Document [original]

Fakultät für

Elektrotechnik, Informatik und Mathematik

Institut für Mathematik

Dissertation

The

Hall Algebra

and the

Composition Monoid

Stefan Wolf

2009

Betreuer: Andrew Hubery, Ph.D.

Prof. Dr. Henning Krause

Abstract

Let Qbe a quiver. M. Reineke and A. Hubery investigated the connection between

the composition monoid CM(Q), as introduced by M. Reineke, and the generic com-

position algebra Cq(Q), as introduced by C. M. Ringel, specialised at q= 0. In this

thesis we continue their work. We show that if Qis a Dynkin quiver or an oriented

cycle, then C0(Q)is isomorphic to the monoid algebra QCM(Q). Moreover, if Qis an

acyclic, extended Dynkin quiver, we show that there exists a surjective homomorphism

Φ: C0(Q)→QCM(Q), and we describe its non-trivial kernel.

Our main tool is a geometric version of BGP reflection functors on quiver Grassman-

nians and quiver flags, that is varieties consisting of filtrations of a fixed representation

by subrepresentations of fixed dimension vectors. These functors enable us to calculate

various structure constants of the composition algebra.

Moreover, we investigate geometric properties of quiver flags and quiver Grassmanni-

ans, and show that under certain conditions, quiver flags are irreducible and smooth. If,

in addition, we have a counting polynomial, these properties imply the positivity of the

Euler characteristic of the quiver flag.

Zusammenfassung

Sei Qein Köcher. M. Reineke und A. Hubery untersuchten den Zusammenhang zwi-

schen dem von M. Reineke eingeführten Kompositionsmonoid CM(Q)und der bei q= 0

spezialisierten Kompositionsalgebra Cq(Q), die von C. M. Ringel definiert wurde. Diese

Dissertation führt diese Arbeit fort. Wir zeigen, dass C0(Q)isomorph zu der Monoidal-

gebra QCM(Q)ist, wenn Qein Dynkin Köcher oder ein orientierter Zykel ist. Wenn Q

ein azyklischer erweiterter Dynkin Köcher ist, so zeigen wir, dass es einen surjektiven

Homomorphismus Φ: C0(Q)→QCM(Q)gibt, und beschreiben dessen nicht trivialen

Kern.

Um dies zu beweisen, müssen wir viele Strukturkonstanten der Kompositionsalgebra

berechnen. Dazu führen wir eine geometrische Version von BGP-Spiegelungsfunktoren

auf Köchergrassmannschen und Köcherfahnen ein. Dies sind Varietäten bestehend aus

Filtrierungen einer festen Darstellung durch Unterdarstellungen fester Dimensionvekto-

ren.

Außerdem untersuchen wir die geometrischen Eigenschaften von Köchergrassmann-

schen und Köcherfahnen. Wir erhalten ein Kriterium, das uns erlaubt festzustellen, wann

eine Köcherfahne irreduzibel und glatt ist. Wenn man zusätzlich noch ein Zählpolynom

hat, so folgt aus diesen Eigenschaften die Positivität der Euler-Charakteristik.

iii

Contents

1. Introduction 1

2. Preliminaries 7

2.1. Quivers, Path Algebras and Root Systems . . . . . . . . . . . . . . . . . . 7

2.2. Representations of Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3. ReflectionFunctors............................... 11

2.4. Dynkin and Extended Dynkin Quivers . . . . . . . . . . . . . . . . . . . . 14

2.5. Canonical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6. Degenerations.................................. 17

2.7. SegreClasses .................................. 18

2.8. Ringel-HallAlgebra .............................. 20

2.9. Generic Composition Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.10. Generic Extension Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.11. Quiver Flags and Quiver Grassmannians . . . . . . . . . . . . . . . . . . . 24

2.12.MainTheorems................................. 24

3. Cyclic Quiver Case 27

4. Composition Monoid 37

4.1. Generalities................................... 37

4.2. Partial Normal Form in Terms of Schur Roots . . . . . . . . . . . . . . . . 39

4.3. ExtendedDynkinCase............................. 42

5. Geometry of Quiver Flag Varieties 47

5.1. BasicConstructions .............................. 47

5.2. Grassmannians and Tangent Spaces . . . . . . . . . . . . . . . . . . . . . 52

5.3. Geometry of Quiver Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6. Reflections on Quiver Flags 65

6.1. Reflections and Quiver Flags . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2. Reflections and Hall Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3. DynkinCase .................................. 75

7. Extended Dynkin Case 77

7.1. BasicResults .................................. 77

7.2. BasisofPBW-Type .............................. 80

7.3. Example..................................... 89

Contents

A. Fitting Ideals 93

B. Tensor Algebras 95

Bibliography 99

Index 103

1. Introduction

The representation theory of quivers has its origin in a paper by P. Gabriel [Gab72],

who showed that a connected quiver is of finite representation type if and only if the

underlying graph is a Dynkin diagram of type A,Dor E. In doing so, he observed that

there is a strong connection to the theory of Lie algebras. Namely, there is a bijection

between isomorphism classes of indecomposable representations of the quiver and the

set of positive roots of the associated complex Lie algebra. In [BGP73] I. Bernšte˘ın, I.

Gel0fand and V. Ponomarev gave a more direct proof of this result. More precisely, they

used sequences of reflection functors to obtain all indecomposable representations in a

similar way to how all roots are obtained by applying reflections in the Weyl group to

the simple roots.

P. Donovan and M. Freislich [DF73] and independently L. Nazarova [Naz73] extended

this work to cover quivers of extended Dynkin, or affine, type. They described the set

of isomorphism classes of indecomposables, therefore showing that these quivers are of

tame representation type. A unified approach, which can also be used for species of

Dynkin or affine type, can be found in [DR74]. Again there is a connection with the

root systems of affine Kac-Moody Lie algebras, namely the dimension vectors of the

indecomposables are exactly the positive roots. This does not extend to a bijection with

the isomorphism classes, since for each imaginary root there is a continuous family of

indecomposables.

V. Kac [Kac80] proved that this correspondence holds in general. That is, for a fi-

nite quiver without vertex loops the set of dimension vectors of indecomposables over

an algebraically closed field is precisely the set of positive roots of the corresponding

(symmetric) Kac-Moody Lie algebra. Finally, A. Hubery [Hub04] established this cor-

respondence in the case of species and general Kac-Moody Lie algebras.

These results hint towards a deep connection between the category of representations

of a quiver and the corresponding Kac-Moody Lie algebra. This was further strengthened

by the theory of Ringel-Hall algebras. The Hall algebra appeared at first in the work of

E. Steinitz [Ste01] and afterwards in the work of P. Hall [Hal59]. C. M. Ringel [Rin90a]

generalised this construction to obtain an associative algebra structure on the Q-vector

space HK(Q)with basis the isomorphism classes of representations of Qover a finite

field K, the Ringel-Hall algebra. The structure constants are given basically by counting

numbers of extensions.

The whole Ringel-Hall algebra is generally too complicated, and therefore one intro-

duces the Q-subalgebra CK(Q)generated by the isomorphism classes of simple represen-

tations without self-extensions. There is a generic version Cq(Q), a Q(q)-algebra, such

that specialising qto |K|recovers CK(Q). C.M. Ringel [Rin90a], for the Dynkin case,

and later J. Green [Gre95], for the general case, showed that, after twisting the multi-

plication with the Euler form of Q, the generic composition algebra Cq(Q)is isomorphic

to the positive part of the quantised enveloping algebra of the Kac-Moody Lie algebra

1. Introduction

corresponding to Q.

When doing calculations in the Hall algebra one often has to decide which representa-

tions are an extension of two other fixed representations. If we fix two representations,

then there is the easiest extension, the direct sum. For the Dynkin case there is also

the most complicated extension, the generic extension. If we work over an algebraically

closed field, then the closure of the orbit of the generic extension contains all other ex-

tensions. M. Reineke [Rei01] was the first to notice that the multiplication by taking

generic extensions is associative. We therefore obtain a monoid structure on the set of

isomorphism classes of representations.

If we work over a non-Dynkin quiver, then, in general, generic extensions do not exist

anymore. M. Reineke [Rei02] showed that one can solve this problem by, instead of

taking individual representations, taking irreducible closed subvarieties of the represen-

tation variety to obtain a similar result. The multiplication is then given by taking all

possible extensions. Again, this yields a monoid, the generic extension monoid M(Q).

Similarly to the Hall algebra, it is in general too complicated, so one restricts itself to the

submonoid generated by the orbits of simple representations without self-extensions, the

composition monoid CM(Q). The elements of the composition monoid are the varieties

consisting of representations having a composition series with prescribed composition

factors in prescribed order.

For the generic composition algebra viewed as a Q(v)-algebra with v2=qthe (twisted)

quantum Serre relations are defining as shown in [Rin96]. M. Reineke showed that the

quantum Serre relations specialised to q= 0 hold in the composition monoid. They are

in general not defining any more if we specialise qto 0in the composition algebra. But

nonetheless one can conjecture, as M. Reineke did in [Rei01] and [Rei02], that there is a

homomorphism of Q-algebras

Φ: C0(Q)→QCM(Q)

sending simples to simples and therefore being automatically surjective.

The first step in this direction was done by A. Hubery [Hub05] showing that for the

Kronecker quiver

• •

Φis a homomorphism of Q-algebras with non-trivial kernel. He did this by calculating

defining relations for C0(K)and CM(K). He also was able to give generators for the

kernel of Φ.

The main aim of this thesis is to extend this result to the Dynkin and extended Dynkin

case. More precisely we show that if Qis a Dynkin quiver or an oriented cycle, then

C0(Q)and QCM(Q)are isomorphic. Then we prove that, if Qis an acyclic, extended

Dynkin quiver, there is such a morphism Φand the kernel of this morphism is given by

the same relations as A. Hubery gave for the Kronecker quiver.

In order to do this we need to calculate many of the structure coefficients in the com-

position algebra. These are given by counting points of quiver flags of a representation

M, i.e. increasing sequences of subrepresentations of fixed dimension vectors, over finite

fields.

For the oriented cycle we show directly that the coefficients at q= 0 are only one or

zero. As an immediate consequence we obtain the desired isomorphism.

To obtain results for the Dynkin and extended Dynkin case we first develop a frame-

work for applying reflection functors to quiver flags. Having done so, we can immediately

show that for the Dynkin case the coefficients at q= 0 are also one or zero and by this

we obtain the desired isomorphism.

In the extended Dynkin case there will be coefficients which are not equal to one

or zero. Therefore, the proof becomes more involved, but by using the framework of

reflection functors on flags it is possible to obtain the morphism. Along the way we

obtain a basis of PBW-type for C0(Q), a normal form for elements of CM(Q)and show

that the second one does not depend on the choice of the algebraically closed field,

making geometric arguments possible.

The varieties which appear, namely quiver Grassmannians and quiver flags, are of

interest of their own. They are interesting projective varieties to study and their Euler

characteristics give coefficients in the cluster algebra as shown in [CC06]. In general,

they are neither smooth nor irreducible or even reduced. We show that under some extra

conditions they are smooth and irreducible. We then use this information to calculate

the constant coefficient of the counting polynomial if it exists. Moreover, if we have a

counting polynomial, we show that the Euler characteristic is positive for rigid modules,

and this implies a certain positivity result for the associated cluster algebra. This works

in the Dynkin and the extended Dynkin case without using G. Lusztig’s interpretation

in terms of perverse sheaves, and therefore gives an independent proof of positivity.

Outline

This thesis is organised as follows: In chapter 2 we recall basic facts about the represen-

tation theory of quivers and related topics. In chapter 3 we show, by a direct calculation

of some Hall polynomials, that the Hall algebra of a cyclic quiver at q= 0 is isomorphic

to the generic extension monoid. Then, in chapter 4, we develop normal forms in terms

of Schur roots for the composition monoid of an extended Dynkin quiver, showing that

it is independent of the choice of the algebraically closed field. These were the direct

results. In chapter 5 we focus our attention on the geometry of quiver flags. We prove

a dimension estimate, generalising the one of M. Reineke, and then show that in cer-

tain cases the quiver flag varieties are smooth and irreducible. If one additionally has a

counting polynomial, we use this to deduce that the constant coefficient is one modulo

qand the Euler characteristic is positive. In chapter 6 we develop the aforementioned

calculus of reflections on quiver flags, culminating in the proof that the composition

algebra of a Dynkin quiver at q= 0 is isomorphic to the composition monoid. Finally, in

chapter 7 we show the result on the composition algebra at q= 0 and the composition

monoid for the extended Dynkin case.

Acknowledgements

First of all, I wish to express my sincere thanks to my supervisors Andrew Hubery

and Henning Krause. In particular, I would like to thank Andrew for coming up with

the topic of the thesis, for always answering my questions and for being open for my

ideas and Henning for his continuing support and his encouragement. I also thank the

representation theory group in Paderborn and their various guests for their mathematical

and non-mathematical support. In particular, I am grateful to Karsten Dietrich, Claudia

Köhler and Torsten Wedhorn for proofreading parts of this thesis. I thank the IRTG

“Geometry and Analysis of Symmetries” for providing financial support during these

three years and the DAAD for funding my research stay in Lyon which Philippe Caldero

made so enjoyable. Last but not least, I would like to thank my parents and Marta for

their unconditional support.

2. Preliminaries

In the beginning the Universe

was created. This has made a lot

of people very angry and has

been widely regarded as a bad

move.

(Douglas Adams)

2.1. Quivers, Path Algebras and Root Systems

For standard notations and results about quivers we refer the reader to [Rin84]. A

quiver Q= (Q0, Q1, s, t)is a directed graph with a set of vertices Q0, a set of arrows

Q1and maps s, t:Q1→Q0, sending an arrow to its starting respectively terminating

vertex. In particular, we write α:s(α)→t(α)for an α∈Q1. A quiver Qis finite

if Q0and Q1are finite sets. In the following, all quivers will be finite. A quiver is

connected if its underlying graph is connected. For a quiver Q= (Q0, Q1, s, t)we define

Qop := (Q0, Q1, t, s)as the quiver with all arrows reversed.

Apath of length r≥0in Qis a sequence of arrows ξ=α1α2· · · αrsuch that

t(αi) = s(αi+1)for all 1≤i<r. We write t(ξ) := t(αr)and s(ξ) := s(α1). Pictorially,

if ij=s(αj) = t(αj−1), then

i1i2i3irir+1.

ξ:α1α2αr

Clearly, the paths of length one are exactly the arrows of Q. For each i∈Q0there is

the trivial path iof length zero starting and terminating in the vertex i. We denote by

Q(i, j)the set of paths starting at the vertex iand terminating at the vertex j.

If Ris a ring, then the path algebra RQ has as basis the set of paths and the

multiplication ξ·ζis given by the concatenation of paths if t(ξ) = s(ζ)or zero otherwise.

In particular, the iare pairwise orthogonal idempotents of RQ, that is ij=δiji. The

path algebra is obviously an associative, unital algebra with unit 1 = Pi∈Q0i.

Let Qrdenote the set of paths of length r. This extends the notation of vertices Q0

and arrows Q1. Then

RQ =M

r≥0

RQr,

where RQris the free R-module with basis the elements of Qr. By construction,

RQrRQs=RQr+s,

thus RQ is an N-graded R-algebra.

From now on let Kbe a field. We have the following.

2. Preliminaries

Lemma 2.1. The iform a complete set of pairwise inequivalent orthogonal primitive

idempotents in KQ. In particular, KQ0=Qi∈Q0Kiis a semisimple algebra and the

modules iKQ are pairwise non-isomorphic indecomposable projective modules.

The root lattice ZQ0is the free abelian group on elements ifor i∈Q0. We define

a partial order on ZQ0by d=Pidii≥0if and only if di≥0for all i∈Q0. An

element d∈ZQ0is called a dimension vector. We endow ZQ0with a bilinear form

h · ,· iQdefined by

hd , e iQ:= X

i∈Q0

diei−X

α:i→j∈Q1

diej.

This form is generally called the Euler form or the Ringel form. We also define its

symmetrisation

(d , e)Q:= hd , e iQ+he , d iQ.

For each vertex a∈Q0we have the reflection

σa:ZQ0→ZQ0

d7→ d−(d , a)a.

If Qhas no loop at a, one easily checks that σ2

ad=d. By definition, the Weyl group

Wis the group generated by the simple reflections σa, which are those reflections corre-

sponding to vertices a∈Q0without loops. Clearly, (·,·)is W-invariant.

We also define reflections on the quiver itself. The quiver σaQis obtained from Qby

reversing all arrows ending or starting in a. If α:a→jis an arrow in Q1, then we call

α∗:j→athe arrow in the other direction and analogously for α:i→a. We have that

(σaQ)0=Q0, therefore we can regard σadas a dimension vector on σaQ. Obviously,

σ2

aQ=Qif we identify α∗∗ with α(and we will do this in the remainder).

A vertex i∈Q0is called a sink (resp. source) if there are no arrows starting (resp.

terminating) in i. An ordering (i1, . . . , in)of the vertices of Qis called admissible, if ip

is a sink in σip−1· · · σi1Qfor each 1≤p≤n. Note that there is an admissible ordering

if and only if Qhas no oriented cycles. Such a quiver will be called acyclic.

Example 2.2.Let

Then

σ1Q=

α∗

β∗

γ∗

and σ2Q=

α∗

Let d= (2,1,1,1). Then σ1d= (2,1,1,1) −(2 + 2 −(1 + 1 + 1))1= (1,1,1,1) and

σ2d= (2,1,1,1) −(1 + 1 −2)2= (2,1,1,1). We have that (1,2,3,4) is an admissible

ordering of Q.

We can define the set ∆⊂ZQ0of roots of Qcombinatorially as follows. We have the

2.2. Representations of Quivers

set of simple roots

Π := {i|i∈Q0,no loop at i}.

The fundamental region is

F:= {d > 0|(d , i)≤0for all i∈Πand suppdconnected },

where suppdis the support of d, i.e. the full subquiver of Qon the vertices isuch that

di6= 0.

Remark 2.3.The fundamental region may be empty.

The sets of real and imaginary roots are now

∆re := W · Πand ∆im := ±W · F.

We define the set of roots ∆ := ∆re ∪∆im. Moreover, each root is either positive or

negative and we write ∆+for the set of positive roots. Let dbe a root. We note that

hd , d i= 1 if dis real and hd , d i ≤ 0if dis imaginary. We call a root disotropic if

hd , d i= 0.

2.2. Representations of Quivers

For notation and results about representations of quivers and algebras we refer to [Rin84]

and for the geometric aspects to [Kac80] and [Sch92]. Let Qbe a quiver and Kbe a

field. A K-representation Mof Qis given by finite dimensional K-vector spaces Mi

for each i∈Q0and K-linear maps Mα:Mi→Mjfor each α∈Q1. If Mand N

are K-representations of Q, then a morphism f:M→Nis given by K-linear maps

fi:Mi→Nifor each i∈Q0such that the following diagram commutes

Mα//



NiNα//Nj

for all α:i→j∈Q1.

For each vertex i∈Q0there is a simple K-representation Sigiven by setting (Si)i:=

K,(Si)j:= 0 for i6=j∈Q0and (Si)α:= 0 for all α∈Q1.

AK-representation Mis called nilpotent if it has a filtration by semisimples involving

only the simples Si,i∈Q0. If not otherwise stated, every representation will assumed

to be nilpotent.

The direct sum of two representations is given by (X⊕Y)i:= Xi⊕Yiand (X⊕

Y)α:= Xα⊕Yαand a representation is called indecomposable if it is non-zero and

not isomorphic to a proper direct sum of two representations. We obtain an additive

category rep(Q, K)in which the Krull-Remak-Schmidt theorem holds.

Theorem 2.4. Every representation is isomorphic to a direct sum of indecomposable

representations and the isomorphism classes and multiplicities are uniquely determined.

2. Preliminaries

The dimension vector of Mis defined by

dimKM:= X

i∈Q0

dimKMii∈ZQ0.

By fixing bases, we see that the representations of dimension vector d(probably also

non-nilpotent) are parametrised by the affine space

RepQ(d, K) = Rep(d) := M

α:i→j

HomK(Kdi, Kdj).

In the following we identify each point x∈RepQ(d, K)with the corresponding K-

representation Mx∈rep(Q, K)and write directly M∈RepQ(d, K).

The group

GLd= GLd(K) := Y

i∈Q0

GL(Kdi)

acts on RepQ(d, K)by conjugation: given g= (gi)∈GLdand M∈RepQ(d, K)we

define (g·M)α:= gjMαg−1

ifor each α:i→j∈Q1, i.e. making the following diagram

commute

KdiMα//



Kdj



Kdi(g·M)α

//Kdj.

There is a 1-1 correspondence between the GLd-orbits in RepQ(d, K)and isomorphism

classes of K-representations of Qof dimension vector d. Denote the orbit of a represen-

tation Munder this action by OM.

AK-representation Xgives rise to a KQ-module M:= LXiwhere the action of

α:i→jis given by ιjXαπi,ιj:Xj→LXk=Mdenoting the canonical inclusion

and πi:M=LXk→Xithe canonical projection. Vice versa, each KQ-module M

gives rise to a K-representation Xwith Xi:= Miand Xαis the restriction of the

multiplication with α:i→jto the domain Xiand the codomain Xj. This yields an

equivalence of modKQ with rep(Q, K). Therefore, rep(Q, K)is abelian and we can

speak of kernel, cokernel, image and exactness. Vector space duality D= HomK(−, K)

gives a duality D: modKQ →mod KQop.

We will use the following notations for a K-algebra Λand two finite dimensional

Λ-modules Mand N:

•(M, N)Λ:= HomΛ(M, N),

•(M, N)i

Λ:= Exti

Λ(M, N),

•[M, N]Λ:= dimKHomΛ(M, N),

•[M, N]i

Λ:= dimKExti

Λ(M, N).

Note that [M, N]0= [M, N]and (M, N)0= (M, N). If Qis a quiver, we define

(M, N)Q:= (M, N)KQ if Mand Nare K-representations and similarly for the other

2.3. Reflection Functors

notations. For dimension vectors dand ewe denote by homKQ(d, e)the minimal value

of [M, N]Qfor M∈RepQ(d, K)and N∈RepQ(e, K)and similarly for exti

KQ(d, e).

More generally, if Aand Bare subsets of RepQ(d, K)and RepQ(e, K)respectively, then

hom(A,B)and exti(A,B)are defined analogously. Whenever the algebra, the field or

the quiver is clear from the context, we omit them from the notation. If our algebra is

hereditary we denote Ext1by Ext.

We can generalise the notion of Euler form to K-algebras Λof finite global dimension.

Namely, for two Λ-modules Mand Nwe define

hM , N iΛ:= X

i≥0

(−1)i[M, N]i.

By appendix B we obtain that KQ is hereditary and that for two K-representations M

and Nof Qwe have that

hM , N iQ=hM , N iKQ = [M, N]−[M, N]1.

2.3. Reflection Functors

The main reference for this section is [Rin84]. For a nice introduction see [Kra07]. If a

is a sink of Q, we define for each K-representation Mof Qthe homomorphism

φM

a:M

α:j→a

(Mα)

−−−−→ Ma.

Dually, if bis a source of Q, we define

φM

b:Mb

(Mα)

−−−−→ M

α:b→j

Mj.

Note that DφDM

a=φM

aand that da−rank φX

a= dim Hom(X, Sa)for aa sink of Qand

a representation Xof dimension vector d.

We define a pair of reflection functors S+

aand S−

b. To this end we fix a K-

representation Mof Qof dimension vector d.

(1) If the vertex ais a sink of Qwe construct

S+

a: rep(Q, K)→rep(σaQ, K)

as follows. We define S+

aM:= Nby letting Ni:= Mifor a vertex i6=aand

letting Nabe the kernel of the map ΦM

a. Denote by ι: KerφM

a→LMjthe

canonical inclusion and by πi:LMj→Mithe canonical projection. Then, for

each α:i→awe let Nα∗: KerφM

a→Mibe the composition πi◦ιmaking the

2. Preliminaries

following diagram commute

0−−−−→ KerφM

a−−−−→

ιLMj−−−−→ Ma



yNα∗



yπi

MiMi.

We obtain a K-representation Nof σaQ. We call this representation S+

aM.

Let s:= da−rank φM

athe codimension of Im φM

ain Ma. Obviously, dim(S+

aM)a=

eawhere

ea=X

α:i→a

di−rank ΦM

a=da−(d, a) + s=da−s−(d−sa, a).

Therefore, dim S+

aM=σa(d−sa) = σa(d) + sa.

For a morphism f= (fi): X→Ywe obtain a morphism S+

af:S+

aX→S+

aYby

letting (S+

af)i=fifor i6=aand letting (S+

af)abe the map induced on the kernels

of ΦX

arespectively ΦY

0→X1→X2→X3→0

is a short exact sequence in rep(Q, K), then we obtain an exact sequence

0→S+

aX1→S+

aX2→S+

aX3→Ss

a→0

in rep(σaQ, K)where s=s1−s2+s3with si= dim Xi

a−rank ΦXi

(2) If the vertex bis a source of Q, we construct

S−

b: rep(Q, K)→rep(σbQ, K)

dually.

