph called marked ribbon
graph A. We refer the reader to [37, Definition 1.8] for details (see also [41]). This
graph can be embedded in a surface SAwith boundary (cf. [33,37]), in such a way
that the vertices of Acorrespond to marked points MAof the boundary, and the edges
12 K. Baur and R. Coelho Simões
are curves in the surface that do not intersect themselves or each other, except at the
endpoints. The authors in [37]definealamination L of this embedding, an algebra
AL=kQL/ILassociated to this lamination and show that AL=A,cf.[37, Definition 1.18
and Proposition 1.21].
Now consider (S,M,P), where S=SA,Mis the set of marked points obtained
from MAby adding a marked point in each boundary segment isotopic to an edge of
the embedding of Ain SAand Pis given by the edges of Aembedded in S.We
claim that the tiling algebra Aassociated to the partial triangulation Pof (S,M)is
AL, thus proving the theorem. The set of vertices of the quivers of Aand ALcoincide,
as it is in one-to-one correspondence with the set of edges of A.Letiand jbe two
edges of Aand γiand γjbe the corresponding laminates in L. There is an arrow
γi→γjin ALif and only if γiand γjhave an endpoint on the same boundary segment
and the endpoint of γjfollows the endpoint of iin the clockwise direction along the
boundary. This is equivalent to the edges iand jsharing an endpoint and jfollowing
iin the clockwise direction around that endpoint, that is, γi→γjin ALif and only
if i→jin A.
Finally, let be two arrows in A. Suppose ab ∈IA.Then
i,j,andkcorrespond to arcs in Athat bound the same region in (SA,MA).By[37,
Proposition 1.12], this region has either exactly one unmarked boundary component or
exactly one boundary segment δ. By the definition of lamination, each laminate γi,γj,γk
has an endpoint in δ, and the order is consistent with the existence of the arrows aand b.
The only case when ab is not a relation in ILis if both endpoints of γjlie in δ.This
implies, in particular, that jdoes not correspond to a loop arc. Note that a lamination
divides SAup into polygons, each polygon containing exactly one marked point in MA
(cf. [37, Lemma 2.6]). Hence, we have that γi,γj,andγkbound a polygon with exactly one
marked point p,andp=pa=pb. But since jdoes not correspond to a loop arc, this
would mean that ab ∈ IA, a contradiction. Therefore, ab ∈IL. A similar argument proves
the converse, that is, that if ab ∈ILthen ab ∈IA.
By [37, Proposition 1.12], any gentle algebra Aarises from a marked surface
(S,M)together with a set of curves with endpoints in Mand with no intersections,
which divides the surface into polygons bounded by curves in and precisely one
boundary segment or polygons bounded entirely by curves in and with exactly one
unmarked boundary component in its interior. Some of the curves of might be isotopic
to a boundary segment, that is, they cut a digon. Since these are not curves allowed in
a partial triangulation, we change the set of marked points Mby adding a marked point
Module Categories of Gentle Algebras 13
Fig. 4. TilesoftypeIandII.
in each boundary segment isotopic to a curve in , creating a new set of marked points
M. Hence, is a partial triangulation of (S,M), and so it makes sense to consider its
tiling algebra, which, as we have seen in the proof of Theorem 2.10, coincides with the
initial gentle algebra A.
If we remove the unmarked boundary components lying inside 1-gons and
2-gons, we get either curves isotopic to points or two isotopic curves in . This changes
the property that is a partial triangulation of (S,M). However, if we remove the
unmarked boundary components lying inside m-gons, for m3, and call the new
surface S, we still have that is a partial triangulation of (S,M). Although these
unmarked boundary components are used in the definition of lamination, they do not
play a role in the notion of tiling algebra associated to partial triangulations. Indeed,
tiling algebras are defined only in terms of the arcs in , the marked points, and their
complete fans. Hence, the tiling algebra associated to (S,M,) coincides with the tiling
algebra associated to (S,M,).
For this reason, throughout the rest of this paper, we will consider tilings
(S,M,P)where the partial triangulation Pdivides Sinto a collection of regions (also
called tiles) of the following list:
•three-gons bounded by two boundary segments and one arc of Pand with no
unmarked boundary components of Sin its interior;
•m-gons, with m3 vertices and whose edges are arcs of Pand at most
one boundary segment, and whose interior contains no unmarked boundary
component of S;
•one-gons, that is, loops, with exactly one unmarked boundary component in
their interior—we call these tiles of type I;
•two-gons with exactly one unmarked boundary component in their interior -
we call these tilesoftypeII(see Figure 4).
