Preprojective algebras and c-sortable words

Abstract

Let $Q$ be an acyclic quiver and $\Lambda$ be the complete preprojective
algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in
the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and
studied in \cite{Bua2} a finite dimensional algebra $\Lambda_w=\Lambda/I_w$. In
this paper we look at filtrations of $\Lambda_w$ associated to any reduced
expression $\mathbf{w}$ of $w$. We are especially interested in the case where
the word $\mathbf{w}$ is $c$-sortable, where $c$ is a Coxeter element. In this
situation, the consecutive quotients of this filtration can be related to
tilting $kQ$-modules with finite torsionfree class.

Full text

Page 1

arXiv:1002.4131v2 [math.RT] 26 May 2010
PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS
CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV
Abstract. Let Q be an acyclic quiver and Λ be the complete preprojective algebra of Q over an
algebraically closed field k. To any element w in the Coxeter group of Q, Buan, Iyama, Reiten
and Scott have introduced and studied in [BIRS09a] a finite dimensional algebra Λw = Λ/Iw.
In this paper we look at filtrations of Λw associated to any reduced expression w of w. We are
specially interested in the case where the word w is c-sortable, where c is a Coxeter element. In
this situation, the consecutive quotients of this filtration can be related to tilting kQ-modules
with finite torsionfree class. This nice description allows us to construct a triangle equivalence
between the 2-Calabi-Yau triangulated category SubΛw and the generalized cluster category
associated with an Auslander algebra.
Contents
Introduction
Notation
Acknowledgements
1. Background
1.1. 2-Calabi-Yau categories associated with reduced words
1.2.Mutation of tilting modules
1.3. Reflections and reflection functors
2. Layers associated with reduced words
2.1. Layers are simples up to autoequivalences
2.2. The dimension vectors of the layers
2.3.Reflection functors and ideals Ii
3. Tilting modules and c-sortable words
3.1. Comparison of three series of kQ-modules
3.2. Tilting modules with finite torsionfree class
3.3.Example
4. Equivalences with cluster categories associated with Auslander algebras
4.1. Standard cluster-tilting objects of SubΛw
4.2.Generalized cluster categories
4.3.Computing endomorphism algebras
4.4. Triangle equivalence
4.5. Example
5.Problems and examples
References
2
3
3
3
3
5
6
6
7
9
11
13
14
17
18
19
20
21
22
25
28
29
32
All authors were supported by the Storforsk-grant 167130 from the Norwegian Research Council.
The second author was supported by JSPS Grant-in-Aid for Scientific Research 21740010 and 60015849.
The fourth author was also supported by the NSA-grant MSPF-08G-228.
1