Let s:= db−rank φM

bthe dimension of kerφM

b⊂Mb. We have that dim(S−

bM)b=

ebwhere

eb=X

α:b→i

di−rank φM

b=db−(d, b) + s=db−s−(d−sb, b).

Therefore, dim S−

bM=σb(d−sb) = σb(d) + sb.

0→X1→X2→X3→0

is a short exact sequence in rep(Q, K), then we obtain an exact sequence

0→Ss

b→S−

bX1→S−

bX2→S−

bX3→0

in rep(σbQ, K)where s=s1−s2+s3and si= dim Xi

b−rank ΦXi

2.3. Reflection Functors

For a sink aof Qwe have that (S−

a, S+

a)is a pair of adjoint functors and that S+

ais left

exact and S−

ais right exact. There is a natural monomorphism ιa,M :S−

aS+

aM→Mfor

M∈rep(Q, K)and a natural epimorphism πa,N :N→S+

aS−

aNfor N∈rep(σaQ, K).

We have the following lemma.

Lemma 2.5. Let abe a sink and Xan indecomposable representation of Q. Then, the

following are equivalent:

1. XSa.

2. S+

aXis indecomposable.

3. S+

aX6= 0.

4. S−

aS+

aX∼

=Xvia the natural inclusion.

5. The map ΦX

ais an epimorphism.

6. σa(dimX)>0.

7. dimS+

aX=σa(dimX).

For any admissible sequence of sinks w= (i1, . . . , ir)define

S+

w:= S+

ir◦ · · · ◦ S+

i1.

If w= (i1, . . . , in)is an admissible ordering of Q, we define the Coxeter functors

C+:= S+

in◦ · · · ◦ S+

i1C−:= S−

i1◦ · · · ◦ S−

in.

Since both reverse each arrow of Qexactly twice, these are endofunctors of rep(Q, K).

Neither functor depends on the choice of the admissible ordering.

AK-representation Pis projective if and only if C+P= 0. Dually, a K-representation

Iis injective if and only if C−I= 0. This motivates the nomenclature in the following

definition.

Definition 2.6. An indecomposable K-representation Mof Qis called preprojective

if (C+)rM= 0 for some r≥0and preinjective if (C−)rM= 0 for some r≥0. If M

is neither preprojective nor preinjective, then Mis called regular.

An arbitrary K-representation is called preprojective (or preinjective or regular) if it

is isomorphic to a direct sum of indecomposable preprojective (or preinjective or regular)

representations.

Let Qbe a connected, acyclic non-Dynkin quiver as introduced in the next sec-

tion. Then an arbitrary representation Mcan be decomposed uniquely into a direct

sum M∼

=MP⊕MR⊕MIsuch that MPis preprojective, MRis regular and MIis

preinjective. This means that the set of isomorphism classes of indecomposable K-

representations indrep(Q, K)decomposes into a disjoint union P∪R∪I where Pdenotes

the set of indecomposable preprojective, Rdenotes the set of indecomposable regular

2. Preliminaries

and Idenotes the set of indecomposable preinjective K-representations. Note that

Hom(R, P) = Hom(I, P ) = Hom(I, R) = 0 and Ext(P, R) = Ext(P, I) = Ext(R, I) = 0

for all representations P∈ P,R∈ R and I∈ I. There is a partial order on P∪I given

by MNfor M, N ∈ P ∪ I if and only if there is a sequence of non-zero morphisms

M→M1→M2→ · · · → Nfor some indecomposable representations Mi∈ P ∪ I. Note

that, if Ext1(M, N)6= 0 for two indecomposables M, N ∈ P ∪ I, then N≺M.

Fix an admissible ordering (a1, . . . , an)of the vertices of Q. For each indecomposable

preprojective representation Mthere is a natural number r=kn+sfor some k≥0and

0≤s<nsuch that

S+

as· · · S+

a1(C+)kM= 0.

Let σ(M)be the minimal such number. Note that σ(M)depends on the choice of the

admissible ordering. We have the following easy

Lemma 2.7. The map

σ:P → N

M7→ σ(M)

is an injection respecting the partial order on P, i.e. for all M, N ∈ P with MN

we have that σ(M)≤σ(N)for the natural ordering on N.

Proof. Choosing an admissible ordering is the same as refining the partial order on

the projectives in rep(Q, K)to a total order. Since the Auslander-Reiten quiver is just a

number of copies of the quiver given by the projectives with morphisms only going from

left to right, the lemma follows.

2.4. Dynkin and Extended Dynkin Quivers

The references for this chapter are [Rin84] and [Gab72]. The representation type of a

quiver Qis governed by its underlying graph Γ. Note that the symmetric bilinear form

(·,·)Qdepends only on Γ.

Theorem 2.8. Suppose Γis connected.

1. Γis Dynkin if and only if (·,·)Qis positive definite. By definition the (simply-

laced) Dynkin diagrams are:

An:• • • ... •(n≥1) E6:• • • • •

•

Dn:• • .... •

•



•?

?(n≥4) E7:• • • • • •

•

(n=number of vertices)E8:• • • • • • •

•

2.4. Dynkin and Extended Dynkin Quivers

2. Γis extended Dynkin if and only if (·,·)Qis positive semi-definite and not

positive definite. We have that rad(·,·)Q=Zδfor some dimension vector δ. By

definition the extended Dynkin diagrams are as below. We have marked each vertex

iwith the value of δi. Note that δis sincere and δ≥0.

An:1



..... 1

1



. . . . .

(n≥0)

Dn:2 2 ... 2



1



(n≥4)

E6:1 2 3 2 1

(n+ 1 = number of vertices)

E7:1 2 3 4 3 2 1

E8:2 4 6 5 4 3 2 1

Note that e

A0has one vertex and one loop and e

A1has two vertices joined by two

edges.

3. Otherwise, there is a dimension vector d≥0with (d , d)<0and (d , i)≤0for

all i.

A vertex iof an extended Dynkin graph Γwith δi= 1 is called an extending vertex.

Removing ifrom Γgives the corresponding Dynkin diagram. We call a quiver of Dynkin

type if it is a disjoint union of quivers with underlying graphs being Dynkin. Similarly

for extended Dynkin.

Gabriel’s theorem states the following.

Theorem 2.9. If Qis a connected, acyclic quiver with underlying graph Γ, then there

are only finitely many isomorphism classes of indecomposable representations of Qif and

only if Γis Dynkin. In this case the assignment X7→ dim Xinduces a bijection between

the isomorphism classes of indecomposable representations and the positive roots ∆+.

More generally, Kac’s theorem [Kac80] yields that the map X7→ dim Xis a surjection

from isomorphism classes of indecomposable representations to the set of positive roots

∆+.

Now let us review the representation theory of connected, acyclic, extended Dynkin

quivers, the main references being [DR74] and [Rin76]. Let Qbe an acyclic extended

2. Preliminaries

Dynkin quiver and Ka field. Let ΓAR be the Auslander-Reiten quiver of Q. Vertices of

ΓAR correspond to indecomposable representations and arrows to irreducible morphisms.

We have a decomposition of ΓAR into the preprojective, the preinjective and the regular

part. The set of regular representations addRis an abelian subcategory of the category

of all representations. We say that Mis regular simple, of regular length k, .. . if M

is simple, of length k, . . . in add R. The regular part is the disjoint union of pairwise

orthogonal tubes Txfor each scheme theoretic, closed point x∈P1

K= ProjK[X, Y ]in

such a way that each regular simple representation Rin the tube labelled by xsatisfies

End(R)∼

=κ(x),κ(x)denoting the residue field at the point x. The degree of xis defined

to be [κ(x) : K]as degree of field extensions. We have that each regular simple Rin the

tube Txis of dimension vector dim R= (deg x)δ. Note that if Kis algebraically closed,

then all tubes have degree one. We define rank Txto be the number of regular simples

in the tube Tx. A tube Txis called homogeneous if rank Tx= 1 and inhomogeneous

otherwise. Each tube Txis equivalent to the category of (nilpotent) representations of

an oriented cycle with rank Txvertices. For a K-representation Mwe define Mxto be

the summand of Mliving in the tube x∈P1

K. Finally, we may also assume that the non-

homogeneous tubes are labelled by some subset of {0,1,∞}, whereas the homogeneous

tubes are labelled by the points of the scheme HK=HZ⊗Kfor some open integral

subscheme HZ⊂P1

For the representation theory of Qthe defect ∂plays a major role. For a dimension

vector dwe define

∂d := hδ , d iQ.

If Mis an indecomposable representation, then

•Mis preprojective if and only if ∂d < 0;

•Mis preinjective if and only if ∂d > 0;

•Mis regular if and only if ∂d = 0.

2.5. Canonical Decomposition

In this section we recall the canonical decomposition of a dimension vector dintroduced

by V. Kac [Kac80, Kac82] and examined by A. Schofield [Sch92]. Let Kbe an alge-

braically closed field and let Qbe a quiver. A dimension vector dis called a Schur

root if the general representation of dimension vector dhas endomorphism ring K, i.e.

there is an open non-empty subset U⊂RepQ(d)such that for all M∈Uwe have that

End(U)∼

=K. We have that all preprojective and preinjective roots are real Schur roots.

A decomposition d=Pfiis called the canonical decomposition if and only if a

general representation of dimension vector dis isomorphic to the direct sum of indecom-

posable representations of dimension vectors f1, . . . , fr.

This is closely related to the question if all representations of a dimension vector d+e

have a subrepresentation of dimension vector d. A. Schofield [Sch92] proved the following

theorem for charK= 0 and W. Crawley-Boevey [CB96c] for charKarbitrary.

Theorem 2.10. Every representation of dimension vector d+ehas a subrepresentation

of dimension vector dif and only if ext(d, e) = 0. Moreover, the numbers ext(d, e)are

2.6. Degenerations

given combinatorially and therefore are independent of the algebraically closed base field.

Remark 2.11.Note that, if Kis not algebraically closed, then the theorem fails. Let Qbe

the Kronecker quiver. Then there is a regular simple K-representation Mof dimension

vector (2,2). The representation Mdoes not have a subrepresentation of dimension

vector (1,1) even though ext((1,1),(1,1)) = 0 as one easily sees by taking two regular

simple representations of dimension vector (1,1) living in different tubes.

By using this theorem, A. Schofield showed the following.

Theorem 2.12. A decomposition d=Pfiis the canonical decomposition if and only

if each fiis a Schur root and ext(fi, fj)=0for all i6=j. Moreover, the decomposition

is independent of the field.

We often write the canonical decomposition as d=Prifisuch that ri>0and the fi

are pairwise different Schur roots. Note that for every Schur root fiappearing in this

sum with multiplicity ri>1we have that ext(fi, fi) = 0. Moreover, for each i6=jwe

have that ext(fi, fj)=0.

We have the following.

Lemma 2.13. Let d1, . . . , dkbe dimension vectors such that ext(di, dj)=0for all i6=j.

Then the canonical decomposition of Pdiis a refinement of the decomposition given by

Pdi, i.e. there are Schur roots fi

jsuch that Pjfi

jis the canonical decomposition of di

and Pi,j fi

jis the canonical decomposition of Pdi.

Proof. Let fi

jbe Schur roots such that for each iwe have that Pjfi

jis the canonical

decomposition of di. A general representation of dimension vector diis therefore a

direct sum of indecomposable representations of dimension vectors fi

j. By A. Schofield

[Sch92, Theorem 3.4] a general representation of dimension vector Pdiis a direct sum of

representations of dimension vector di. Therefore, a general representation of dimension

vector Pdiis a direct sum of indecomposable representations of dimension vectors fi

Hence, Pfi

jis the canonical decomposition of Pdi.

2.6. Degenerations

The basic reference for the following is [Bon96]. Let Kbe an algebraically closed field.

Let Mand Nbe K-representations of a quiver Q. We say that M≤deg Nif ON⊆ OM.

The degeneration order ≤deg on the representation variety has been investigated by

various authors.

C. Riedtmann [Rie86] and G. Zwara [Zwa00] were able to describe ≤deg in purely

representation theoretic terms.

Theorem 2.14. Let Mand Nbe K-representations of Q. The following are equivalent:

•M≤deg N.

•There is representation Yand a short exact sequence

0→Y→Y⊕M→N→0.

2. Preliminaries

•There is representation Zand a short exact sequence

0→N→M⊕Z→Z→0.

There are two other orders on the isomorphism classes of representations. Firstly,

the Hom-order ≤is given by M≤Nif [M, X]≤[N, X]for all representations X. A

result of Auslander yields that ≤is a partial order. Moreover, M≤Nif and only if

[X, M]≤[X, N]for all representations X. Therefore, the definition of ≤is symmetric.

Secondly, we have the Ext-order ≤ext. We define M≤ext Nif there are representations

Mi,Uiand Qiand short exact sequences

0→Ui→Mi→Qi→0

for 0≤i<rsuch that M=M0,Mi+1 =Ui⊕Qiand N=Mr. This also yields a

partial order and we have the following.

Theorem 2.15 ([Bon96]).Let Mand Nbe K-representations. Then we have the

following implications:

M≤ext N⇒M≤deg N⇒M≤N.

It is interesting to investigate when the orders agree. We have the following theorem.

Theorem 2.16. If Qis Dynkin or extended Dynkin, then

M≤ext N⇔M≤deg N⇔M≤N.

This theorem is due to K. Bongartz [Bon96] for Dynkin quivers and to G. Zwara

[Zwa97, Zwa98] for extended Dynkin quivers.

If Kis not algebraically closed we have the obvious definitions of ≤and ≤ext. We

define M≤deg Nif M⊗K≤deg N⊗K,Kbeing the algebraic closure of K.

2.7. Segre Classes

For this chapter the main references are [BD01], [Bon89] and [Hub07]. Let

•

be the Jordan quiver, having one vertex and one loop, and let Kbe a field.

In order to describe the isomorphism classes of Qwe use partitions. A partition

λ= (λ1≥λ2≥ · · · ≥ λr)of nis a sequence of decreasing positive integers λi∈Nsuch

that n=Pr

i=1 λi. Each λiis called a part of size λiof λ. The length l(λ)of a partition

λis the number of parts. An alternative way to write a partition is in exponential form

λ= (1l12l2· · · mlm)for an integer m∈Nand non-negative integers l1, . . . , lm∈N. This

means that λhas exactly liparts of size i. Therefore, λis a partition of Pii·li.

2.7. Segre Classes

Example 2.17.For example λ= (3,3,2) is a partition of 8.λcan be written in expo-

nential form as (102132). The partition λhas two parts of size 3and one part of size

Now we come back to the Jordan quiver Q. Note that RepQ(d, K) = End(Kd)for a

dimension d. We know that KQ =K[t]is a principal ideal domain, so finite dimensional

modules are described by their elementary divisors. In particular, we can associate to a

finite dimensional module Mthe data {(λ1, p1),...,(λr, pr)}consisting of partitions λi

and distinct monic irreducible polynomials pi∈K[t]such that

M∼

=MM(λi, pi)

where, for a partition λ= (1l1· · · nln)and a monic irreducible polynomial p, we write

M(λ, p) = M

ik[t]/(pi)li.

Clearly, the primes pidepend on the field, but we can partition the set of isomorphism

classes by considering only their degrees. We call this the Segre decomposition. More

precisely, a Segre symbol is a multiset σ={(λ1, d1),...,(λr, dr)}of pairs (λ, d)con-

sisting of a partition λand a positive integer d. The corresponding Segre class S(σ, K)

consists of all modules of isomorphism type {(λ1, p1),...,(λr, pr)}where the pi∈K[t]

are distinct monic irreducible polynomials with deg pi=di.

Theorem 2.18 ([BD01]).Let Kbe an algebraically closed field. Then the Segre classes

stratify the variety RepQ(d, K) = End(Kd)into smooth, irreducible, GLd(K)-stable sub-

varieties, each admitting a smooth and rational geometric quotient. Moreover, the sta-

bilisers of any two matrices in the same Segre class are conjugate inside GLd(K).

Let Qbe a connected, acyclic, extended Dynkin quiver and Ka field. The indecompos-

able preprojective and preinjective representations are all exceptional, as are the regular

simple representations in the non-homogeneous tubes. Hence, the isomorphism class of

a representation without homogeneous regular summands can be described combinato-

rially, whereas homogeneous regular representations are determined by pairs consisting

of a partition together with a point of the scheme HK.

Now we can define the decomposition of K. Bongartz and D. Dudek [BD01] or more

precisely the generalisation given by A. Hubery [Hub07]. A decomposition symbol

is a pair α= (µ, σ)such that µspecifies a representation without homogeneous regular

summands and σ={(λ1, d1),...,(λr, dr)}is a Segre symbol. Given a decomposition

symbol α= (µ, σ)and a field K, we define the decomposition class SQ(α, K) =

S(α, K)to be the set of representations Xof the form X∼

=M(µ, K)⊕Rwhere M(µ, K)

is the K-representation determined by µand

R=R(λ1, x1)⊕ · · · ⊕ R(λr, xr)

for some distinct points x1, . . . , xr∈HKsuch that deg xi=diand R(λ, x)is the

representation associated to the partition λliving in the tube Txof rank one. We call µ

the discrete part and σthe continuous part of α. If σ=∅, we say that αis discrete.

2. Preliminaries

Let abe a sink of Qand let αbe a decomposition symbol. Let M∈ SQ(α, K)and let β

be the decomposition symbol of S+

aM. Then S+

agives a bijection between isomorphism

classes in SQ(α, K)and isomorphism classes in SσaQ(β, K). This is obvious since S+

ais

additive and gives a bijection from RQto RσaQwith inverse S−

awhich deals with the

continuous part and for the discrete part there is only one choice.

2.8. Ringel-Hall Algebra

For this section the main reference is [Rin90a] or the lecture notes [Sch06] and [Hub].

Let Abe a skeletally small abelian category such that for two objects M, N ∈ A the

sets Exti

A(M, N)are finite for all i≥0. Such a category is called finitary. For three

objects M, N, X ∈ A define

MN := #{U≤X|U∼

=N, X/U ∼

=M}.

Let H(A)be the Q-vector space with basis u[X]where [X]is the isomorphism class of

X. For convenience we write uXinstead of u[X]. Define

uMuN:= X

MN uX.

Then (H(A),+,)is an associative Q-algebra with unit 1 = u0, the Ringel-Hall alge-

bra or just Hall algebra. The composition algebra is the subalgebra C(A)of H(A)

generated by the simple objects without self-extensions. Note that H(A)and C(A)are

naturally graded by the Grothendieck group of A.

Let Qbe a quiver and qa prime power. Then the finite dimensional Fq-representations

of Qform a finitary abelian category. Define

HFq(Q) := H(rep(Q, Fq))

and

CFq(Q) := C(rep(Q, Fq)).

Note that HFq(Q)and CFq(Q)are naturally graded by dimension vector. We set ui:= uSi

for each i∈Q0. If w= (i1, . . . , ir)is a word in vertices of Q, we define

uw:= ui1 · · ·  uir.

By definition, there are numbers FX

wfor each Fq-representation Xof Qsuch that

uw=X

wuX.

2.9. Generic Composition Algebra

For the following let Qbe a finite quiver with vertex set Q0and arrow set Q1. We

consider only finite dimensional and nilpotent representations and modules. We define

the generic composition algebra via Hall polynomials.

2.9. Generic Composition Algebra

The main references for this section are C. M. Ringel [Rin90a, Rin90b] for the rep-

resentation finite case, J. Guo [Guo95] and C. M. Ringel [Rin93] for the oriented cycle

and A. Hubery [Hub07] for the acyclic, extended Dynkin case.

Now let Qbe Dynkin or extended Dynkin. Then, there is a partition of the isomor-

phism classes of representations of each dimension vector given by some combinatorial

set Σ = SdΣdsuch that Σdis finite. This means, for each dimension vector dand each

field Kwe have subsets S(α, K)⊂RepQ(d, K)for each α∈Σdsuch that

RepQ(d, K) = [

α∈Σd

S(α, K)

and that there are polynomials aα, nα∈Q[q]such that for each finite field Kwe have

that aα(|K|) = #Aut(M)for all M∈ S(α, K)and that #[S(α, K)] = nα(|K|).

Following [Hub07], we say that Hall polynomials exist with respect to this decom-

position if there are polynomials fγ

αβ ∈Q[q]such that for each finite field Kwe have

that X

[A]∈[S(α,K)]

[B]∈[S(β,K)]

AB =fγ

αβ(|K|)for all C∈ S(γ, K)

and further that

nγ(|K|)fγ

αβ(q) = nα(|K|)X

[B]∈[S(β,K)]

[C]∈[S(γ,K)]

AB for all A∈ S(α, K)

=nβ(|K|)X

[A]∈[S(α,K)]

[C]∈[S(γ,K)]

AB for all B∈ S(β, K).

For the Dynkin case the isomorphism classes of indecomposable representations are

in bijection with the positive roots ∆+of the corresponding Lie algebra, which are

independent of the field K. Therefore, we can take Σto be the set of all α: ∆+→N

with finite support. For each such αand any field Kthere is a unique isomorphism class

such that the indecomposable representation corresponding to the root ρappears α(ρ)

times. Choose an element M(α, K)of this isomorphism class. C. M. Ringel showed that

there are polynomials fγ

αβ(q)∈Z[q]such that for each finite field Kwe have

FM(γ,K)

M(α,K)M(β,K)=fγ

αβ(|K|).

If Qis an oriented cycle, we have that Hall polynomials exist if we take Σdto be the

set of isomorphism classes of representations of dimension vector dand S(α, K)then to

be the set of representations Mlying in this isomorphism class. See chapter 3 for more

on this.

For Qan acyclic, extended Dynkin quiver we have to take decomposition symbols as

defined in section 2.7. See [Hub07].

Note that for all three choices of Σwe have that each simple representation Sigives

a class S(α, K)for some α∈Σand that the classes [S(α, K)] are stable under S+

aand

S−

a, meaning that for an M∈ SQ(α, K)such that S+

aM∈ SσaQ(β, K)we have that S+

2. Preliminaries

induces a bijection from [SQ(α, K)] to [SσaQ(β, K)].

We define the generic Hall algebra Hq(Q)to be the free Q[q]-module with basis

{uα|α∈Σ}

and multiplication given by

uαuβ=X

fγ

αβ(q)uγ.

The generic composition algebra Cq(Q)is then the subalgebra of Hq(Q)generated by

the simple representations without self-extensions, i.e. the elements of Σcorresponding

to those. If the quiver is fixed, then we often write Hqand Cqinstead of Hq(Q)and

Cq(Q). Note that the definition of the generic Hall algebra seems to be non-standard.

Again, Hq(Q)and Cq(Q)are graded by dimension vector.

We can then specialise Cq(Q)to any n∈Qby evaluating the structure constants given

by the polynomials fγ

αβ at n. We call this algebra Cn(Q). By definition, we have that

Cq0(Q)∼

=CFq0(Q)

for any prime power q0∈N. In the following, we identify these two algebras.

For a word win vertices of the quiver Qwe define uw∈ Cq(Q)in the obvious way.

There are polynomials fα

wfor each class α∈Σsuch that

uw=X

fα

wuα.

Note that we have for each finite field Kand each X∈ S(α, K)that FX

w=fα

w(|K|).

This thesis mainly deals with the algebras C0(Q)for QDynkin or extended Dynkin.

Let Q(v)be the function field in one variable. Consider it as a Q[q]-algebra via v2=q.

Denote the twisted composition algebra by

Cq(Q) := Cq(Q)⊗Q[q]Q(v)

with the multiplication given by

uα∗uβ:= vhα,βiuαuβ.

Let gbe the Kac-Moody Lie algebra associated to the Cartan datum given by Q(or

by (., .)Q). C. M. Ringel [Rin96], J. Green [Gre95] and G. Lusztig [Lus93] showed that

Cq(Q)∼

=U+

q(g)where U+

q(g)is the positive part of the quantised enveloping algebra of

G. Lusztig [Lus93] showed that the Q(v)-dimension of the d-th graded part U+

q(g)d

is equal to the Q-dimension of the d-th graded part U+(Q)dof the positive part of

the universal enveloping algebra of g. Therefore, dim e

Cq(Q)dis equal to dim U+(g)d.

The twist does not change the dimension of the graded parts and Cq(Q)is free as a

Q[q]-module, as a submodule of a free module. Therefore, we obtain that the Q[q]-

rank of the d-th graded part Cq(Q)dof the generic composition algebra is equal to

dim U+(g)d. Finally, specialisation does not change the rank of a free module, therefore

2.10. Generic Extension Monoid

dim C0(Q)d= dim U+(g)d.

Note that dim U+(g)dis the number of ways of writing das a sum of positive roots

with multiplicities, in the sense that we count each positive root fwith multiplicity

dim gfof the corresponding root space.

2.10. Generic Extension Monoid

In [Rei02], M. Reineke introduced the generic extension monoid and the composition

monoid. We recall briefly how this is done. Fix an algebraically closed field K.

For two arbitrary sets U⊆Rep(d), V ⊆Rep(e)we define

E(U, V ) := {M∈Rep(d+e)| ∃ A∈U, B ∈Vand a short exact sequence

0→B→M→A→0}.

The multiplication on closed irreducible GLd-stable respectively GLe-stable subvari-

eties A ⊆ Rep(d),B ⊆ Rep(e)is defined as:

A∗B:= E(A,B).