14 K. Baur and R. Coelho Simões
Fig. 5. The surfaces associated to the gentle algebra A. The lamination is given by the red curves.
Fig. 6. The gentle algebra B. The relations are represented by dotted lines.
From now on, APdenotes the tiling algebra associated to a tiling (S,M,P)
satisfying the conditions described above.
Example 2.11. The following examples illustrate the differences between the surfaces
considered in this paper and in [37].
(1) Let Abe the algebra .In[37], the authors
describe the derived category of Aby considering the laminated surface on the left-
hand side of Figure 5, while we consider the partial triangulation of the surface on the
right-hand side of Figure 5. In this case, the surface is the same, but the surface on the
right-hand side has two more marked points (in yellow).
(2) Let Bbe the algebra given by the quiver with relations given in Figure 6.
In [37], the derived category of Bis described by considering the laminated surface
on the left-hand side of Figure 7, while we consider the partial triangulation of the
surface on the right-hand side of Figure 7. In this case, our surface is obtained by
removing one unmarked boundary component, and we have two more marked points
(in yellow).
Module Categories of Gentle Algebras 15
Fig. 7. The surfaces associated to the gentle algebra B.
3 The geometric model of the module category of a tiling algebra
We are now ready to give a geometric model for the module category of a gentle algebra
via the tilings described at the end of Section 2. This section is divided into three
subsections. The 1st subsection is devoted to the indecomposable modules, the 2nd one
concerns irreducible morphisms and in the last one we consider any morphism between
indecomposable modules.
3.1 Permissible arcs and closed curves
Not every curve in the surface with endpoints in Mwill correspond to an inde-
composable AP-module. Moreover, non-isotopic curves can correspond to the same
indecomposable module. Hence, we need the notion of permissible curves and of
equivalence between these curves.
Definition 3.1.
(1) A curve γin (S,M,P)is said to consecutively cross a1,a2∈Pif γcrosses a1
and a2in the points p1and p2, and the segment of γbetween the points p1
and p2does not cross any other arc in P.
(2) Let Bbe an unmarked boundary component of S,γ: [0, 1] →Sbe an
arc in (S,M),andwriteγ=γ1γγ2, where γ1(γ2, resp.) is the segment
between γ(0)(γ(1), resp.) and the 1st (last, resp.) crossing with P.The
winding number of γiaround B, where i=1, 2, is the minimum number
of times βtravels around Bin either direction, with βlying in the isotopy
class of γi.
(3) An arc γin (S,M,P)is called permissible if it satisfies the following two
conditions:
16 K. Baur and R. Coelho Simões
Fig. 8. γis a permissible arc.
Fig. 9. Examples of arcs not satisfying Condition (3)(a).
(a) The winding number of γiaround an unmarked boundary component
is either zero or one, for i=1, 2.
(b) If γconsecutively crosses two (possibly not distinct) arcs xand yof
P,thenxand yhave a common endpoint p∈M, and locally we have a
triangle, as shown in Figure 8.
(4) A permissible closed curve is a closed curve γthat satisfies condition (3)(b).
Note that boundary segments and arcs in Pare considered to be permis-
sible arcs.
Remark 3.2. It follows from the definition that any two consecutive crossings of a
permissible arc with arcs in Pis associated to an arrow of the tiling algebra.
Example 3.3. Figures 9,10,and11 show some examples of permissible and non-
permissible arcs.
Remark 3.4. In the case when Pis a triangulation of the surface, every arc is
permissible. Furthermore, if Pis a cut of a triangulation, and hence the corresponding
algebra is a surface algebra in the sense of [20], then our notion of permissible arcs
Module Categories of Gentle Algebras 17
Fig. 10. Examples of arcs not satisfying Condition (3)(b).
Fig. 11. Examples of permissible arcs.
coincides with that in [20]. Indeed, Condition (3)(a) is superfluous in this case, since
there are no unmarked boundary components, and Condition (3)(b) boils down to saying
that a permissible arc cannot cross two non-adjacent arcs of a quasi-triangle.
