Mutation graphs of maximal rigid modules over finite dimensional preprojective algebras

0.19 MB·English

by  Hongbo Yin

Mutation graphs of maximal rigid
modules over finite dimensional
3
preprojective algebras⋆
1
0
2

n
a Hongbo Yin, Shunhua Zhang∗
J

7 School of Mathematics, Shandong University, Jinan 250100, P. R. China
1

]
T
R Abstract Let Q be a finite quiver of Dynkin type and Λ = ΛQ be
. the preprojective algebra of Q over an algebraically closed field k. Let
h
t T be the mutation graph of maximal rigid Λ modules. Geiss, Leclerc
a Λ
m
and Schro¨er conjectured that T is connected, see [C.Geiss, B.Leclerc,
Λ
[
J.Schro¨er, Rigid modules over preprojective algebras, Invent.Math.,

1
165(2006), 589-632]. In this paper, we prove that this conjecture is
v
3 true when Λ is of representation finite type or tame type. Moreover, we
8
also prove that T is isomorphic to the tilting graph of End T for each
9 Λ Λ
3 maximal rigid Λ-module T if Λ is representation-finite.
.
1
0
3 Key words and phrases: Preprojective algebras; maximal rigid module; mu-
1
: tation graph of maximal rigid modules; tilting graph.
v
i
X
r
a
MSC(2000): 16E10,16G20.
⋆
Supported by the NSF of China (Grant No. 11171183).
∗
Corresponding author.
Email addresses: [email protected](H.Yin), [email protected](S.Zhang).
1
1 Introduction
Let Q be a finite quiver without oriented cycles and kQ be the path algebra of Q
over an algebraically closed field k. The preprojective algebra Λ = Λ of Q was
Q
introduced by Gelfand and Ponomarev in [16] such that Λ contains kQ as a sub-
algebra, and when considered as a left kQ module, Λ decomposes as a direct sum
of the indecomposable preprojective kQ modules with one from each isomorphism
class. Now, preprojective algebras play important roles in representation theory
and other areas of mathematics, such as resolutions of Kleinian singularities, quan-
tum groups, quiver varieties, and cluster theory, see [8, 11, 12, 13, 14, 17, 18] for
details.
By using mutations of maximal rigid modules and their endomorphism algebras
over preprojective algebras of Dynkin type, Geiss, Leclerc and Schro¨er studied the
cluster algebra structure on the ring C[N] of polynomial functions on a maximal
unipotent subgroup N of a complex Lie group of Dynkin type, and obtained that
all cluster monomials of C[N] belong to the dual semicanonical basis, see [11].
Let Q be a Dynkin quiver, and Λ be the preprojective algebra of Q. Recall from
[11], T denotes the mutation graph of maximal rigid modules of Λ. Fix a basic
Λ
maximal rigid Λ-module T, then the contravariant functor FT = Hom (−,T) :
Λ
mod Λ → mod End T yields an anti-equivalence of categories
Λ
mod Λ → P(mod End T)
Λ
where P(mod End T) ⊂ mod End T denotes the full subcategory of all End T-
Λ Λ Λ
modules of projective dimension at most one. Moreover, the functor FT induces
an embedding of graphs ψ : T → T whose image is a union of connected
T Λ EndΛ T
components of T , where T is the tilting graph of the algebra End T.
EndΛ T EndΛ T Λ
Each vertex of T (and therefore each vertex of the image of ψ ) has exactly r−n
Λ T
neighbours.
In [11], Geiss, Leclerc and Schro¨er conjectured that the graph T is connected.
Λ
Inthispaper, we prove thatthis conjecture istruewhen Λis ofrepresentation finite
type or tame type. Moreover, we also prove that ψ is an isomorphism whenever
T
Λ is representation finite. The following theorems are our main results.
2
Theorem 1. Let Λ be a preprojective algebra of type A with n ≤ 4, and T
n
be a maximal rigid Λ-module. Then the functor FT = Hom (−,T) induces an
Λ
isomorphism of graphs ψ : T → T .
T Λ EndΛ T
Corollary 2. Let Λ be a preprojective algebra of type A with n ≤ 4. Then for
n
each maximal rigid Λ-module T, the tilting graph T of End T is isomorphic
EndΛ T Λ
to the mutation graph T of maximal rigid modules of Λ.
Λ
Remarks. Let Λ be a preprojective algebra with finite representation type.
The above corollary implies that for all maximal rigid Λ-modules, their endomor-