M. Reineke [Rei02] showed that then A∗B is again closed, irreducible and GLd+e-stable.

Moreover, he showed that ∗is associative and has a unit: Rep(0). The set

M(Q) := a

dnA ⊂ RepQ(d)Ais a closed, irreducible and GLd-stable subvariety o

with this multiplication is therefore a monoid, the generic extension monoid M(Q).

The composition monoid CM(Q)is the submonoid generated by the orbits of simple

representations without self-extensions. Note that M(Q)and CM(Q)are graded by

dimension vector.

For any word w= (N1, . . . , Nr)in semisimples we define Aw:= ON1∗ · · · ∗ ONr. This

is an element of M(Q). If N1, . . . , Nrare simple, then Aw∈ CM(Q). Since there is a

simple for each vertex of a quiver, we can similarly define Awfor wa word in the vertices

of Q. Note that the definition of Awmakes sense even if Kis not algebraically closed,

at least as a set. Therefore, we obtain a set Awconsisting of all K-representations of Q

having a filtration of type w.

If Ais a closed, irreducible, GLd-stable subvariety of some Rep(d), define [A] :=

A/GLdas the set of orbits. Hence elements of [A]correspond to isomorphism classes in

More generally, if Ais any subset of Rep(d, K)for any field Kdenote by [A]the set

of isomorphism classes in A.

If Mis any monoid and Rany ring we denote by RMthe monoid algebra of Mgiven

formal R-linear combinations of elements of Mand the obvious multiplication induced

by the multiplications in Mand R.

2. Preliminaries

2.11. Quiver Flags and Quiver Grassmannians

Let Mbe a finite dimensional K-vector space and let

d= (d0, d1, . . . , dν)

be a (ν+1)-tuple of integers. Then one usually defines FlKM

dto be the set of sequences

of subspaces 0 = U0⊂U1⊂ · · · ⊂ Uν=Msuch that dimK(Ui) = di. If this set is

non-empty, we have that d0= 0,di≤di+1 and dν= dim M. We call such a sequence a

filtration of Mor of dimension dim M. If we have the first two conditions we call da

filtration.

Let Λbe a K-algebra and Ma finite dimensional Λ-module. Then it is natural to

define

FlΛ M

d!:= (U0⊂U1⊂ · · · ⊂ Uν∈FlK M

d!Uiis a Λ-submodule of M).

Let Qbe a quiver, Ka field and Λ := KQ. Each module Udecomposes, as a K-vector

space, into the direct sum of Ui,i∈Q0. The flag of submodules FlΛM

dis therefore

the disjoint union into open subvarieties

FlQ M

d!:= (U0⊂U1⊂ · · · ⊂ Uν∈FlΛ M

d!dim(Ui) = di)

such that Pj∈Q0di

j=di. We call a sequence of dimension vectors d= (d0, . . . , dν)a

filtration of a K-representation M, or of a dimension vector dim M, if d0= 0,di≤di+1

and dν= dim M. If we only have the first two conditions we call da filtration.

If dis a dimension vector and MaK-representation of dimension vector d+e, then

we define the quiver Grassmannian as

GrQ M

d!:= FlQ M

(0, d, d +e)!.

For an algebraically closed field Kand a filtration dwe denote by Ad(K)the closed,

irreducible and GLdν-stable subvariety of Rep(dν, K)consisting of all K-representations

Msuch that FlQM

dis non-empty. For the geometric statements see chapter 5.

2.12. Main Theorems

We want to investigate the relationship between the composition algebra at q= 0 and the

composition monoid. For Qa Dynkin or extended Dynkin quiver we obtain a complete

answer.

2.12. Main Theorems

Theorem 1. Let Qbe a Dynkin quiver or an oriented cycle. Then the map

Ψ: QM(Q)→ H0(Q)

A 7→ X

M∈[A]

is an isomorphism of graded Q-algebras.

Proof. This is the combination of theorems 3.12 and 6.24.

Theorem 2. Let Qbe an acyclic, extended Dynkin quiver. Then the map

Φ: C0(Q)→QCM(Q)

sending uSito OSiis a graded Q-algebra homomorphism with kernel generated by the

relations

(uδ)r=urδ ∀r∈N.

Proof. This is corollary 7.16.

Moreover, we obtain a basis of PBW-type for the generic composition algebra.

On a more geometric side, we can prove that the quiver flag variety is irreducible

under certain circumstances.

Theorem 3. Let Kbe an algebraically closed field. Assume that there is an M∈ Ad(K)

such that dim Ext1

Q(M, M) = codim Ad(K). Then FlQM

dis smooth and irreducible.

Proof. This is theorem 5.34.

Remark 2.19.We prove this statement for arbitrary fields K.

3. Cyclic Quiver Case

Don’t disturb my circles.

(Archimedes)

Consider the cyclic quiver Q=Cnof type e

An(all arrows in one direction):

Let Kbe a field and Λ := KCn. We want to examine rep(Cn, K), the category of

nilpotent K-representations over Cn. Note that rep(Cn, K)is equivalent to modΛ, the

category of nilpotent Λ-modules. Let S0, S1, . . . , Sndenote the simple representations

in rep(Cn, K)corresponding to the vertices of Q. We have that Ext(Si, Si+1)6= 0 (now

and in the remainder of this section always count modulo n+1). For generalities on the

cyclic quiver see [DD05].

Up to isomorphism there is exactly one indecomposable K-representation Si[l]of

length lwith socle Si. For a partition λ= (λ1≥λ2≥ · · · ≥ λr)we set

Si[λ] :=

k=1

Si[λk].

The set of isomorphism classes of representations in rep(Cn, K)is therefore in bijection

with

Π := n(π(0), . . . , π(n))π(i)is a partition ∀io,

where each partition π(i)describes the indecomposable summands with socle Si. Hall

polynomials exist with respect to Π, as shown in [Guo95].

Now let Mbe an arbitrary representation of isomorphism class π∈Π. Then we denote

by uMor uπits symbol in the generic Hall algebra Hq(Cn). Most times we will write a

partition λof m0in exponential form, i.e. λ= (1s12s2· · · msm)such that m0=Pm

i=1 sii.

When calculating Hall polynomials, certain quantum numbers appear. Let Rbe some

commutative ring and let q∈R. Usually Rwill be Q[q], the polynomial ring in one

3. Cyclic Quiver Case

variable. We define for r, n ∈N,0≤r≤n:

[n]q:= 1 + q+· · · +qn−1

[n]q! :=

i=1

[i]q

r#q

:= [n]q!

[r]q![n−r]q!.

Obviously, "n

r#0

= 1.

Now we want to calculate the Hall polynomial between direct sums of a simple and

two arbitrary representations. We use the following well-known lemma which describes

the structure of the Grassmannian of subspaces.

Lemma 3.1. Let V, W be two K-vector spaces and denote by πW:V⊕W→Wthe

second projection. Then the map

φV,W : Gr V⊕W

r!→a

s+t=r

Gr V

s!×Gr W

U7→ (U∩V, πW(U))

is surjective and the fibre over (A, B)is isomorphic to HomK(B, W/A).

Proof. See for example [CC06] for a generalisation to quiver representations.

Let λ= (1s12s2· · · tst)be a partition and let Sbe a simple representation. Then

socS[λ]∼

=SPsiand every inclusion of Sinto S[λ]corresponds to a one dimensional

subspace of socS[λ]. More generally, every inclusion of Sk,k∈N, into S[λ]corresponds

to a kdimensional subspace of socS[λ].

Lemma 3.2. Let λ= (1s12s2· · · msm)be a partition, Sa simple representation, M:=

S[m]smand N:= Lm−1

i=1 S[i]si. So X:= S[λ] = M⊕N. Fix r∈N. Let Ube a subspace

of socXof dimension r. Then X/U ∼

=M/A ⊕N/B where (A, B) = φsoc M,soc N(U)and

the fibre of φsoc M,soc Nover (A, B)has dimension (sm−dim A)dim B.

Proof. For any i≤jthere is an inclusion S[i],→S[j]which restricts to an isomor-

phism of the one dimensional socles. Therefore, each vector space homomorphism

socN→soc Mextends to a homomorphism N→Mof K-representations. Note that

this extension is not unique.

We have U∈Gr soc M⊕soc N

r. Let (A, B) := φsoc M,soc N(U). We can choose com-

plements A0of Ain socMand B0of Bin soc N. Then, by the preceding lemma, Uis

given by a homomorphism g∈HomK(B, A0)and we denote its extension by 0on Aand

B0by g0: socN→soc M. Let ˜g:N→Mbe any homomorphism of K-representations

induced by g0. Now the following diagram commutes.

A⊕B idA0

0 idB

0 0 !//A⊕A0⊕B⊕B0//M⊕N

idM˜g

0 idN

∼

A⊕B idA0

0 0

0 idB

0 0 !//A⊕A0⊕B⊕B0//





idA0 0 0

0 idA0g0

0 idB0

0 0 0 idB0



∼

M⊕N

Hence the cokernel of the map in the top row is the same as the cokernel of the map in

the bottom row, therefore (M⊕N)/U ∼

=M/A ⊕N/B.

Proposition 3.3. Let s= (s1, . . . , sm)and t= (t1, . . . , tm)∈Nmsuch that ti≤si. Let

Sand Tbe simples such that Ext(T, S)6= 0. Set

λ:= (1s12s2· · · msm)µ:= (1s1−t12s2−t2· · · msm−tm)

ν:= (1t22t3· · · (m−1)tm)r:= Xti

X:= S[λ]Y(s, t) := S[µ]⊕T[ν].

Then

Y(s,t)Sr(q) = Y

i"si

ti#q

qPj<i tj(si−ti).

Moreover, all quotients of Xby Srare of the form of Y(s, t)for some choice of t.

Proof. We proof this by induction on m. For m= 0 the claims are true. Now let m > 0.

Then X∼

=M⊕N, where M:= S[m]smand N:= Lm−1

i=1 S[i]si, hence, without loss of

generality, we can assume X=M⊕N. Now, a subrepresentation of Xisomorphic to

Sris given by an element U∈Gr soc M⊕soc N

r. Let (A, B) := φsoc M,soc N(U)as in the

previous lemma. Then X/U ∼

=M/A ⊕N/B. This yields by induction that all quotients

are of the desired form. One has that M/A ∼

=S[m]sm−dim A⊕T[m−1]dim Aand no

quotient of Nby Bhas a summand isomorphic to S[m]. Therefore, if X/U ∼

=Y(s, t),

dim Ahas to be equal to tm.

Set s0:= (s1, s2, . . . , sm−1),t0:= (t1, t2, . . . , tm−1)and analogous as before λ0,µ0,ν0,

X0,Y(s0, t0)and r0. The number of subspaces Usuch that φsoc M,soc N(U)=(A, B)is

given by q(sm−dim A) dim B=q(sm−tm)Pj<m tj. Therefore, we have:

Y(s,t)Sr(q) = q(sm−tm)Pj<m tj"sm

tm#q

fX0

Y(s0,t0)Sr0(q).

By induction this is equal to

=q(sm−tm)Pj<m tj"sm

tm#q

m−1

i=1 "si

ti#q

qPj<i tj(si−ti)=

i=1 "si

ti#q

qPj<i tj(si−ti).

3. Cyclic Quiver Case

Definition 3.4. Let Sand Tbe simples such that Ext(T, S)6= 0. Let X=S[λ]

for a partition λ= (1s12s2· · · msm)and k∈Nsuch that k≤Pisi=l(λ). Define

t= (t1, . . . , tm)recursively by

tm:= min{sm, k},

ti:= min



si, k −X

j>i

tj



for all 0≤i≤m−1.

We define

Q(X, Sk) := Y(s, t) =

i=1

S[i]si−ti⊕

m−1

i=1

T[i]tm+1 .

More generally, if X=Ln

i=0 Si[π(i)]for a π∈Πand N=LSki

ifor some ki∈N

such that ki≤l(π(i)), define

Q(X, N) :=

i=0

Q(Si[π(i)], Ski

i).

We will prove in corollary 3.10 that Q(X, N)is maximal with respect to the degener-

ation order, hence the quotient of Xby Nwith the smallest orbit dimension.

We obtain the following.

Corollary 3.5. Let X=S[λ]for some λ= (1s12s2· · · msm). Let Mbe a quotient of X

by Sk,k∈Nwith k≤l(λ). Then

MSk(0) = (1if M∼

=Q(X, Sk)

0otherwise.

Proof. By proposition 3.3, M=Y(s, t)for some t= (t1, . . . , tm)∈Nmwith ti≤siand

Pti=k. We have that "si

ti#0

= 1.

Therefore, fX

MSk(0) 6= 0 if and only if Pj<i tj(si−ti) = 0 for all 1≤i≤m. This is the

case if either si=tior that for all j < i we have that tj= 0. This is exactly the way

we chose Q(X, Sk)in the definition and in this case we have fX

MSk(0) = 1.

Now we are able to describe the coefficients modulo qfor an extension with a semisim-

ple.

Lemma 3.6. Let N=Ln

i=0 Ski

i,ki∈N, be a semisimple representation. Let X=

i=0 Si[π(i)],π∈Π, be arbitrary and let M∈rep(Cn, K)be a quotient of Xby N.

Then

MN (0) = (1if M∼

=Q(X, N)

0otherwise.

Proof. Since Hom(Ski

i, Sj[π(j)]) = 0 for i6=jevery short exact sequence 0→N→X→

M→0is the direct sum of short exact sequences of the form

0→Ski

i→Xi→Mi→0,

where Xi=Si[π(i)]and for some representations Misuch that Ln

i=0 Mi∼

=M. So we

have

MN =X

(M0,...,Mn):

LMi∼

i=0

fXi

MiSki

where Xi:= Si[π(i)]. But now fXi

MiSki

(0) is non-zero if and only if

Mi∼

=Q(Xi, Ski

by lemma 3.6. Moreover, the same lemma yields fXi

Q(Xi,Ski

i)Ski

(0) = 1. Hence we are

done.

Lemma 3.7. Let w= (N1, N2, . . . , Nr)be a word in semisimples. Let M∈rep(Q, K)

and M0∈rep(Q, L)for two arbitrary fields Kand Lsuch that Mand M0are of iso-

morphism type π∈Π. Then Mhas a filtration of type wif and only M0has. In other

words, the sets [Aw]can be considered as subsets of Πand do not depend on the field we

are working over.

Proof. We prove the claim by induction on r. If r= 1, then the claim is trivial. Now

let r > 1and set w0:= (N1, N2, . . . , Nr−1). A representation Mhas a filtration of type

wif and only if it has a subrepresentation Uisomorphic to Nrsuch that the quotient

M/U ∈ Aw0. Since the polynomials of proposition 3.3 have positive coefficients and by

a similar argumentation as in lemma 3.6 we have that the possible isomorphism classes

of the quotients only depend on the isomorphism classes of Mand Nr. By induction we

have that the isomorphism classes in Aw0do not depend on the field. This finishes the

proof.

Now we show that Q(X, N)is maximal with respect to the degeneration order among

the quotients of Xby N.

Lemma 3.8. Let N,X1, X2be arbitrary representations, f1:N→X1and f2:N→X2

two injections. Moreover, let g:X1→X2be a morphism such that gf1=f2.

Then we have a short exact sequence

0//X1//X2⊕X1/f1(N)//X2/f2(N)//0

and therefore X2⊕(X1/f1(N)) ≤deg X1⊕(X2/f2(N)).

Proof. We construct an extension degeneration. Let π1:X1→X1/f1(N)and π2:X2→

X2/f2(N)be the canonical projections and let g:X1/f1(N)→X2/f2(N)be the map

3. Cyclic Quiver Case

induced by π2g, which exists since f1(N)⊆ker(π2g). By construction we have the

following commutative diagram:

0−−−−→ Nf1

−−−−→ X1

π1

−−−−→ X1/f1(N)−−−−→ 0





yg



y¯g

0−−−−→ Nf2

−−−−→ X2

π2

−−−−→ X2/f2(N)−−−−→ 0.

This is a pullback, therefore

0//X1

π1)//X2⊕X1/f1(N)(π2−g)

//X2/f2(N)//0

is the desired short exact sequence.

Corollary 3.9. Let M,Nbe arbitrary representations and let h:N→Mbe an injec-

tion. Let g∈End(M)be any endomorphism such that gh(N)∼

=N. Then

M/h(N)≤deg M/gh(N).

Proof. From lemma 3.8 we have an extension of the form

0→M→M⊕M/h(N)→M/gh(N)→0.

By a result of C. Riedtmann [Rie86, prop. 4.3] this yields that

M/h(N)≤deg M/gh(N).

Corollary 3.10. Let X, M ∈rep(Cn, K)and Na semisimple representation such that

there is a short exact sequence 0→N→X→M→0. Then Mdegenerates to

Q(X, N).

Proof. As before, the short exact sequence is a direct sum of short exact sequences of

the form

0→Ski

i→Xi→Mi→0,

where X∼

=LXiand M∼

=LMifor some Xiwith socle in addSi.

It is obviously enough to show the claim for every short exact sequence of this form.

We can therefore assume that X=S[λ],M=S[µ]⊕T[ν]and N=Skfor k∈N,Sand

Tsimples such that Ext(T, S)6= 0 and partitions λ= (λ1,· · · , λm),µand ν. Let Ube

the k-dimensional subspace of socX=LsocS[λi]induced by the inclusion of Skinto

X. We have that X/U ∼

=M. We want to apply the previous corollary to X, so we have

to find an endomorphism fwhich maps Uto U0, where U0is a subspace of socXsuch

that X/U0∼

=Q(X, Sk).

By repeated application of the construction used in the proof of lemma 3.2 we obtain

an automorphism φof Xsuch that

φ(U) =

j=1

socS[λij]⊂X,

for a sequence of integers 1≤i1< i2<· · · < ik≤m, since socS[λij]∼

=k. Note that

X/Φ(U)∼

=M, we can therefore assume that U=φ(U). Let

U0:=

j=1

socS[λj]⊂X.

By definition we have that X/U0=Q(X, Sk).

For i≤jthere is an injection ϕi,j from S[i]to S[j]which restricts to an isomorphism

on the one dimensional socles. Therefore, the map

j=1

S[λij]→

j=1

S[λj]

(x1, . . . , xk)7→ (ϕλi1,λ1(x1), . . . , ϕλik,λk(xk))

induces an isomorphism from Uto U0and can be extended by 0on the remaining

summands to an endomorphism of X. By applying the previous lemma we obtain that

M=X/U ≤deg X/U0=Q(X, Sk),

and this yields the desired result.

Now we are able to describe the monomial elements of H0(Cn), the generic Hall algebra

of the cyclic quiver specialised at q= 0.

Theorem 3.11. Let w= (N1, . . . , Nr)be a sequence of semisimple representations.

Then

uw:= u[N1]u[N2] · · ·  u[Nr]=X

[M]∈[Aw]

u[M]∈ H0(Cn).

Proof. By lemma 3.7 the expression on the right hand side is well-defined. We prove

the theorem by induction on r. For r= 1 the statement is trivial. Now let w0=

(N1, . . . , Nr−1). We need to show that P[M]∈[Aw]u[M]=uw∈ H0(Cn). First, it is obvi-

ous that every isomorphism class appearing in uw=uw0u[Nr]with non-zero coefficient

is in [Aw].

Now let Xbe in Aw. We have to show that the coefficient of u[X]in uw=uw0u[Nr]∈

H0(Cn)is 1. This coefficient is

[M]∈[Aw0]

MNr(0).

3. Cyclic Quiver Case

By lemma 3.6 this sum is 1if and only if [Q(X, Nr)] is in [Aw0]and 0otherwise. So it

remains to show that [Q(X, Nr)] ∈[Aw0]. By lemma 3.7 we can assume that we work over

an algebraically closed field Kto check this. Now, since X∈ Aw, we know that there is

at least one M∈ Aw0such that there is a short exact sequence 0→Nr→X→M→0.

Moreover, Aw0is closed as a variety over K. Via lemma 3.10 we know that Mdegenerates

to Q(X, Nr)and therefore Q(X, Nr)∈ Aw0.

By using this we obtain the following result.

Theorem 3.12. The map

Ψ: QM(Cn)→ H0(Cn)

A 7→ X

[M]∈[A]

u[M]

is an isomorphism of graded rings.

Proof. The sets ONfor Nsemisimple generate M(Cn)and therefore we can apply lemma

3.7 to show that the map is well-defined. The map Ψobviously maps ONto u[N]for N

a semisimple representation. Applying theorem 3.11 yields that Ψis a homomorphism

of rings. More precisely, if wand vare words in semisimples, we have that

Ψ(Aw∗ Av) = Ψ(Awv) = X

[M]∈[Awv]

u[M]=uwv

=uwuv=X

[M]∈[Aw]

u[M]X

[M]∈[Av]

u[M]= Ψ(Aw)Ψ(Av).

Moreover, Ψis obviously a graded homomorphism.

Now we show that Ψis an isomorphism by showing that it is an isomorphism of

Q-vector spaces for every graded component. Note that every element A∈M(Cn)is

given by the orbit closure of some representation since we are only considering nilpotent

representations. Therefore, the Q-dimension of the graded components of M(Cn)and

H0(Cn)agree, since both are equal to the number of isomorphism classes of the given

dimension vector.

The degeneration order is a partial order on the isomorphism classes of representations

and we have that Ψ(OX) = uX+PX<degYuY. Therefore, on the graded components,

Ψis given by a unipotent matrix and so is an isomorphism.

Corollary 3.13. The isomorphism Ψrestricts to an isomorphism

QCM(Cn)∼

=C0(Cn).

Proof. Everything follows from the theorem since Ψmaps OSito u[Si]and these are the

generators of QCM(Cn)respectively C0(Cn).

Remark 3.14.We call an isomorphism class π= (π(0), . . . , π(n))∈Πseparated if for each

k≥1there is some ik∈ {0, . . . , n}such that π(ik)

j6=kfor all j≥1. In other words,

π(ik)has no part of size k. We denote by Πsthe set of all separated isomorphism classes.

We call a representation separated if its isomorphism class is separated. B. Deng and J.

Du [DD05, Theorem 4.1] show that

CM(Cn) = nOMMis separated o.

Therefore, an element of the generic extension monoid is in the composition monoid if

and only if it is the orbit closure of a separated representation.

4. Composition Monoid

I love hearing my relations

abused. It is the only thing that

makes me put up with them at

all.

(Oscar Wilde)

4.1. Generalities

Let Qbe an arbitrary quiver and Kan algebraically closed field. M. Reineke proved the

following.

Theorem 4.1 ([Rei02]).Let char K= 0. Let A ⊆ Rep(d, K),B ⊆ Rep(e, K)be closed

irreducible subvarieties. Then we have:

codim(A∗B)≤codim A+ codim B+ ext(B,A).

If ext(A,B)=0, or A= Rep(d)and B= Rep(e), equality holds.

Remark 4.2.A similar theorem holds for elements of the composition monoid in arbitrary

characteristic. We prove this in chapter 5.

From this we obtain the following.

Corollary 4.3. Let Mand Nbe representations such that [M, N]1= 0. Then

OM∗ ON=OM⊕N.

We can prove this corollary, without using the theorem, in arbitrary characteristic.

For this, we need the following.

Lemma 4.4. Let A,Bbe two sets in the generic extension monoid and let U⊆ A, V ⊆ B

be two open subsets contained in those. Then A∗B=E(U, V ).

Proof. To prove this we use the setup in [Rei02]. The subvariety U×Vis open, thus

dense in A×B. Let Z(U, V )be the set of x∈Rep(d+e)such that

xα= uαζα

0vα!

for every arrow α∈Q1, where u∈U, v ∈Vand ζarbitrary. Now, Z(U, V )is an

open subvariety of the irreducible variety Z(A,B), thus dense. Therefore, the GLd+e-

saturation of Z(U, V )is dense in the GLd+e-saturation of Z(A,B). But the former is

equal to E(U, V )whereas the latter is equal to A∗B, hence the claim follows.

4. Composition Monoid

Proof of corollary 4.3. We know that OMis open in OMand ONis open in ON. By

using lemma 4.4 we have OM∗ON=E(OM,ON). But E(OM,ON) = OM⊕Nsince every

extension is split and the corollary follows.

We have the following fundamental relation in CM(Q).

Proposition 4.5. Let dand ebe two dimension vectors such that ext(e, d)=0. Then

Rep(d)∗Rep(e) = Rep(d+e).

Note that ext(e, d)only depends on the dimension vectors and the quiver and not on the

field we are working over.

Proof. This is an immediate consequence of theorem 2.10.

We will deduce from this relation some relations on multiples of Schur roots.

Lemma 4.6. Let d,ebe two dimension vectors with ext(e, d)=0. Then

Rep(d)∗Rep(e) = Rep(r1f1)∗ · · · ∗ Rep(rlfl)

where Prifiis the canonical decomposition of d+e. Moreover, if ais any other dimen-

sion vector, then

ext(a, d) + ext(a, e)≥Xext(a, rifi)

and

ext(d, a) + ext(e, a)≥Xext(rifi, a).

Proof. Using the previous proposition we obtain Rep(d)∗Rep(e) = Rep(d+e).