Definition 3.5. Two permissible arcs γ,γin (S,M)are called equivalent,whichwe
denote by γγ, if one of the following condition holds:
(1) There is a sequence of consecutive sides δ1,...,δkof a tile of (S,M,P)that
isnotoftypeIsuchthat
(a) γis isotopic to the concatenation of γand δ1,...,δk,and
(b) γstarts at an endpoint of δ1(δk, resp.), γstarts at an endpoint of δk
(δ1, resp.), and their 1st crossing with Pis with the same side of .
(2) The starting points of γand γare marked points of a tile of type
I or II; their 1st crossing (at point p)withPis with the same side of
and the segments of γand γbetween pand their ending points are
isotopic.
The equivalence class of a permissible arc γwill be denoted by [γ].
Example 3.6. Figures 12 and 13 show examples of permissible arcs that are equivalent.
18 K. Baur and R. Coelho Simões
Fig. 12. Equivalences of permissible arcs.
Fig. 13. Equivalences of permissible arcs when is of Type II.
Remark 3.7. In the case when Pis a triangulation, equivalence classes of (permissible)
arcs coincide with isotopy classes. Furthermore, when Pis a cut of a triangulation, our
notion of equivalence of permissible arcs coincide with that in [20]. Indeed, there are
no tiles of Type I or II, so Condition (2) in our definition does not apply. Moreover, the
tiles of Pare either triangles or quasi-triangles, and Condition (1) only applies to the
latter, in which case the sequence of consecutive sides must have length 1 in order to
satisfy (1)(b).
Given two curves γ,γin (S,M), we denote by I(γ ,γ)the minimal number of
transversal intersections of representatives of the isotopy classes of γand γ.Notethat
here, we are only considering crossings in the interior of S, not at its boundary. Hence,
we might have I(γ ,γ)=0 for arcs γ,γsharing an endpoint. Moreover, if γ∈P,then
I(γ ,γ)=0, for every arc γ∈P.
Given a curve γin (S,M,P),theintersection vector IP(γ ) of γwith respect to P
is the vector (I(γ ,ai))ai∈P.Theintersection number |IP(γ )|of γwith respect to Pis given
by a∈PI(γ ,a). We say that a permissible arc is trivial if its intersection number with
respect to Pis zero. We associate the zero string to any trivial permissible arc. Note that
this includes boundary segments and arcs in P.
Module Categories of Gentle Algebras 19
Fig. 14. The curve γ1. Here i= j∈{1, 2}.
Theorem 3.8. Let Pbe a partial triangulation of the surface (S,M). There is a bijection
between the equivalence classes of non-trivial permissible arcs in (S,M)and the non-
zero strings of AP. Under this bijection, the intersection vector corresponds to the
dimension vector of the corresponding string module.
Proof. Suppose is a non-zero string in
AP. Here, the double-headed arrows indicate a fixed but arbitrary orientation of
the arrows αi.
We define a curve γ(w)in (S,M)as follows: since there is an arrow α1between
v1and v2, the arcs v1,v2in Pshare a common endpoint p∈M, and there are no other
arcs of Pincident with psitting between v1and v2. Hence, v1and v2are sides of a
unique tile 1. We choose one point q1of the arc v1and one point q2of the arc v2and
connect them by a curve γ1in the interior of 1in such a way that we have a triangle as
in Figure 14.
Proceeding in the same way with the remaining arcs v2,...,vk, we obtain curves
γ2,...,γk−1. Now, the arc v1is incident with two tiles: 1considered above and 0,the
tile on the other side of v1. Note that these two tiles may coincide.
Case 1: p is the only marked point of 0.
This means v1is a loop and there is an unmarked boundary component σ0in
the interior of 0,thatis,0is of type I. Let γ0be a curve in the interior of 0with
endpoints pand q1and winding number 1 around σ0. Note that if the winding number
were zero, then the concatenation of γ0with γ1would be isotopic to a curve that does
not intersect v1.
Case 2: 0has exactly two marked points: pand p.
These points must be the endpoints of v1.Sincev1is not isotopic to a boundary
segment, 0must be of type II. Choose γ0to be, for instance, a curve in the interior of
0with endpoints q1and pin such a way that we have a triangle with vertices p,q1,
and p(see Figure 15).
20 K. Baur and R. Coelho Simões
Fig. 15. γ0when 0is of type II.
Case 3: 0has at least three marked points.
Choose γ0to be a curve in the interior of 0with endpoints q1and q, where qis
one of the marked points in 0that is not an endpoint of v1.