phism algebras have same tilting graphs up to isomorphism. However, this kind of
algebras are very different, such as some of them is strongly quasi-hereditary and
most of them is even not quasi-hereditary, see [14] for details.
Theorem 3. Let Λ be a preprojective algebra of representation finite or tame
type. Then the mutation graph T of the maximal rigid Λ-modules is connected.
Λ
This paper is organized as follows: in Section 2, we recall some definitions and
facts needed for our research, in Section 3, we prove Theorem 1 and Corollary 2,
in Section 4, we prove Theorem 3.
2 Preliminaries
Let k be an algebraically closed field, and let A be a finite dimensional algebra over
k. We denote by mod A the category of all finitely generated left A-modules, and
by ind A the full subcategory of mod A consisting of one representative from each
isomorphism class of indecomposable modules. For a A-module M, we denote by
add M the full subcategory of mod A whose objects are the direct summands of
finite direct sums of copies of M. The projective dimension of M is denoted by
pd M, and the Auslander Reiten translation of A by τ .
A
T ∈ mod A is called a classical tilting module if the following conditions are
satisfied:
(1) pd T ≤ 1;
3
(2) Ext1(T,T) = 0;
A
(3) There is an exact sequence 0 −→ A −→ T −→ T −→ 0 with T ∈ add T for
0 1 i
0 ≤ i ≤ 1.
Let T be the set of all basic classical tilting A-modules up to isomorphism.
A
According to [11, 15], the tilting graph T is the defined as following: the vertices
A
are the non-isomorphic basic tilting moduels, there is an edge between T and T
1 2
if T = T′ ⊕T′ and T = T′ ⊕T′ for some A-module T′ and some indecomposable
1 1 2 2
A-modules T′ and T′ with T′ 6≃ T′.
1 2 1 2
Let Q = (Q ,Q ) be a connected quiver, where Q is the set of vertices and
0 1 0
Q is the set of arrows. Given an arrow α, we denote by s(α) the starting vertex
1
of α and by t(α) the ending vertex of α. Let Q be the double quiver of Q, which
is obtained from Q by adding an arrow α∗ : j → i whenever there is an arrow
α : i → j in Q . Let Q∗ = {α∗|α ∈ Q } and Q = Q ∪ Q∗. The preprojective
1 1 1 1 1 1
algebra of Q is defined as
Λ = Λ = kQ/(ρ)
Q
where ρ is the relation with
ρ = X[α,α∗],
α∈Q1
and kQ is the path algebra of Q. See [20].
NotethatthepreprojectivealgebraΛisindependent oftheorientationofQ, and
that Λ is finite dimensional if and only if Q is a Dynkin quiver. Moreover, Λ is also
self-injective if it is finite dimensional. In particular, Λ is of finite representation
type if and only if Q is of type A with n ≤ 4, and it is of tame representation
n
type if and only if Q is of type A or D , see [9, 12].
5 4
Let d,e ∈ Zn be two dimension vectors. The symmetry bilinear form is defined
as (d,e) = 2 d e − d e . The following lemma is proved in [8].
Pi∈Q0 i i Pa∈Q1 s(a) t(a)
Lemma 2.1. Let Λ be a preprojective algebra and X,Y be Λ-modules. Then we
have
dim Ext1(X,Y) = dim Hom (X,Y)+dim Hom (Y,X)−(dim X,dim Y).
Λ Λ Λ
4
In particular, dim Ext1(X,Y) = dim Ext1(Y,X).
Λ Λ
From now on, we always assume that Λ is a preprojective algebra of Dynkin
type. A Λ-module T is called rigid if Ext1(T,T) = 0. T is called Maximal rigid
Λ
if for any Λ-module M with Ext1(T ⊕M,T ⊕M) = 0, then we have M ∈ add T.
Λ
Note that each maximal rigid Λ-module T is also a generator-cogenerator. Let
FT = Hom (−,T). A short exact sequence 0 → X → E → Y → 0 of Λ-modules
Λ
is called FT-exact if 0 → FT(Y) → FT(E) → FT(X) → 0 is an exact sequence of
End T-modules. We denote by FT(Y,X) the equivalent classes of all the FT-exact
Λ
sequences as above.
Let χ be a subcategory of mod Λ whose objects admit an add T-resolution.
T
Namely, X ∈ χ if and only if there is an exact sequence
T
0 //X //T // T // T //···
0 1 2
with all T ∈ add T, which is still exact by applying the functor Hom (T,−). Let
i Λ
Exti (Y,X) be the cohomology group by applying the functor Hom (Y,−) to an
FT Λ
add T-resolution of X.
The following lemma is proved in [3, 4].
Lemma 2.2. Assume that X ∈ χ and Y ∈ mod Λ. Then there are following
T
functorial isomorphisms:
(1) Ext1 (Y,X) ∼= FT(Y,X);