Let a,band cbe three arbitrary dimension vectors. We have

ext(a, b) + ext(a, c)≥ext(a, b +c).

To see this take representations A∈Rep(a),B∈Rep(b)and C∈Rep(c)such that

[A, B]1= ext(a, b)and [A, C]1= ext(a, c). This is possible since the set of representa-

tions taking minimal Ext values is open in Rep(a). Now we have

ext(a, b) + ext(a, c)=[A, B]1+ [A, C]1= [A, B ⊕C]1≥ext(a, b +c).

Therefore, if ext(a, b) = 0 = ext(a, c)for three arbitrary dimension vectors a,band

c, then ext(a, b +c) = 0. If Prifiis the canonical decomposition of d+e, then, by

definition, ext(fi, fj)=0for all 1≤i6=j≤l. Therefore, we can iteratively apply the

previous proposition to obtain that

Rep(r1f1)∗ · · · ∗ Rep(rlfl) = Rep(d+e)

and this proves the first claim.

Now we prove the second claim. We can choose representations C0∈Rep(a)and

Ai∈Rep(rifi)such that

[C0, A1⊕ · · · ⊕ Al]1= ext(a, r1f1+· · · +rlfl) = ext(a, d +e),

4.2. Partial Normal Form in Terms of Schur Roots

since a general representation in Rep(d+e)will have a decomposition into summands

like this. So we have

ext(a, d) + ext(a, e)≥ext(a, d +e)

= ext(a, r1f1+· · · +rlfl)=[C0, A1⊕ · · · ⊕ Al]1

i=1

[C0, Ai]1≥

i=1

ext(a, rifi).

The other statement is proved dually.

4.2. Partial Normal Form in Terms of Schur Roots

For the following let Qbe a connected acyclic quiver. We say that a root dis prepro-

jective/regular/preinjective if one (and therefore all) indecomposable representation of

dimension vector dis. If Qis Dynkin, then all roots are preprojective and preinjective,

but for convenience we set them to be preprojective and not preinjective. We also choose

a total order ≺ton the preprojective and preinjective Schur roots refining the order ≺

on P ∪ I.

We define a new monoid SR(Q), the Schur root monoid. We will use it to obtain a

partial normal form in the composition monoid. For QDynkin we will show that it is

isomorphic to CM(Q). Moreover we will later show that the relations of SR(Q)also

hold in C0(Q)for Qan extended Dynkin quiver.

Definition 4.7. The monoid SR(Q)is given by the generators

{ {rd} | dis a Schur root, r ∈N}

and the relations

{sd}∗{te}={r1f1}∗···∗{rlfl}if ext(e, d)=0, d or ereal,(4.1)

where Prifiis the canonical decomposition of sd +te and

{sd}∗{te}={te}∗{sd}if ext(d, e) = ext(e, d)=0.(4.2)

If w= (i1, i2, . . . , ir)is a word in vertices of Qwe write

{w}:= {i1}∗···∗{ir} ∈ SR(Q).

Remark 4.8.Note that for real Schur roots dwe have that

{sd}∗{td}={(s+t)d}

by relation (4.1).

We have the following observation.

4. Composition Monoid

Proposition 4.9. The map

Θ: SR(Q)→ CM(Q)

{sd} 7→ Rep(sd)

is an epimorphism of monoids.

Proof. Since Qis acyclic we have that Rep(d)∈ CM(Q)for each dimension vector

d. The map is well-defined since the defining relations of SR(Q)hold in CM(Q)by

proposition 4.5 and lemma 4.6.

Definition 4.10. We say that an element of SR(Q)is in partial normal form if it is

equal to P ∗ R ∗ I where

P={r1d1}∗···∗{rldl}with ri>0, dipreprojective and di≺tdjfor all i<j;

R={s1e1}∗···∗{smem}with si>0, eiregular;

I={t1f1}∗···∗{tnfn}with ti>0, fipreinjective and fi≺tfjfor all i<j

where di, eiand fiare Schur roots.

Remark 4.11.Note that if M≺tN, for two indecomposable representations M, N ∈

P ∪ I, then [M, N]1= 0. Therefore, this is a similar partial normal form as in theorem

5.8 of [Rei02].

Lemma 4.12. Let d1, . . . , drbe Schur roots such that

d1or dris not regular,

ext(d1, dr)6= 0 and

ext(di, dj)=0 for all 1≤i<j≤rwith (i, j)6= (1, r).

Let s1, . . . , sr∈Nbe positive integers. Then there is a permutation πof {1, . . . , r}such

that such that

{s1d1}∗···∗{srdr}={sπ(1)dπ(1)}∗···∗{sπ(r)dπ(r)} ∈ SR(Q),

ext(dπ(i), dπ(j))=0 for all 1≤i<j≤rwith (i, j)6= (π−1(1), π−1(r)) and

π−1(r) = π−1(1) + 1.

In other words, we can interchange the roots with each other such that {s1d1}is next to

{srdr}.

Proof. We prove the claim by induction on r. If r= 2 we are done. Now assume r > 2.

If there exists a 1< l < r such that either ext(dl, di) = 0 for all i < l or ext(dj, dl) = 0

for all j > l, then we can use relation (4.2) to move {sldl}either to the left of {s1d1}or

to the right of {srdr}and we are done by induction.

Assume now that for each 1< l < r there is an i<lsuch that ext(dl, di)6= 0 and a

j > l such that ext(dj, dl)6= 0. We therefore have a sequence 1 = i0< i1<· · · < is=r

such that ext(dij+1 , dij)6= 0 for all 1≤j < s.

4.2. Partial Normal Form in Terms of Schur Roots

This yields a path in the Auslander-Reiten quiver from d1to dr.1Moreover, since

ext(d1, dr)6= 0, we have a path from drto d1. This yields a cycle in the Auslander-

Reiten quiver which involves at least one non-regular Schur root, a contradiction.

Theorem 4.13. Now let Xbe any element of SR(Q). Then Xcan be written in partial

normal form, i.e.

X={s1d1}∗···∗{srdr} ∈ SR(Q),

where the right hand side is in partial normal form.

Proof. By definition, X={s1d1}∗···∗{srdr}for some Schur roots di.

Let n:= Pi<j ext(sidi, sjdj). We proof the claim by induction on n. For n= 0 we

can reorder the roots in the desired way by using relation (4.2), because there are no

extensions between regular and preinjective, preprojective and preinjective and prepro-

jective and regular Schur roots. After reordering we probably have to use relation (4.1)

to obtain {rd} ∗ {sd}={(r+s)d}for every preprojective or preinjective Schur root d

to end up with an expression in partial normal form.

Let n≥1. If

X={s1d1}∗···∗{srdr}

cannot be reordered in the desired way by relation (4.2), then there exist 1≤i0< j0≤r

such that di0or dj0is not regular, ext(di0, dj0)6= 0 and ext(di, dj) = 0 for all i0≤i <

j≤j0with (i, j)6= (i0, j0). By lemma 4.12 we can assume that j0=i0+ 1.

Now we can apply relation (4.1) to obtain that

{si0di0}∗{sj0dj0}={t1e1}∗···∗{trer},

where Ptjejis the canonical decomposition of si0di0+sj0dj0. Replacing {si0di0} ∗

{sj0dj0}by {t1e1}∗···∗{trer}makes nsmaller, as the following calculation shows, and

we are done by induction.

i<j,{i,j}∩{i0,j0}=∅

ext(sidi, sjdj) + X

i<i0,j

ext(sidi, tjej) + X

j>j0,i

ext(tiei, sjdj)

4.12

≤X

i<j,{i,j}∩{i0,j0}=∅

ext(sidi, sjdj) + X

i<i0ext(sidi, si0di0) + ext(sidi, sj0dj0

j>j0ext(si0di0, sjdj) + ext(sj0dj0, sjdj)

i<j

ext(sidi, sjdj)−ext(si0di0, sj0dj0)< n.

As a corollary we obtain a partial normal form in CM(Q).

Corollary 4.14. Let Aw∈ CM(Q)for a word win vertices of Q. Then Awcan be

1More precisely, from the indecomposables corresponding to each.

4. Composition Monoid

written as P ∗ R ∗ I where

P= Rep(r1d1)∗ · · · ∗ Rep(rldl)with ri>0, dipreprojective and di≺tdj∀i<j;

R= Rep(s1e1)∗ · · · ∗ Rep(smem)with si>0, eiregular;

I= Rep(t1f1)∗ · · · ∗ Rep(tnfn)with ti>0, fipreinjective and fi≺tfj∀i < j.

Proof. By definition of the morphism Θ: SR(Q)→ CM(Q)we have that Θ({w}) = Aw.

Since {w}can be written in partial normal form and Θis a homomorphism sending {rd}

to Rep(rd)for each Schur root dand each r∈Nwe obtain the result.

Corollary 4.15. If Qis Dynkin, then

Θ: SR(Q)→ CM(Q)

{sd} 7→ Rep(sd)

is an isomorphism and the partial normal form is a normal form. In particular CM(Q)

is independent of the base field.

Proof. By definition, there are no preinjective or regular Schur roots, therefore each

element of SR(Q)can be written as {r1d1} ∗ · · · ∗ {rldl}with ri>0,dipreprojective

and di≺tdjfor all i<j. For each dithere is a unique indecomposable representation

Miwithout self-extensions such that OMi= Rep(di). Therefore, Rep(ridi) = OMri

i. By

the condition on the partial normal form we have that [Mi, Mj]1= 0 for all i<j. We

can therefore apply lemma 4.3 and obtain

Θ({r1d1}∗···∗{rldl}) = Rep(r1d1)∗ · · · ∗ Rep(rldl) = OMr1

1∗ · · · ∗ OMrl

l=OLMri

Since every representation in CM(Q)is uniquely given by the orbit closure of some

representation M, and the isomorphism class of Mis uniquely given by some Schur

roots with multiplicities we have that Θis bijective.

4.3. Extended Dynkin Case

Let Kbe an algebraically closed field and let Qbe a connected, acyclic, extended Dynkin

quiver. We show that CMK(Q)is a quotient of SR(Q)by two more classes of relations

which are independent of K. As a consequence we will obtain that CM(Q)does not

depend on the base field K.

For a connected, acyclic, extended Dynkin quiver the tubes are indexed by P1

K. There

is exactly one isotropic Schur root, δ. Let HK⊆P1

Kbe the set of indices of the ho-

mogeneous tubes, which is an open subset. For each x∈HKthere is a unique (up to

isomorphism) regular simple representation in the corresponding tube Tx. Let us call

this representation Rx.

Let x∈P1

Ksuch that Txis inhomogeneous. Denote by Πxthe set of isomorphism

classes in this tube. Since Txis equivalent to rep(Ck−1, K)where k= rank Txwe have

that

Πx∼

=nπ= (π(0), . . . , π(k−1))π(i)is a partition o.

4.3. Extended Dynkin Case

In the following we identify Πxwith this set.

K. Bongartz and D. Dudek show in [BD01] the following.

Theorem 4.16. Let Mbe a representation without homogeneous direct summands and

d= dim M+rδ for δbeing the isotropic Schur root and some r∈N. Then the set

D(M, r) := (N∈Rep(d)

N∼

=M⊕Lr

j=1 Rxjwith xj

pairwise different elements of HK)

is a smooth, locally closed subset of Rep(d).

Note that D(M, r)is the decomposition class corresponding to (µ, σ), where µis the

isomorphism class of Mand σ={((1),1),((1),1),...,((1),1)}.

First, we look at an inhomogeneous tube of rank kand use our result about the

composition monoid of a cyclic quiver. So let T1, . . . , Tkbe the regular simples of an in-

homogeneous tube. We know that dim Tjis a Schur root and OTjis open in Rep(dim Tj).

Moreover, the tube is closed under extensions. By work of K. Bongartz [Bon95] and G.

Zwara [Zwa98] we also know that the Hom-order and the Ext-order agree on rep(Q, K).

We have the following.

Lemma 4.17. Let Mand Nbe K-representations of Qin the same inhomogeneous

tube Txof rank k. Then M≤deg Nif and only if M0≤deg N0in rep(Ck−1, K), where

M0and N0are the images of Mand Nunder the equivalence Tx∼

=rep(Ck−1, K).

Proof. If M0≤deg N0, then this degeneration is given by successive extensions. These

can be transformed to successive extensions in rep(Q, K)to obtain a degeneration from

Mto N.

Now, if M≤deg N, then [M, B]≤[N, B]for all representations B∈rep(Q, K).

But then, by the equivalence, we also have [M0, B0]≤[N0, B0]for all representations

B0∈rep(Ck−1, K). Hence M0≤deg N0since in rep(Ck−1, K)the degeneration order

agrees with the Hom-order.

Theorem 4.18. Let Mand Nbe K-representations in an inhomogeneous tube Txof

rank k. Then there is a representation E∈ Txsuch that

OM∗ ON=OE,

i.e. for an inhomogeneous tube we have generic extensions. Moreover, the isomorphism

class π∈Πxof Eonly depends on the isomorphism classes π0, π00 ∈Πxof Mand N

and not on the field K.

Proof. Let M0, N0∈rep(Ck−1, K)be the images of Mand Nunder the equivalence

Tx∼

=rep(Ck−1, K). In rep(Ck−1, K)we have generic extensions. Therefore, there is an

E0∈rep(Ck−1, K)such that E0∈ E(OM0,ON0)and E0≤deg X0for all X0∈ E(OM0,ON0).

Let E∈rep(Q, K)be the representation corresponding to E0under the equivalence. By

lemma 4.17 E≤deg Xfor all X∈ E(OM,ON)since E(OM,ON)⊂ Tx. Therefore, OEis

dense in E(OM,ON). Now we can apply lemma 4.4:

OM∗ ON=E(OM,ON) = OE.

4. Composition Monoid

Moreover, the set of isomorphism classes [E(OM0,ON0)] ⊂Πxonly depends on the

isomorphism classes π0of M0and π00 of N0by lemma 3.7 since OM0and ON0are elements

of the generic extension monoid M(Ck−1). Therefore, the isomorphism class π∈Πx

of E0only depends on the isomorphism classes π0, π00 ∈Πxsince it is the isomorphism

class in [E(OM0,ON0)] with the smallest endomorphism ring dimension. Now πis by

definition the isomorphism class of Eand this proves the claim.

Corollary 4.19. Let e1, . . . , elbe regular Schur roots living in one inhomogeneous tube

Txand r1, . . . , rl∈N. Then there is a regular representation M∈ Txwhose isomorphism

class π∈Πxonly depends on r1e1, . . . , rleland not on Ksuch that

Rep(r1e1)∗ · · · ∗ Rep(rlel) = OM.

Proof. Each Rep(riei)is given by OMri

ifor some indecomposable K-representation Mi∈

Txwhose isomorphism class only depends on ei. Iteratively applying theorem 4.18 yields

the result.

We therefore have additional relations in CM(Q)which seem not to be a consequence

of the relations of proposition 4.5. It would be interesting to make this precise. We add

them to the defining relations of the Schur root monoid and call the resulting monoid

ER(Q).

Definition 4.20. We define ER(Q)as the quotient of SR(Q)by the following type of

relation. For each x∈P1

Ksuch that Txis inhomogeneous we identify

{r1d1}∗···∗{rldl}={s1e1}∗···∗{smem}(4.3)

if d1, . . . , dland e1, . . . , emlive in the inhomogeneous tube Txand there is a representation

M∈ Txsuch that

{r1d1}∗···∗{rldl}=OM={s1e1}∗···∗{smem}.

In this case we denote {r1d1} ∗ · · · ∗ {rldl}by {(π, x)},π∈Πs

xbeing the isomorphism

class of M. Note that for every π∈Πs

xwe have that {(π, x)} ∈ ER(Q)and that this

relation does not depend on K.

Remark 4.21.We will later show that, for Qacyclic, extended Dynkin, QER(Q)∼

=C0(Q).

Proposition 4.22. The morphism Θ: SR(Q)→ CM(Q)factors via ER(Q). We denote

the induced morphism ER(Q)→ CM(Q)again by Θ.

Proof. By corollary 4.19 the additional relations of ER(Q)hold in CM(Q)and the claim

follows.

Remark 4.23.Note that, for an inhomogeneous tube Tx,Θ({(π, x)}) = OMfor M∈ Tx

aK-representation of isomorphism class π∈Πs

4.3. Extended Dynkin Case

Lemma 4.24. Let x1, . . . , xr∈P1

Kbe the indices of the inhomogeneous tubes. Then

every element X ∈ ER(Q)can be written in the form X=P ∗ C1∗ · · · ∗ Cr∗ R ∗ I where

P={s1p1}∗···∗{skpk}with si>0, pipreprojective and pi≺tpjfor all i<j;

Ci={(π, xi)}for a π∈Πs

xi;

R={λ1δ}∗···∗{λlδ}with λ= (λ1≥ · · · ≥ λl)a partition;

I={t1q1}∗···∗{tmqm}with ti>0, qipreinjective and qi≺tqjfor all i < j.

Proof. By lemma 4.13 Xcan be written as P ∗ R0∗ I where Pand Iare already in the

desired form and R0={i1e1} ∗ · · · ∗ {inen}for some regular Schur roots e1, . . . , en. For

each jwe have that either ej∈ Txfor some x∈ {x1, . . . , xr}or that ej=δ. Since there

are no extensions between different tubes we can use relation (4.2) to reorder them in

ER(Q)in such a way that roots from the same inhomogeneous tube are next to each

other as are multiples of δ. Using relation (4.3) we obtain the desired form.

Remark 4.25.We will show in chapter 7 that this is a normal form.

We finally define the last monoid, g

ER(Q), which will be isomorphic to CM(Q).

Definition 4.26. We define g

ER(Q)as the quotient of ER(Q)by the relations

{sδ}∗{tδ}={(s+t)δ}for all s, t > 0.(4.4)

Proposition 4.27. The morphism Θ: ER(Q)→ CM(Q)induces a morphism

Θ: g

ER(Q)→ CM(Q).

Proof. Relation (4.4) holds in CM(Q)by proposition 4.5 and the claim follows.

Now we obtain a normal form in g

ER(Q).

Theorem 4.28. Every element X ∈ g

ER(Q)can be uniquely written as P ∗ C1∗. . . Cr∗

{lδ}∗I where

P={s1p1}∗···∗{skpk}with si>0, pipreprojective and pi≺tpjfor all i < j;

Ci={(πi, xi)}for a πi∈Πs

xi;

l≥0;

I={t1q1}∗···∗{tmqm}with ti>0, qipreinjective and qi≺tqjfor all i < j.

Moreover, Θ: g

ER(Q)→ CM(Q)is an isomorphism. Since the relations of g

ER(Q)do

not depend on Kwe have that CM(Q)is independent of K.

Proof. By using lemma 4.24 and relation (4.4) we have that every element can be written

in this way. We have to show that this form is unique. Let X=P ∗ C1∗ · · ·∗Cr∗{lδ}∗I

as above. Then for each ithere is a unique (up to isomorphism) indecomposable pre-

projective K-representation Piof dimension vector pi. We have that OPsi

i= Rep(sipi).

4. Composition Monoid

Dually, for each ithere is unique indecomposable preinjective K-representation Iiof

dimension vector qiand OIti

= Rep(tiqi). Finally, for each 1≤i≤rthere is a unique

K-representation Mi∈ Txiof isomorphism class πi. Let P:= LPsi

i,I:= LIti

iand

M:= LMi. By the Krull-Remak-Schmidt theorem P,C1, . . . , Crand Iare uniquely

determined by [P],[M]and [I].

By applying corollary 4.3 and using that Θis a homomorphism we have that

Θ(X) = Θ(P)∗Θ(C1)∗ · · · ∗ Θ(Cr)∗Rep(lδ)∗Θ(I)

=OP∗ OM∗Rep(lδ)∗ OI=OP⊕M∗Rep(lδ)∗ OI.

If we show that [P],[M],[I]and lare uniquely determined by the element A:= Θ(X),

then we have that Xcan be uniquely written as P ∗ C1∗ · · · ∗ Cr∗ {lδ} ∗ I and that Θis

injective.

Assume therefore that there is a preprojective representation P0, a regular inhomoge-

neous representation M0, a preinjective representation I0and an integer l0such that

OP⊕M∗Rep(lδ)∗ OI=OP0⊕M0∗Rep(l0δ)∗ OI0.

By theorem 4.16 we have that the set D(0, l)is smooth and locally closed in Rep(δ)l.

Moreover, it is open in Rep(δ)l. Now we can apply lemma 4.4 to obtain that D(P⊕M⊕

I, l)is dense in A. Since it is also locally closed by theorem 4.16 we have that it is open

in A. With the same arguments we have that D(P0⊕M0⊕I0, l0)is open in A. Since

Ais irreducible these two open sets have to intersect. Hence there are x1, . . . , xl∈HK

and y1, . . . , yl0∈HKsuch that

P⊕M⊕I⊕

i=1

Rxi∼

=P0⊕M0⊕I0⊕

i=1

Ryi.

By the Krull-Remak-Schmidt theorem this yields P∼

=P0,R∼

=R0,I∼

=I0and l=l0.

Therefore, we have uniqueness of the expression and that Θis injective. Since it is

also surjective by definition we have that it is an isomorphism.

Corollary 4.29. Every element Aof the composition monoid CM(Q)can be written in

the form

OP⊕M∗Rep(δ)l∗ OI,

where Pis preprojective, Mis regular having no homogeneous summand, Iis preinjective

and l∈N. Moreover, P,R,Iand lare, up to isomorphism, uniquely determined by A.

5. Geometry of Quiver Flag Varieties

For every complex problem, there

is a solution that is simple, neat,

and wrong.

(H.L. Mecken)

In this chapter we will talk about the geometry of quiver Grassmannians and quiver

flag varieties. Most times when we use algebraic geometry we take the functorial view-

point, i.e. we identify a scheme Xwith its functor of points R7→ Hom(SpecR, X)from

the category of commutative rings to sets. A good reference for this is [DG70].

5.1. Basic Constructions

We want to define some functors from commutative rings to sets which are schemes,

ending up with the quiver Grassmannian. These should turn out to be useful to define

morphisms between schemes naturally coming up in representation theory of quivers.

All of this section should be standard. We say that a scheme Xover some field Kis a

variety if it is separated, noetherian and of finite type over K.

Definition 5.1. Let d, e ∈N. We define the scheme Hom(d, e)via its functor of points

Hom(d, e)(R) := HomR(Rd, Re).

For any I⊂ {1, . . . , d +e}with |I|=c≤d+ewe have a morphism

ρI: Hom(d, d +e)→Hom(d, c)

given by removing the rows not indexed by Iand a morphism

δI: Hom(d+e, d)→Hom(c, d)

given by removing columns not indexed by I. For f∈Hom(d, d +e)(R)we write fIfor

ρI(f)and for g∈Hom(d+e, d)(R)we write gIfor δI(g).

The scheme Hom(d, e)is an affine space since it is obviously isomorphic to

SpecZ[Xij ]1≤i≤e

1≤j≤d

The maps ρIand δIare natural transformations, hence morphism of schemes.

Now we want to define open subschemes resembling the injective and surjective, or

more general rank relements of the homomorphism spaces.

5. Geometry of Quiver Flag Varieties

Definition 5.2. Let d, e, r ∈N. We define the schemes Hom(d, e)r,Inj(d, d +e)and

Surj(d+e, d)via their functors of points

Hom(d, e)r(R) := {f∈Hom(d, e)(R)|Im fis a direct summand of Reof rank r},

Inj(d, d +e)(R) := {f∈Hom(d, d +e)(R)|fis a split injection }

= Hom(d, d +e)d(R)

and

Surj(d+e, e)(R) := {g∈Hom(d+e, e)(R)|gis surjective }

= Hom(d+e, e)e(R).

Remark 5.3.All elements of Surj(d+e, e)(R)are automatically split since Reis a pro-

jective module.

Proposition 5.4. Let d, e, r ∈N. We have that Hom(d, e)ris a locally closed subscheme

of Hom(d, d +e). Moreover, Inj(d, d +e)is an open subscheme of Hom(d, d +e)and

Surj(d+e, e)is an open subscheme of Hom(d+e, e).

Proof. The functor Hom(d, e)ris a locally closed subscheme of Hom(d, e)if for each

f∈Hom(d, e)(R)there are two ideals N1and N2such that for each R-algebra S,

f⊗S∈Hom(d, e)r(S)if and only if (N1⊗S)S=Sand N2⊗S= (0) ([DG70, I, §1,

3.6] and [DG70, I, §2, 4.1]).

Let f∈Hom(d, e)(R). Let N1:= Fe−r(Re/Im(f)) and N2:= Fe−r+1(Re/Im(f)),

the Fitting ideals of the finitely presented module Re/Im(f).1Then (N1⊗S)S=Sand

N2⊗S= (0) if and only if Se/Im(f⊗S)is projective of rank e−r([CSG79]), if and

only if Im(f⊗S)is a direct summand of Seof rank r.

The subschemes Inj(d, d +e)and Surj(d+e, e)are open since N2will be always (0)

and only the open condition remains.

Definition 5.5. Let d, e ∈N. We define the scheme Ex(d, e)via its functor of points

Ex(d, e)(R) := {(f, g)∈(Inj(d, d +e)×Surj(d+e, e))(R)|g◦f= 0 }.

