Content uploaded by Kaveh Mousavand
Author content
All content in this area was uploaded by Kaveh Mousavand on Mar 04, 2018
Content may be subject to copyright.
arXiv:1707.07665v2 [math.RT] 28 Feb 2018
ON THE COMBINATORICS OF GENTLE ALGEBRAS
THOMAS BR ¨
USTLE, GUILLAUME DOUVILLE, KAVEH MOUSAVAND, HUGH THOMAS,
EMINE YILDIRIM
Abstract. For Aa gentle algebra, and Xand Ystring modules, we con-
struct a combinatorial basis for Hom(X, τ Y ). We use this to describe support
τ-tilting modules for A. We give a combinatorial realization of maps in both
directions realizing the bijection between support τ-tilting modules and func-
torially finite torsion classes. We give an explicit basis of Ext1(Y, X ) as short
exact sequences. We analyze several constructions given in a more restricted,
combinatorial setting by McConville [McC], showing that many but not all of
them can be extended to general gentle algebras.
1. Introduction
In this paper, we study the combinatorics of gentle algebras. Suppose that
Xand Yare string modules for a gentle algebra A=kQ/I. (Terminology not
explained in the introduction will be defined in the next section.) An explicit
basis of Hom(X, Y ) has been known for a long time [CB]. We give an analogous
construction for Hom(X, τAY).
Our construction proceeds by embedding A=kQ/I into a larger gentle algebra
ˆ
A=kˆ
Q/ ˆ
I, which we call the fringed algebra of A. The reason for embedding A
in ˆ
Ais that, although τAis well-understood thanks to work of Butler and Ringel
[BR], its behaviour is somewhat complicated. It turns out that the behaviour of τˆ
A
on mod Ais more uniform and thus easier to analyze than that of τA.
Using our construction, in Section 5, we give an explicit description in terms of
combinatorics of strings of the support τ-tilting modules for A, as certain maximal
non-kissing collections of strings. Support τ-tilting modules are in bijection with
functorially finite torsion classes; in Section 7, we show explicitly in terms of the
combinatorics of strings how to pass from a maximal non-kissing collection to its as-
sociated functorially finite torsion class and how to return from a functorially finite
torsion class to the maximal non-kissing collection. If there are only finitely many
functorially finite torsion classes, we give, in Section 6, a combinatorial description
of the poset of functorially finite torsion classes as a combinatorially-defined poset
on the maximal non-kissing collections.
By Auslander-Reiten duality, Ext1(Y, X ) is dual to a quotient of Hom(X, τ Y ).
In Section 8, we use our construction of Hom(X, τ Y ) to find a basis for Ext1(Y, X ),
and we realize that basis as a collection of extensions of Yby X.
Our paper was inspired throughout by work of McConville [McC]. He was work-
ing in a combinatorial context, studying certain generalizations of the Tamari lat-
tice, but he uncovered many phenomena which extend practically verbatim to ar-
bitrary gentle algebras.
One phenomenon which does extend beyond McConville’s setting, but not to all
gentle algebras, is his observation that when two maximal non-kissing collections
1
2 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
are related by a single mutation, the exchanged strands kiss exactly once. In Section
9, we give an example (originally due to [GLS]) showing that this does not hold for
all gentle algebras. We then show that it holds for all gentle algebras such that all
τ-rigid modules are bricks.
While working on this project, we were informed of two other papers which
overlap with our results. [PPP] also constructs the fringed algebra (which they
call the “blossoming algebra”). They apply this to realize and study the support
τ-tilting fan for gentle algebras. [CPS] studies Hom groups in the derived category
of representations of a gentle algebra. From this, they also deduce an explicit
description of the extensions between string modules. Another recent paper which
is relevant is [EJR]: they also study Hom(X, τ Y ), but because they work in the
more general setting of string algebras, the combinatorics which they analyze is
more complicated.
Acknowledgements. T.B. was partially supported by an NSERC Discovery Grant.
G.D. was partially supported by an NSERC Alexander Graham Bell scholarship.
K.M. and E.Y. were partially supported by ISM scholarships. H.T. was partially
supported by NSERC and the Canada Research Chairs program. This paper was
developed in the context of the LaCIM Representation Theory Working Group,
and benefited from discussions with its other participants, including in particular
Mathieu Guay-Paquet and Amy Pang.
2. Preliminaries and Background
Notations and Conventions. Throughout this paper kdenotes an arbitrary
field. A quiver Q= (Q0, Q1, s, e) is a directed graph, which we always assume
to be finite and connected, with Q0the vertex set, Q1the set of arrows, and
s, e :Q1→Q0two functions which respectively send each arrow γ∈Q1to its
source s(γ) and its target e(γ). We use lower case Greek letters α,β,γ,... to
denote arrows of Q.
Apath of length m≥1 in Qis a finite sequence of arrows γm···γ2γ1where
s(γj+1) = e(γj), for every 1 ≤j≤m−1. By kQ we denote the path algebra of Q.
Azero (monomial) relation in Qis given by a path of the form γm···γ2γ1, where
m≥2. A two sided ideal Iis called admissible if Rm
Q⊂I⊆R2
Qfor some m≥2,
where RQdenotes the arrow ideal in kQ. In what follows, Ialways denotes an
admissible ideal generated by a set of zero relations in Q. We use capital letters A
and Bto denote quotients of kQ by such ideals.
Gentle algebras. B=kQ/I is called a string algebra if the following conditions
hold:
(S1) There are at most two incoming and two outgoing arrows at every vertex of
Q.
(S2) For every arrow α, there is at most one βand one γsuch that αβ /∈Iand
γα /∈I.
A string algebra is called gentle if the following additional conditions are satisfied:
(G1) The ideal Iis quadratic, i.e. there is a set of monomials of length two that
generate I.
(G2) For every arrow α, there is at most one βand one γsuch that 0 6=αβ ∈I
and 0 6=γ α ∈I.
ON THE COMBINATORICS OF GENTLE ALGEBRAS 3
Strings and Bands. To introduce the notion of string, we need the following
definitions:
For a given quiver Q, let Qop be the opposite quiver, obtained from Qby reversing
its arrows. We consider the arrows in Qop as the formal inverses of arrows in Qand
denote them by γ−1, for every γ∈Q.
Let Qsbe the double quiver of Q, which has Q0as the vertex set and Q1∪Qop
1
as the arrow set. A string in Qis a path C=γn···γ1in Qswith the following
properties:
(A1) No γiis followed by its inverse.
(A2) Neither Cnor C−1contains a subpath in I.
In such a case, C=γn···γ1is called a string of length nwhich starts at
s(C) = s(γ1) and ends at e(C) = e(γn). In addition, to every vertex x∈Q0we
associate a zero-length string exstarting and ending at xwhich is its own formal
inverse.
The set of all strings in Qis denoted by str(Q). It is clear that str(Q) really
depends on Qand I, but we suppress Ifrom the notation, because it will be clear
from context.
We call C=γn···γ1adirect string if γi∈Q1for every 1 ≤i≤n, and dually
Cis called inverse if C−1is direct. A band in Qis a string C=γn···γ1such that
Cnis well-defined for each n∈Nand, furthermore, Citself is not a strict power of
another string of a smaller length.
Definition 2.1 For every string C=γn···γ1, the associated diagram of Cis the
illustration by a sequence of up and down arrows, from right to left, by putting a
left-down arrow starting at the current vertex for an original arrow and a right-down
arrow ending at the vertex for an inverse arrow.
Example 2.2 In the following quiver Q:
•2
α1
−−−−→ •4
β1x
x
β2
•1−−−−→
α2
•3
since R3
Q= 0, the zero ideal I= 0 is admissible. It is easy to see A=kQ/I =kQ is
a gentle algebra. Following the description above, the diagram of C=α2β−1
1α−1
1β2
in str(Q) is the following:
•1
β1
❇
❇
❇
❇
❇
❇
❇
❇
α2
~~⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
•3•2
α1
❇
❇
❇
❇
❇
❇
❇
❇•3
β2
~~⑤
⑤
⑤
⑤
⑤
⑤
⑤
⑤
•4
String Modules. Considering a string algebra B, to every string Cwe associate
an indecomposable B-module M(C), constructed as follows: in the diagram of C,
as defined in Definition 2.1, put a one dimensional k-vector space at each vertex
and the identity map for each arrow connecting two consecutive vertices of the
4 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
diagram. In the representation M(C), the dimension of the vector space associated
to each vertex of the quiver is given by the number of times that the string C
passes through the aforementioned vertex. Consequently, the dimension vector of
M(C) has an explicit description. Provided it does not cause confusion, we do not
distinguish between a string Cand the associated string module M(C) and we use
Cto refer to both. For details see [CB, BR].