Page 2

2CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV
Introduction
Attempts to categorify the cluster algebras of Fomin and Zelevinsky [FZ02] have led to the
investigation of categories with the 2-Calabi-Yau property (2-CY for short) and their cluster-
tilting objects. Main early classes of examples were the cluster categories associated with finite
dimensional path algebras [BMR+06] and the preprojective algebras of Dynkin type [GLS06].
This paper is centered around the more general class of stably 2-CY and triangulated 2-CY
categories associated with elements in Coxeter groups [BIRS09a] (the adaptable case was done
independently in [GLS08]), and their relationship to the generalized cluster categories from
[Ami09a] (see Section 4 for definition).
Let Q be a finite connected quiver with vertices 1,...,n, and Λ the complete preprojective
algebra of the quiver Q over a field k. Denote by s1,...,sn the distinguished generators in
the corresponding Coxeter group WQ. To an element w in WQ, there is associated a stably
2-CY category SubΛwand a triangulated 2-CY category SubΛw. The definitions are based on
first associating an ideal Ii in Λ to each si, hence to any reduced word by taking products.
This way we also get a finite dimensional algebra Λw:= Λ/Iw. Objects of the category SubΛw
are submodules of finite dimensional free Λw-modules. The cluster category is then equivalent
to SubΛw with w = c2, where c is a Coxeter element such that c2is a reduced expression
[BIRS09a, GLS08]. When Λ is a preprojective algebra of Dynkin type, then the category modΛ
as investigated in [GLS06] is also obtained as SubΛwwhere w is the longest element [BIRS09a,
III 3.5].
Using the construction of ideals we get for each reduced expression w = su1su2...sula chain
of ideals
Λ ⊃ Iu1⊃ Iu2Iu1⊃ ... ⊃ Iw,
which gives rise to an interesting set of Λ-modules:
L1
w:=
Λ
Iu1
, L2
w:=
Iu1
Iu2Iu1
,...,Ll
w:=Iul−1...Iu1
Iw
which all turn out to be indecomposable and to lie in SubΛw.
The investigation of this set of modules, which we call layers, from different points of view,
including connections with tilting theory, is one of the main themes of this paper, especially for
a class of words called c-sortable.
The modules L1
ciated with the reduced expression w = su1...sul(see Section 1). These modules can be used
to show that the endomorphism algebras EndΛ(Mw) are quasi-hereditary [IR10]. Here we show
that these modules are rigid (Theorem 2.2), that is Ext1
sion vectors are real roots (Theorem 2.6), so that there are unique associated indecomposable
kQ-modules (Lj
w)Q(which are not necessarily rigid).
w,...,Ll
wprovide a natural filtration for the cluster-tilting object Mwasso-
Λ(Lj
w,Lj
w) = 0 and that their dimen-
The situation is especially nice when all layers are indecomposable kQ-modules, so that Lj
(Lj
w)Q. This is the case for c-sortable words. An element w of WQis c-sortable when there exists
a reduced expression of w of the form w = c(0)c(1)...c(m)with c(m)⊆ ... ⊆ c(1)⊆ c(0)⊆ c,
where c is a Coxeter element, that is, a word containing each generator siexactly once, and in
an order admissible with respect to the orientation of Q.
Starting with the tilting kQ-module kQ (when c(0)= c), there is a natural way of performing
exchanges of complements of almost complete tilting modules, determined by the given reduced
expression. We denote the final tilting module by Tw, and the indecomposable kQ-modules used
in the sequence of constructions by Tj
w=
wfor j = 1,...,l. We show that Lj
w≃ Tj
wfor all j (Theorem