Proposition 5.6. Let d, e ∈N.Ex(d, e)is a closed subscheme of Inj(d, d+e)×Surj(d+

e, e)and therefore a locally closed subscheme of Hom(d, d +e)×Hom(d+e, e).

Proof. Let Rbe a commutative ring and (f, g)∈Inj(d, d +e)(R)×Surj(d+e, e)(R).

Let Ibe the ideal generated by the entries of the matrix of g◦f. Then (f⊗S, g ⊗S)∈

Ex(d, e)(S)if and only if I⊗S= (0) for all R-algebras S.

Proposition 5.7. Let d, e ∈Nand let Rbe a commutative ring. Then the elements

(f, g)∈Ex(d, e)(R)are exactly the short exact sequences 0→Rdf

→Rd+eg

→Re→0.

Proof. Obviously, every such short exact sequence given by (f, g)is in Ex(d, e)(R). So let

(f, g)∈Ex(d, e)(R). We have to show that the sequence 0→Rdf

→Rd+eg

→Re→0is

1For more on Fitting ideals see appendix A.

5.1. Basic Constructions

exact. The R-homomorphisms fand gare split by definition, so we can choose sections:

0//Rdf//Rd+eg//

jjRe//

ll0.

Now the map (f s ) : Rd+e→Rd+eis a split monomorphism since it has r−rsg

gas a

section. Therefore, the cokernel of (f s )is projective of rank 0, hence equal to 0and

the sequence is a short exact sequence.

Definition 5.8. Let n∈N. We define the group scheme GLnvia its functor of points

GLn(R) := GL(Rn).

GLnis a smooth group scheme ([DG70, II, §1, 2.4]) over Z.

Proposition 5.9. Let d, e ∈N. Then GLd×GLeacts on Ex(d, e)via the action on

R-valued points

(σ, τ)·(f, g) := (f◦σ−1, τ ◦g)

for any σ∈GLd(R),τ∈GLe(R)and (f, g)∈Ex(d, f)(R).

Proof. obvious.

We recall the definition of a principal G-bundle in our setting, which is just a bundle

being trivial for the Zariski topology.

Definition 5.10. Let Xbe a scheme, Pa scheme with the action µof a group scheme

Gand a projection π:P→Xwhich is G-invariant. Then (P, X, µ, π), or simply π,

is a principal G-bundle if for each ring Rand for every morphism SpecR→Xthere

are elements s1, . . . , sk∈Rwhich generate RRsuch that Spec Rsi×XPis trivial, i.e.

isomorphic to SpecRsi×G, for each i.

Theorem 5.11. Let d, e ∈N. The natural projection π: Ex(d, e)→Inj(d, d +e)is a

principal GLe-bundle.

Proof. Let Rbe a ring and f∈Inj(d, d+e)(R). For every I⊂ {1, . . . , d+e}with |I|=d

let sI:= ∆I(f), where ∆I(f) := det fI. The elements ∆I:= ∆I(f)generate RR. We

have to show that

D(∆I)×Inj Ex(d, e)

is trivial, where D(∆I) = SpecR∆i. We construct an isomorphism to D(∆I)×GLe.

For I⊂ {1, . . . , d +e}denote by θI: Hom(d, e)→Hom(d+e, e)the section of δI

replacing the columns not indexed by Iwith −idRe. Let f∈D(∆I). Then fIis

invertible. Let φ:D(∆I)→Surj(d+e, e)be the morphism sending fto θI(fICf−1

I).

Then φ(f)f=fICf−1

IfI−fIC= 0. Here, ICdenotes the complement of Iin {1, . . . , d+

e}. Moreover, φ(f)is surjective since ∆IC(φ(f)) = det(−idRe)=(−1)e. Let ψ:

D(∆I)×GLe→D(∆I)×Inj Ex(d, e)be the morphism given by ψ(f, σ)=(f, σφ(f)).

This is an isomorphism. Therefore, the claim follows.

5. Geometry of Quiver Flag Varieties

Definition 5.12. Let d, e ∈N. We define the scheme Gr d+e

dvia its functor of points

Gr d+e

d!(R) := nP⊂Rd+ePis a direct summand of rank do.

This is a projective scheme, the Grassmannian, smooth over Z.

Proof. See [DG70].

More generally, we can define the flag scheme.

Definition 5.13. Let d= (d0, . . . , dν)be a sequence of integers. Define Fl(d)⊂

QGr dν

dias

Fl(d)(R) := ((U0, . . . , Uν)∈

i=0

Gr dν

di!(R)Ui⊂Ui+1 for all 0≤i≤ν−1).

This is a closed subscheme of the product of Grassmannians and therefore projective.

Theorem 5.14. Let d, e ∈N. The projection Inj(d, d +e)→Gr d+e

dsending each f

to Im(f)is a principal GLd-bundle.

Proof. See, for example, [CB96a].

Let Qbe a quiver, dand edimension vectors and d= (d0, . . . , dν)a filtration. By

taking fibre products we can generalise the previous constructions and define Hom(d, e),

Inj(d, d+e),Surj(d+e, e),Ex(d, e),Gr d+e

dand Fl(d)pointwise. For all above schemes

we can do base change to any commutative ring K, and we will denote these schemes

by e.g. Hom(d, e)K.

Definition 5.15. Define the representation scheme

RepQ(d) := Y

α:i→j

Hom(di, dj).

This is isomorphic to an affine space.

Remark 5.16.For each scheme defined for a quiver Qwe often omit the index if Qis

obvious, e.g. Rep(d) = RepQ(d).

More generally we have the module scheme.

Definition 5.17. Let Kbe a field, Λa finitely generated K-algebra and

ρ:Khx1, . . . , xmi → Λ

a surjective map from the free associative algebra to Λ. The affine K-scheme ModΛ(d)

is defined by

ModΛ(d)(R) = n(M1, . . . , Mm)∈(End(Rd))mf(M1, . . . , Mm)=0∀f∈Ker ρo.

There is a natural GLd-action on ModΛ(d)given by conjugation.

5.1. Basic Constructions

Remark 5.18.We have that ModΛ(d)is isomorphic to the functor

R7→ HomR−alg(Λ ⊗R, End(Rd)).

Therefore, another choice of ρgives an isomorphic scheme.

Assume that Λis finite dimensional and Λ = S⊕Jas K-module with Jbeing

the Jacobson radical and S∼

=Knbeing a semisimple subalgebra. Let ei,1≤i≤

n, be the canonical basis of Sand f1, . . . , fmelements in the union of the eiJej,

1≤i, j ≤n, such that the residue classes form a basis of the direct sum of the

eiJ/J2ej. Then the eitogether with the figenerate Λ. We can choose a presenta-

tion ρ:Khx1, . . . , xn, y1, . . . , ymi → Λsending xito eiand yito fi. Then a point Min

ModΛ(d)(R)starts with some matrices M1, . . . , Mncorresponding to the eiand these

define a point in ModS(d)(R)so that we have a functorial GLd-equivariant morphism

pfrom ModΛ(d)to ModS(d). In this case, one has the following lemma, see [Bon91,

lemma 1] and [Gab74, 1.4]. Note that both authors do the proof only for Kalgebraically

closed, but if one has a decomposition of the algebra as above, the same proof works.

Lemma 5.19. For any vector d∈Nnsuch that Pn

i=1 di=d, there is a connected

component ModS(d)of ModS(d)characterised by the fact that ModS(d)(K)consists of

the GLd(K)-orbit of the semisimple module containing Seiwith multiplicity di. Every

connected component of ModS(d)is of this type and smooth. The connected components

ModΛ(d)of ModΛ(d)are the inverse images of the ModS(d)under p.

Let Qbe a quiver, Ka field, dan integer and da dimension vector such that Pdi=d.

Then, after choosing an isomorphism Kd→LKdi, there is a natural immersion

RepQ(d)→ModKQ(d)⊂ModKQ(d).

We define now quiver flags and quiver Grassmannians as varieties. Choose a K-

representation M∈RepQ(d+e)(K). We set

GrQ M

d!(R) := (U∈Gr d+e

d!(R)Uis a subrepesentation of M⊗KR)

for each K-algebra R.GrQM

dis a closed subscheme of Gr d+e

dKand therefore pro-

jective. In a similar way we obtain a closed subscheme FlQM

dof Fl(d)Kfor a filtration

dof dim M.

Let Λbe a K-algebra. Let dand ebe two integers. For a Λ-module M∈ModΛ(d+e)

we define

GrΛ M

d!(R) := (U∈Gr d+e

d!(R)Uis a submodule of M⊗KR).

If Λ∼

=S⊕Jas before, then we have that

GrΛ M

d!=aGrΛ M

d!,

5. Geometry of Quiver Flag Varieties

each GrΛM

dbeing open in GrΛM

d. Moreover, for Λ = KQ we obtain that

GrQ M

d!∼

=GrKQ M

via the immersion RepQ(d)→ModKQ(d).

We have the following well-known result.

Proposition 5.20. Let X, Y be two schemes, Ybeing irreducible. Let f:X→Ybe

a morphism of schemes such that fis open and for each z∈Ythe fibre f−1(z)is

irreducible. Then Xis irreducible.

Proof. Take ∅ 6=U, V ⊂Xand U, V open. We need to show that U∩Vis non-empty.

We know that f(U)and f(V)are non-empty and open in Y. Therefore, they intersect

non-trivially. Let y∈f(U)∩f(V). By definition, we have that U∩f−1(y)6=∅and

V∩f−1(y)6=∅and both are open in f−1(y). By assumption, the fibre f−1(y)is

irreducible and therefore U∩V∩f−1(y)6=∅and we are done.

Remark 5.21.The same is true for arbitrary topological spaces, since the proof relies

purely on topology.

5.2. Grassmannians and Tangent Spaces

Let Kbe a field and Λa finitely generated K-algebra. We want to calculate the tangent

space at a point of GrΛM

d.

Lemma 5.22. Let d, e ∈N,Ka field, Λa finitely generated K-algebra and M∈

ModΛ(d+e)(K). Let U∈GrΛM

d(F), for a field extension Fof K. Then

TUGrΛ M

d!∼

=HomΛ⊗KF(U, (M⊗KF)/U).

Proof. In this proof we use left modules since the notation is more convenient. For the

tangent space we use the definition of [DG70, I, §4, no 4]. By base change, we can assume

that F=K. We prove the claim by doing a K[ε]valued point calculation (ε2= 0). The

short exact sequence

0//Uι//Mπ//

iiM/U //

jj0

is split in the category of K-vector spaces, therefore there is a retraction pof ιand a

section jof π. We consider elements of Uas elements of Mvia the inclusion ι.

The map K[ε]→Kgiven by s+r 7→ sinduces a map θ:V⊗K[ε]→Vfor each

K-vector space V.

For each homomorphism f∈HomΛ(U, M/U)we define

Sf:= {u+vε |u∈U, π(v) = f(u)} ⊂ M⊗K[ε].

5.3. Geometry of Quiver Flags

Note that θ(Sf) = U. We need to show that Sf∈GrΛ⊗K[ε]M⊗K[ε]

dand that every

element Sof the Grassmannian with θ(S) = Uarises in this way.

First, we show that Sfis a Λ⊗K[ε]-submodule. Let u+vε ∈Sfand r+sε ∈Λ⊗K[ε].

Then we have

(r+sε)(u+vε) = ru + (rv +su)ε.

Since π(v) = f(u),u∈Uand fis a Λ-homomorphism we obtain that

π(rv +su) = rπ(v) = rf(u) = f(ru).

Therefore, Sfis a Λ⊗K[ε]submodule.

Now we show that Sfis, as a K[ε]-module, a summand of M⊗K[ε]of rank d. Let

f∈HomK(U, M) := j◦fbe a K-linear lift of f.Let

φ:U⊗K[ε]→M⊗K[ε]

u+vε 7→ u+ ( ˜

f(u) + v)ε.

Obviously, φis K[ε]-linear and Im φ=Sf. Moreover, φis split with retraction

ψ:M⊗K[ε]→U⊗K[ε]

x+yε 7→ p(x) + p(y−˜

f◦p(x))ε.

Therefore, Sfis a summand of M⊗K[ε]of rank d.

On the other hand, let S∈GrΛ⊗K[ε]M⊗K[ε]

dsuch that θ(S) = U. Then, Uε is a K-

subspace of Swith dimKUε =d, therefore dimKS/(Uε) = d. The map sending u+vε ∈

Sto u∈Uis surjective, therefore the induced map from S/(Uε)is an isomorphism.

Hence, for each u∈Uthere is a v∈M, such that u+vε ∈Sand vis unique up to

a element in Uε. Set f(u) := π(v)and, by the discussion before, this does not depend

on the choice of vand we have that S=Sf. Moreover, f∈HomΛ(U, M/U): Let

r∈Λ,u∈Uand v∈M, such that u+vε ∈S. Then, ru +rvε ∈S. By definition,

f(ru) = π(rv) = rπ(v) = rf(u).

5.3. Geometry of Quiver Flags

Now we come back to quiver flags. Let Kbe a field. We can consider RepQ(d)as an

affine scheme over Kwith the obvious functor of points. More generally, we work in the

category of schemes over K. Fix a filtration d= (d0= 0 ≤d1≤ · · · ≤ dν).

Let Λ := (KQ)Aν+1. Then modΛ is the category of sequences of K-representations of

Qof length ν+ 1 and chain maps between them, i.e. a morphism between two modules

M=M0→M1→ · · · → Mν

and

N=N0→N1→ · · · → Nν

5. Geometry of Quiver Flag Varieties

is given by a commuting diagram

M0−−−−→ M1−−−−→ · · · −−−−→ Mν



y



y



N0−−−−→ N1−−−−→ · · · −−−−→ Nν.

For Λit is easy to calculate the Euler form of two modules M,N∈modΛ by using

theorem B.3:

hM,NiΛ=

i=0 DMi, NiEKQ −

r−1

i=0 DMi, Ni+1EKQ .

We show that GrΛM

d∼

=FlQM

d, where M= (M=M=· · · =M), and then use

the previous results to calculate the tangent space.

Lemma 5.23. Let dbe a filtration and M∈RepQ(dν)(K). Let U= (U0, U1, . . . , Uν)∈

FlQM

d(F)for a field extension Fof K. Then we have that

TUFlQ M

d!∼

=HomΛ⊗F(U,(M⊗F)/U),

where Λ=(KQ)Aν+1 and M= (M=M=· · · =M)∈ModΛ(dν, dν, . . . , dν)(K).

Proof. For a submodule

U= (U0→U1→ · · · → Uν)∈GrΛ M

d!(R)

we have automatically that the maps Ui→Ui+1 are injections. Therefore, such a

submodule Ugives, in a natural way, rise to a flag U∈FlQM

d(R)and vice versa. This

yields an isomorphism GrΛM

d∼

=FlQM

d. Since GrΛM

dis open in GrΛM

d, where

d=Pk,i dk

i, we have for a point U∈FlQM

d(F)that

TUFlQ M

d!∼

=TUGrΛ M

d!=TUGrΛ M

d!= HomΛ⊗F(U,(M⊗F)/U).

We define the closed subscheme RepFlQ(d)of RepQ(dν)×Fl(d)by its functor of

points

RepFlQ(d)(R) := ((M, U)∈RepQ(dν)(R)×Fl(d)(R)U∈FlQ M

d!).

We have the following.

Lemma 5.24. Let dbe a filtration. Consider the two natural projections from the fibre

5.3. Geometry of Quiver Flags

product restricted to RepFlQ(d).

RepFlQ(d)π1//

π2



RepQ(dν)

Fl(d)

Then π1is projective and π2is a vector bundle of rank

k=1 X

α:i→j

j(dk

i−dk−1

i).

Therefore, RepFlQ(d)is smooth and irreducible of dimension

ν−1

k=1 Ddk, dk+1 −dkEQ+ dim RepQ(dν).

Finally, the (scheme-theoretic) image Ad:= π1(RepFlQ(d)) is a closed, irreducible sub-

variety of RepQ(dν).

Proof. π1is projective since it factors as a closed immersion into projective space times

RepQfollowed by the projection to RepQ.

For I= (Ii)i∈Q0, each Ii⊂ {1, . . . , dν

i}, we set WIto be the graded subspace of Kdν

with basis {ej}j∈Iiin the i-th graded part Kdν

iand |I|:= (|Ii|)i∈Q0∈NQ0. For a

sequence I= (I0, I1, . . . , Iν)such that Ik

i⊃Ik+1

iand |Ik|=dν−dkwe set

WI:= (WI0, . . . , WIν)

to be the decreasing sequence of subspaces associated to I. We show that π2is trivial

over the open affine subset UIof Fl(d)given by

UI(R) := nU∈Fl(d)(R)Uk⊕(WIk⊗R) = Rdνo.

Without loss of generality we assume Ik

i={dk

i+ 1, . . . , dν

i}. Each element U∈UI(R)

is given uniquely by some matrices Ak

i∈Mat(dν

i−dk

i)×(dk

i−dk

i−1)such that

i= Im

idd1

i0· · · 0

idd2

i−d1

....

...0

...iddk

i−dk−1

· · · Ak













5. Geometry of Quiver Flag Varieties

Let Vk:= W({1,...,dk

i}). Let Xbe the closed subscheme of RepQ(dν)given by the

functor of points

X(R) := nM∈RepQ(dν)Vk⊗Ris a subrepresentation of M∀0≤k≤νo.

Note that Xis an affine space of dimension

k=1 X

α:i→j

j(dk

i−dk−1

i).

Let gU:= (gi)i∈Q0where

gi=

idd1

i0· · · 0 0

idd2

i−d1

....

...0.

...iddν−1

i−dν−2

· · · Aν−1

iiddν

i−dν−1













i∈GLdν

i(K).

Then, the map from X×UIto UI×FlRepFlQ(d)given by sending (M, U)to (gU·M, U)is

an isomorphism which induces an isomorphism of vector spaces on the fibres. Therefore,

we have that π2is a vector bundle.

Finally, we prove the claim on dimension. Since Fl(d)is smooth, we have that

dim Fl(d) = X

i∈Q0

ν−1

k=1

l=k+1

(dk

i−dk−1

i)(dl

i−dl−1

i∈Q0



ν−1

k=1

l=k+1

i(dl

i−dl−1

i)−

ν−2

k=1

l=k+2

i(dl

i−dl−1

i)



i∈Q0 dν−1(dν

i−dν−1

i) +

ν−2

k=1

i(dk+1

i−dk

i)!=X

i∈Q0 ν−1

k=1

i(dk+1

i−dk

i)!.

Since RepFlQ(d)is smooth and π2is a vector bundle we obtain

dim RepFlQ(d) = dim Fl(d) + X

α:i→j

k=1

(dk

i−dk−1

i)dk

i∈Q0 ν−1

k=1

i(dk+1

i−dk

i)!+X

α:i→j

k=1

(dk

i−dk−1

i)dk

5.3. Geometry of Quiver Flags

ν−1

k=1 

X

i∈Q0

i(dk+1

i−dk

i) + X

α:i→j

i(dk

j−dk+1

j)

+X

α:i→j

dν

idν

ν−1

k=1 Ddk, dk+1 −dkEQ+ dim RepQ(dν).

Remark 5.25.Note that if Mis a K-valued point of Adfor Knot algebraically closed,

then Mdoes not necessarily have a flag of type d. This only becomes true after a finite

field extension.

We now can give an estimate for the codimension of Adin RepQ(dν). For this we use

Chevalley’s theorem.

Theorem 5.26 (Chevalley).Let X, Y be irreducible schemes over a field Kand let

f:X→Ybe a dominant morphism. Then for every point y∈Yand every point

x∈f−1(y), the scheme theoretic fibre, we have that

dimxf−1(y)≥dim X−dim Y.

Moreover, on an open, non-empty subset of Xwe have equality.

Proof. See [Gro65, §5, Proposition 5.6.5].

Theorem 5.27. Let dbe a filtration, Fa field extension of K,Λ := (FQ)Aν+1 and

(M, U)∈RepFlQ(d)(F). Let M= (M=· · · =M)as a Λ-module. Then we have that

codimAd≤dim Ext1

Λ(U,M/U)≤dim Ext1

Λ(U,M)≤dim Ext1

F Q(M, M).

Proof. Since dim Adis stable under flat base change we can assume K=F. Let V:=

M/U. Then we have the following short exact sequence of Λ-modules:

0 : 0 −−−−→ 0−−−−→ · · · −−−−→ 0



y



y



y



U:U0−−−−→ U1−−−−→ · · · −−−−→ Uν



y



y



y



M:M M · · · M



y



y



y



V:V0−−−−→ V1−−−−→ · · · −−−−→ Vν



y



y



y



0 : 0 −−−−→ 0−−−−→ · · · −−−−→ 0.

5. Geometry of Quiver Flag Varieties

We already know that HomΛ(U,V)is the tangent space of Fl M

dat the point U. Using

Chevalley’s theorem, we have that

dim HomΛ(U,V)≥dimUFl M

d!≥dim RepFlQ(d)−dim Ad

and therefore

dim Ad≥dim RepFlQ(d)−dim HomΛ(U,V).

We now calculate

hU,ViΛ=

k=0 DUk, V kEQ−

ν−1

k=0 DUk, V k+1EQ

ν−1

k=1 Ddk, dν−dkEQ−

ν−1

k=1 Ddk, dν−dk+1EQ=

ν−1

k=1 Ddk, dk+1 −dkEQ.

Recall that

dim RepFlQ(d) =

ν−1

k=1 Ddk, dk+1 −dkEQ+ dim RepQ(dν).

In total

codimAd≤dim Rep(d) + dimHomΛ(U,V)−dim RepFlQ(d)

= dim HomΛ(U,V)−

ν−1

k=1 Ddk, dk+1 −dkEQ= dim HomΛ(U,V)− h U,ViΛ

= dim Ext1

Λ(U,V)−dim Ext2

Λ(U,V).

Here we have the last equality since gldim Λ ≤2. Since gldim KQ = 1 and P=P=

· · · =Pis projective in modΛ for every projective Pin mod KQ we see that pdΛM≤1

and similarly idΛM≤1. Consider, as before, the short exact sequence

0→U→M→V→0.

Applying (−,V)yields (U,V)2= 0. Applying (−,M)gives a surjection (M,M)1→

(U,M)1. Applying (−,U)yields that (U,U)2= 0. Therefore, applying (U,−)yields

a surjection (U,M)1→(U,V)1. Hence the above result simplifies to

codim Aa≤dim Ext1(U,V)≤dim Ext1(U,M)≤dim Ext1(M,M).

Obviously, Ext1

Λ(M,M)∼

=Ext1

KQ(M, M)and the claim follows.

Remark 5.28.Note that if the characteristic of Kis 0, then, by generic smoothness,

there is a point M∈ Adand an U∈Fl M

dsuch that Fl M

dis smooth in U. In this

case we have that

codim Ad= dim Ext1

Λ⊗F(U,(M⊗F)/U).

5.3. Geometry of Quiver Flags

We also construct an additional vector bundle.

Definition 5.29. Let dbe a filtration. Let RepQ,Aν+1 (d)be the scheme given via its

functor of points

RepQ,Aν+1 (d)(R) :=

((U,f)∈

i=0

RepQ(di)(R)×

ν−1

i=0

Hom(di, di+1)(R)fi∈HomRQ(Ui, Ui+1)).

Let IRepQ,Aν+1 (d)be the open subscheme of RepQ,Aν+1 (d)given by its functor of points

IRepQ,Aν+1 (d)(R) := n(U,f)∈RepQ,Aν+1 (d)(R)fi∈Inj(di, di+1)(R)o.

Remark 5.30.Note that RepQ,Aν+1 (d)(K)consists of sequences of representations of Q.

Therefore, these are modules over (KQ)Aν+1. Vice versa, every (KQ)Aν+1-module of

dimension vector dis isomorphic to an element of RepQ,Aν+1 (d)(K).

For shortness, we will often write Uinstead of (U,f)for an (U,f)∈RepQ,Aν+1 (d)(R).

Lemma 5.31. Let dbe a filtration. Then the projection

π: IRepQ,Aν+1 (d)→

ν−1

i=0

Inj(di, di+1)

given by sending

(U,f)7→ f

is a vector bundle and therefore flat. In particular, IRepQ,Aν+1 (d)is smooth and irre-

ducible.

Proof. The first part is analogously to lemma 5.24. Irreducibility then follows by the

fact that flat morphisms are open and proposition 5.20.

Before we continue, we give the following easy lemma, stated by K. Bongartz in

[Bon94], which gives rise to a whole class of vector bundles.

Lemma 5.32. Let Xbe a variety over a ground ring K. Let m, n ∈Nand f:X→

Hom(m, n)Ka morphism. Then for any r∈N, the variety X(r)given by the functor of

points

X(r)(R) := {x∈X(R)|f(x)∈Hom(m, n)m−r(R)}

is a locally closed subvariety of X. Moreover, the closed subvariety

Ur(R) := {(x, v)∈X(r)(R)×Rm|f(x)(v)=0}

of X(r)×Kmis a sub vector bundle of rank rover X(r).

Proof. The claim follows easily by using Fitting ideals.

5. Geometry of Quiver Flag Varieties

Example 5.33.Let (M,g)∈RepQ,Aν+1 (e)(K).