We proceed in the same way on the other end of the string, constructing a curve
γk. The curve γ(w)is then defined to be the concatenation of the curves γ0,...,γk.By
construction, we have the following:
•Since wisastring,vi= vi+1unless viis a loop. Either way, none of the γiis
isotopic to a piece of an arc in P.
•The crossing points of γ(w)with the arcs in Pare indexed by the vertices of
the string w, and the intersections are transversal.
•γ(w)is a permissible arc.
Thus, the intersection numbers are minimal and IP(γ (w)) =dim M(w), where
dim M(w)denotes the dimension vector of the module M(w).
Note that we have a choice of curves γ0and γs. But if we keep the winding
number around unmarked boundary components at zero or one, in order to obtain a
permissible arc, every possible choice would give a permissible arc equivalent to the
one built above.
Conversely, let γ: [0, 1] →Sbe a permissible arc in (S,M)with |IP(γ )| = 0,
belonging to an equivalence class [γ]. We assume that the arc is chosen (in its isotopy
class) such that it intersects the arcs aof Ptransversally and such that the intersections
are minimal.
Orienting γfrom p=γ(0)∈Mto q=γ(1)∈M, we denote by v1the 1st internal
arc of Pthat intersects γ,byv2the 2nd arc, and so on. We thus obtain a sequence
v1,...,vkof (not necessarily different) arcs in P. Since the intersections with arcs in P
are minimal, we know that vi= vi+1, unless viis a loop. Either way, there are arrows
αi:vi→vi+1or αi:vi+1→viin Q(AP), by Condition (3)(b) of the definition. Thus, we
Module Categories of Gentle Algebras 21
have a walk , which avoids relations and such
that αi+1= α−1
i. Therefore, w(γ ) is a non-zero string in AP.
If γ∈[γ], it follows by the definition of equivalent arcs that w(γ )=w(γ ),so
the map from equivalent classes of permissible arcs to strings is well defined. It follows
from their construction that the two maps defined above are mutually inverse.
To finish the characterisation of indecomposable modules in terms of curves in
the surface, we have to consider the bands.
Theorem 3.9. Let Pbe a partial triangulation of the surface (S,M). Then there is
a bijection between the isotopy classes of permissible closed curves cin (S,M)with
|IP(c)|2 and powers of bands of AP. Moreover, the permissible closed curve associated
to the Auslander–Reiten translate of an indecomposable band module Mis the closed
curve associated to Mitself.
Proof. Suppose is a power of a band
in AP. We define a closed curve γ(b)in (S,M)in a similar way to the construction of
the curve γ(w)in the proof of Theorem 3.8. Namely, γ(b)is the concatenation of curves
γ1,γ2,...,γk, where each γiconnects a point qiof arc viwith a point qi+1of arc vi+1in
such a way that we have triangles as the one in Figure 14.Notethatqk+1=q1, meaning
that the curve is closed, and making the description of the curve slightly easier than the
one in the proof of Theorem 3.8.
Since bis a power of a band in AP, it must correspond, in particular, to an
undirected cycle in the quiver QP, otherwise APwould not be finite dimensional. The
closed curve γ(b)divides Sinto two regions, its interior R1and its exterior R2. It follows
from the fact that bis undirected and has no relations that all the marked points
corresponding to paths in bin one orientation (depending on the chosen orientation
of γ(b)) lie in R1and all the marked points corresponding to the paths in bin the other
orientation lie in R2.Sincebhas at least one arrow and one inverse arrow, there is
at least one marked point in each region, and so γ(b)is not contractible to a point.
Moreover, all the crossings between γ(b)and arcs in Pare transversal, there are at least
two such crossings, and γ(b)is permissible, by construction.
Conversely, let γbe a permissible closed curve with |IP(γ )|2. We assume that
the curve is chosen (in its isotopy class) such that it intersects the arcs of Ptransversally
and such that the intersections are minimal.
22 K. Baur and R. Coelho Simões
Pick an intersection point v1of γwith Pand choose an orientation for the
curve γ. Denote by v2the 2nd internal arc of Pthat intersects γ,byv3the 3rd arc,
and continue in this manner until you reach v1again. Since γis permissible, there are
arrows αi:vi→vi+1or αi:vi+1→viin Q(AP). In order to prove that the cyclic walk
is a power of a band in AP, we need to check
that αiαi+1is not a relation, for each i=1, ...,k(here we have αk+1=α1).