For a string C, by means of the equivalence relation introduced in [BR], we
always identify M(C) and M(C−1), which are isomorphic as B-modules. Two
string modules are isomorphic if and only if their strings are equivalent.
The indecomposable modules which are not string modules are associated to
bands, and are called band modules. Since our focus is on string modules, we do
not discuss the details of band modules here.
As shown in [BR], the Auslander-Reiten translation of string modules over a
string algebra admits an explicit combinatorial description. In order to state it, we
will need to introduce some further combinatorial notions.
Lemma 2.3 Let Wbe a string of positive length, and let us pick an orientation
ǫwhere ǫ∈ {direct,inverse}. There is at most one way to add an arrow preceding
s(W)whose orientation agrees with ǫ, such that the resulting path is still a string.
Similarly, there is at most one way to add such an arrow following e(W).
Proof. The lemma follows immediately from conditions (S1) and (S2).
In these cases we say that Wcan be extended at its start, or at its end, by an
inverse arrow, or a direct arrow, as applicable. Note that the lemma is not true as
stated if Wis of length zero.
If a string Wcan be extended at its start by an inverse arrow, then we consider
the result of adding an inverse arrow at s(W), followed by adding as many direct
arrows as possible. We call this adding a cohook at s(W). We denote the result of
this operation Wc. It is well-defined by Lemma 2.3.
Symmetrically, if Wcan be extended by a direct arrow at e(W), then we consider
the result of adding a direct arrow at e(W) followed by as many inverse arrows as
possible, and we call this adding a cohook at e(W). We denote the result of this
by cW.
If it is possible to add a cohook at each end of W, we write cWcfor the result of
doing so.
If Wis of length zero, i.e., a single vertex, there may be two arrows pointing
towards W, and in that case, two cohooks can be added to W. We define cWcto
be the result of adding both cohooks.
We now describe the operation of removing a hook from W. To remove a hook
from the start of W, we look for the first direct arrow in W, and we remove it,
together with all its preceding inverse arrows. In other words, we find a factorization
W=XθI, with Ian inverse string and θa direct arrow. The result of removing
a hook from the beginning of Wis X. We write X=W/h. Note that this is not
defined if Wcontains no direct arrows.
Similarly, to remove a hook from the end of W, we look for the last inverse arrow
of Wand remove it, together with all subsequent direct arrows. In other words,
we find a factorization of Was W=Dγ−1Y, with Ddirect; then the result of
removing a hook from the end of Wis Y. We write Y=h\W.
ON THE COMBINATORICS OF GENTLE ALGEBRAS 5
Theorem 2.4 [BR] Let B=kQ/I be a string algebra, and let Wbe a string
module. At either end of W, if it is possible to add a cohook, add a cohook. Then,
at the ends at which it was not possible to add a cohook, remove a hook. The result
is τBW.
Note that if it is possible to add a cohook at exactly one endpoint of W, then
after having done so, it will be possible to remove a hook from the other end. If it
is impossible to add a cohook at either end, then either it is possible to remove a
hook from each end, or else the module was pro jective, in which case we interpret
the result of removing a hook from each end as the zero module (which is consistent
with the fact that the Auslander-Reiten translation of a projective module is zero).
Morphisms between string modules. A basis for the space of all homomor-
phisms between two strings was given by W. Crawley-Boevey in [CB] in a more
general setting. In [S], J. Schr¨oer reformulated the aforementioned basis for string
modules. In this paper, we mainly use Schr¨oer’s reformulation and notation for the
description of Hom-space.
Definition 2.5 For C∈str(Q), the set of all factorizations of Cis defined as
P(C) = {(F, E, D)|F, E , D ∈str(Q)and C=F E D}.
Moreover, if (F, E, D)∈ P(C), we write (F, E , D)−1= (D−1, E−1, F −1)∈ P (C−1).
A triple (F, E , D)∈ P(C) is a called a quotient factorization of Cif the following
hold:
(i) D=es(E)or D=γD′with γin Qop;
(ii) F=ee(E)or F=F′θwith θin Q.
A quotient factorization (F, E , D) induces a surjective quotient map from Cto
E. The set of all quotient factorizations of Cis denoted by F(C).
A factor of Cis generally of the following form, where Dand Fcan also be lazy
paths.
θγ
Es(E)e(E)••
DF
Dually, (F, E, D)∈ P(C) is called a submodule factorization of Cif the following
hold:
(i) D=es(E)or D=γD′with γin Q;
(ii) F=ee(E), or F=F′θwith θin Qop.
The set of all submodule factorizations of Cis denoted by S(C). A submodule
factorization (F, E , D) induces an inclusion of Einto C.
Definition 2.6 For C1and C2in str(Q), a pair ((F1, E1, D1),(F2, E2, D2)) ∈
F(C1)× S(C2) is called admissible if E1∼E2(i.e, E1=E2or E1=E−1
2). The
collection of all admissible pairs is denoted by A(C1, C2).
For each admissible pair T= ((F1, E1, D1),(F2, E2, D2)) in A(C1, C2), there
exists a homomorphism between the associated string modules
fT:C1→C2,
6 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
defined as the composition of the projection from C1to E1, followed by the identi-
fication of E1with E2, followed by the inclusion of E2into C2. We refer to these
homomorphisms fTas graph maps.
The following theorem plays a crucial role in this paper:
Theorem 2.7 [CB] If A=kQ/I is a string algebra and C1and C2are string
modules, the set of graph maps {fT|T∈ A(C1, C2)}is a basis for HomA(C1, C2).
Auslander-Reiten duality. Write HomB(M, N ) = HomB(M, N )/P (M, N ) for
the quotient of HomB(M, N ) by the morphisms factoring through a projective
module.
Similarly, write I(M, N ) for the morphisms from Mto Nfactoring through
injectives, and HomB(M, N ) for HomB(M, N )/I(M, N ).
The following functorial isomorphisms, known as the Auslander-Reiten formulas,
play an important role in the rest of this paper:
Theorem 2.8 For every pair of B-modules Xand Y, we have the following:
•HomB(X, τBY)≃DExt1
B(Y, X );
•DHomB(Y, X )≃Ext1
B(X, τBY).
τ-tilting theory. τ-tilting theory was recently introduced by Adachi, Iyama, and
Reiten [AIR]. An A-module module is said to be τ-rigid if Hom(X, τ X) = 0. The
module Xis said to be τ-tilting if it is τ-rigid and it has as many non-isomorphic
indecomposable summands as A. The module Xis said to be support τ-tilting if
Xis τ-rigid and the number of indecomposable summands of Xequals the number
of non-isomorphic simple modules in its composition series.
Atorsion class in mo d Ais a full subcategory closed under quotients and ex-
tensions. If Xis an A-module, we write Fac Xfor the full subcategory of mo d A
consisting of quotients of sums of copies of A. A torsion class is called functorially
finite if it can b e written as Fac Xfor some A-module X.
An Ext-projective module in a subcategory Tof mod Ais a module Xsuch that
Ext1
A(X, −) vanishes on all modules in T.
Theorem 2.9 ([AIR, Theorem 2.7]) There is a bijection from basic support τ-
tilting modules to functorially finite torsion classes, which sends the support τ-tilting
module Xto Fac X, and sends the functorially finite torsion class Tto the direct
sum of its Ext-projective indecomposables.
3. Fringed Algebras
Definition 3.1 For a gentle algebra A=kQ/I, we call a vertex of Qdefective if
it does not have exactly two incoming arrows and two outgoing arrows.
Our strategy is to define a larger quiver ˆ
Qcontaining Q, and an ideal ˆ
Iin kˆ
Q
containing I, such that:
•all the vertices of Qwill be non-defective in ˆ
Q, and
•kˆ
Q/ ˆ
Iis still a gentle algebra.
We call this process fringing. We begin by describing ˆ
Q.
Definition 3.2 For a gentle algebra A=kQ/I , let ˆ
Qbe the quiver obtained by
adding up to two new vertices with arrows pointing towards vand up to two new
vertices with arrows from v, for each defective vertex vof Q, such that in ˆ
Q, none
ON THE COMBINATORICS OF GENTLE ALGEBRAS 7
of the original vertices of Qare defective anymore. We call ˆ
Qthe fringed quiver of
Q, and ˆ
Q0\Q0the fringe vertices.
We now define ˆ
I.
Definition-Proposition 3.3 Let A=kQ/I be a gentle algebra. Let ˆ
Ibe the
ideal of kˆ
Qwhich contains I, such that ˆ
A=kˆ
Q/ ˆ
Iis a gentle algebra. ˆ
Iis well-
defined up to relabelling the vertices of kˆ
Q. We call ˆ
Ithe fringed ideal and ˆ
Athe
fringed algebra.
