On the structure of triangulated category with finitely many indecomposables

0.42 MB·English

by  Claire Amiot

ON THE STRUCTURE OF TRIANGULATED CATEGORIES
WITH FINITELY MANY INDECOMPOSABLES
CLAIREAMIOT
7
0 Abstra
t. Westudytheproblemof
lassifyingtriangulated
ategories with
0 (cid:28)nite-dimensional morphism spa
es and (cid:28)nitely many inde
omposables over
2 an algebrai
ally
losed (cid:28)eld. We obtain a new proof of the following result
n due to XiZa∆o/aGnd Zhu:∆the Auslander-Reiten quiver of su
h a
ategory is of
a thefGorm where isadisjointunionofsimplyla
Zed∆Dynkindiagrams
and a weakly admissible group of automorphisms of . Then we prove
J G T k
thatfor`most'groupsDb,(mthoed
ka∆te)g/oΦry isstandard,i.e. -linearlyequivaTlent
2 to an orbit
ategory . This happens in parti
ular when is
d d≥2 T
1 maximal -Calabi-Yau with . Moreover, if is standard and algebrai
,
T
we
aneven
onstru
tatriangleequivalen
ebetween andthe
orresponding
] orbit
ategory. Finallywegiveasu(cid:30)
ient
onditionforthe
ategoryofproje
-
T
tives of a Frobenius
ategory to be triangulated. This allows us to
onstru
t
C 1
non standard -Calabi-Yau
ategories using deformed preproje
tive algebras
. ofgeneralized Dynkintype.
h
t
a
m
[ Introdu
tion

k T k
2 Let beanalgebrai
ally
losed(cid:28)eldand asmall -lineartriangulated
ategory
v
(see [31℄) with split idempotents. We assume that
1 T Hom HomT(X,Y)
a) is -(cid:28)nite, i.e. the spa
e is (cid:28)nite-dimensional for all
4 X Y T
1 obje
ts , of .
T
2 It follows that inde
omposable obje
ts of have lo
al endomorphism rings and
T
1 that ea
h obje
t of de
omposes into a (cid:28)nite dire
t sum of inde
omposables [9,
6
3.3℄. We assume moreoverthat
0 T X T
b) is lo
ally (cid:28)nite, i.e. for ea
h inde
omposable of , there are at most
/ Y Hom (X,Y)6=0
h (cid:28)nitely many iso
lasses of inde
omposables su
h that T .
t b) b)
a It was shown in [32℄ that
ondition implies its dual. Condition holds in
m parti
ular if we have
T
: b') isadditively(cid:28)nite,i.e. thereareonly(cid:28)nitelymanyisomorphism
lasses
v T
of inde
omposables in .
i T
X The study of parti
ular
lasses of su
h triangulated
ategories has a long
r history. Let us brie(cid:29)y re
all some of its highlights:
A
a
(1)If isarepresentation-(cid:28)nitesel(cid:28)nje
tivealgebra,thenthestable
ategory
T A
of(cid:28)nite-dimensional(right) -modulessatis(cid:28)esourassumptionsandisadditively
k T
(cid:28)nite. The stru
ture of the underlying -linear
ategory of was determined by
C. Riedtmann in [24℄, [25℄, [26℄ and [27℄.
(2) In [10℄, D. Happel showed that the bounded derived
ategory of the

ategory of (cid:28)nite-dimensional representations of a representation-(cid:28)nite quiver is
k
lo
ally (cid:28)nite and des
ribed its underlying -linear
ategory.
CM(R)
(3)Thestable
ategory ofCohen-Ma
aulaymodulesovera
ommuta-
R d Hom
tive
omplete lo
alGorensteinisolatedsingularity ofdimension isa -(cid:28)nite
(d−1)
triangulated
ategory whi
h is -Calabi-Yau (
f. for example [17℄ and [34℄).
In [2℄, M. Auslander and I. Reiten showed that if the dimension of R is 1, then the
CM(R)

ategory is additively (cid:28)nite and
omputed the shape of
omponents of its
Auslander-Reiten quiver.
1
2 CLAIREAMIOT
C Q
Q
(4) The
lusteQr
ategory of a (cid:28)nite quiver without orienAted
y
les was
introdu
ed in [7℄ if is an orientationof a Dynkin diagramof type and in [6℄ in
C Q
Q
the general
ase. The
ategory is triangulated [19℄ and, if is representation-
a) b′)
(cid:28)nite, satis(cid:28)es and .
In a re
ent arti
le [32℄, J. Xiao and B. Zhu determined the stru
ture of the
Auslander-Reiten quiver of a lo
ally (cid:28)nite triangulated
ategory. In this paper, we
obtain the same result with a new proof in se
tion 4, nameTly that ea
h
onZn∆e
/tGed