FT
(2) Exti (Y,X) ∼= Exti (Hom (X,T),Hom (Y,T)) for all i ≥ 1.
FT EndΛ(T) Λ Λ
Let Λ be a finite dimensional preprojective algebra, and let T be a maximal
rigid Λ-module. Then χ = mod Λ since every Λ-module has an add T-resolution
T
[11, Corollary 5.2].
Recall from [11, section 6], the mutation graph T of maximal rigid modules is
Λ
defined as following. The vertex set of T is the set of the isomorphism classes of
Λ
basic maximal rigid Λ-modules, and there is an edge between vertices T and T if
1 2
5
and only if T = T ⊕ T′ and T = T ⊕ T′ for some T and some indecomposable
1 1 2 2
modules T′ and T′ with T′ 6≃ T′.
1 2 1 2
Lemma 2.3. Let T be a basic maximal rigid Λ-module. The functor FT :
mod Λ → mod End (T) induces an embedding of graphs ψ : T → T whose
Λ T Λ EndΛ(T)
image is a union of connected components of T .
EndΛ(T)
We follow the standard terminology and notation used in the representation
theory of algebras, see [1, 2, 19].
3 The mutation graph and the tilting graph of
representation finite preprojective algebras
In this section, we assume that Λ is a preprojective algebra of representation finite
type. Namely, Λ is of type A with n ≤ 4. For the AR-quivers of this kind
n
of preprojective algebras we refer to [12, section 20.1]. Here we give the stable
AR-quivers of Λ and Λ for convenience.
A3 A4
◦ ◦ Y
(cid:0)(cid:0)(cid:0)(cid:0)@@ BBBB!! (cid:127)(cid:127)(cid:127)?? ;;;;(cid:29)(cid:29) (cid:127)(cid:127)(cid:127)(cid:127)?? CCC!!
◦ Z ◦ Z
===(cid:30)(cid:30) }}}>> 1>>>>(cid:31)(cid:31) (cid:4)(cid:4)(cid:4)(cid:4)BB >>>(cid:31)(cid:31) |||== 2
X ◦ Z
3
The stable quiver of Λ
A3
◦ ◦ • ◦ ◦ ◦ ◦
@@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>>
◦ • ◦ ◦ ◦ ◦
~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@
◦ ◦ • ◦ ◦ ◦ ◦
@@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>>
◦ //◦ // ◦ // • // • // ◦ //◦ //◦ // ◦ // ◦ //◦ // ◦ //◦
~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@ ~~~>> @@@
◦ ◦ • ◦ ◦ ◦ ◦
The stable quiver of Λ
A4
6
Definition. Two AR-sequences are called centrally connected if they have
common indecomposable summands in the middle terms. A column in the AR-
quiver is a set consist of the indecomposable summands of the middle terms in the
centrally connected AR-sequences.
A path from X to Y in the AR-quiver is a chain of irreducible morphisms
X = M → M → M → ··· → M → M = Y. We say that Z is between X
0 1 2 n−1 n
and Y if there is a chain
X = M → M → M → ··· → M → M = Y
0 1 2 n−1 n
such that all M is not in the same column with Y for 0 < i < n and that Z is in
i
the same column with some one of M with 0 < i < n.
i
A class Σ of pairwise non-isomorphic indecomposable Λ-modules in the stable
quiver above is called a complete slice if it satisfies the following conditions:
(1) the indecomposable modules in Σ lie in different τ-orbits;
(2) Σ is convex. Namely, if X and Y belong to Σ and there is a path from X
to Z and a path from Z to Y, then Z belongs to Σ.
A complete slice is called standard if it lies in two adjacent columns.
For example, in the stable quiver of Λ , Z is between X and Y while Z is
A3 1 2
between Y and X. The complete slice which consists of • in the stable quiver of
Λ is standard.
A4
Lemma 3.1. Given a communicative diagram of exact sequences
i
0 // X //E // Y //0
h
f (cid:15)(cid:15) (cid:15)(cid:15)
0 // X //F //Z //0
with the bottom sequence non-split. Then the top sequence is non-split if and only
if h cannot factor through F.
Proof. Apply the functor Hom (Y,−) to the bottom sequence, we get an exact
Λ
sequence
0 // Hom (Y,X) //Hom (Y,F) α // Hom (Y,Z) β //Ext1(Y,X) .
Λ Λ Λ Λ
7
Then h is in the kernel of β if and only if it is in the image of α. Namely, the top
sequence is the zero element in Ext1(Y,X) if and only if h factors through F. This
Λ
2
complete the proof.
Lemma 3.2. Let Λ be a preprojective algebra of type A , n ≤ 4. Let X,
n
Y and Z be non-isomorphic indecomposable Λ-modules with Ext1(Y,X) 6= 0 and
Λ
Ext1(Z,X) 6= 0. If Y is between X and Z with Hom (Y,Z) 6= 0, then there is a
Λ Λ