Let ϕ: RepQ,Aν+1 (d)→Hom(m, n)be the morphism given by

(U,f)7→ h= (hk

i)7→ (hk

jUk

α−Mk

αhk

i)α:i→j

0≤k≤ν

,(hk+1

ifk

i−gk

ihk

i)i∈Q0

0≤k≤ν!!

for every (U,f)∈RepQ,Aν+1 (d)(R), where

m=X

i∈Q0

k=0

iek

iand n=

k=0 X

α:i→j

iek

ν−1

k=0 X

i∈Q0

iek+1

Then h∈kerϕ(U,f)if and only if h∈Hom(RQ)Aν+1 (U,M⊗R).

Set

RepHomQ,Aν+1 (d,M)r:= Ur(R)

from the previous lemma. Note that elements (U,h)∈RepHomQ,Aν+1 (d,M)r(R)are

all pairs consisting of a representation U∈RepQ,Aν+1 (d)(R)and a morphism h∈

Hom(RQ)Aν+1 (U,M)such that rank Hom(RQ)Aν+1 (U,M) = r.

The lemma yields that the projection

RepHomQ,Aν+1 (d,M)r→RepQ,Aν+1 (d)(r)

(U,h)7→ U

is a vector bundle of rank r.

It also stays a vector bundle if we restrict it to the open subset IRepQ,Aν+1 (d)(r)

of RepQ,Aν+1 (d)(r). We denote the preimage under the projection to this variety by

IRepHomQ,Aν+1 (d,M)r.

We obtain the following.

Theorem 5.34. Let dbe a filtration and Fa field extension of K. Assume that there

is an M∈ Ad(F)such that dim Ext1

F Q(M, M) = codim Ad. Then FlQM

dis smooth

over Fand geometrically irreducible.

Proof. Smoothness is immediate, since by the last theorem we have that dim TUFlQM

d

is constant and smaller or equal to the dimension at each irreducible component living

in U. Since Fis a field this implies smoothness. See [DG70, I, §4, no 4].

Now we prove irreducibility. By base change we can assume that Fis algebraically

closed. Consider all the following schemes as F-varieties. We construct the following

diagram of varieties.

IRepHomQ,Aν+1 (d,M)r

vector bundle



IRepInjQ,Aν+1 (d,M)r

open



IRepQ,Aν+1 (d) IRepQ,Aν+1 (d)(r)

open

ooFlQM

d,

rbeing equal to hd,MiΛ+ [M, M]1. Since open subvarieties and images of irreducible

5.3. Geometry of Quiver Flags

varieties are again irreducible and by application of proposition 5.20 we then obtain that

FlQM

dalso is irreducible.

Consider the minimal value rof dim Hom(U,M)for U∈IRep(d)(F). Denote by

π: IRep(d)→RepQ(dν)

U7→ Uν.

Since OMis open in Adand IRep is irreducible, the intersection of the two open sets

π−1(OM)and IRep(r)is non-empty. For all elements Uof π−1(OM)we have, by theorem

5.27, that [U,M]1

Λ= [M, M]1

Q. We already saw that [U,M]2= 0, therefore

[U,M]Λ=hU,MiΛ+ [U,M]1

Λ=hd,MiΛ+ [M, M]1

This means that the dimension of the homomorphism space is constant on π−1(OM)

and we obtain that r=hd,MiΛ+ [M, M]1

Q. Moreover, IRepris irreducible as an open

subset of IRep.

We then have that IRepHomQ,Aν+1 (d,M)ris irreducible, since it is a vector bundle on

IReprby example 5.33. Take the open subvariety IRepInj(d,M)rof IRepHom(d,M)r

where the morphism to Mis injective. It is irreducible as an open subset of an irreducible

variety. The projection from this variety to FlQM

dis surjective since π−1(OM)is

contained in IRep(d)(r), and therefore FlQM

dis irreducible.

We now want to interpret theorem 5.34 in terms of Hall numbers. Let X0be a variety

defined over a finite field Fq, where q=pnfor a prime n. Denote by Fqthe algebraic

closure of Fqand by X:= X0⊗Fqthe variety obtained from X0by base change from

Fqto Fq. Let Fbe the Frobenius automorphism acting on X. Denote by Hi(X, Q`)

the `-adic cohomology group with compact support for a prime `6=p, see for example

[FK88]. Denote by F∗the induced action of Fon cohomology H∗(X, Q`). P. Deligne

proved the following theorem.

Theorem 5.35 (P. Deligne [Del80], 3.3.9).Let X0be a proper and smooth variety over

Fq. For every i, the characteristic polynomial det(Tid−F∗, Hi(X, Q`)) is a polynomial

with coefficients in Z, independent of `(`6=p). The complex roots αof this polynomial

have absolute value |α|=qi

Moreover, the Lefschetz fixed point formula yields that

#X0(Fqn) = X

i≥0

(−1)iTr((F∗)n, Hi(X, Q`)).

Assume now that there is a polynomial P∈Q[t]such that, for each finite field

extension L/Fqwe have that #X0(L) = P(|L|),|L|being the number of elements of the

finite field. We call Pthe counting polynomial of X0. Then we have the following.

Theorem 5.36. Let X0be proper smooth over Fqwith counting polynomial P. Then

5. Geometry of Quiver Flag Varieties

odd cohomology of Xvanishes and

P(t) =

dim X0

i=0

dim H2i(X, Q`)ti.

Proof. See [CBVdB04, Lemma A.1].

Assume now that Yis a projective scheme over Zand set YK:= Y⊗Kfor any field

K. Note that for Y `-adic cohomology agrees with `-adic cohomology with compact

support. Assume furthermore that there is a counting polynomial P∈Q[t]such that,

for each finite field K, we have that #YK(K) = P(|K|). By the base change theorem

[FK88, Theorem 1.6.1] we have

Hi(YQ,Q`)∼

=Hi(YC,Q`).

By the comparison theorem [FK88, Theorem 1.11.6] we have

Hi(YC,Q`)∼

=Hi(YC(C),Q`),

where on the right hand side we consider the usual cohomology of the complex analytic

manifold attached to YC.

Moreover, there is an open, non-empty subset Uof Spec Zsuch that Hi(Yκ(v),Q`)∼

Hi(YQ,Q`)for all v∈U, where κ(v)denotes the residue field at v. This means that for

almost all primes pwe have that

Hi(YFp,Q`)∼

=Hi(YQ,Q`)∼

=Hi(YC(C),Q`).

Therefore, if we know the Betti numbers of YC(C), then we know the coefficients of

the counting polynomial. In order to apply this to our situation we use the following

theorem of W. Crawley-Boevey [CB96b].

Theorem 5.37. Let Mbe an K-representation without self-extensions. Then there is

aZ-representation Nsuch that M=N⊗Kand for all fields Fwe have that Ext(N⊗

F, N ⊗F)=0.

Putting all this together, we obtain the following.

Theorem 5.38. Assume that there is an M∈ Ad(Q), being a direct sum of exceptional

representations, such that

dim Ext1

QQ(M, M) = codim Ad.

Let Nbe a Z-representation and P∈Q[t]a polynomial, such that N⊗Q∼

=Mand

#FlQN⊗Fq

d=P(q). Then P(0) = 1 and P(1) = χFlQM⊗C

d>0.

Moreover, if Qis Dynkin or extended Dynkin, then there is a representation Nand a

polynomial Pwith the required properties.

Proof. Let X:= FlQN

das a scheme over Z. Using theorem 5.34 we obtain that XK

is smooth and irreducible for every field K. By the previous discussion we have then

5.3. Geometry of Quiver Flags

that the i-th coefficient of Pis exactly dim H2i(XC(C),Q`)and that odd cohomology

vanishes. Therefore,

0<Xdim H2i(XC(C),Q`) = χ(XC) = P(1).

By irreducibility we have that

P(0) = dimH0(XC(C),Q`)=1

and this proves the first claims.

If Qis Dynkin or extended Dynkin, then let Nbe the Z-representation given by

theorem 5.37. We have the polynomial Psince we have Hall polynomials and there is a

decomposition symbol α= (µ, ∅)such that N⊗Kis in S(α, K)for any field K, since

it is a direct sum of exceptional representations and therefore discrete.

6. Reflections on Quiver Flags

Always be wary of any helpful

item that weighs less than its

operating manual.

(Terry Pratchett)

Let Kbe an arbitrary field and Qa quiver.

6.1. Reflections and Quiver Flags

Let Mbe a K-representation of Qand da filtration of dim M. We want to define

reflections on a flag U∈FlQM

d. Let abe a sink and let U∈FlQM

d. Then, for each

i, we have the following commutative diagram with exact rows.

0−−−−→ (S+

aUi−1)a−−−−→ LUi−1

φUi−1

−−−−→ Im φUi−1

a−−−−→ 0



yf



y



0−−−−→ (S+

aUi)a−−−−→ LUi

j−−−−→

φUi

Im φUi

a−−−−→ 0

By definition, gand the map in the middle are injective. Therefore, fis injective. This

immediately yields that S+

aUis a new quiver flag of S+

aM=S+

aUν. The problem

is that the dimension of S+

aUiis dependent on the rank of φUi

a. This motivates the

next definition. Recall that da−rank φX

a= dim Hom(X, Sa)for a representation Xof

dimension vector d.

Definition 6.1. Let abe a sink, da dimension vector and san integer. Define

RepQ(d)hais:= nM∈RepQ(d)dim Hom(M, Sa) = so.

Let dbe a (ν+1)-tuple of dimension vectors and r= (r0, r1, . . . , rν)be a (ν+1)-tuple

of integers. For each representation Mdefine

FlQ M

d!hair:= (U∈FlQ M

d!dim Hom(Ui, Sa) = ri).

Moreover, let

RepQ(d)hai:= RepQ(d)hai0

and

FlQ M

d!hai:= FlQ M

d!hai0.

6. Reflections on Quiver Flags

Remark 6.2.Recall that a filtration dof dim Mis a sequence of dimension vectors such

that d0= 0,dν= dim Mand that di≤di+1. In order to know a filtration dit is enough

to know the terms d1, . . . , dν−1, since d0is always 0and dνis always dim M. Therefore,

we identify the (ν−1)-tuple (d1, . . . , dν−1)with the (ν+ 1)-tuple d.

Example 6.3.Let

1 2

Consider the representation Mgiven by M1=M2=K2and Mα= (1 0

0 0 ). We have that

M∈RepQ((2,2))h2i1. Now consider flags of type ((0,0),(1,1),(2,2)), i.e. subrepresen-

tations Nof dimension vector (1,1). We need two injective linear maps f1, f2:K1→K2

making the following diagram commutative.

KNα//



K2(1 0

0 0 )//K2

We have the following situations.

•N∈GrQM

(1,1)h2i1: This means that Nα= 0. Therefore, we need that the image

of f1is in the kernel of Mα, which is 1-dimensional. Hence, a subrepresentation in

GrQM

(1,1)h2i1is given by f1= (0

1)and f2being an arbitrary inclusion. The point

f1=f2= (0

1)is special, since for this inclusion we have that M/N ∼

=S1⊕S2and

otherwise M/N ∼

=K1

→K.

•N∈GrQM

(1,1)h2i0: This means that Nα6= 0. Therefore, we need that the image

of f1is not in the kernel of Mα, which is 1-dimensional. Hence, a subrepresentation

in GrQM

(1,1)h2i0is given by f1=f2= ( 1

x)for any x∈K.

The variety GrQM

(1,1)consists therefore of two P1

Kglued together at one point. Graph-

ically,

GrQ M

(1,1)!=GrQ M

(1,1)!h2i1qGrQ M

(1,1)!h2i0∼

Note that the Grassmannian is neither irreducible nor smooth.

In order to get rid of rwe define the following maps and then look at the fibres.

Definition 6.4. Let abe a sink, da dimension vector, s∈Nand M∈RepQ(d)hais.

We have that M∼

=M0⊕Ss

afor some element M0∈RepQ(d−sa)hai. Without loss of

generality we can assume that M=M0⊕Ss

aand we set πaM:= M0. Obviously, πaM

is unique up to isomorphism.

6.1. Reflections and Quiver Flags

Now let dbe a filtration and r= (r0, . . . , rν)a(ν+ 1)-tuple of integers. Define

πr

a: FlQ M

d!hair→FlQ πaM

d−ra!hai

U7→ Vwhere Vi

j:= (Ui

jif j6=a,

Im φUi

aif j=a.

Remark 6.5.Note that πaMis ιa,M S−

aS+

aM.

Example 6.6.Coming back to example 6.3 we see that π1

2collapses GrQM

(1,1)h2i1to

the point

GrQ K2(1 0 )

→K

(1,0) !h2i.

The fibre of π1

2over this point is the vector space Grassmannian Gr K2

1, being isomor-

phic to P1

We now introduce a little bit more notation. If dis a sequence, then denote by ←−

dthe

sequence given by (←−

d)i=dν−i. Moreover, we define the sequence eby ei:= dν−dν−i.

Therefore, if dis a filtration of dν, then eis a filtration of dν.

The fibre of the map πr

ais a set of the following type.

Definition 6.7. Let e= (e0, e1, . . . , eν)and r= (r0, r1, . . . , rν)be sequences of non-

negative integers such that e+←−

ris a filtration. Let Aν+1 be the quiver

0→1→2→3→ · · · → ν.

Then define

Xr,e:= Keν+r0////Keν−1+r1////· · · ////Ke1+rν−1////Ke0+rν.

Remark 6.8.The K-representation Xr,e∈rep(Aν+1, K)is injective and its isomorphism

class does not depend on the choice of the surjections.

Lemma 6.9. Let e= (e0, e1, . . . , eν)and r= (r0, r1, . . . , rν)be sequences of non-

negative integers such that e+←−

ris a filtration. Then Xr,ehas a subrepresentation of

dimension vector rif and only if eis a filtration of eν. Moreover, if Kis a finite field

with qelements, then the number of K-subrepresentations is given by

#GrAν+1 Xr,e

r!=

i=0 "eν−i−eν−i−1+ri

ri#q

In particular, this number is equal to 1modulo qif and only if the set of subrepresenta-

tions is non-empty.

Proof. We prove this by induction on ν.

6. Reflections on Quiver Flags

ν= 0 There is a subspace of dimension r0of Kr0+e0if and only if e0≥0and, for Ka

finite field of cardinality q, the number of those is obviously "e0+r0

r0#q

ν≥1If (U0, U1, U2, . . . , Uν)is a subrepresentation of dimension vector rof Xr,e, then

(U1, U2, . . . , Uν)is a subrepresentation of X(r1,r2,...,rν),(e0,e1,...,eν−1)of dimension

vector (r1, r2, . . . , rν). Therefore, by induction, ei≤ei+1 for 0≤i < ν −1

and 0≤e0. The preimage Vof U1under the surjection from U0has dimension

r1+ ((eν+r0)−(eν−1+r1)) = eν−eν−1+r0. Since Uis a subrepresentation, we

must have that U0⊂V. Therefore, r0≤eν−eν−1+r0or equivalently eν−1≤eν.

On the other hand, if e0≥0and ei≤ei+1 for all 0≤i<ν, then, by induc-

tion, there is a subrepresentation (U1, U2, . . . , Uν)of X(r1,r2,...,rν),(e0,e1,...,eν−1)of

dimension vector (r1, r2, . . . , rν). As before, the dimension of the preimage Vof

U1under the surjection from U0has dimension eν−eν−1+r0≥r0. If we choose

now any subspace U0of dimension r0in V, then we obtain a subrepresentation of

Xr,eof dimension vector r.

If Kis a finite field of cardinality q, then, by induction, we have that the number of

subrepresentations of X(r1,r2,...,rν),(e0,e1,...,eν−1)of dimension vector (r1, r2, . . . , rν)

is equal to ν

i=1 "eν−i−eν−i−1+ri

ri#q

To complete such a subrepresentation to a subrepresentation of Xr,ewe have to

choose an r0-dimensional subspace of an (eν−eν−1+r0)-dimensional space. There-

fore, the number of subrepresentations is equal to

"eν−eν−1+r0

r0#q

i=1 "eν−i−eν−i−1+ri

ri#q

This yields the claim.

Theorem 6.10. Let abe a sink of Q,da filtration, ra(ν+ 1)-tuple of non-negative

integers and M∈RepQ(dν). Then

πr

a: FlQ M

d!hair→FlQ πaM

d−ra!hai

is surjective and the fibre (πr

a)−1(U)over any U∈FlQπaM

d−rahaiis isomorphic to

GrAνXr,ea

r, where eν−i:= dν−di. In particular the number of points in the fibre only

depends on rand dand not on U.

Proof. Fix a flag V∈FlQπaM

d−rahai. Let U∈FlQM

dhair. The flag Uis in (πr

a)−1(V)

if and only if Ui

j=Vi

jfor all j6=ain which case Im(ΦUi

a) = Vi

a. Therefore, we only

have to choose Ui

a⊂Masuch that Vi

a⊆Ui

a,Ui−1

a⊂Ui

aand dim Ui

a=di

afor all i. This

6.1. Reflections and Quiver Flags

is the same as choosing Ui

a⊂Ma/V i

asuch that θi(Ui−1

a)⊂Ui

aand dim Ui

a=riif we

denote by θi:Ma/V i−1

a→Ma/V i

athe canonical projection. This is equivalent to finding

a subrepresentation of

Ma/V 0

a→Ma/V 1

a→ · · · → Ma/V ν

of dimension vector r. All the maps in this representation are surjective since Vis a

flag, therefore this representation of Aν+1 is isomorphic to Xr,e. Since dis a filtration

of Mwe have that eis a filtration. Therefore, GrAνXr,ea

ris non-empty by lemma 6.9

and πr

ais surjective.

Now we are nearly ready to do reflections. The only thing left to define is what

happens on a source. If bis a source in Q, then bis a sink in Qop, so we just dualise

everything.

Definition 6.11. Let U∈FlQM

dand let eν−i=dν−di. Then define

D: FlQ M

d!→FlQop DM

U7→ (ˆ

D(U))i:= ker(DM →D(Ui)) = D(M/Ui).

Remark 6.12.Obviously, ˆ

D2= id and the map ˆ

Dis an isomorphism of varieties.

Definition 6.13. Let bbe a source, da dimension vector and san integer. Define

RepQ(d)hbis:= nM∈RepQ(d)dim Hom(Sb, M) = so.

Let dbe a (ν+1)-tuple of dimension vectors and r= (r0, r1, . . . , rν)be a (ν+1)-tuple

of integers. For each representation Mdefine

FlQ M

d!hbir:= (U∈FlQ M

d!dim Hom(Sb, M/Ui) = ri).

Moreover, let

RepQ(d)hbi:= RepQ(d)hbi0

and

FlQ M

d!hbi:= FlQ M

d!hbi0.

Remark 6.14.Note that U∈FlQM

dhbirif and only if ˆ

DU∈FlQop DM

ehbir.

Now we state the main result on reflections.

Theorem 6.15. Let abe a sink of Q,dbe a filtration and M∈RepQ(dν)hai. The map

S+

a: FlQ M

d!hai → FlσaQ S+

σad!hai

U7→ S+

6. Reflections on Quiver Flags

is an isomorphism of varieties with inverse ˆ

D◦S+

a◦ˆ

D=S−

Proof. First, we show that S+

aUlies in the correct set. Let eν−i=dν−di. For each i,

we have the following commutative diagram with exact columns.

0 0 0



y



y



0−−−−→ (S+

aUi)a−−−−→ L

j→a

φUi

−−−−→ Ui

a−−−−→ 0



y



y



0−−−−→ (S+

aM)a−−−−→ L

j→a

φM

−−−−→ Ma−−−−→ 0



y



y



0−−−−→ (S+

aM/S+

aUi)a−−−−→ L

j→a

(M/Ui)j

φM/Ui

−−−−→ (M/Ui)a−−−−→ 0



y



y



0 0 0

The two top rows are exact since U∈FlQM

dhai. By the snake lemma, we have that

the bottom row is exact. Therefore, the map

(S+

aM/S+

aUi)a→M

j→a

(M/Ui)j

is injective and hence S+

a(M)/S+

a(Ui)∈RepσaQ(σa(eν−i))hai. The diagram also yields

that ˆ

D◦S+

a◦ˆ

D◦S+

a= id. Since S+

ais a functor and all choices where natural, we

have that S+

agives a natural transformation between the functors of points of these two

varieties. Therefore, it is a morphism of varieties.

Dually, S+

a◦ˆ

D◦S+

a◦ˆ

D= id. This concludes the proof.

6.2. Reflections and Hall Numbers

Let Fqbe the finite field with qelements and Qa quiver. Let w= (ir, . . . , i1)be a word

in vertices of Q. Recall that

uw=uir · · ·  ui2ui1=X

wuX.

Define a filtration d(w)by letting

d(w)k:=

j=1

ik.

6.2. Reflections and Hall Numbers

Then we obviously have FX

w= #FlQX

d(w). Therefore, coefficients in the Hall algebra

are closely related to counting points of quiver flags over finite fields. In the following,

we will use reflection functors to simplify the problem of counting the number of points

modulo q. As an application, we will show that for a preprojective or preinjective

representation Xwe have that #FlQX

d= 1 mod qif FlQX

dis non-empty.

Lemma 6.16. Let abe a sink of Q,Ka field, da filtration and M∈RepQ(dν, K).

Then

#FlQ M

d!=X

r≥0

#GrAν Xr,ea

r!#FlQ πaM

d−ra!hai

(on both sides we possibly have ∞).

We further note that, for each sequence of non-negative integers r, if FlQπaM

d−rahai

is non-empty, then so is GrAνXr,ea

r.

Proof. We have that

FlQ M

d!=a

r≥0

FlQ M

d!hair.

By theorem 6.10, we have for each sequence of non-negative integers rthat

#FlQ M

d!hair= #GrAν Xr,ea

r!#FlQ πaM

d−ra!hai.

The same theorem yields that, if FlQπaM

d−rahaiis non-empty, then so is GrAνXr,ea

r.

Lemma 6.17. Let abe a sink of Q,Ka field, da filtration and M∈RepQ(dν, K)hais.

Let r+=r+(d)be given as follows:

+:= 0;

+:= max{0,(σa(di−1−di))a+ri−1

+}for 0< i < ν;

rν

+:= s.

Now let rbe a sequence of integers. If FlQM

dhairis non-empty, then r≥r+.

Proof. Let U∈FlQM

dhair. By definition, Hom(Ui, Sa) = ri. We have

ri= codim Im ΦUi

a= dim kerΦUi

a+di

a−X

j→a

j= dim kerΦUi

a−(σadi)a.

We prove ri≥ri

+by induction on i. For i= 0 the claim is obviously true. Now let

0≤i≤ν−2. Obviously, dim ker ΦUi

a≤dim ker ΦUi+1

a. Therefore,

++ (σadi)a≤ri+ (σadi)a= dim ker ΦUi

a≤dim ker ΦUi+1

a=ri+1 + (σadi+1)a.

Hence, ri+1 ≥max{0,(σa(di−di+1))a+ri

+}=ri+1

6. Reflections on Quiver Flags

For rν

+note that, by definition, Uν=Mand therefore

rν= codim Im ΦUν

a= codim Im ΦM

a=s.

Remark 6.18.Note that r+(d−r+(d)a)=0since

σa(di−di−1)a+ri

+(d)−ri−1

+(d)≥ri

+(d)−ri

+(d)=0.

For any filtration dof some representation Mit is enough to remember the terms

(d1, . . . , dν−1)

since we always have d0= 0 and dν= dim M. Note that the rule to construct ri

+for

0< i < ν depends neither on d0nor on dν. Therefore, we can define

S+

ad=S+

a(d1, . . . , dν−1) := (σad1+r1

+a, . . . , σadν−1+rν−1

+).

If dis a filtration of M, then S+

adis a filtration of S+

aMif and only if (S+

ad)ν−1≤

dim S+

aM.

Corollary 6.19. Let abe a sink, Ka field, da filtration and M∈RepQ(dν, K)hais.

Then

#FlQ M

d!=X

r≥0

#GrAν Xr+r+,ea

r+r+!#FlσaQ S+

S+

ad+ra!hai.

In particular, if Kis a finite field of cardinality q, we have

#FlQ M

d!=X

r≥0

#FlσaQ S+

S+

ad+ra!haimod q.

Proof. By lemmas 6.16 and 6.17 we obtain that

#FlQ M

d!=X

r≥0

#GrAν Xr+r+,ea

r+r+!#FlQ πs

d−(r+r+)a!hai.

Note that σa(di−(ri+ri

+)a) = (S+

ad)i+riafor all 0< i < ν. Therefore, theorem

6.15 yields that

FlQ πs

d−(r+r+)a!hai∼

=FlσaQ S+

S+

ad+ra!hai.

This proves the first claim.

Now let Kbe a finite field of cardinality q. If GrAνXr+r+,ea

r+r+is non-empty, then

its number is one modulo qby lemma 6.9. The second part of lemma 6.16 yields that

whenever FlQπs

d−(r+r+)ahaiis non-empty, then GrAνXr+r+,ea

r+r+is non-empty. This

finishes the proof.

6.2. Reflections and Hall Numbers

We obtain the following.

Theorem 6.20. Let abe a sink, Ka field, da filtration and M∈RepQ(dν, K)hais.