Suppose that αiαi+1∈IP, for some i. Then, by definition, either pαi= pαi+1,
or pαi=pαi+1and vi+1is a loop arc. In the 1st case, we have an isotopy between
γand a closed curve which does not cross vi+1, contradicting the minimality of the
intersections. In the latter case, since the intersections are minimal and transversal,
we have that either γcrosses more arcs in Pbetween viand vi+1or the segment of the
curve γbetween the two intersections with the loop arc vi+1would not satisfy ondition
3(b) from Definition 3.1. Either way, we reach a contradiction, which finishes the proof
of the 1st statement.
The 2nd statement follows from the fact that the Auslander–Reiten translate
acts on band modules as the identity morphism.
Note that there can be permissible closed curves that do not intersect any arc
in P, namely when they go around an unmarked boundary component or intersect just
one arc of Pat only one point. Clearly these curves do not correspond to bands. However,
if Pis a cut of a triangulation, such curves do not occur. This explains why the condition
on the intersection number is required in Theorem 3.9 but it is not needed in [20].
The following result is a straightforward consequence of Theorem 3.9 and it is
a natural generalisation of [20, Corollary 5.10].
Corollary 3.10. A tiling algebra APis of finite representation type if and only if every
permissible simple closed curve cis such that |IP(c)|1.
Example 3.11. Consider the algebras (r,n,m)given by the quiver with relations
given in Figure 16. Up to derived equivalence, these are all the derived discrete
algebras that are not of Dynkin type (cf. [7]), and they are known to be of finite
representation type. Such algebras can be seen as tiling algebras associated to the
partial triangulations of an annulus given in Figure 17.
As we can see in the figure, the simple closed curve cis such that either
|IP(c)|1(ifn−r=0, 1) or |IP(c)|2(ifn−r2) and it is not permissible.
Module Categories of Gentle Algebras 23
Fig. 16. The algebra (r,n,m).
Fig. 17. Tiling associated to (r,n,m).
Remark 3.12. In order to pass from a module to the corresponding object in
the homotopy category we take the projective resolution of the module. To see the
correspondence between the arcs in our model and in the geometric model of the derived
category in [37] representing a given module, one needs to interpret [15, Corollary 2.12]
in terms of arc combinatorics. On the other hand, the homotopy category has many
more objects and identifying those corresponding to modules is non-trivial; one has to
interpret [15, Section 2] in terms of arc combinatorics.
The following examples show the arc representing a module in our model and
the arc representing its projective resolution in the model of [37], illustrating the
fact that there is not a direct match-up. The examples also illustrate the necessity of
having the extra marked points in our model and the fact that the unmarked boundary
components inside m-gons, for m3, are not needed in our model but are essential in
the model of [37].
Example 3.13.
1. Consider the algebra ,andthe
A-module M(a). Its projective resolution is . Therefore,
24 K. Baur and R. Coelho Simões
Fig. 18. The arcs of the A-module M(a)and of the corresponding object in the homotopy category.
Fig. 19. The gentle algebra B.
the arcs corresponding to M(a)in our model and in the model in [37] are as
in Figure 18.
2. Consider the algebra Bgiven by the quiver with relations given in Figure 19.
The B-module M(a)is projective and so its projective resolution is
simply P1. The arcs corresponding to M(a)in both models are as in
Figure 20.
3. Consider the algebra Bas above and the simple B-module S1. Its projective
resolution is .
The arcs corresponding to M(a)in both models are as in Figure 21.
3.2 Pivot elementary moves
Now we want to give a description of the irreducible morphisms between string modules
in terms of pivot elementary moves.
Given a permissible arc γ, we denote by M([γ])the string module M(w(γ )), where
w(γ ) is the string associated to γ.
Definition 3.14. Let γbe a permissible arc corresponding to a non-empty string, and
let s(t, resp.) be its starting point (ending point, resp.). Denote by the tile in the surface
such that the segment γof γbetween sand the 1st intersection point with Plies in the
interior of .Thepivot elementary move ft([γ])of [γ] is defined to be the equivalence
Module Categories of Gentle Algebras 25
Fig. 20. ThearcsoftheB-module M(a)and of the corresponding object in the homotopy category.