non-split exact sequence
(1) 0 → X → E → Y → 0
which is induced from a non-split exact sequence
f
(2) 0 → X −→ F → Z → 0.
Proof. Let (3) 0 → X −→i M → τ−1X → 0 be the AR-sequence start at X.
Then we have the following communicative diagram:
0 //X i // M //τ−1X // 0.
h
f (cid:15)(cid:15) (cid:15)(cid:15)
0 //X // F // Z //0
By using AR-formula Hom (τ−1X,Z) ≃ DExt1(Z,X), we know that different
Λ Λ
sequences of the form (2) corresponds to different homomorphisms from τ−1X to
Z in the stable category modΛ. According to Lemma 3.1, we know that h can’t
factor through F.
LetX,Y andZ benon-isomorphicindecomposableΛ-moduleswithExt1(Y,X) 6=
Λ
0 6= Ext1(Z,X). If Y is between X and Z with Hom (Y,Z) 6= 0, then by reading
Λ Λ
the pictures given in [12, section 20.4] we know that there is a path from τ−1X to
Z which induces a nonzero morphism from τ−1X to Z in modΛ factoring through
Y. Hence there exists a morphism g from Y to Z which cannot factor through F.
Then we have a pull-back diagram:
0 // X //E // Y //0 ,
g
f (cid:15)(cid:15) (cid:15)(cid:15)
0 // X //F //Z //0
8
by Lemma 3.1 again, we have a non-split sequence of the form (1) which is induced
2
from (2).
Remark. We should mention that Lemma 3.2 is not true without the assump-
tion that Y is between X and Z. The following example is pointed out to us by
C.M.Ringel. Let
α1 α2 α3
// // //
1 2 3 4
oo oo oo
α∗ α∗ α∗
1 2 3
2
4 2 4
be the quiver of Λ . Take X = , Y = 1 3 , Z = 2, V = . Then
A4
3 3
2
Ext1(Y,X) = Ext1(Z,X) = k, Hom (Y,Z) = k. 0 → X → P(2) → Y → 0
Λ Λ Λ
and 0 → X → V → Z → 0 are the corresponding exact sequences. But the
first sequence cannot be induced by the second one because the inclusion map
0 → X → V cannot factor through P(2), since there is no map from P(2) to the
simple module 4.
Lemma 3.3. Let Λ be a preprojective algebra of type A , n ≤ 4. Let X and Y
n
be non-isomorphic indecomposable Λ-modules with Ext1(X,Y) 6= 0. Let N be an
Λ
indecomposable non-projective Λ-module which is between X and Y or in the same
column with X. Then any exact sequence
(∗) 0 → Y → M → X → 0
is FN-exact. Moreover, if N is in the same column with X, then any exact sequence
(∗∗) 0 → X → M → Y → 0
is also FN-exact.
Proof. We choose a standard complete slice which contains X and extend it to
a maximal rigid Λ module T by adding all the indecomposable projective-injective
modules. Then it follows from the stable quiver of Λ that every non-zero map from
Y to N factors through T since Y 6∈ add T.
Note that Ext (X,T) = 0, by applying Hom (−,T) to the exact sequence (∗),
Λ Λ
we get an exact sequence
0 → Hom (X,T) → Hom (M,T) → Hom (Y,T) → 0.
Λ Λ Λ
9
Thus every map from Y to T factors through M, which implies that every map
from Y to N factors through M. Namely, the sequence
0 → Hom (X,N) → Hom (M,N) → Hom (Y,N) → 0
Λ Λ Λ
is exact.
Now, we assume that N is in the same column with X. Then any map from X
to N in the stable quiver factors through the maximal rigid module obtained from
the standard complete slice which contains X. Repeat the proof above we see that
any exact sequence
(∗∗) 0 → X → M → Y → 0
is also FN-exact. This completes the proof. 2
Lemma 3.4. Let Λ be a preprojective algebra of type A , n ≤ 4. Let X and
n
Y be non-isomorphic indecomposable Λ-modules with Ext1(X,Y) 6= 0. Let N be
Λ
an indecomposable non-projective Λ-module. Then there exists a non-split exact
sequence
(∗) 0 → Y → E → X → 0
or
(∗∗) 0 → X → M → Y → 0
which is FN-exact.
Proof. If N is between X and Y or in the same column with X, then any
exact sequence (∗) 0 → Y → M → X → 0 is FN-exact by Lemma 3.3. If N is
between Y and X or in the same column with Y, then any exact sequence (∗∗)
0 → X → E → Y → 0 is FN-exact by Lemma 3.3 again. 2
Lemma 3.5. Let Λ be a preprojective algebra of type A , n ≤ 4. Let X and
n
Y be non-isomorphic indecomposable Λ-modules with Ext1(X,Y) 6= 0. Let N and
Λ 1
N be two non-isomorphic indecomposable Λ-module with Ext1(N ,N ) = 0. Then
2 Λ 1 2
there exists a non-split exact sequence
(∗) 0 → Y → E → X → 0
or
(∗∗) 0 → X → M → Y → 0
10