Then FlQM

dis empty if and only if FlσaQS+

S+

adis.

Moreover, if K=Fqis a finite field, then

#FlQ M

d!= #FlσaQ S+

S+

ad!mod q.

Proof. By corollary 6.19 we obtain that

#FlQ M

d!=X

r≥0

#GrAν Xr+r+,ea

r+r+!#FlσaQ S+

S+

ad+ra!hai.

Note that S+

adis a filtration of S+

aMif and only if (S+

ad)ν−1≤dim S+

aM. Therefore,

if S+

adis not a filtration of S+

aM, then each FlσaQS+

S+

ad+rahaiis empty for all r≥0

and hence, so is FlQM

d. In this case, we also have that FlσaQS+

S+

adis empty. Both

claims follow.

Assume now that S+

adis a filtration of S+

aM. We have that

FlσaQ S+

S+

ad!∼

=FlσaQop DS+

dim S+

aM−

←

S+

ad!

via ˆ

D. Let f:= dim S+

aM−

←

S+

ad. By using lemma 6.16 we obtain that

#FlσaQop DS+

f!=X

r≥0

#GrAν X

←

r,(S+

ad)a

←

r!#FlσaQop DS+

f−←

ra!hai.

Moreover, the same lemma yields that for each r≥0we have that if FlσaQop DS+

f−←

rahai

is non-empty, then GrAνX

←

r,(S+

ad)a

←

ris non-empty. Using ˆ

Dyields

FlσaQop DS+

f−←

ra!hai∼

=FlσaQ S+

S+

ad+ra!hai.

Combining these equalities, we have that

#FlσaQ S+

S+

ad!=X

r≥0

#GrAν X

←

r,(S+

ad)a

←

r!#FlσaQ S+

S+

ad+ra!hai.

Therefore, FlσaQS+

S+

adis empty if and only if for all r≥0the variety FlσaQS+

S+

ad+rahai

is empty. The same is true for FlQM

dand this proves the first claim.

Now let Kbe a finite field with qelements. We already saw that if FlσaQS+

S+

ad+rahai

6. Reflections on Quiver Flags

is non-empty, then GrAνX

←

r,(S+

ad)a

←

ris non-empty. Therefore, lemma 6.9 yields that

#FlσaQ S+

S+

ad!=X

r≥0

#FlσaQ S+

S+

ad+ra!haimod q.

By corollary 6.19 this is equal to #FlQM

d. This finishes the proof.

Remark 6.21.The Coxeter functor C+is by definition the composition of reflection

functors associated to an admissible ordering (a1, . . . , an)of Q. The action on a filtration,

which we also denote by C+, is given by C+d:= S+

an. . . S+

a1d. It is not clear that C+

on a filtration does not depend on the choice of the admissible ordering.

We immediately obtain the following.

Corollary 6.22. Let Mbe a preprojective K-representation and let dbe a filtration of

dim M. Take r≥0such that (C+)rM= 0.

Then FlQM

dis non-empty if and only if we have that (C+)rd= 0 and that for every

intermediate sequence wof admissible sink reflections S+

wdis a filtration of S+

wM. In

particular, this depends only on the isomorphism class of Mand the filtration d, but not

on the choice of Mor the field K.

Moreover, if Kis a finite field with qelements, then FlQM

dnon-empty implies that

#FlQ M

d!= 1 mod q.

Proof. Using remark 6.18 we obtain that for each reflection at a sink aof Qwe have

that S+

adis again a filtration of S+

aMif and only if (S+

ad)ν−1≤dim S+

aM. If this is

not the case, then the quiver flag is empty by theorem 6.20. Therefore, if the quiver flag

is non-empty, then for every intermediate sequence wof admissible sink reflections we

have that S+

wdis a filtration of S+

wM. We call this condition (*).

Assume that (*) holds. Iteratively applying theorem 6.20 we have that FlQM

dis

empty if and only if FlQ(C+)rM

(C+)rd= FlQ0

(C+)rdis empty. There is only one filtration of

the 0representation, namely (0,0,...,0). Therefore, FlQM

dis non-empty if and only

if (C+)rd= 0. This proves the first part since we already have seen that if (*) does not

hold, then FlQM

dis empty.

Assume now that Kis a finite field with qelements. If (*) does not hold, then the

quiver flag is empty and the claim holds. Assume therefore that (*) holds. As before,

applying theorem 6.20 yields that

#FlQ M

d!= #FlQ (C+)rM

(C+)rd!= #FlQ 0

(C+)rd!mod q.

There is only one filtration of the zero representation, namely (0,0,...,0), and the

number of flags of this type is obviously equal to one. This concludes the proof.

6.3. Dynkin Case

In this section let Qbe a Dynkin quiver. Then every representation is preprojective,

and there are Hall polynomials with respect to isomorphism classes. We can use the

machinery we just developed to prove that, for Q, the generic composition algebra

specialised at q= 0 and the composition monoid are isomorphic.

Proposition 6.23. Let Xbe a K-representation of Qand wa word in vertices of Q.

Then the condition that Xhas a filtration of type wonly depends on wand [X]and not

on the choice of Xor the field K.

Moreover, we have that

uw=X

[X]∈[Aw]

u[X]∈ H0(Q).

Proof. Since all representations of Qare preprojective, the first part of the statement

follows directly from corollary 6.22. Therefore, the sum in the second part is well-defined

(i.e. the set [Aw]does not depend on the field).

If Kis a finite field with qelements, corollary 6.22 also yields that

w= #FlQ X

d(w)!=(1 mod qif X∈ Aw,

0else.

Since FX

w=f[X]

w(q)and we just showed that this is one modulo qfor all prime powers

qwe have that

f[X]

w(0) = (1if [X]∈[Aw],

0else.

This yields the claim.

We obtain the main theorem for the Dynkin case.

Theorem 6.24. The map

Ψ: QM(Q)→ H0(Q)

A 7→ X

[M]∈[A]

u[M]

is an isomorphism of Q-algebras.

Proof. Note that for QDynkin we have that M(Q)∼

=CM(Q)and Hq(Q)∼

=Cq(Q).

Therefore, for each A∈M(Q)there is a word win vertices of Qsuch that A=Aw. In

the previous proposition we showed that the map sending Awto

Ψ(Aw) = X

[M]∈[Aw]

u[M]=uw

is well-defined. Therefore, Ψis a homomorphism, since

Ψ(Aw∗ Av) = Ψ(Awv) = uwv =uwuv= Ψ(Aw)Ψ(Av).

6. Reflections on Quiver Flags

Ψis surjective since it is a homomorphism, and the generators uiof H0(Q)are in

the image of Ψ. More precisely, Ψ(OSi) = ui. Obviously, Ψis a graded morphism of

graded algebras. The dimension of the d-th graded part of QM(Q)is the same as the

dimension of the d-th graded part of H0(Q), namely the number of isomorphism classes

of representations of dimension vector d. Since each graded part is finite dimensional

and Ψis surjective, we have that Ψis an isomorphism.

7. Extended Dynkin Case

When all you have is a hammer,

everything starts to look like a

nail.

(Proverb)

In this chapter we examine the relation between the generic composition algebra and

the composition monoid of an extended Dynkin quiver. In the following, let Qbe a con-

nected, acyclic, extended Dynkin quiver. Fix again a total order ≺ton the preprojective

and preinjective Schur roots refining the order ≺on P ∪ I.

7.1. Basic Results

In this section we prove some general facts on the number of points of quiver flag varieties.

First, we want to prove the following.

Theorem 7.1. Let Mbe any Fq-representation and let dbe a filtration of dim M. Then

the number of points of FlQM

dis equal to the number of points of FlQMR

d0modulo q

for a sequence d0such that the defect of each d0iis equal to zero. Moreover, FlQM

dis

empty if and only if FlQMR

d0is empty.

Proof. By using the Coxeter functors we can reduce to the case where M=MR. More

precisely, eliminate MPby using C+r≥0times and then apply C−rtimes to return to

Mwithout the preprojective part. In the same way eliminate MI. If, at some point, the

reflection of the filtration dis not a filtration any more, then the flag variety is empty.

In this case set d0to any sequence which is not a filtration of dim MR.

Therefore, we can assume that Mis purely regular. If the defect ∂diis positive for

some 0<i<ν, then FlQM

dis empty, since every representation of dimension vector

dihas a preinjective summand and can therefore not be a subrepresentation of Mwhich

is purely regular. Therefore, ∂di≤0for all i.

Now prove the claim by induction on h=−P∂di. If h=0 we are done. Assume that

h > 0and jis minimal with the property that ∂dj<0. Again, if FlQM

dis non-empty,

then there is a representation of dimension vector djhaving at least one preprojective

and possibly some regular summands. Therefore, the C+-orbit (here just reflections on a

dimension vector) of djtends to −∞ and, applying our machinery of reflections on flags,

at some point we have to add some r+a,r+>0, where ais a source of Q, increasing

the defect. Therefore, by induction and inverting all reflections, we are done.

We obtain the following.

7. Extended Dynkin Case

Corollary 7.2. Let Mbe an Fq-representation such that dim Mx≥δfor at most one

x∈P1

Fqand let dbe a filtration of dim M. Then # FlQM

dis equal to one modulo qif

and only if it is non-empty.

Proof. Using the last theorem we can assume that the defect of each diis 0and M

is purely regular with the condition. If Uiis a subrepresentation of Mof dimension

vector di, then Uiis purely regular, too. Assume that this is not the case, then it must

have a non-zero preinjective summand eliminating the defect of a non-zero preprojective

summand. But this cannot be a subrepresentation of M, since Mis purely regular

and therefore has no preinjective subrepresentations. If we decompose di=Px∈P1dx,i,

where dx,i = dim Ui

x, then one has that ext(dx,i, dy,i)=0for x6=y. Therefore, the

canonical decomposition of diis a refinement of Pdx,i by lemma 2.13. Since the canonical

decomposition is unique, and, if δappears, the condition on Myields that there is only

one possibility in which tube it can occur, we know that for each subrepresentation Vi

of Mwith dim Vi=dithat dim Vi

x=dx,i. Hence, choosing a flag of type dof Mis the

same as choosing flags of type dxof Mx, i.e.

FlQ M

d!=Y

FlQ Mx

dx!.

By the result on the cyclic quiver the number of points of each of these is equal to one

modulo q, and therefore, so is the number of points of the product.

The next few lemmas deal with quiver Grassmannians.

Lemma 7.3. Let dbe the dimension vector of a preprojective representation and let M

be an arbitrary Fq-representation of dimension d+efor some dimension vector e. If

Gr M

dis non-empty, then #Gr M

d= 1 mod q.

Dually, if eis the dimension vector of an indecomposable preinjective, Man arbitrary

Fq-representation of dimension d+eand Gr M

dis non-empty, then # Gr M

d= 1

mod q.

Moreover, in both cases Gr M

dis non-empty if and only if Gr M⊗F

dis non-empty,

where Fis the algebraic closure of Fq.

Proof. Let w= (a1, . . . , an)be an admissible ordering of the vertices of Q. Recall the

map σdefined in section 2.3. For each indecomposable preprojective representation P

there is a natural number r=kn +sfor some k≥0and 0≤s<nsuch that

S+

as· · · S+

a1(C+)kP= 0.

Let σ(P)be the minimal such number. The number σ(P)depends on the choice of the

admissible ordering.

The basic idea is now to use S+

ato reduce σ(P)and then use induction.

Let dbe the dimension vector of a preprojective representation P=LPifor some

indecomposable preprojective representations Pi. For each Pilet ti:= σ(Pi)for a fixed

admissible ordering (a1, . . . , an).

We proceed by induction on t= max{ti}. If t= 0, then we have that each Piis

0. Therefore, P= 0 and d= 0. Obviously, # Gr M

d= #Gr M

0= 1 and Gr M

0is

7.1. Basic Results

non-empty if and only if Gr M⊗F

0is non-empty.

Now let t > 0. If d > dim M, then Gr M

dis empty and so is Gr M⊗F

d. We are done

in this case. Assume therefore that d≤dim M.

Let a=a1. By definition, σ(S+

aPi) = σ(Pi)−1for each PiSaif we calculate σ

with respect to the admissible ordering (a2, . . . , an, a)on σaQ. We also have

σad=Xσadim Pi=X

PiSa

dim S+

aPi−sa,

where sis the number of Piisomorphic to Sa.

By our algorithm we have that

#Gr M

d!= # Gr S+

S+

ad!mod q

and the right Grassmannian is non-empty if and only if the left Grassmannian is. The

same is true for Gr M⊗F

dand Gr S+

aM⊗F

S+

adsince reflection functors commute with field

extension. Therefore, it is enough to prove the claim for Gr S+

S+

ad.

We have that

S+

ad=σad+r+a=X

PiSa

dim S+

aPi+ (r+−s)a≥0,

where r+= max{0,−(σad)a}. Since PPiSadim S+

aPi≥0, we have that r+≤s.

Since ais a source in σaQ, we obtain that Sais a simple injective and X:= LS+

aPiis a

representation having (dim X)a≥s−r+. Therefore, there is a surjection π:X→Ss−r+

and the kernel is again preprojective, say isomorphic to LP0

i. Since for each P0

ithere

is at least one Pjsuch that P0

iS+

aPj, we have that σ(P0

i)≤σ(S+

aPj)by lemma 2.7.

Therefore, max{σ(P0

i)} ≤ max{σ(S+

aPi)}< t. We are done by induction.

If eis the dimension vector of an indecomposable preinjective, then eis the dimen-

sion vector of an indecomposable preprojective on Qop, and the claim follows from the

preprojective case since

GrQ M

d!∼

=GrQop DM

e!.

Lemma 7.4. Let dbe a dimension vector having da= 0 for some a∈Q0and Man

Fq-representation of Q. If Gr M

d6=∅, then its number is one modulo q.

Moreover, Gr M

dis non-empty if and only if Gr M⊗F

dis, where Fis the algebraic

closure of Fq.

Proof. Note that there is an admissible series of source reflections S−

b1· · · S−

brsuch that

each bi6=aand ais a source in σb1· · · σbrQ. Since bi6=a, we have that (S−

b1· · · S−

brd)a= 0

and neither the number of points modulo qnor whether it is empty changed. Therefore,

we can assume that ais a source in Q.

7. Extended Dynkin Case

Let Q0be the quiver obtained from Qby deleting the vertex a. Subrepresentations

Uof dimension vector dof Mhave to make the following diagram commute for each

α:a→i.

0Uα

−−−−→ Ui



y



Mα

−−−−→ Mi

Therefore, Uis a subrepresentation of Mif and only if U|Q0is a subrepresentation of

M|Q0. The quiver Q0is obviously a union of Dynkin quivers. The restriction M|Q0is

therefore preprojective, and whether it is empty is independent of the field. Hence the

claims follow.

7.2. Basis of PBW-Type

In this section, we will construct a basis of PBW-type for C0(Q)consisting of monomial

elements, for an acyclic, extended Dynkin quiver Q. This basis then lifts to a basis of

Cq(Q). We prove this by showing first that in C0(Q)the relations of ER(Q)hold and

then that the partial normal form in ER(Q)is a normal form in C0(Q). This will be the

desired basis.

Let Abe a closed irreducible GLd-stable subvariety of Rep(d)defined over Z. For

example, take a word win vertices of Qand consider Aw. Then we define

uA:= X

α∈A

uα∈ C0(Q),

where we say that α∈ A for an α∈Σif for all finite fields Kand all X∈ S(α, K)we

have that X⊗K∈ A,Kbeing the algebraic closure of K. Note that for some finite

fields Kthe set S(α, K)may be empty.

Assume that the vertices Q0={1, . . . , n}of Qare ordered in such a way that we have

Ext(Si, Sj)=0for i≥j.

Definition 7.5. Let dbe a dimension vector. Define ud:= ud1

1ud2

2 · ··  udn

n∈ Cq(Q).

Then we have in the composition algebra of Q:

ud=

Y

i∈Q0

[di]q!

X

α∈Rep(d)

uα.

Therefore, specialising qto zero yields

ud=X

α∈Rep(d)

uα=uRep(d)∈ C0(Q).

Let d= (d0= 0, d1, . . . , dν)be a filtration dν. Then

udν−dν−1 · · ·  ud2−d1ud1−d0=Yhdi

j−di−1

jiq!X

α∈Ad

fα

d(q)uα=X

α∈Ad

fα

d(0)uα∈ C0(Q)

7.2. Basis of PBW-Type

for some polynomials fα

dfor each α∈ Adsuch that for each finite field Fqand each

X∈ S(α, Fq)we have that #FlQX

d=fα

d(q). In the sum only α∈ Adappear since

if the polynomial is non-zero, then the flag variety will be non-empty over the algebraic

closure. On the other hand, if α∈ Adwe do not necessarily have that fα

d6= 0.

In this section we will often deal with quiver Grassmannians. Therefore, we define

fα

e d := fα

(0,d,d+e).

We want to say something about f(0) for a Hall polynomial f. For this we often use

the following.

Lemma 7.6. Let f∈Q[x]be a polynomial. Assume that there is a c∈Zsuch that

f(q)∈Zand f(q) = cmod q, for infinitely many integers q∈Z. Then f(0) = c.

Proof. We have that f=1

Ngfor a polynomial g∈Z[x]and a positive integer N∈N.

Therefore,

c·N=f(q)·N=g(q) = g(0) mod q

for infinitely many integers q. Hence, g(0) = c·Nand therefore f(0) = 1

Ng(0) = c.

In the composition algebra C0(Q)we consider the following elements:

•udfor da Schur root,

•usδ for s > 0.

Note that in the composition monoid we have that

Rep(sδ)∗Rep(tδ) = Rep((s+t)δ) = Rep(tδ)∗Rep(sδ).

In the composition algebra this will not be true anymore, but at least we will prove that

one has

usδ utδ =utδ usδ.

In the following we will denote by ext(d, e) := extF Q(d, e), where Fis any algebraically

closed field. This number is independent of Fby work of A. Schofield and W. Crawley-

Boevey.

We will first prove that all relations of ER(Q)hold in C0(Q), replacing {sd}by usd.

In the following all elements udwill be in C0(Q)if not otherwise stated.

Lemma 7.7. Let dbe a preprojective Schur root, ean arbitrary dimension vector such

that ext(d, e)=0and r, s ∈N. Then

use urd =urd+se.

Dually, let ebe a preinjective Schur root, dan arbitrary dimension vector such that

ext(d, e)=0and r, s ∈N. Then

use urd =urd+se.

7. Extended Dynkin Case

Proof. Since ext(rd, se)=0, we have, for any Fq-representation Mof dimension vector

rd +se, that Gr M⊗F

rd is non-empty, where Fis the algebraic closure of Fq.

By lemma 7.3 we have that fα

se rd(q) = 1 mod qfor all prime powers qand all α∈

Rep(rd +se). Therefore, fα

se rd(0) = 1 and the claim follows.

Lemma 7.8. Let dbe a preinjective Schur root, ea preprojective Schur root such that

ext(d, e)=0and r, s ∈N. Then

use urd =urd+se.

Proof. We first prove the claim for s=t= 1.

Since there is an indecomposable preprojective representation Pof dimension vector e

and an indecomposable preinjective representation Iof dimension vector d, the minimal

value of Ext is taken on those. Moreover, since Hom(I, P )=0, we have that

0 = ext(d, e) = Ext(I, P) = Ext(I, P)−Hom(I, P) = − h d , e i.

Therefore, d < δ since, otherwise, d=d0+δwith d0being a positive preinjective root

and

hd , e i=d0, e +∂e < 0,

in contradiction to hd , e i= 0. Dually, we have that e < δ.

Summing up, we have that d+e < 2δ. Therefore, for each Fq-representation Mof

dimension vector d+ewe have that dim Mx≥δfor at most one x∈P1

Fqand for each

tube Txwith Mx6= 0 we have that deg x= 1. Over the algebraic closure Fof Fqwe have

that every representation of dimension vector d+ehas a subrepresentation of dimension

vector d. Let αbe the decomposition symbol of M. Since Mlives only in tubes of

degree one by the restriction on the dimension, we have that the decomposition symbol

of M⊗Fis also α. Therefore, the polynomial fα

e d is non-zero. We conclude that fα

e d(q)

modulo qis equal to one for an infinite number of prime powers q, namely all qwhich

are not a zero of fα

e d, by corollary 7.2. Hence, the constant coefficient is equal to one

and the claim follows.

Therefore, we have proved that ueud=ud+e. We automatically have that ext(e, d) =

0, since dis preinjective and eis preprojective. By the previous lemma, we have that

udue=ud+e. Therefore, udue=ueud. Again by the previous lemma, we have that

(ud)r=urd and (ue)r=ure. Using the lemma once more, we obtain

use urd =us

eur

d=ur

dus

e=urd use =urd+se.

Lemma 7.9. Let d,ebe dimension vectors, one being a real Schur root of defect 0, such

that ext(d, e)=0and let r, s ∈N. Then

use urd =urd+se.

7.2. Basis of PBW-Type

Proof. First assume that dis a real Schur root and has defect 0. Then d < δ and there

is a regular simple representation Tof dimension vector fsuch that d+f≤δ. There is

a series of sink reflections S+

a1· · · S+

aksuch that (S+

a1· · · S+

ak(rf))j=rδjfor an extending

vertex jof Q. Therefore, (S+

a1· · · S+

ak(rd))j= 0.

Let Mbe an Fq-representation of Qof dimension vector rd +se and let Fbe the

algebraic closure of Fq. We have that M⊗Fhas a subrepresentation of dimension

vector rd, since ext(d, e)=0. Therefore, for each j≥1

dim S+

aj· · · S+

ak(M⊗F)≥S+

aj· · · S+

ak(rd).

Since

dim S+

aj· · · S+

ak(M⊗F) = dim S+

aj· · · S+

akM,

the same is true for M. Applying lemma 7.4 yields that the number modulo qis one

since it is non-empty over the algebraic closure.

If eis a real Schur root with defect zero, then apply ˆ

Dto Gr M

rdfor M∈Rep(rd +se)

to reduce to first case.

Lemma 7.10. For s, t ∈Nwe have that

usδ utδ =utδ usδ ∈ C0(Q).

Proof. We need to show that for all finite fields Fqand each M∈Rep((s+t)δ, Fq)we

have that #Gr M

tδ = # Gr M

sδmod q. Since both Grassmannians are non-empty over

the algebraic closure, any sequence of reflections applied to sδ or tδ will yield a filtration

of the reflection of M.

Assume that Mhas a preprojective summand. We want to show that

#Gr M

tδ != # Gr M

sδ!= 1 mod q.

Since the defect of Mis zero and it has a preprojective summand, it will automatically

have a preinjective summand. Let S+

a1· · · S+

akbe a minimal sequence of admissible sink

reflections eliminating a preprojective summand of M. Let M0:= dim S+

a1· · · S+

akMand

Q0:= σa1· · · σakQ. Then, at this point, we have that dim M0= dim M+ra1, where

r > 0is the number of times Mhas the eliminated preprojective as a summand. We

have that σa1· · · σaktδ =tδ, and therefore the cokernel of M0by a subrepresentation of

dimension vector tδ has dimension vector dim M0−tδ =sδ +ra1. Moreover, the vertex

a1is a source in Q0. If we apply ˆ

D, we obtain that

#GrQ0 M0

tδ != #GrQ0op DM0

sδ +ra1!.

Note that a1is a sink in Q0op. Applying sink reflections we obtain that each entry of

(C+)ia1tends to −∞ for i→ ∞. Therefore, at some point while using the algorithm

of lemma 6.20 to do sink reflections, we will have that one component of the reflected

7. Extended Dynkin Case

dimension vector will become zero. But then, applying lemma 7.4 yields that # Gr M

tδ =

1 mod qsince the Grassmannian is non-empty over the algebraic closure of Fq. The

same argument yields that #Gr M

sδ= 1 mod q.

Assume now that Mis purely regular. Consider a subrepresentation Uof Mof

dimension vector tδ. Then U=UR. Since Ext(Ux, Uy)=0for x6=y∈P1

Fqwe

have that ext(dim Ux,dim Uy)=0. Therefore, the canonical decomposition of tδ is a

refinement of Px∈P1dim Uxby lemma 2.13. But the canonical decomposition of tδ is

ttimes δ. Therefore, there are integers tx∈Nsuch that dim Ux=tx(deg x)δwith

P(deg x)tx=t. The regular representation Mhas its regular part only in finitely many

tubes, say Tx1,...,Txk. By a similar argument using the canonical decomposition we

have that dim Mxi=mi(deg xi)δfor some integers mi>0. Hence,

#Gr M

tδ !=X

(t1,...,tk)

Pti(deg xi)=t

i=1

#Gr Mxi

ti(deg xi)δ!mod q

(t1,...,tk)

Pti(deg xi)=t

ti≤mi

It is easy to see that Mxihas a subrepresentation of dimension vector ti(deg xi)δif and

only if ti≤mi. The last equality follows by theorem 3.11 and using that Txis equivalent

to modκ(x)Cr−1,rbeing the rank of Tx. Note that κ(x)has qdeg xelements.