Fig. 21. ThearcsoftheB-module S1and of the corresponding object in the homotopy category.
class of the permissible arc ft(γs)obtained from γby fixing the ending point tand
moving sin the following way:
Case 1: is not of Type I or II.
Move sin the counterclockwise direction around up to the vertex sfor which
γγs, with endpoints tand s,andγ γ, where γis isotopic to the concatenation
of γswith the side of connecting swith its counterclockwise neighbour. Note that γs
might be γ(and so s=s).
Case 2: is of Type I.
Let γsbe the permissible arc equivalent to γ, with starting point s=sand such
that it wraps around the unmarked boundary component in the interior of in the
counterclockwise direction.
Case 3: is of Type II.
Let γsbe the permissible arc equivalent to γ, with starting point given by one
of the two marked points sof tile , and such that the winding number around the
unmarked boundary component σin the interior of is zero and σis to the left of γs.
26 K. Baur and R. Coelho Simões
In either case, ft(γs)is defined to be the concatenation of γswith the boundary
segment connecting sto its counterclockwise neighbour in the boundary component.
The pivot elementary move fs([γ])=[fs(γt)] is defined in a similar way.
Theorem 3.15. Let M(w)be a string module over the tiling algebra AP,with
corresponding string w.
(1) Each irreducible morphism in mod(AP)starting at M(w)is obtained by a
pivot elementary move on an endpoint of the corresponding permissible arc
γ=γ(w).
(2) All Auslander–Reiten sequences between string modules in mod(AP)are of
the form 0 →M(γ ) →M(fs(γ )) ⊕M(ft(γ )) →M(fs(ft(γ ))) →0, for some
permissible arc γ.
Proof. By Proposition 1.2, we need to prove that w=w(ft(γs)) and wr=w(fs(γt)).We
will only prove the former, as the latter is similar.
Firstly, we need to define the starting and ending points s,tof the arc
γcorresponding to w. Fix some sign functions σ,ε(recall the definition from
Section 1).
If wis not a trivial string, write ,with
k2. Then we orient γin such a way that it crosses the arcs v1,...,vkin this order.
Now suppose wis a trivial string, and write w=1x
v, for some vertex vand
x∈{+,−}.Ifvis a source, then M(w)is an injective module, and w=0=wr.Onthe
other hand, one can easily see that, independently of the orientation of γ, the pivot
elementary moves at either endpoint give rise to permissible arcs with intersection
number zero, which finishes the proof for this case. Thus, suppose there is at least
one arrow in Qwhose target is v.
Suppose vis the side of two different tiles 0and 1. If there is an arrow αwith
target vsuch that ε(α) =x(and so αwis a string), assume, without loss of generality,
that the arc corresponding to the source of αisasideof0. Then orient γsuch that sis
a marked point of 0and tis a marked point of 1. If there is no such arrow, then there
is only one arrow βwith target vand ε(β) =−x. In this case, if s(β) corresponds to a
side of 0, we orient γsuch that s∈1and t∈0.Ifthetwotiles0and 1coincide,
we can use a similar argument to orient the arc γ.
Now that we have determined the orientation of γ,letx1,...,xkbe the arcs of P
that γcrosses in order. The arc x1and the marked point scompletely determine a tile of
the surface, which shall be denoted by .
Module Categories of Gentle Algebras 27
Fig. 22. Pivot elementary move which adds a hook—Case 1.
Fig. 23. Pivot elementary move which adds a hook—Case 2.
Suppose there is an arrow αin QAPsuch that αwis a string. Note that this arrow
is unique, given the sign functions σ,εand the notion of composition of strings.
Case 1: is not of Type I or II.
By assumption, there is one side x0of , which is not a boundary segment and
it has one endpoint in common with x1: the marked point corresponding to α. Moreover,
the arc γsis the permissible arc obtained from the concatenation of γwith the sides
of when travelling counterclockwise around from sto s, the endpoint of x0not in
common with x1. Clearly, γsγ.Figure22 illustrates this case.
The arc ft(γs)will then cross the fan at sstarting at x0, together with x1,...,xk,
as Figure 22 shows. Hence, ft(γs)is clearly permissible and the corresponding string is
w(ft(γs)) =β−1αw, where βis the path corresponding to the fan at sstarting at x0.
Thus, w(ft(γs)) =w.
Case 2: is of Type I.