The well-definedness of ˆ
Iis immediate from the fact that the relations added
to ˆ
Iconsist of certain length two paths through vertices that were defective in Q,
and there is no choice about which relations to add up to relabelling the fringe
vertices. It is obvious that Ais a subalgebra of ˆ
A. Also, A=ˆ
A/(eF), where
eF=Pf∈ˆ
Q0\Q0ef, is the sum of all the idempotents associated to the fringe
vertices of ˆ
Q. We refer to eFas the fringe idempotent of ˆ
A.
Example 3.4 Let Qbe the quiver below consisting of the black vertices and the
arrows between them. All the vertices of Qare defective. The corresponding fringed
quiver ˆ
Qis obtained by adding the white vertices and arrows, with the result that
every black vertex has exactly two incoming and two outgoing arrows.
◦ ◦
γ1x
γ2x
◦θ1
−−−−→ •1
α1
−−−−→ •2
θ2
−−−−→ ◦
β1x
β2x
◦θ3
−−−−→ •3
α2
−−−−→ •4
θ4
−−−−→ ◦
γ3x
γ4x
◦ ◦
If we consider the admissible ideal Iin kQ generated by α1β1and β2α2, we have
that A=kQ/I is a gentle algebra. The corresponding ideal ˆ
Iin kˆ
Qis generated by
{β2α2, α1β1, γ2α1, α2γ3, θ2β2, β1θ3, θ4γ4, γ1θ1}. Note that we could have swapped
the roles of, for example, γ3and θ3in ˆ
I, but that is equivalent to relabelling their
sources. The associated fringed algebra is ˆ
A=kˆ
Q/ ˆ
I.
The previous example admits a natural generalization, which is closely related
to the work of McConville [McC].
8 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
Example 3.5 Let Qk,n be the following quiver, which is an orientation of a kby
n−kgrid:
•1,1
α1,1
−−−−→ •1,2
α1,2
−−−−→ · · · α1,n−k−1
−−−−−−→ •1,n−k
β1,1x
β1,2x
β1,n−kx
•2,1
α2,1
−−−−→ •2,2
α2,2
−−−−→ · · · α2,n−k−1
−−−−−−→ •2,n−k
β2,1x
β2,2x
β2,n−kx
.
.
..
.
.....
.
.
βk−1,1x
βk−1,2x
βk−1,n−kx
•k,1
αk,1
−−−−→ •k,2
αk,2
−−−−→ · · · αk,n−k−1
−−−−−−→ •k,n−k
Consider the ideal Igenerated by all relations of the form αβ and βα.
The fringed quiver ˆ
Qk,n is obtained by adding the white vertices and incident
arrows, so that none of the black vertices is defective:
◦ ◦ · · · ◦
β0,1x
β0,2x
β0,n−kx
◦α1,0
−−−−→ •1,1
α1,1
−−−−→ •1,2
α1,2
−−−−→ · · · α1,n−k−1
−−−−−−→ •1,n−k
α1,n−k
−−−−→ ◦
β1,1x
β1,2x
β1,n−kx
◦α2,0
−−−−→ •2,1
α2,1
−−−−→ •2,2
α2,2
−−−−→ · · · α2,n−k−1
−−−−−−→ •2,n−k
α2,n−k
−−−−→ ◦
β2,1x
β2,2x
β2,n−kx
.
.
..
.
..
.
.....
.
..
.
.
βk−1,1x
βk−1,2x
βk−1,n−kx
◦αk,0
−−−−→ •k,1
αk,1
−−−−→ •k,2
αk,2
−−−−→ · · · αk,n−k−1
−−−−−−→ •k,n−k
αk,n−k
−−−−−→ ◦
βk,1x
βk,2x
βk,n−kx
◦ ◦ · · · ◦ .
The new ideal ˆ
Iis generated by all relations of the form αβ and βα, including the
new arrows.
This class of examples is sufficiently complicated to illustrate most of the phe-
nomena which will be of interest to us in this paper. We will therefore sometimes
draw examples of string modules in a shape as in Example 3.5, suppressing the
underlying grid-shaped quiver, and drawing only the strings.
Definition 3.6 For a string Xin Q, we write cohook(X) for the result of adding
cohooks to both ends of Xin the fringed quiver ˆ
Q. We call this the cohook com-
pletion of X.
The diagram of cohook(X) is illustrated as follows:
ON THE COMBINATORICS OF GENTLE ALGEBRAS 9
X
βα
Here we refer to the arrows αand βadjacent to Xas the shoulders of cohook(X)
and the sequence of direct arrows on the right and the sequence of inverse arrows
on the left as the arms of cohook(X).
In the following proposition we give the description of τˆ
A(X) for Xa string in
mod A.
Proposition 3.7 For a pair of strings Xand Yin A, we have the following:
(1) τˆ
AY= cohook(Y);
(2) Ext1
A(Y, X )≃Ext1
ˆ
A(Y, X ).
Proof. (1) Because Yis an A-module, it is possible to add cohooks to both
ends of it when we think of it inside ˆ
Q. Therefore, Theorem 2.4 tells us
that τˆ
AYis obtained by adding cohooks to both ends of Y.
(2) Recall that eFis the fringe idempotent in ˆ
A, and that A=ˆ
A/(eF). Hence,
Ext1
A(Y, X )≃Ext1
ˆ
A(Y, X ).
The following lemma relates τAYand τˆ
AY.
Lemma 3.8 Let Ybe a string for Q. Then τAYis a submodule of τˆ
AY.
In fact, it is known that if Bis an algebra, B/I is a quotient algebra, and M
is a B/I -module, then τB/I Mis a submodule of τBM, see [ASS, Lemma VIII.5.2].
We include a proof of the special case we state because it is an easy consequence of
the combinatorics of strings.
Proof. This follows from the fact that τˆ
AYis defined by adding cohooks in ˆ
Qto
Y, while τAYis defined by, at each end, either adding a cohook in Q(which will
coincide with the cohook in Yexcept that it will be missing the final arrow) or
subtracting a hook; the result is that τAYis a substring of τˆ
AYand at each end,
the first arrow missing from τAYpoints towards τAY. Thus, τAYis a submodule
of τˆ
AY.
4. Kisses
In this section we give an interpretation of Hom ˆ
A(X, τ ˆ
AY) for a pair of strings X
and Yin A. Our technique is inspired by work of Schr¨oer [S] and the combinatorial
framework developed by McConville [McC].
In order to apply Theorem 2.4 to analyze Hom ˆ
A(X, τ ˆ
AY), we need to introduce
a new notion and fix some notation, as follows:
Definition 4.1 For a pair of strings Xand Y, with factorizations X= (X′′ , Z, X ′)
and Y= (Y′′, Z, Y ′), we say there exists a kiss from Xto Yalong Z, provided:
(i) (X′′, Z, X ′) is a quotient factorization,
(ii) (Y′′, Z, Y ′) is a submodule factorization,
(iii) all of X′′, X′, Y ′′ , Y ′have strictly positive length.
Such a kiss is denoted by hY′′
X′′ ZY′
X′i.
10 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
We emphasize that the notion of kiss is directed. A kiss from Xto Ycan be
illustrated as follows. Note that the four arrows γ,ζ,θ, and σmust all appear, and
must be oriented as shown.
Z
γ
Y′′ Y′
θ
ζ
X′′ X′
σ
By kiss(X, Y ) we denote the number of kisses from Xto Y, whereas we use
Kiss(X, Y ) for the set of all kisses from Xto Y, thought of as a set of pairs of
triples, as above. This is a generalization of a notion introduced by McConville
[McC].
Lemma 4.2 Let Xand Ybe strings in Q. Let hY′′
X′′ ZY′
X′ibe a kiss from cohook(X)
to cohook(Y). Then
(1) Zis a quotient of X.
(2) Zis a submodule of τAY.
Note that Zis by definition a factor of cohook(X); the point of (1) is that it is
really a quotient of X(though not necessarily a proper quotient: it might be equal
to X). Note also that Zis by definition a submodule of τˆ
AY, but, as depicted in
the following example, Zis not necessarily a submodule of Y. In this example, the
cohooks are drawn dashed. Note that αand βbelong to the kiss from cohook(X)
to cohook(Y), but they are not in Y:
X
Y
α
β
Note that in this example we are following the convention of considering our
strings to lie in a grid quiver as in Example 3.5, without drawing the underlying
grid.
Proof. (1) If the start of Zwere before the start of Xin cohook(X) then X′would
consist only of direct arrows, which is contrary to the definition of a kiss. A similar
argument shows that the end of Zcannot be later than the end of X, so Zis a
substring of X. Because of the direction of the arrows just outside Zin cohook(X)
(which are part of the definition of a kiss), Zis a quotient of X.
(2) Recall from Theorem 2.4 that τAYis obtained from Yby adding cohooks if
possible, and otherwise removing hooks.