Page 3

PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS3
3.5) and that the indecomposable modules in the torsionfree class Sub(Tw) are exactly the Tj
(Theorem 3.8). In particular this gives a one-one correspondence between c-sortable words and
torsionfree classes, as first shown in [Tho] using different methods.
There is another sequence U1
wof indecomposable kQ-modules, defined using restricted
reflection functors, which coincide with the above sequences. This is both interesting in itself,
and provides a method for proving Lj
wfor j = 1,...,l.
In another paper [AIRT], we give a description of the layers from a functorial point of view.
When the c-sortable word is cm, and c = s1...sn, then the successive layers are given by
P1,...,Pn,τ−P1,...,τ−Pn,τ−2P1,...,τ−mPn
w
w,...Ul
w≃ Tj
for the indecomposable projective kQ-modules Pi, where τ denotes the AR-translation. In the
general case we will give a description of the layers using specific factor modules of the above
modules.
The generalized cluster categories CAfor algebras A of global dimension at most two were
introduced in [Ami09a]. It was shown that for a special class of words w, properly contained in
the dual of the c-sortable words, the 2-CY category SubΛwis triangle equivalent to some CA.
We show that the procedure for choosing A works more generally for any (dual of a) c-sortable
word (Theorem 4.10), with a simpler proof due to developments in the meantime.
The paper is organized as follows. We start with some background material on 2-CY categories
associated with reduced words, on complements of almost complete tilting modules and on
reflection functors. In Section 2 we show that for any reduced word w, the associated layers
are indecomposable rigid modules, which also are real roots. Hence there are unique associated
indecomposable kQ-modules. In Section 3 we show that our three series of indecomposable
modules {Lj
w} coincide in the c-sortable case. The description of the layers
as specific factor modules of the τ−iP for P indecomposable projective is given in Section 4.
In Section 5 we show the relationship with generalized cluster categories in the c-sortable case.
Section 6 is devoted to examples and questions beyond the c-sortable case.
Some of this work was presented at a conference in Trondheim in August 2009.
w}, {Tj
w} and {Uj
Notation. Throughout k is an algebraically closed field. The tensor product − ⊗ −, when not
specified, will be over the field k. For a k-algebra A, we denote by modA the category of finitely
presented right A-modules, and by f.l.A the category of finite length right A-modules. For a
quiver Q we denote by Q0the set of vertices and by Q1the set of arrows, and for a ∈ Q1we
denote by s(a) its source and by t(a) its target.
Acknowledgements. This work was done when the first author was a postdoc of NTNU,
Trondheim. She would like to thank the Research Council of Norway for financial support. Part
of this work was done while the second author visited NTNU during March and August 2009.
He would like to thank the people in Trondheim for their hospitality.
1. Background
1.1. 2-Calabi-Yau categories associated with reduced words. Let Q be a finite connected
quiver without oriented cycles and with vertices Q0= {1,...,n}. For i,j ∈ Q0we denote by
mijthe positive integer
mij:= ♯{a ∈ Q1| s(a) = i,t(a) = j} + ♯{a ∈ Q1| s(a) = j,t(a) = i}.
The Coxeter group associated to Q is defined by the generators s1,...,snand relations
• s2
i= 1,
• sisj= sjsiif mij= 0,

Page 4

4 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV
• sisjsi= sjsisjif mij= 1.
In this paper w will denote a word (i.e. an expression in the free abelian group generated by
si,i ∈ Q0), and w will be its equivalence class in the Coxeter group WQ.
An expression w = su1...sulis reduced if l is smallest possible. An element c = su1...sulis
called Coxeter element if l = n and {u1,...,ul} = {1,...,n}. We say that a Coxeter element
c = su1...sunis admissible with respect to the orientation of Q if i < j when there is an arrow
ui→ uj.
The preprojective algebra associated to Q is the algebra
kQ/?
?
a∈Q1
(aa∗− a∗a)?
where Q is the double quiver of Q, which is obtained from Q by adding for each arrow a : i → j
in Q1an arrow a∗: i ← j pointing in the opposite direction. We denote by Λ the completion of
the preprojective algebra associated to Q and by f.l.Λ the category of right Λ-modules of finite
length.
The algebra Λ is finite-dimensional selfinjective if Q is a Dynkin quiver. Then the stable
category modΛ satisfies the 2-Calabi-Yau property (2-CY for short), that is, there is a functorial
isomorphism
DHomΛ(X,Y ) ≃ HomΛ(Y,X[2]),
where D := Homk(−,k) and [1] := Ω−1is the suspension functor [AR96, CB00] (see [GLS07,
Rei06] for a complete proof).
When Q is not Dynkin, then Λ is infinite dimensional and of global dimension 2. In this
case the triangulated category Db(f.l.Λ) is 2-CY [CB00, GLS07, Boc08] (see [SY] for a complete
proof).
We now recall some work from [IR08, BIRS09a]. For each i = 1,...,n we have an ideal
Ii:= Λ(1 − ei)Λ in Λ, where eiis the idempotent of Λ associated with the vertex i. We write
Iw:= Iul...Iu2Iu1when w = su1su2...sulis a reduced expression of w ∈ WQ. We denote by
Si:= Λ/Iithe simple bimodule corresponding to the vertex i.
We collect the following information which is useful for Section 2:
Proposition 1.1. [BIRS09a] Let Λ be a preprojective algebra.
(a) If w = su1...suland w′= sv1...svlare two reduced expression of the same element in
the Coxeter group, then Iw= Iw′.
(b) If w = w′si with w′reduced, then Iw ⊆ Iw′. Moreover w is reduced if and only if
Iw? Iw′. And for j ?= i we have ejIw= ejIw′.
If Λ is not of Dynkin type we have moreover:
(c) Any finite product I of the ideals Ijis a tilting module of projective dimension at most
one, and EndΛ(I) ≃ Λ.
(d) If S is a simple Λ-module and I is a tilting module of projective dimension at most one,
then S ⊗ΛI = 0 or TorΛ
1(S,I) = 0.
(e) If Si= Λ/Iiand TorΛ
1(Si,I) = 0, then Ii
projective dimension at most one.
L
⊗ΛI = Ii⊗ΛI = IiI for a tilting module I of
By (a) the ideal Iwdoes not depend on the choice of the reduced expression w of w. Thefore
we write Iwfor the ideal Iwand Λw:= Λ/Iw. This is a finite dimensional algebra. We denote by
SubΛwthe category of submodules of finite dimensional free Λw-modules. This is a Frobenius
category, that is, an exact category with enough projectives and injectives, and the projectives