By construction, the loop arrow at x1is not the 1st arrow in string w.Given
the orientation of γ,αmust be the loop at x1.Letγsbe as in Definition 3.14,andlet
y1,...,ysbe the sequence of arcs of Pincident with s=ssuch that there is a string
β=x1→y1→ ··· → ys. Thus, w(ft(γs)) =β−1αw=w(see Figure 23).
Case 3: is of Type II.
28 K. Baur and R. Coelho Simões
Fig. 24. Pivot elementary move which adds a hook—Case 3.
In this case, γγs, where γsis as described in Definition 3.14. Denote by x0the
other arc of ,andletp1be the starting point of γsand p2be the other marked point of
.Thenαis the arrow from x0to x1with corresponding marked point p2.Letβbe the
path associated to the fan at p1starting at x0. Then, w(ft(γs)) is again w. See Figure 24.
Finally, suppose there is no arrow α∈QAPsuch that αwis a string. Then, in
particular, cannot be of Type I nor of Type II.
Let p1and p2be the endpoints of the arc x1such that p1,p2,s,andsfollow
each other in the counterclockwise order around the border of , where sis the
starting point of γs. By assumption, has a side with endpoints sand p1,which
is a boundary segment. Thus, ft(γs)has endpoints p1and t, and it only crosses
x,x+1,...,xk, where xis the 1st arc of the sequence x1,...,xkfor which the substring
is direct and α−1:x→x−1. In particular, if wis
direct, then ft(γs)does not cross any arc in P. Hence, w(ft(γs)) =w.Figure25 illustrates
this argument.
To finish the proof, we need to show that w(fs(ft(γs))t)=(w)r. But, by the
1st part of the proof (and its dual), we have that w(ft(γs)) =wand w(fs(ft(γs))t)=
(w(ft(γs))t)r=(w)r.
Remark 3.16. Given a permissible arc, denote by γs,tthe arc (γs)t. Note that the
construction of γs(γt, resp.) does not affect the ending point (starting point, resp.) of
γ. Hence,(γt)s=(γs)t. Moreover, the ending points (starting points, resp.) of ft(γs)and γs
(fs(γt)and γt, resp.) coincide. Therefore, ft(γs)=ft(γs,t)and fs(γt)=fs(γs,t).
Given a permissible arc γ,wedefineτ−1([γ])to be the equivalence class of the
arc, which we denote by τ−1(γs,t), obtained from γs,tby simultaneous rotation in the
counterclockwise direction of its starting and ending points to the next marked points
at the boundary.
Module Categories of Gentle Algebras 29
Fig. 25. Pivot elementary move which removes a cohook.
Fig. 26. Arcs γ,γs,tand their images under τ−1.
Corollary 3.17. Let M([γ])be a string module. We have that M([γ])is non-injective if
and only if |IP(τ −1(γs,t))| = 0. Moreover, in this case, we have τ−1(M([γ])) =M(τ −1([γ])).
Proof. It follows from the notions of pivot elementary moves that τ−1([γ])=
[fsft(γs,t)]=[ftfs(γs,t)]. By Theorem 3.15 (2), we have M([ftfs(γs,t)])=M(τ −1([γ])),in
the case when M([γ])is not injective. On the other hand, if M([γ])is injective then
the corresponding string wis such that there are no arrows α,βin APfor which
αwor wβare strings. One can easily deduce that wr,is the empty string and that
|IP(τ −1(γs,t))|=0.
Remark 3.18. We need to use the arc γs,tto compute the Auslander–Reiten translate
of a module, as we may have [τ−1(γ )]= [τ−1(γs,t)], where τ−1(γ ) is defined in the same
way as τ−1(γs,t). See Figure 26 for an example.
30 K. Baur and R. Coelho Simões
3.3 Reading morphisms from curves
In the previous subsection, we completely described the irreducible morphisms between
string modules in terms of pivot elementary moves. We recall that arbitrary morphism
spaces between indecomposable modules for gentle algebras have been described, cf.
[19,31]. In this subsection we will translate this result in terms of our combinatorial
model.
Let wbe a string or a band. A decomposition of wof the form w=w1a−1ebw2,
where a,b∈Q1,andw1,e,w2are strings, is called a factor string. The set of factor
strings of wis denoted by Fac(w). Similarly, a decomposition of wof the form
w=w1ced−1w2, where c,d∈Q1,andw1,e,w2are strings, is called a substring.The
set of substrings of wis denoted by Sub(w).