Repeating the same argument for sδ, we obtain that

#Gr M

sδ!=X

(s1,...,sk)

Psi(deg xi)=s

i=1

#Gr Mxi

si(deg xi)δ!mod q

(s1,...,sk)

Psi(deg xi)=s

si≤mi

There is an obvious bijection between the two sets we are summing over, namely sending

(t1, . . . , tk)7→ (m1−t1, . . . , mk−tk), and therefore the two numbers agree.

We can now use the previous lemmas to prove the following.

Theorem 7.11. The following relations hold in C0(Q).

use urd =urd use for all s, t ≥0, d, e Schur roots such that

ext(d, e) = ext(e, d) = 0;

use urd =urd+se for all s, t ≥0, d, e Schur roots such that

ext(d, e)=0and not both roots are imaginary;

(ud)r=urd for all r > 0, d a real Schur root.

Proof. If both roots are imaginary, this is lemma 7.10. In the other cases at least one

root is real, and it is enough to prove the second statement since then the first statement

follows by

use urd =urd+se =urd use ∈ C0(Q).

The third statement follows since for a real Schur root done always has ext(d, d)=0.

7.2. Basis of PBW-Type

Now we prove the second statement. If d=δ, then eis not preprojective. Therefore,

lemmas 7.7 and 7.9 yield the claim. If e=δ, then dis not preinjective, and again lemmas

7.7 and 7.9 yield the claim. If one of them is real of defect zero, then apply lemma 7.9.

Finally, if both are real not of defect zero, then lemmas 7.7 and 7.8 finish the proof.

Now we obtain the following.

Theorem 7.12. Let dand ebe Schur roots, at least one real, such that ext(d, e) = 0

and r, s ∈N. Let Pk

i=1 tifibe the canonical decomposition of rd +se, i.e. the fiare

pairwise different Schur roots with ext(fi, fj)=0and ti>0.

Then

use urd =ut1f1 · · ·  utkfk.

In particular, the product on the right hand side does not depend on the order.

Proof. We already proved in theorem 7.11 that

use urd =usd+re.

We have to show that the right hand side equals the same.

By the previous theorem, the product on the right hand side does not depend on the

order. We can therefore assume that we have the preinjective roots on the left and the

preprojective roots on the right.

At most one fiis equal to the isotropic Schur root δ. We start multiplying the

expression together, starting with utδ. Then the left factor will be a multiple of a

preinjective or a real regular Schur root, or the right factor will be a multiple of a

preprojective or a real regular Schur root. The claim then follows by lemmas 7.7 and

7.9.

As the last step before proving the main theorem we need to cope with Schur roots

living in only one inhomogeneous tube.

Lemma 7.13. Let c1, . . . , ckbe real Schur roots such that the general representations of

cilive in a single inhomogeneous tube, say Tx. Let Mbe such that

OM= Rep(c1)∗ · · · ∗ Rep(ck).

This representation exists and we have that

uc1 · · ·  uck=uOM.

Proof. We can assume that each ckis the dimension vector of a regular simple represen-

tation in Txby lemma 7.9. We already showed in the part on the composition monoid

that there is a generic extension Mwith this property living in the inhomogeneous tube

Tx. Therefore, Mis given by a decomposition symbol α= (µ, ∅). A representation

over an algebraically closed field Fhas a filtration of type c1, . . . , ckif and only if it is a

degeneration of M. Therefore, we only have to consider decomposition symbols β∈ OM.

7. Extended Dynkin Case

We use the same method as in lemma 7.3, but now we deal with a flag and not only

with a Grassmannian. In order to make the proof more readable, for a flag dwe set

a= (a1, . . . , aν−1) := (d1−0, d2−d1, . . . , dν−1−dν−2)

and define ˆ

FlM

a:= Fl M

d. Note that we do not loose any information since dν=

dim M. Moreover, S+

aapplied to a flag in the new notation is given by

S+

aa= (σaa1+r1

+a, σaa2+ (r2

+−r1

+)a, . . . , σaaν−1+ (rν−1

+−rν−2

+)a).

We show that for any decomposition symbol β∈ OMand any Fq-representation

N∈ S(β, Fq)we have that

#ˆ

Fl N

(ck, ck−1, . . . , c2)!= 1 mod q.

This then yields the claim.

Now let Nbe such an Fq-representation. First, apply sink reflections in the admissible

ordering to the flag until the reflected Nhas no preprojective summand left. Note that

σwci≥0for each admissible word in vertices of Q. Therefore, in each step r+= 0. This

means that we end up with a flag of type (σwck, . . . , σwc2)and dim S+

wN=σwdim N+

dim I, where Iis some preinjective representation. Applying ˆ

D, we have to consider a

flag of the following type of DS+

wN:

(σwc1+ dim I, σwc2, . . . , σwck−1).

DS+

wNhas no preinjective summands and dim Iis the dimension vector of a preprojec-

tive representation of σwQop. Note that DS+

wN⊗Fqhas a flag of the given type, since

this is invariant under under ˆ

Dand, by theorem 6.20, sink reflections. We are left with

showing that in this situation

#ˆ

Fl DS+

(σwc1+ dim I, σwc2, . . . , σwck−1)!= 1 mod q.

The flag we have at this point has the following properties, formulated for a filtration

aand an Fq-representation N0:

Fl N0⊗Fq

(a1, a2, . . . , ak−1)!

is non-empty, N0has no preinjective direct summand, each ai= dim Ri+ dim Piwhere

Riis 0or a regular simple representation living in one inhomogeneous tube Txand Pi

is 0or a preprojective representation. If we can show that, in this situation, we have

#ˆ

Fl N0

(a1, a2, . . . , ak−1)!= 1 mod q,

7.2. Basis of PBW-Type

then the claim follows. We want to prove this more general statement by induction.

For a preprojective representation Pwe define

σ(P) := max{σ(P1), . . . , σ(Pr)},

where P1, . . . , Prare the indecomposable direct summands of Pand σ(0) := 0.

Let rbe the number of Rinot equal to 0and t:= max{σ(Pi)}for a fixed admissible

ordering (b1, . . . , bn)of the vertices of Q. To prove the claim we do induction on (r, t),

ordered lexicographically, i.e. (r, t)<(r0, t0)if r < r0or r=r0and t<t0.

We start the induction at t= 0. In this case dim Pi= 0 for all i, therefore we can

proceed as in the beginning and eliminate the preprojective summand of N0by doing

sink reflections. While doing this, tremains 0and rconstant. We can therefore assume

that N0is purely regular since N0did not have a preinjective summand.

The only regular representation of dimension vector dim Riis Riitself, living in the

inhomogeneous tube Tx, and therefore all regular representations having a flag of type

(dim R1,dim R2,...,dim Rk−1)of dimension vector Pdim Rilive in Tx, too. Since the

tubes are orthogonal, we have that

#ˆ

Fl N0

(dim R1,dim R2,...,dim Rk−1)!= # ˆ

Fl N0

(dim R1,dim R2,...,dim Rk−1)!.

The representation N0

xlives in an inhomogeneous tube which is equivalent to represen-

tations of the cyclic quiver Cn. By lemma 3.7, we have that

Fl N0

(dim R1,dim R2,...,dim Rk−1)!6=∅.

By the result on the cyclic quiver we finally conclude that

#ˆ

Fl N0

(dim R1,dim R2,...,dim Rk−1)!= 1 mod q.

Now let t > 0. The vertex b=b1is a sink of Qand σbai=σbdim Ri+σbdim Pi. We

know that σbdim Ri= dim S+

bRi≥0. Let us assume that Pihas the simple Sbsi-times

as a direct summand. Then σbdim Pi= dim S+

bPi−sib. Therefore, ri

+−ri−1

+≤si. Note

that σ(S+

bPi)< σ(Pi)calculated with respect to the admissible ordering (b2, . . . , bn, b)

of σbQ. Let u, v ∈Nsuch that 0≤si−ri

++ri−1

+=u+v,σbdim Ri−ub≥0and

dim S+

bPi−vb≥0. Since Sbis simple injective in σbQ, there is a surjection from the

regular simple representation S+

bRito Su

b. If u > 0, then the kernel will be preprojective

since S+

bRiis regular simple, and therefore there is a preprojective representation Xi

such that dim Xi= dim S+

bRi−ub. If u= 0, set Xi:= Ri.

With a similar argument we have that there is a preprojective representation P0

isuch

that dim P0

i= dim S+

bPi−vb. By lemma 2.7 we also have that σ(P0

i)≤σ(S+

bPi)<

σ(Pi). Therefore,

S+

ba= (dim X1+ dim P0

1,dim X2+ dim P0

2,...,dim Xk−1+ dim P0

k−1)

7. Extended Dynkin Case

is of the same form as aand if rdid not decrease, then tdid. Therefore, we are done

by induction.

Proposition 7.14. There is a surjection of Q-algebras

Ξ: QER(Q)→ C0(Q)

{sd} 7→ usd.

Proof. We just showed that the defining relations (4.1), (4.2) and (4.3) of ER(Q)are

satisfied by the elements usd in C0(Q). Therefore, the given map is well-defined. It is

surjective since the generators uiare in the image of Ξ.

Now we can finally prove the main result.

Theorem 7.15. Let Qbe a connected, acyclic, extended Dynkin quiver with rinhomo-

geneous tubes indexed by x1, . . . , xr∈P1

Z. Let wbe a word in vertices of Q. Then uw

can be uniquely written as PC1· · · CrRI ∈ C0(Q), where

P=us1p1 · · ·  uskpkwith si>0, pipreprojective Schur roots and

pi≺tpjfor all i<j;

Ci=uOMifor a separated [Mi]∈[Txi];

R=uλ1δ · · ·  uλlδwith λ= (λ1≥ · · · ≥ λl)a partition;

I=ut1q1 · · ·  utmqmwith ti>0, qipreinjective Schur roots and

qi≺tqjfor all i < j.

Moreover, the set of elements of this form gives a basis of C0(Q)and Ξ: QER(Q)→

C0(Q)is an isomorphism.

Proof. By lemma 4.24 every element of ER(Q)can be written as above. Since Ξis a

homomorphism we have that every monomial element uw∈ C0(Q)can be written in this

form.

The composition algebra at q= 0,C0(Q), is naturally graded by dimension vector

and each graded part has the same dimension as the corresponding graded part of the

positive part U+(g)of the universal enveloping algebra of the Kac-Moody Lie algebra

ggiven by the Cartan datum associated to Q. Therefore, dim C0(Q)dis the number of

ways of writing das a sum of positive roots with multiplicities, in the sense that we

count mδ for m > 0with multiplicity n−1, where nis the number of vertices of Q, and

every other root with multiplicity one.

For each inhomogeneous tube Txof rank land each dimension vector dwe have

that dim C0(Tx)dis equal to dim U+(ˆ

sll)d, the dimension of the d-th graded part of the

positive part of the universal enveloping algebra of the Lie algebra ˆ

sll, by [Rin93]. The

roots of ˆ

sllcorrespond to the dimension vectors of indecomposables in Tx. Moreover,

the multiplicity of mδ in ˆ

sllis l−1. Note that dim(U+(ˆ

sll))dis equal to the number of

separated isomorphism classes in Πxof dimension vector d.

Theorem 4.1 in [DR74] yields that the sum P(li−1) of the ranks of the inhomogeneous

tubes minus one is equal to n−2, the number of vertices of Qminus two.

7.3. Example

Since the elements in the above form which are homogeneous of degree dgenerate

C0(Q)d, being of the right number, they have to be linearly independent. Hence, they

are a basis and every element uwcan be written uniquely in such a form. Since every

element in ER(Q)can be written in this partial normal form, we have that the d-th

graded part of QER(Q),QER(Q)d, has dimension at most dim C0(Q)d. Since Ξis

surjective, this yields that it is an isomorphism and that the partial normal form is a

normal form.

Corollary 7.16. The map

Φ: C0(Q)→QCM(Q),

sending uSito OSifor each i∈Q0, is a surjective Q-algebra homomorphism with kernel

generated by urδ = (uδ)r.

Proof. Since C0(Q)∼

=QER(Q),CM(Q)∼

ER(Q)and g

ER(Q)arises from ER(Q)by

dividing out the relation {rδ}={δ}r, the claim follows.

Corollary 7.17. Let Qbe a connected, acyclic, extended Dynkin quiver with rinho-

mogeneous tubes indexed by x1, . . . , xr∈P1

Z. For every separated partition πi∈Πs

let u(πi,xi)∈ Cq(Q)be a lift of uOMi∈ C0(Q)for a representation Mi∈ Txiof iso-

morphism class πi. Then the elements X ⊗Q[q]Q(q)∈ Cq(Q)⊗Q[q]Q(q)of the form

X=PC1· · · CrRI ∈ Cq(Q), where

P=us1p1 · · ·  uskpkwith si>0, pipreprojective Schur roots and

pi≺tpjfor all i<j;

Ci=u(πi,xi)for a separated πi∈Πs

xi;

R=uλ1δ · · ·  uλlδwith λ= (λ1≥ · · · ≥ λl)a partition;

I=ut1q1 · · ·  utmqmwith ti>0, qipreinjective Schur roots and

qi≺tqjfor all i < j.

are a basis of Cq(Q)⊗Q[q]Q(q).

Proof. Since the images of the set of elements of the form above under the specialisation

to q= 0 are a basis, they are linearly independent. Since each graded part Cq(Q)⊗Q[q]

Q(q)dis a Q(q)-vector space of dimension dim C0(Q)d, we have that the elements are a

basis.

7.3. Example

If Qis a Dynkin quiver, we have shown that

fα

w(0) = (1fα

w6= 0,

0otherwise,

7. Extended Dynkin Case

Figure 7.1.: Degeneration order on the quiver e

M∈ CM(Q)∩ {test symbols}

M∈ CM(Q)\{test symbols}

M∈ {test symbols}\CM(Q)

2δ

δ⊕S2

Tδ⊕T

δ⊕S2⊕T

⊕T

⊕S2

I(2,2,1) ⊕S3

I(2,1,1) ⊕P(0,1,1)P(1,2,2) ⊕S1P(1,1,2) ⊕I(1,1,0)

I(2,1,1) ⊕S2⊕S3

P(1,1,2) ⊕S2⊕S1δ⊕I(1,1,0) ⊕S3

δ⊕S1⊕P(0,1,1) S2

T⊕T

I(1,1,0) ⊕T⊕P(0,1,1)

δ⊕S1⊕S2⊕S3

T⊕I(1,1,0) ⊕S3

T⊕S2⊕T

T⊕S1⊕P(0,1,1) T

S2⊕I(1,1,0) ⊕S3

S2⊕S2⊕T

S2⊕S1⊕P(0,1,1)

⊕S1⊕S3

I(1,1,0) ⊕S2⊕T⊕S3

S2⊕T⊕S1⊕P(0,1,1)

T⊕S1⊕S2⊕S3

S2⊕S1⊕S2⊕S3

I(1,1,0) ⊕S1⊕P(0,1,1) ⊕S3

I(1,1,0)2⊕S2

2⊕T2

1⊕P(0,1,1)2

I(1,1,0) ⊕S1⊕S2⊕S2

P(0,1,1) ⊕S2

1⊕S2⊕S3T⊕S1⊕S2

2⊕S3

1⊕S2

2⊕S2

7.3. Example

for an arbitrary decomposition symbol αand all words win vertices of Q. We already

saw that this is not true anymore in the extended Dynkin case.

Definition 7.18. A decomposition symbol αis called a test symbol if, for all words

win vertices of Q,

fα

w(0) = (1fα

w6= 0,

0otherwise.

If it would we possible to distinguish between the sets Awof the composition monoid

only by using test symbols, then it would be possible to prove that there is a homomor-

phism

Φ: C0(Q)→ CM(Q)

by a similar method as we used in the Dynkin case. We will now illustrate that this

is not possible in general, namely if we have at least one inhomogeneous tube. The

smallest example of this is the quiver e

A2.

A2:=

It has one inhomogeneous tube with regular simples S2and T, where dim T= (1,0,1).

We wrote down the degeneration graph of representations of dimension vector 2δin

figure 7.1. In there, we write classes of representations on the vertices. A connecting line

means, that the lower class is a subset of the closure of the GL2δ-saturation of the upper

class. We write P(d1, d2, d3)for the unique indecomposable preprojective representation

of dimension vector (d1, d2, d3)and similarly I(d1, d2, d3).

One easily sees that only decomposition symbols α= (µ, σ)having their regular

part in one single tube are test symbols. Moreover, just using test symbols, we cannot

distinguish between the following elements

and Rep(δ)⊕T

S2,or between

and Rep(δ)⊕S2

since every test symbol which occurs in one class also occurs in the other class. Similarly

for

Rep(δ)⊕S1⊕S2⊕S3and

⊕S1⊕S3.

This immediately yields that this method cannot work in the extended Dynkin case,

therefore our hard work was necessary.

In general, it would be interesting to obtain the morphism C0(Q)→QCM(Q)in

a global way, i.e. without using generators and relations. This could then possibly

generalise to the wild case, where it is hopeless to apply our method.

A. Fitting Ideals

We define Fitting ideals for morphisms and finitely presented modules and give some

results for them. For proofs see [Nor76]. Let Rbe any commutative ring. A projective R-

module Pis called of constant rank rif for every prime ideal p∈SpecRthe localisation

Ppis a free Rp-module of rank r.

Definition A.1. Let f:Rd→Rebe a homomorphism, which is given by a matrix

A∈Re×d. For each r∈Ndefine Fr(f)to be the ideal of Rgenerated by the e−r

minors of A. For r≥eset Fr(f) := R.

Definition A.2. Let Mbe a finitely presented R-module and

Rdf

→Re→M→0

a presentation. Define Fr(M) := Fr(f).

Remark A.3.This definition does not depend on the choice of the presentation f.

Here are the first basic properties of the Fitting ideals Fr.

Proposition A.4. Let Mbe a finitely presented module. Then

1. Fr(M)⊂ Fr+1(M)for all r∈N.

2. If Mis a free module of rank r, then

(0) = F0(M) = F1(M) = · · · =Fr−1(M)⊂ Fr(M) = R.

3. If Sis an R-algebra, then Fr(M⊗S) = Fr(M)⊗Sfor all r.

More generally, one has the following theorem ([CSG79]).

Theorem A.5. Let Mbe a finitely presented R-module. Then Mis projective of con-

stant rank rif and only if Fr−1(M) = (0) and Fr(M) = R.

One can generalise this to finitely generated modules, but we do not need this.

B. Tensor Algebras

Let Λ0be a ring and Λ1aΛ0-bimodule. Define the tensor ring T(Λ0,Λ1)to be the

N-graded Λ0-module

Λ := M

r≥0

Λr,Λr:= Λ1⊗Λ0· · · ⊗Λ0Λ1(rtimes),

with multiplication given via the natural isomorphism Λr⊗Λ0Λs∼

=Λr+s. If λ∈Λris

homogeneous, we write |λ|=rfor its degree.

The graded radical of Λis the ideal Λ+:= Lr≥1Λr. Note that Λ+∼

=Λ1⊗Λ0Λas

right Λ-modules.

Lemma B.1. Let Rand Sbe rings. Let AR,RBSand CSbe modules over the corre-

sponding rings. Assume that RBSis S-projective and R-flat. Then

Extn

S(A⊗RB, C)∼

=Extn

R(A, HomS(B, C))

Proof. Choose a projective resolution P•of AR. The functor −⊗RBis exact, and for any

projective module PRthe functor HomS(P⊗RB, −)∼

=HomR(P, −)◦HomS(BS,−)is

exact since BSis projective. Therefore, P•⊗RBgives a projective resolution of A⊗RBS

as an S-module. We obtain

Extn

S(A⊗RB, C) = HnHomS(P•⊗RB, C)∼

∼

=HnHomR(P•,HomS(B, C)) = Extn

R(A, HomS(B, C)).

Theorem B.2. Let Λbe a tensor ring and M∈Mod Λ. Then there is a short exact

sequence

0→M⊗Λ0Λ+

δM

−−→ M⊗Λ0ΛM

−−→ M→0,

where, for m∈M,λ∈Λand µ∈Λ1,

M(m⊗λ) := m·λ

δM(m⊗(µ⊗λ) := m⊗(µ⊗λ)−m·µ⊗λ.

Moreover, if Λ0Λ1is flat, then pd M⊗Λ0Λ+≤gldim Λ0and pd M⊗Λ0Λ≤gldim Λ0.

Proof. It is clear that Mis an epimorphism and that MδM= 0. To see that δMis a

monomorphism, we decompose M⊗Λ = Lr≥0M⊗Λrand similarly M⊗Λ+as Λ0-

modules. Then δMrestricts to maps M⊗Λr→(M⊗Λr)⊕(M⊗Λr−1)for each r≥1and

moreover acts as the identity on the first component. In particular, if Pt

r=1 xr∈Ker(δM)

with xr∈M⊗Λr, then xt= 0. Thus δMis injective.

B. Tensor Algebras

Next we show that M⊗Λ = (M⊗Λ0)⊕Im(δM). Let x=Pt

r=0 xr∈M⊗Λ. We

show that x∈(M⊗Λ0) + Im(δM)by induction on t. For t= 0 this is trivial. Let

t≥1. Then xt∈M⊗Λ+and x−δM(xt) = Pt−1

r=0 x0

r. So we are done by induction. If

x∈Im(δM)∩M⊗Λ0, then there is a y∈M⊗Λ+such that δM(y) = x∈M⊗Λ0. But

this is only the case if y= 0 and therefore x= 0.

Now let Λ0Λ1be flat. Then Λ0Λis flat. If Nis any Λ0-module, then pdΛN⊗Λ≤

pdΛ0Nsince

Extn

Λ(N⊗Λ, L)∼

=Extn

Λ0(N, HomΛ(Λ, L)) = Extn

Λ0(N, L)

for any L∈ModΛ by the lemma. Therefore, the second part follows since Λ+=

Λ1⊗Λ.

Now let Qbe a quiver and Λ0=RQ0for a fixed ring R. Let Λ1:= RQ1be the free

Λ0-bimodule given by the arrows. The tensor ring T(Λ0,Λ1)is then equal to RQ. For

an RQ-module Mdenote Miby Mi. As R-modules we have that M=LMi.

Theorem B.3. Let Rbe a ring with gldim R=n < ∞. Let Mand Nbe two modules

of finite length over RQ. Then the Euler form is given by

hM, NiRQ =X

i∈Q0

hMi, NiiR−X

α:i→j

hMi, NjiR.

Proof. Let Λ0=RQ0. There is a natural Λ0-bimodule structure on Λ1=RQ1. Let

Λ = T(Λ0,Λ1) = RQ. Let 0→M⊗Λ0Λ1⊗Λ0Λ→M⊗Λ0Λ→M→0be the short

exact sequence of the previous theorem. Apply Hom(−, N)to it and consider the long

exact sequence

0//HomΛ(M, N)//HomΛ(M⊗Λ0Λ, N)//HomΛ(M⊗Λ0Λ1⊗Λ0Λ, N)EDBC

@A //Ext1

Λ(M, N)//Ext1

Λ(M⊗Λ0Λ, N)//Ext1

Λ(M⊗Λ0Λ1⊗Λ0Λ, N)

· · ·

//Extn

Λ(M, N)//Extn

Λ(M⊗Λ0Λ, N)//Extn

Λ(M⊗Λ0Λ1⊗Λ0Λ, N)

//Extn+1

Λ(M, N)//0.

Here we obtain the last 0since the pdM⊗Λ≤gldim Rby the previous theorem.

Now, since Λ0ΛΛis projective as a Λ0-module and as a Λ-module, we obtain

Exti

Λ(M⊗Λ0Λ, N) = Exti

Λ0(M, N) = M

i∈Q0

Exti

R(Mi, Ni)

and

Exti

Λ(M⊗Λ0Λ1⊗Λ0Λ, N) = Exti

Λ0(M⊗Λ0Λ1, N) = M

α:i→j

Exti

R(Mi, Nj).

This yields the claim.

Remark B.4.Note that, for a field K, we recover C. M. Ringel’s result, namely that

i∈Q0

dim Midim Ni−X

α:i→j

dim Midim Nj= [M, N]KQ −[M, N]1

for two K-representations Mand Nof a quiver Q.

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Index

acyclic, 8

admissible ordering, 8

canonical decomposition, 16

composition algebra, 20

generic, 20

composition monoid, 23

Coxeter functors, 13

decomposition class, 19

decomposition symbol, 19

defect, 16

dimension vector, 10

Dynkin

diagram, 14

extended, 15

Euler form, 8

finitary, 20

Fitting ideals, 93

fundamental region, 9

generic composition algebra, 20

generic extension monoid, 23

Hall polynomials, 20

indecomposable, 9

Krull-Remak-Schmidt theorem, 9

partition, 18

preinjective, 13

preprojective, 13

quiver, 7

acyclic, 8

arrow, 7

representation, 9

vertices, 7

reflection

on a quiver, 8

on dimension vectors, 8

reflection functors, 11

regular, 13

representation, 9

dimension vector, 10

indecomposable, 9

Ringel-Hall algebra, 20

root lattice, 8

roots

fundamental region, 9

imaginary, 9

positive, 9

real, 9

Schur, 16

simple, 9

Schur root, 16

Segre class, 19

Segre symbol, 19

simple roots, 9

sink, 8

source, 8

support

of a dimension vector, 9

tube, 16

Weyl group, 8

103