Suppose that it was possible to add a cohook to the start of Yin A. In this case,
both τAYand τˆ
AYadd a cohook at the start of Y. The only difference is that τˆ
AY
will include one additional inverse arrow from a fringe vertex. By the definition of
a kiss, Zcannot include that additional arrrow.
ON THE COMBINATORICS OF GENTLE ALGEBRAS 11
Now suppose that it was not possible to add a cohook at the start of Y. In
this case, τAYwas obtained by removing up to the first direct arrow of Y. By the
definition of a kiss, there is a direct arrow before Z, so what was removed does not
intersect Z, and thus the start of Zis not before the start of τAY.
We have therefore established that the start of Zis after the start of τAY.
Similarly, the end of Zis before the end of τAY. Because of the directions of the
arrows just outside Zin τAY,Zis a submodule of τAY.
By the previous lemma, a kiss from cohook(X) to cohook(Y) determines a non-
zero morphism from Xto τAYand from Xto τˆ
AY.
Theorem 4.3 (1) The elements of Hom(X, τ ˆ
AY)corresponding to kisses from
cohook(X)to cohook(Y)form a basis.
(2) The elements of Hom(X, τAY)corresponding to kisses from cohook(X)to
cohook(Y)form a basis.
Proof. By the description of morphisms between string modules in terms of graph
maps, Lemma 4.2 implies that the kisses define a linearly independent collection
inside each of the Hom spaces, and also span the Hom spaces.
Theorem 4.4 Let Xand Ybe strings in Q.
kiss(cohook(X),cohook(Y)) = dim Hom ˆ
A(X, τ ˆ
AY) = dim HomA(X, τAY).
Proof. This theorem is immediate from the stronger Theorem 4.3 above.
Proposition 4.5 Let Xand Ybe strings in Q. The canonical map from τAYto
τˆ
AYinduced by the fact that the former is a submodule of the latter, induces an
isomorphism from Hom(X, τAY)to Hom(X, τ ˆ
AY).
Proof. This is immediate from Theorem 4.3 and Lemma 3.8.
5. Maximal non-kissing collections
In this section, we show that support τ-tilting modules for a gentle algebra
correspond to certain collections of strings in its fringed quiver which can be com-
binatorially characterized.
Let Bbe an algebra with nsimples. We consider the set of τ-rigid indecompos-
able modules together with nformal objects Pi[1], on which we define a relation
of compatibility. Two B-modules Mand Nare compatible if Hom(M, τ N) = 0 =
Hom(N, τ M ). A B-module Mand Pi[1] are compatible if Hom(Pi, M ) = 0. Pi[1]
and Pj[1] are always compatible. There is a bijective correspondence from maximal
compatible collections to basic support τ-tilting modules (see [AIR]): one takes the
sum of the modules in the collection, and throws away whatever Pi[1] appear.
Let A=kQ/I be a gentle algebra, and let ˆ
A=kˆ
Q/I be its fringed algebra.
For va vertex of ˆ
Q, define the injective string of vto be the string obtained by
adding to the lazy path at vboth maximal sequences of arrows oriented towards v
in ˆ
Q. We denote it Iv. The corresponding ˆ
A-module is an indecomposable injective
module.
We define the set of long strings of ˆ
Qto consist of {cohook(X)|X∈str(Q)} ∪
{Iv|v∈Q0}. We call these long strings because they run between two fringe
vertices, so they are in a sense maximally long.
12 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
Theorem 5.1 There is a bijective correspondence from maximal compatible collec-
tions of A-modules to maximal non-kissing collections of long strings of ˆ
Q, induced
by the correspondence:
X∈str(Q)→cohook(X)
Pv[1] for v∈Q0→Iv.
Proof. We have to check that the bijection above takes the compatibility relation
to the non-kissing relation.
Two A-modules are compatible if and only if the corresponding long strings do
not kiss, by Theorem 4.4.
Any two injective strings are compatible, which is consistent with the fact that
Pi[1] and Pj[1] are compatible.
An A-module Mis compatible with Pv[1] if and only if HomA(Pv, M ) = 0, if
and only if Mdoes not pass through vertex v. It suffices to show that there is a
kiss from cohook(M) to Ivif and only if Mpasses through v, and there is never a
kiss from Ivto cohook(M). We now verify this.
If Mpasses through v, then in each direction from v, cohook(M) eventually
leaves the injective string from v(at the endpoint of Mif not before), and it leaves
it along an arrow pointing away from the injective string. Thus there is a kiss from
cohook(M) to Iv.
If Mdoes not pass through v, then any common substring of Mand Ivmust
consist of arrows all oriented in the same direction, with the arrow of Ivon either
side of the substring also having the same orientation. Since this means that the
arrows of Ivon either side of the common substring point in opposite directions, it
cannot be a kiss. The same argument applies to a common substring of an arm of
cohook(M) and Iv, unless vlies on an arm of cohook(M). But in that case, there is
a direction in which cohook(M) and Ivnever separate, so this is not a kiss either.
Because all the arrows of Ivare oriented towards vin the middle of the string,
it is impossible for there to be a kiss from Ivto any string.
6. Poset of functorially finite torsion classes
It is natural to order the functorially finite torsion classes by inclusion. Since
by Theorem 2.9 there is a bijection between functorially finite torsion classes and
support τ-tilting modules, this can also be thought of as a poset structure on
support τ-tilting modules. Having shown in the previous section that we can give a
combinatorial description of the support τ-tilting modules as maximal non-kissing
collections, we proceed in this section to interpret this poset structure on maximal
non-kissing collections.
The following theorem combines a few different results from [AIR]: Theorem
2.18, the discussion following Definition 2.19, and Corollary 2.34.
Theorem 6.1 ([AIR]) If Tand Uare two functorially finite torsion classes with
Tproperly contained in U, and with no functorially finite torsion class properly
between them, then their corresponding maximal compatible collections are S ∪{P},
S ∪ {R}, and conversely, given two maximal compatible collections S ∪ {P}and
S ∪ {R}, they correspond to functorially finite torsion classes which form a cover
in the poset of torsion classes.
Suppose we have two maximal nonkissing collections of the form S∪{cohook(X)}
and S ∪ {cohook(Y)}. Clearly cohook(X) and cohook(Y) kiss; otherwise, this
ON THE COMBINATORICS OF GENTLE ALGEBRAS 13
would violate maximality of the collections. By Theorem 6.1, these two collec-
tions correspond to a pair of torsion classes which form a cover. Suppose that the
torsion class corresponding to S ∪ {cohook(X)}contains the torsion class corre-
sponding to S ∪ {cohook(Y)}. It follows that Yis also in the first torsion class,
while Xis Ext-projective in the torsion class, so Hom(Y, τAX) = 0 by [AS, Propo-
sition 5.8]. The kiss(es) between cohook(X) and cohook(Y) are therefore from
cohook(X) to cohook(Y). Similarly, if we have two maximal nonkissing collections
S ∪ {cohook(X)}and S ∪ {Iv}, then we see that there is a kiss from cohook(X) to
Iv. This establishes the following theorem.
Theorem 6.2 The cover relations in the poset of functorially finite torsion classes
of Acan be described in terms of their maximal non-kissing collections as follows:
they correspond to pairs of maximal non-kissing collections of the form S ∪{C},S ∪
{D}, and S ∪ {C}>S ∪ {D}if the kisses go from Cto D.
An infinite poset is not necessarily characterized by its cover relations, but a
finite poset is. We therefore have the following corollary:
Corollary 6.3 If Ahas only finitely many functorially finite torsion classes, then
the poset of torsion classes is isomorphic to the poset of maximal non-kissing col-
lections, ordered by the transitive closure of S ∪ {C}>S ∪ {D}where the kisses go
from Cto D.
7. Combinatorics of torsion classes
In this section we consider how the bijection between functorially finite torsion
classes and support τ-tilting module plays out for gentle algebras in terms of the
combinatorics we have been developing.
We begin with the following theorem, describing the strings in the category of
quotients of copies of a collection of strings.
Theorem 7.1 Let A=kQ/I be a gentle algebra. Let X=LXiwith each Xia
string. A string Yis in Fac Xif and only if Ycan be written as a union of strings
each of which is a substring of Yand a factor of some Xi.
Here, when we write that Yis a union of a certain set of strings, we mean that
the strings may overlap, but each arrow of Yoccurs in at least one of the strings.
Proof. Yis in Fac Xif and only if there is a quotient map from a sum of copies of
Xonto Y. This means that, for each arrow of Y, we have to be able to find some
Xiwhich maps to Yand hits that arrow. The map from Xito Ycorresponds to a
quotient of Xiand a submodule of Y.
Let Sbe a collection of strings in Q, and let αbe an arrow of ˆ
Q. We will define
Mc(S, α) to be a certain long string of ˆ
Q. We will construct it arrow by arrow in
both directions from α. (We use the symbol Mc for this map to emphasize that
this is the algebraic version of a map defined by McConville [McC, Section 8].)