Band modules of the form M(b,1,ϕ) lie at the mouth of homogeneous tubes, and
are therefore called quasi-simple band modules.
Theorem 3.19. [19,31] Let v, w be strings or bands, and M(v),M(w)be the cor-
responding string or quasi-simple band modules. Then dimkHomA(M(v),M(w)) =
|{(v1a−1ebv2,w1cfd−1w2)∈Fa c(v)×Sub(w)|f=eorf=e−1}|.
Pairs in Fac(v)×Sub(w)satisfying the condition stated in the previous theorem
are called admissible pairs.
Definition 3.20. Let γbe a permissible arc or closed curve corresponding to a string
or band w.Writeγas the concatenation of segments γ=γ1γγ
2,andletx(y, resp.) be the
connecting point of γ1(γ2, resp.) with γ, and assume xlies in the interior of the surface,
but does not belong to an arc in P.Letv1,v2,...,vkbe the arcs in Pthat γcross, in this
order. Assume |IP(γ)| = 0andletvi,...,vjbe the arcs crossed by γ.
(2) The segment γis said to be a clockwise admissible segment if it satisfies
the following two conditions:
•Either vi=v1or there is an arrow α:vi−1→viin Q1and a
triangle of the form:
Module Categories of Gentle Algebras 31
Fig. 27. Left: γis clockwise admissible. Right: γis anticlockwise admissible.
where γ
1is the segment of γbetween the intersection points of γ
with vi−1and vi.
•Either vj=vkor there is an arrow β:vj+1→vjin Q1and a
triangle of the form:
where γ
2is the segment of γbetween the intersection points of γ
with vjand vj+1.
(2) The notion of anticlockwise admissible segment is obtained from the notion
above by changing the orientation of the arrows αand β.
Figure 27 gives an illustration of this concept.
The next result describes the dimensions of Hom-spaces between two inde-
composable modules in terms of their corresponding curves. The proof requires a
straightforward interpretation of Theorem 3.19 in terms of our geometric model and
so is omitted.
Proposition 3.21. The dimension of the Hom-space HomA(M(w),M(v)) is given by the
number of pairs (
γ(w),
γ(v)) of isotopic segments such that IP(
γ(w)) =IP(
γ(v)),
γ(w)is
anticlockwise admissible and
γ(v)is clockwise admissible.
Note that the intersection of
γ(w)with Pgives rise to the string ein
Theorem 3.19.
32 K. Baur and R. Coelho Simões
Fig. 28. Tiling of A.
Fig. 29. Left: e=w.Right:e=b−1c.
Example 3.22. Consider the tiling algebra
given by the tiling in Figure 28.
Let w=b−1cdc−1b.ThendimHomA(M(w),M(w)) =2, as the admissible
pairs in Fac (w)×Sub(w)are (e=w,e=w)and (ed(c−1b),(b−1c)de−1), where
e=b−1cin the latter. Figure 29 shows the corresponding isotopic admissible
segments.
Remark 3.23. Let M(w)and M(v)be two string modules. If there is a non-zero
morphism between M(v)and M(w), then the corresponding arcs (γ (w))s,tand (γ (v))s,t
cross each other, either in the interior of Sor at one of the endpoints. However, the
converse is not true.
3.4 Example
We finish this section with an example of our model for a representation finite gentle
algebra. Consider the following tiling algebra:
Module Categories of Gentle Algebras 33
Fig. 30. Auslander–Reiten quiver in terms of our geometric model.
The tiling algebra is the path algebra over the quiver on the right, subject to the relations
ca =ab =0. The Auslander–Reiten quiver of this gentle algebra is as follows:
In terms of arcs in the annulus, the Auslander–Reiten quiver of the algebra is as
in Figure 30, where we chose the arc γs,tas representative for each equivalence class [γ].
34 K. Baur and R. Coelho Simões
Funding
This work was supported by Fundação para a Ciência e Tecnologia [SFRH/BPD/90538/2012 and
project UID/MAT/04721/2013 to R.C.S.] and FWF [30549-N26 and W1230 to K.B.].
Acknowledgments
K.B. would like to thank the University of Leeds for support during a research stay. The authors
are grateful to Mark J. Parsons for discussions at an early stage of this project. The authors would
also like to thank the referee for a careful reading and useful comments and suggestions.
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