Page 5

PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS5
and injectives coincide. Its stable category SubΛwis a triangulated category which satisfies the
2-Calabi-Yau property [BIRS09a]. The category SubΛwis then said to be stably 2-Calabi-Yau.
Recall that a cluster-tilting object in a Frobenius stably 2-CY category C with finite dimen-
sional morphisms spaces is an object T ∈ C such that
• Ext1
C(T,T) = 0
• Ext1
C(T,X) = 0 implies that X ∈ addT.
For any reduced word w = su1...sul, we write Mj
w:= euj
Λ
Iuj...Iu1
.
Theorem 1.2. [BIRS09a, Thm III.2.8] For any reduced expression w = su1...sulof w ∈ WQ,
the object Mw:=?l
For any reduced word w = su1...sul, we have the chain of ideals
j=1Mj
w is a cluster-tilting object in the stably 2-CY category SubΛw.
Λ ⊃ Iu1⊃ Iu2Iu1⊃ ... ⊃ Iw,
which is strict by Proposition 1.1 (b). For j = 1,...,l we define the layer
Lj
w:=Iuj−1...Iu1
Iuj...Iu1
.
Using Proposition 1.1 (b) it is immediate to see the following
Proposition 1.3. We have isomorphisms in f.l.Λ:
Lj
w≃ eujLj
w≃ euj
Iui...Iu1
Iuj...Iu1
≃ Ker(Mj
w
?? ??Mi
w),
where i is the greatest integer satisfying ui= ujand i < j.
Therefore the layers L1
w,...,Ll
wgive a filtration of the cluster-tilting object Mw.
1.2. Mutation of tilting modules. Let Q be a finite connected quiver with vertices {1,...,n}
and without oriented cycles.
Definition 1.4. A basic kQ-module T is called a tilting module if Ext1
n non-isomorphic indecomposable summands.
kQ(T,T) = 0 and it has
For each indecomposable summand Tiof T, it is known that there is at most one indecompos-
able T∗
iis a tilting module [RS91, Ung90], and that there is exactly
one if and only if T/Tiis a sincere kQ-module [HU89]. We then say that Ti(and possibly T∗
is a complement for the almost complete tilting module T/Ti. The (possibly) other complement
of T/Tican be obtained using the following result:
i≇ Tisuch that (T/Ti)⊕T∗
i)
Proposition 1.5. [RS91]
(a) If the minimal left add(T/Ti)-approximation Ti
is a complement for T/Ti.
f
??B is a monomorphism, then Cokerf
(b) If the minimal right add(T/Ti)-approximation B′
is a complement for T/Ti.
g
??Ti is an epimorphism, then Kerg
There is a one-one correspondence between tilting modules T and contravariantly finite tor-
sionfree classes F = SubT containing the projective modules.