Let Γ0be the lazy path at e(α), and γ0=α. We will define a sequence of arrows
γi, and strings Γi=γiΓi−1, for i= 1,2,.... (Note that we do not include αin Γ0.)
Suppose we have already constructed γ1,...,γi, and let u=e(γi) = e(Γi). If uis
a fringe vertex, we set imax =iand stop. Otherwise, we divide into cases:
•If γiis a direct arrow and Γi∈ S, define γi+1 to be the unique direct arrow
such that γi+1γi6∈ I.
14 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
•If γiis a direct arrow and Γi6∈ S , define γi+1 to be the unique inverse arrow
starting at uwhich is not γ−1
i.
•If γiis an inverse arrow and Γi∈ S, define γi+1 to be the unique direct
arrow from uother than γ−1
i.
•If γiis an inverse arrow and Γi6∈ S , define γi+1 to be the unique inverse
arrow such that γ−1
iγ−1
i+1 6∈ I.
We continue in this way until we reach a fringe vertex.
We now extend the string in the opposite direction. Define Θ0to be the lazy
path at s(α). We will proceed to define a sequence of arrows γ−jand strings Θj=
Θj−1γ−jfor j= 1,2,.... Suppose that we have already constructed γ−1, γ−2...γ−j,
and let v=s(γ−j). If vis a fringe vertex, we set jmax =jand stop. Otherwise, we
define γ−j−1using the previous rule, but reversing the roles of direct and inverse
arrows throughout. Explicitly, we divide into cases as follows:
•If γ−jis an inverse arrow and Θj∈ S , define γ−j−1to be the unique inverse
arrow such that γ−1
−j−1γ−1
−j6∈ I.
•If γ−jis an inverse arrow and Θj6∈ S , define γ−j−1to be the unique direct
arrow starting at vwhich is not the inverse of γ−j.
•If γ−jis a direct arrow and Θj∈ S , define γ−j−1to be the unique inverse
arrow ending at vother than γ−1
−j.
•If γ−jis a direct arrow and Θj6∈ S , define γ−j−1to be the unique direct
arrow such that γ−jγ−j−16∈ I.
Now define Mc(S, α) to be the concatenation of Γimax ,α, and Θjmax .
Theorem 7.2 Let Tbe a functorially finite torsion class in mod A. Let STbe the
strings in T. The collection of modules Mc(ST, α), as αruns through the arrows of
ˆ
Q, yields:
•cohook(M)for Man Ext-projective of T(each appearing for two choices
of arrow α),
•the injective string at each vertex of Qover which no module in Tis sup-
ported (each appearing twice), and
•the injective string at each fringe vertex which is a sink (each appearing
once).
We divide the proof of the theorem into the next four propositions.
Let Mbe a string in Q. Let cohook(M) = γr...γ0, and let M=γb...γafor
some 0 < a ≤b < r. Define
X(M) = {i|b+ 1 ≥i≥1, γiis direct, γb. . . γi+1 ∈ ST}.
We consider the condition γb...γi+1 ∈ STto be vacuous when i=b+ 1, and γb+1
is a direct arrow since it is a shoulder of cohook(M), so b+ 1 ∈X(M). For i=b,
the condition that γb...γi+1 ∈ STis interpreted as meaning that the lazy path at
e(γb) is in ST.
Proposition 7.3 Let Mbe Ext-projective in T, with cohook(M) = γr...γ0. Let
xbe the minimum element of X(M). Then Mc(ST, γx) = cohook(M).
Proof. Let us write Ω for the string γb. . . γx+1 which is, by assumption, in ST. (If
x=b+ 1, then Ω is not defined.)
The proof is by induction. Suppose that, in the construction of Mc(ST, γx),
we have constructed Γiand it is a substring of cohook(M), say Γi=γy. . . γx+1.
ON THE COMBINATORICS OF GENTLE ALGEBRAS 15
Suppose now that γy+1 is direct. By our choice of x,γb. . . γx+1 ∈ ST, and since
γy+1 is direct, Γiis a quotient substring of γb...γx+1, so it is also in ST. Thus our
algorithm chooses a direct arrow, necessarily γy+1.
Suppose next that γy+1 is inverse. There is a kiss from cohook(Γi) to cohook(M).
This implies that Γicannot be in ST. Thus, our algorithm chooses an inverse arrow,
necessarily γy+1.
It follows by induction that Γimax agrees with γr...γx+1, i.e., the part of cohook(M)
after γx.
Now let us consider the part of cohook(M) before γx. Suppose that we have
already constructed Θj, and it is a substring of cohook(M), say Θj=γx−1...γz.
Suppose now that γz−1is direct. If Θjwere in ST, then since, by assumption, Ω
is in ST, so is their extension, which contradicts the minimality of x. (If x=b+ 1,
so Ω is not defined, then Θj∈ STand γz−1direct contradicts the minimality of x.)
Thus, our algorithm chooses a direct arrow, necessarily γz−1.
Suppose that γz−1is inverse. Then Θjis a quotient of M, so it is in ST. Thus
our algorithm chooses an inverse arrow, necessarily γz−1. It follows by induction
that Θjmax agrees with γx−1...γ0, i.e., the part of cohook(M) before γx. This
completes the proof.
The above proposition shows that we can construct any Ext-projective of T
in two ways: once as in the proposition, and once applying the proposition to the
reverse string. Note that the orientations of the two chosen edges will be opposite, so
this does indeed yield two distinct arrows α1, α2of ˆ
Qsuch that M= Mc(ST, α1) =
Mc(ST, α2).
We now consider the simpler case of injective strings.
Proposition 7.4 Let Tbe a torsion class in mod A. Let vbe a vertex such that
no module of Tis supported at v, and let Iv=γr...γ0with γjdirect for i≥j≥0,
and γjinverse for r≥j≥i+ 1. Let xbe minimal such that γi−1...γx+1 ∈ ST.
(Again, this condition is vacuous for x=i.) Then Mc(ST, γx) = Iv.
Proof. The proof is essentially the same as for Proposition 7.3.
Let us write Ω for the string γi−1...γx+1 which is, by assumption, in ST. (If
x=i+ 1, then Ω is not defined.)
Since Ω ∈ ST, it also follows that γj...γx+1 ∈ STfor all i−1≥j≥x. It
follows that Mc(ST, γx) proceeds along direct arrows to v. From that point on, by
the defining condition for v, the string constructed so far will not lie in ST, and so
it will continue by inverse arrows away from v.
Now consider the extension of γxin the opposite direction. If the segment
γx−1···γywere in ST, then the extension of it and Ω would be too, violating the
minimality of x. (If Ω is not defined, then γx−1···γyitself violates the minimality
of x.) Thus, the extension from s(γx) proceeds entirely by direct arrows. This
completes the proof.
Again, as for the Ext-pro jectives of T, this shows that for each injective string Iv
corresponding to a vertex voutside the support of T, there are two arrows α1, α2
of opposite orientations such that Iv= Mc(ST, α1) = Mc(ST, α2).
Finally, we consider the injective strings at sink fringe vertices.
16 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
Proposition 7.5 Let vbe a sink fringe vertex. Let Iv=γr...γ0be the injective
at v, with e(γr) = v, which is composed entirely of direct arrows. Let xbe minimal
such that γr−1. . . γx+1 is contained in ST. Then Iv= Mc(ST, γx).
Proof. The proof is the same as for Proposition 7.4, except that since vis a fringe
vertex, once the string reaches v, it ends.
Proposition 7.6 No strings are constructed with greater multiplicity than given
in Theorem 7.2. In particular, those strings not listed there, are not Mc(ST, α)for
any arrow αof ˆ
Q.
Proof. The number of arrows in ˆ
Qcan be determined by counting the number of
arrows going to each vertex. This is the number of sink fringe vertices of ˆ
Qplus
twice the number of vertices of Q. Since the Ext-pro jectives of Tare support τ-
tilting, the number of Ext-projectives of Tequals the number of vertices in the
support of T; this plus the number of vertices not in the support of Tequals the
number of vertices of Q. Therefore, the number of strings listed in Theorem 7.2
equals the total number of arrows in ˆ
Q. Therefore, each string is constructed with
exactly the multiplicity given in Theorem 7.2.
This completes the proof of Theorem 7.2.
Example 7.7 Consider the gentle algebra A=kQ/I, where Q=A2and I= 0,
with the fringed bound quiver ( ˆ
Q, ˆ
I) as below.
2 1
α
◦
◦
◦
◦
◦
◦
For the (functorially finite) torsion class T= mod A, the set of strings ST=
{e1, e2, α}is drawn in bold.
• •
We now verify that as αruns through the arrows of ˆ
Q, we indeed obtain exactly the
strings given in Theorem 7.2. The strings of the form cohook(M) for Man Ext-
projective indecomposable, are as follows. Each is generated twice, by the arrows
drawn as double lines.
Moreover, the injective strings corresponding to the fringe sink vertices are each
generated once, as indicated by the double lines:
ON THE COMBINATORICS OF GENTLE ALGEBRAS 17
.
8. Extension groups of strings
The aim of this section is to study the Ext1-groups of string modules over gentle
algebras. By the Auslander-Reiten formula (see Theorem 2.8), this amounts to
studying the quotient of a Hom-space by morphisms factoring through an injective.
Although HomA(X, τAY) for a gentle algebra Aadmits a nice basis, given by
graph maps as in Theorem 2.7, it appears difficult to decide which of these could
form a basis for the quotient by IA(X, τAY). This question turns out to be easier
to answer in the fringed algebra case, and since we have the following isomorphism
(8.1) Ext1
A(Y, X )≃Ext1
ˆ
A(Y, X )
from Proposition 3.7, it is enough to study fringed algebras.
In the following, we first study graph maps starting in an injective string. We
give an explicit combinatorial description of the graph maps in Hom ˆ
A(X, τ ˆ
AY)
that factor through injectives in Subsection 8.2. Then we use that information to
determine a basis for Ext1
A(Y, X ) in terms of short exact sequences in Subsection
8.3.
8.1. Graph maps from an injective module. We first study graph maps start-
ing in an injective string Itover A. Such an injective string module can be written as
It= (α−1
n+m···α−1
n+1)·t(βn···β1), for a vertex t∈Q0and some arrows αi, βj∈Q1.
Here we write ·trather than ·to indicate that the paths we compose meet at t.
The following notion turns out to be central: A graph map fTgiven by an
admissible pair T= ((F1, E, D1),(F2, E , D2)) is called two-sided if at least one of
D1and D2has positive length, and the same holds for F1and F2.
Lemma 8.1 A graph map fT:It→Ystarting in an injective string Itis not
two-sided.
Proof. There is no quotient factorization (F′
1θ, E, γ−1D′
1) of Itwith arrows γ , θ in Q
because an inverse arrow cannot precede a direct arrow in It. Thus, in any quotient
factorization (F1, E, D1) of It, one of D1or F1has length zero. We consider, without
loss of generality, the case where D1has length zero, thus It=F1E. Since Itis
injective, there is no arrow βin Qsuch that Eβ is a string. Hence Ycannot admit a
submodule factorization of the form (F2, E , βD′
2). We conclude that D2has length
zero in any submodule factorization of Yof the form (F2, E , D2). So, fTis not
two-sided.
8.2. Graph maps factoring through an injective string module. For a pair
of strings Xand Yin A, we now investigate which graph maps in Hom ˆ
A(X, τ ˆ
AY)
factor through an injective string Itin the fringed algebra ˆ
A. From Proposition 3.7,
we know that the Auslander-Reiten translation of a string Yin ˆ
Ais obtained by
cohook completion:
cohook(Y) = τˆ
AY=c
Yc=I · β·Y·α−1· D
18 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
where Iand Drespectively denote inverse and direct paths in ˆ
Q:
Y
β
I
α
D
We say a string Elies on one of the arms of cohook(Y) if Eis a substring
(possibly of length zero) of Ior D. Moreover, we call Yconnectable to Xif there
exists an arrow αsuch that Y α−1Xis a string in Q, or dually, XαY is a string in
Q:
X
α
Y
Note that if Yis connectable to X, there exists a graph map fT∈Hom ˆ
A(X, τ ˆ
AY)
given by an admissible pair T= ((F1, E, D1),(F2, E, D2)) where Elies on one of
the arms of cohook(Y): Denote by Ethe longest direct string at the end of X
and write X=ED1. This gives, with F1of length zero, the quotient factorization
(F1, E, D1) of X. By the cohook construction, Dis the longest direct string in ˆ
A
ending in e(α).But Eis a direct string ending in e(α). Thus Eis a submodule of
D, and we can write D=ED2. Setting F2=I · β·Y·α−1gives the submodule
factorization (F2, E, D2) of τˆ
AY.
We refer to this graph map fTas a connecting map. The following is an illus-
tration of the factorization Twhere the cohooks of are drawn dashed.
X
E
α
Y
•
e(α)
Theorem 8.2 For strings Xand Yin Q, let fT∈Hom ˆ
A(X, τ ˆ
AY)be a graph
map given by an admissible pair T= ((F1, E, D1),(F2, E, D2)). If Elies on one
of the arms of cohook(Y)and fTis not a connecting map, then fTfactors through
an injective string of mod- ˆ
A.
Proof. Consider a graph map fT∈Hom ˆ
A(X, τ ˆ
AY) where Elies on one of the arms
of cohook(Y). Without loss of generality, assume Eis a substring of D. Since
(F2, E, D2) is a submodule factorization of cohook(Y), the string Elies on the
end of the direct string D, so we have D=ED2. We want to find an injective
string Itand graph maps fT′∈Hom ˆ
A(X, It) and fT′′ ∈Hom ˆ
A(It, τ ˆ
AY) such that
fT=fT′′ ◦fT′.
In the factorization (F1, E , D1) of the string X, write F1=F′
1γc···γ1where F′
1
does not start with a direct arrow. The assumptions of the theorem imply that such
a direct string γc···γ1always exists: If F1has length zero, then Yis connectable
to Xvia the arrow α, and fTis the corresponding connecting map. Otherwise, if
ON THE COMBINATORICS OF GENTLE ALGEBRAS 19
F1starts with an inverse arrow, then Eis not a factor module of Xand (F1, E, D1)
not a quotient factorization. Therefore, the direct arrow γcexists in X. Denote by
tits end point, see the following figure:
γ1γc•It
Eα
β
τˆ
AY
Set E′=I′γc· · · γ1E, where I′is the longest inverse substring in Ximmediately
following t. Then E′is a factor module of Xand a submodule of It, therefore there
exists a graph map fT′∈Hom ˆ
A(X, It) with E′as middle term of the factorizations.
Similarly, E′′ =Dis a factor module of Itand a submodule of τˆ
AY, therefore there
exists a graph map fT′′ ∈Hom ˆ
A(It, τ ˆ
AY) with E′′ as middle term. By construction,
we get fT=fT′′ ◦fT′.
In the following corollary we describe a basis of Iˆ
A(X, τ ˆ
AY) in terms of graph
maps:
Corollary 8.3 Let Xand Ybe two strings in Q. Then a basis for Iˆ
A(X, τ ˆ
AY)is
formed by the graph maps fT∈Hom ˆ
A(X, τ ˆ
AY), with T= ((F1, E, D1),(F2, E, D2)),
such that Elies on one of the arms of cohook(Y)and fTis not a connecting map.
Proof. Theorem 8.2 assures that each graph map fTbelongs to Iˆ
A(X, τ ˆ
AY). We
show that every morphism f∈Hom ˆ
A(X, τ ˆ
AY) that factors through an injective
Iis a linear combination of the graph maps as described in the hypothesis. Since
graph maps are linearly independent, we therefore get a basis for Iˆ
A(X, τ ˆ
AY).
Thus, assume f=g◦hwhere h∈Hom ˆ
A(X, I ) and g∈Hom ˆ
A(I, τ ˆ
AY). Write
the maps gand hin the basis of graph maps, thus g=Paigiand h=Pbjhj
with scalars ai, bj∈kand graph maps gi, hj. Each fij =gi◦hjis either zero or a
composition of graph maps (which is again a graph map). It suffices to show that
the fij satisfy the conditions stated above.
The graph map fij factors via hjand githrough some injective string It(which is
a direct summand of I). In general, if a composition of graph maps, given by factor-
izations T′= ((F′
1, E′, D′
1),(F′
2, E′, D′
2)) and T′′ = ((F′′
1, E′′ , D′′
1),(F′′
2, E′′ , D′′
2)),
yields a graph map given by the factorization T= ((F1, E, D1),(F2, E , D2)), one
necessarily has E⊂E′and E⊂E′′ . Since gistarts in It, lemma 8.1 implies
that the factorization T′′ defining the graph map gicannot be two-sided. There-
fore, without loss of generality, we can assume that both D′′
1and D′′
2have length
zero, that is, we have It=F′′
1E′′ and τˆ
AY=F′′
2E′′. If E′′ is a proper quotient
of It, we conclude that E′′ is a direct string, properly contained in the arm Dof
τˆ
AY= cohook(Y). This implies that Elies on one of the arms of cohook(Y) and
fij is not a connecting map. Otherwise, if E′′ =It, we get that τˆ
AY=F′′
2E′′
contains e(E′′) as internal vertex, which is impossible since this does not belong
to Q, or else τˆ
AY=E′′ =Itwhich is also impossible since an injective is not an
image under the Auslander-Reiten translate.
20 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
8.3. The vector space Ext1
A(Y, X ).Recall from Theorem 2.8 and Proposition 3.7
that we have the following isomorphisms:
(8.2) Ext1
A(Y, X )≃Ext1
ˆ
A(Y, X )≃D(Hom ˆ
A(X, τ ˆ
AY)/I ˆ
A(X, τ ˆ
AY))
In Subsection 8.2 we determined a basis for this space given by the graph maps in
Hom ˆ
A(X, τ ˆ
AY) that do not belong to Iˆ
A(X, τ ˆ
AY). In Corollary 8.3 we gave a precise
description of which graph maps have to be excluded. We aim in this subsection
for a description of a basis for Ext1
A(Y, X ) in terms of short exact sequences.
Clearly, every connecting graph map fTgives rise to a non-zero extension
(8.3) ǫT: 0 →X→Y α−1X→Y→0
Moreover, a two-sided graph map
fT∈HomA(X, Y ) with T= ((F1, E , D1),(F2, E, D2))
yields a non-zero extension ǫTas follows, see [S]:
(8.4) ǫT: 0 →X→F1ED2⊕F2ED1→Y→0
Lemma 8.4 For strings Xand Yin Q, there is a bijection between the two-sided
graph maps fT∈HomA(X, Y )and the non-connecting graph maps in Hom ˆ
A(X, τ ˆ
AY)
that do not belong to Iˆ
A(X, τ ˆ
AY)
Proof. Assume fT∈HomA(X, Y ) is a two-sided graph map given by the factor-
ization T= ((F1, E, D1),(F2, E , D2)). The following construction associates to fT
a graph map fˆ
T∈Hom ˆ
A(X, τ ˆ
AY) given by ˆ
T= (( ˆ
F1,ˆ
E, ˆ
D1),(ˆ
F2,ˆ
E, ˆ
D2)): As be-
fore, we denote τˆ
AY=IβY α−1D. If F2has positive length, we define ˆ
F2=IβF2
and leave Eunchanged. Otherwise, since Tis two-sided, we know that F1has
positive length. As Eis a submodule, the string F1starts with a direct arrow,
and this must be the arrow βsince in the gentle algebra Athere is only one way
to extend the string E. We subdivide F1as F1=ˆ
F1I′β, where I′is the longest
inverse substring in Ximmediately following β, and extend Eat the end by I′β,
that is, we put ˆ
E=I′βE. We proceed in the same way for the Dside. These basis
elements fˆ
T∈Hom ˆ
A(X, τ ˆ
AY) are clearly distinct, non-connecting and do not lie in
Iˆ
A(X, τ ˆ
AY), by the description given in corollary 8.3. It is moreover easy to see that
every such basis element fˆ
T∈Hom ˆ
A(X, τ ˆ
AY) is obtained by this construction.
Theorem 8.5 For strings Xand Yin Q, the extensions ǫTgiven in Equation
(8.3) and (8.4), with Tconnecting or two-sided, form a basis for Ext1
A(Y, X ).
Proof. We know from the isomorphism in (8.2) and the bijection in Lemma 8.4
that we listed the correct number of elements ǫTin Ext1
A(Y, X ). It is therefore
sufficient to show that they are linearly independent. We do so by showing that
the ǫTcorrespond to linearly independent elements in an isomorphic space.
Consider an injective envelope Iof Xand extend to a short exact sequence:
0//Xι//Iπ//Z//0
Applying the functor Hom(Y, −) yields the exact sequence
ON THE COMBINATORICS OF GENTLE ALGEBRAS 21
0//Hom(Y, X )ι∗
//Hom(Y, I )π∗
//Hom(Y, Z )δ//Ext(Y, X )//0
where we use Ext(Y, I ) = 0 since Iis injective. Thus, Ext(Y , X) is isomorphic
to Coker π∗, with isomorphism induced by the the connecting homomorphism δ.
To study the map δ, denote by s1,...,snthe sinks of the string X. Then I=
Is1⊕ · · · ⊕ Isn.
Assume the extension ǫTis given by a connecting graph map as in Equation (8.3).
Denote by Ethe longest inverse substring of Ysuch that Eα−1is a string. Then
Eis a factor module of Y, and it also a factor module of Isnand a submodule of
the summand of Zinduced by Isn. The corresponding graph map fE∈Hom(Y, Z )
clearly satisfies δ(fE) = ǫT.
Moreover, denote by t1,...,tmthe sources of the string Xthat have two arrows
in Xattached to it. Then the socle of the module Zcontains the same vertices
t1,...,tm. Let now fT∈HomA(X, Y ) be a two-sided graph map given by the
factorization T= ((F1, E, D1),(F2, E, D2)), and let ti,...,tjbe the sources of X
that are contained in the string E. Note that different factorizations T , T ′give rise
to different intervals [i, j],[i′, j ′]. We let fE∈HomA(Y, Z ) be the map identifying
the elements tkof Ywith the corresponding elements tkin Z, for k∈[i, j]. It is
an exercise in linear algebra to see that δ(fE) = ǫT.
The maps fEstemming from a two-sided graph map are linearly independent
from the maps induced by connecting graph maps, and they are linearly indepen-
dent amongst each other: this comes from the fact that the middle factors E , E′of
two-sided graph maps fT, fT′cannot properly overlap. In fact, the only way that
Eand E′intersect non-trivially is that one is contained in the other.
Moreover, the classes of fEare linearly indep endent in Coker π∗, since graph
maps factoring through an injective Iare described in Corollary 8.3, and these
cannot alter the linear independence for the graph maps fE. Therefore the same
holds for the short exact sequences ǫT.
9. Uniqueness of kisses between exchangeable modules
Let Abe a gentle algebra. Let Zbe an almost complete support τ-tilting A-
module. Then by [AIR, Theorem 2.18] we know that there exist exactly two support
τ-tilting A-modules, say Tand T′, having Zas a direct summand. We recall from
Section 6 that in this case, one of Fac Tand Fac T′covers the other in the poset
of functorially finite torsion classes. Supposing Fac Tcovers Fac T′, we say that
T′is the left mutation of T, and we write T′=µ−
X(T). Let us write T=X⊕Z
and T′=Y⊕Zwhere Xis a τ-rigid indecomposable module, and Yis a τ-rigid
indecomposable module which is not isomorphic to Xor else zero.
Consider the maximal non-kissing collections corresponding to Tand T′. There
is a string C= cohook(X) in the maximal non-kissing collection corresponding
to T. In the maximal non-kissing collection corresponding to T′, this string is
replaced by another string, D. We call such a pair of strings exchangeable. As
shown in Section 6, there will be one or more kisses from Cto D. In his more
restricted setting, McConville showed that exchangeable strings kiss exactly once
[McC, Theorem 3.2(3)]. It is natural to ask whether this, like so many of the other
results from [McC], extends verbatim to the general gentle setting.
22 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
It turns out that this is not the case. For example, consider the gentle algebra
given by the quiver
1
α
2
oo
with the relation α2= 0. Then, the direct sum of the projectives, P1⊕P2, is a
τ-rigid module, its mutation at P1is the module τ−(P1)⊕P2and one has that
dimkHom(P1, τ (τ−P1)) = dimkHom(P1, P1) = 2. This example comes from [GLS,
Section 13.6]. We thank Gustavo Jasso for pointing out its relevance to our situa-
tion.
However, when the τ-rigid indecomposable Xis replaced by a module Ywhich
is a brick, the next theorem will show that the corresponding strings kiss once, i.e.,
that the dimension of Hom(X, τ Y ) is indeed one.
Theorem 9.1 If Y⊕Zis obtained by a left mutation from T=X⊕Z, that is,
Y⊕Z=µ−
X(T), and Yis a brick, then dimkHomA(X, τ Y ) = 1.
Before proving the theorem we will explain the necessary background and a
lemma which will be used in the proof of the theorem. Let
P1
p1//P0
p0//X//0
Q1
q1//Q0
q0//Y//0
R1
r1//R0
r0//Z//0
be minimal projective presentations, and we denote by P, Q, and Rthe correspond-
ing two-term complexes of projective modules in Kb(pro j A).
We are going to prove some results in this setting before proving the main the-
orem of this section.
Lemma 9.2 We have the following properties;
(1) dimkHomKb(proj A)(Q, Q) = 1,
(2) HomKb(proj A)(Q, R[1]) = 0,
(3) dimkHomKb(proj A)(Q, P [1]) = 1.
Proof. (1) Let f•, g•∈HomKb(proj A)(Q, Q). By the universal property of cokernels,
they induce the morphisms f, g ∈HomA(Y, Y ) such that f q0=q0f0and gq0=q0g0.
Since Yis a brick module, we know that dimkHomA(Y , Y ) = 1, so there exists a
λ∈ksuch that f−λg = 0.
Denote by Q•the projective resolution of Yobtained by completing Q. We get
the following commutative diagram
Q
q0
f•−λg•//Q•
q0
Yf−λg = 0//Y .
ON THE COMBINATORICS OF GENTLE ALGEBRAS 23
Finally, by homotopy uniqueness of projective resolutions [W, Theorem 2.2.6]
one can easily prove that f•−λg•is null-homotopic.
(2) Following [AIR, Lemma 3.4], this is true if and only if HomA(Z, τ Y ) = 0.
Now, since Y⊕Zis τ-tilting, HomA(Y⊕Z, τ (Y⊕Z)) = 0, which yields the result.
(3) By [AIR, Theorem 3.2 and Corollary 3.9] we can write the exchange of X
and Yon the level of two-term complexes. By [AIR, Definition-Proposition 1.7],
this guarantees that there exists a triangle in Kb(pro j A) :
P//R′//Q//P[1]
with R′∈add R.
Applying the functor HomKb(proj A)(Q, −), we get the long exact sequence
... //Hom(Q, Q)//Hom(Q, P [1]) //Hom(Q, R′[1]) //... .
Using parts (1) and (2) and the fact that R′[1] ∈add R[1], we now get that
dimkHomKb(proj A)(Q, P [1]) is either 0 or 1. Assume it is 0, then by [AIR, Lemma
3.4] we have that Hom(X, τY ) = 0. This leads a contradiction: since Xand Yare
not compatible, HomA(X, τ Y ) or HomA(Y, τ X) is nonzero. However, HomA(Y, τ X) =
0. This follows from the definition of a left mutation [AIR, Definition-Proposition
2.28] which guarantees that Fac Y⊆Fac Z⊆Fac T, the fact that Xis Ext-
projective in Fac Tand [AIR, Proposition 1.2(a)] which states that HomA(Y , τX ) =
0 if and only if Ext1
A(X, Fac Y) = 0. Thus, dimkHomKb(pro j A)(Q, P [1]) has to be
1.
Proof of Theorem 9.1. Rewriting slightly [AIR, Proposition 2.4], we get an exact
sequence
HomA(Q0, X)(q1,X)//HomA(Q1, X )//D HomA(X, τ Y )//0.
This gives that dimkHomA(X, τ Y ) is equal to the dimension of the quotient of
HomA(Q1, X) by Im(q1, X).
We now show that the dimension of this quotient is one.
Let f, g ∈HomA(Q1, X). Since Q1is projective, there exists f , g ∈HomA(Q1, P0)
such that p0f=fand p0g=g.
These functions give chain morphism in HomKb(pro j A)(Q, P [1]), which is of di-
mension one by Lemma 9.2(3). It follows that there exists h0∈HomA(Q0, P0),
h1∈HomA(Q1, P1) and λ∈ksuch that f−λg =p1h1+h0q1.
Composing the last equality with p0, we get that
f−λg =p0h0q1= (q1, X)(p0h0),
which is to say that f−λg = 0 in the quotient space.
Since HomA(X, τ Y ) is nonzero we conclude the desired result.
This result supposes that the mutation produces a nonzero module Y. In general,
the support of Zcan be smaller than the support of X⊕Z, in which case Zis itself
a support τ-tilting module, and left mutation of X⊕Zat Xyields Z. In this case,
let vbe the vertex in the support of Xwhich is not in the support of Z. Denote
by Pvand Ivthe projective and injective modules at vertex v, respectively.
Proposition 9.3 Let Zbe a support τ-tilting module obtained by a left mutation
from T=X⊕Z, and let vbe the vertex in the support of Xwhich is not in the
support of Z. Suppose that Pvis a brick. Then dimkHomA(X, Iv) = 1.
24 BR ¨
USTLE, DOUVILLE, MOUSAVAND, THOMAS, YILDIRIM
Proof. As before, define Pand Rto be the two-term complexes corresponding
to minimal projective presentations of Xand Z. Let Qbe the two-term com-
plex Pv→0. We now check that the statements of Lemma 9.2 still hold. (1)
follows from the fact that Pvis a brick. (2) follows from Q⊕Rbeing a silting
complex [AIR, Definition 1.5]. We conclude as in the proof of Lemma 9.2(3) that
dimkHomKb(proj A)(Q, P [1]) ≤1. This means that
dimkHom(Pv, P0)/p1(Hom(Pv, P1)) ≤1.
Since Hom(Pv, P0)/p1Hom(Pv, P1) = Hom(Pv, X),we conclude that
dimkHom(Pv, X)≤1.
Since Xis supported over the vertex v, the dimension is exactly one. Finally, since
we have dimkHomA(Pv, X) = dimkHomA(X, Iv), this yields the desired result.
From the previous results in this section, we deduce a combinatorial corollary.
Theorem 9.4 Suppose that Ais a gentle algebra such that every τ-rigid indecom-
posable A-module is a brick. If two maximal non-kissing collections of long strings
in ˆ
Qdiffer by replacing one string by another, then these two strings kiss exactly
once.
Proof. By Theorem 5.1, we know that there is a bijective correspondence between
maximal non-kissing collections of long strings in ˆ
Qand basic support τ-tilting
modules for A. Let the basic support τ-tilting modules corresponding to the given
maximal non-kissing collections be Tand T′. Since the two maximal non-kissing
collections differ by replacing a single string by another, the corresponding support
τ-tilting modules differ by a single mutation. Without loss of generality, let T′be
the left mutation of T. Let Cand Dbe the corresponding strings. Theorem 6.2
shows that there are no kisses from Dto C, but there is at least one kiss from C
to D. We wish to show that there is in fact exactly one kiss.
Let Xbe the A-module associated to the long string C(i.e., C= cohook(X)).
There are two possibilities regarding D: either it also corresponds to an A-module,
or else it is an injective string in ˆ
Q. Suppose first that it corresponds to an A-
module, say Y. Since Yis τ-rigid, it is a brick by assumption, and we can apply
Theorem 9.1 to conclude that dim Hom(X, τ Y ) = 1. By Theorem 4.4, it follows
that Cand Dkiss exactly once.
Now, we consider the possibility that Dis an injective string in ˆ
Q, corresponding
to the vertex v∈Q0. In this case, Proposition 9.3 tells us that dim HomA(X, Iv) =
1. Each kiss from Cto Dgives rise to such a morphism, so there is only one kiss
between Cand Din this case as well.
This recovers the uniqueness of kisses shown in [McC], since in that setting,
bricks, strings, and τ-rigid indecomposable modules all coincide.
References
[AIR] T. Adachi, O. Iyama, and I. Reiten, τ-tilting theory, Compos. Math. 150 (2014), no. 3,
415-452.
[ASS] I. Assem, D. Simson, A. Skowro´nski, Elements of the representation theory of associative
algebras, Volume 1, Cambridge University Press, Cambridge, 2006.
[AS] M. Auslander and S. Smalø, Almost split sequences in subcategories, J. Algebra 69
(1981) 426-454. Addendum: J. Algebra 71 (1981), 592-594.
ON THE COMBINATORICS OF GENTLE ALGEBRAS 25
[BR] M. C. R. Butler, C. M. Ringel, Auslander-Reiten sequences with few middle terms,
Comm. in Algebra. 15 (1987), 145-179.
[CPS] ˙
I. C¸ anak¸cı, D. Pauksztello, and S. Schroll, On extensions for gentle algebras,
arXiv:1707.06934.
[CB] W. Crawley-Boevey, Maps between representations of zero-relation algebras, J. Algebra
126 (1989), 259-263.
[DIJ] L. Demonet, O. Iyama, and G. Jasso, τ-tilting finite algebras and g-vectors,
arXiv:1503.00285.
[EJR] F. Eisele, G. Janssens, and T. Raedschelders, A reduction theorem for τ-rigid modules,
arXiv:1603.04293.
[GLS] C. Geiss, B. Leclerc, and J. Schr¨oer, Quivers with relations for symmetrizable Cartan
matrices I: Foundations, Inventiones mathematicae 209 (2017), no. 1, 61-158.
[McC] T. McConville, Lattice structure of Grid-Tamari orders, arXiv:1504.05213.
[S] J. Schr¨oer, Modules without self-extensions over gentle algebras, J. Algebra 216 (1999),
no. 1, 178-189.
[PPP] Y. Palu, V. Pilaud, P.G. Plamondon, Non-kissing complexes and τ-tilting for gentle
algebras, arXiv:1707.07574.
[W] C.A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced
Mathematics (1995), Cambridge University Press, 450 p.
[Zh] Y. Zhang, On mutation of τ-tilting modules, arXiv:1604.00127.