omponent ofthe Auslander-Reitenquiverof the
ategory isof the form ,
∆ G
where is a simply la
ed Dynkin diagram and is trivial or a weakly admissible
k T
group of automorphisms. WTe are interested iZn∆the -linear stru
ture of . IfTthe
Auslander-Reiten quiver of is of the form , wek(sZh∆ow) that the
ategory is
standard, i.Te. it is equivalent to the mesh
ategory Γ=Z.∆T/hGen in se
tion 6, we
prove that is standard if the number of verti
es of is stri
tly greater
modk∆
than the number of iso
lasses of inde
omposables of . In the last se
tion,
uΓs=ingZ∆[4℄/τwe
onstru
t examples of non standard triangulated
ategories su
h that
.
Finally, in the standard
ases, we are interested in the triangulated stru
ture
T T
of . For this, we need to mZak∆e additionTal assumptions on . If the Auslander-
Reiten quiver is of the form , and if is the base of a tower of triangulated
T

ategories [18℄, we show that there is a triangle equivalen
e between and the
Db(modk∆)
derived
ategory . For the additively (cid:28)nite
ases, we have to assume
T T
that is standard and algebrai
in the sense of [20℄. We then show that is
Db(modk∆)
(algebrai
ally) triangle equivalent to the orbit
ategory of under the
a
tion of a weakly admissible group of automorphisms. In parti
ular, for ea
h
d ≥ 2
, the algebrai
triangulated
ategories with (cid:28)nitely many inde
omposables
d
whi
h are maximal Calabi-Yau of CY-dimension are parametrizedby the simply
la
ed Dynkin diagrams.
modA
Ourresultsapplyinparti
ulartomanystable
ategories ofrepresentation-
A
(cid:28)nite sel(cid:28)nje
tive algebras . These algebras were
lassi(cid:28)ed up to stable equiva-
len
e by C. Riedtmann [25℄ [27℄ and H. Asashiba [1℄. In [5℄, J. Biaªkowski and A.
Skowro«skigiveane
essaryand su(cid:30)
ient
onditiononthesealgebrassothattheir
modA
stable
ategories areCalabi-Yau. In[15℄and[16℄,T.HolmandP.Jørgensen
modA d
prove that
ertain stable
ategories are in fa
t -
luster
ategories. These
results
an also be proved using our
orollary 7.0.6.
T
Thispaperisorganizedasfollows: In se
tion1,weprovethat hasAuslander-
Reiten triangles. Se
tion 2 is dedi
ated to de(cid:28)nitions about stable valued trans-
lation quivers and admissible automorphisms groups [12℄, [13℄, [8℄. We show in
T
se
tion 3 that the Auslander-Reiten quiver of is a stable valued quiver and in
se
tion 4, we reprove the result of JZ.∆X/iaGo and B.∆Zhu [32℄: The Auslander-ReAit,eDn
quiEverisaGdisjointunionofquivers ,where isaDynkinquiveroftype
or , and a weakly admissible group of automorphisms. In se
tion 5, we
on-
Db(modk∆)→T
stru
t a
overingfun
tor using Riedtmann's method [24℄. Then,
T
in se
tion 6, we exhibit some
ombinatorial
ases in whi
h has to be standard,
T d d ≥ 2
in parti
ular when is maximal -Calabi-Yau with . Se
tion 7 is dedi
ated
T
to the algebrai

ase. If is algebrai
and standard, we
an
onstru
t a triangle
T P k modP
equivalen
ebetween and an orbit
ategory. If isa -
ategorysu
hthat
isaFrobenius
ategorysatisfying
ertain
onditions,wewillproveinse
tion8that
P
hasnaturallyatriangulatedstru
ture. Thisallowsustodedu
einse
tion9that
projPf(∆)
the
ategory of the proje
tive modules over a deformed preproje
tive
algebra of generalized Dynkin type [4℄ is naturally triangulated and to redu
e the
1

lassi(cid:28)
ationoftheadditively(cid:28)nitetriangulated
ategorieswhi
hare -Calabi-Yau
to that of the deformed preproje
tive algebras in the sense of [4℄.
ON THE STRUCTURE OF TRIANGULATED CATEGORIES 3
A
knowledgments
IwouldliketothankmysupervisorB.Kellerforhisavailabilityandmanyhelpful
dis
ussions, and P. Jørgensen for his interest in this work and for suggesting some

lari(cid:28)
ations to me in the proof of theorem 7.0.5.
Notation and terminology
k
We work over an algebrai
ally
losed (cid:28)eld . By a triangulated
ategory, we
k T S
mean a -linear triangulated
ategory . We write for the suspension fun
tor
u v w
T U //V // W //SU
of and for a distinguished triangle. We say that
T Hom X Y T Hom (X,Y)
T
is -(cid:28)nite if for ea
h pair , of obje
ts in , the spa
e is
k T
(cid:28)nite-dimensional over . The
ategory will be
alled a Krull-S
hmidt
ategory
if ea
h obje
t is isomorphi
to a (cid:28)nite dire
t sum of inde
omposable obje
ts and
the endomorphism ring of an inde
omposable obje
t is a lo
al ring. This implies
T T
that idempotents of split [11, I 3.2℄. The
ategory will be
alled lo
ally (cid:28)nite
X T
if for ea
h inde
omposable of , there are only (cid:28)nitely many iso
lasses of inde-
Y Hom (X,Y)6= 0
T

omposables su
h that . This property is selfdual by [32, prop
1.1℄.
ν
The Serre fun
tor will be denoted by (see de(cid:28)nition in se
tion 1). The
τ
Auslander-Reiten translation will alwaysbe denoted by (se
tion 1).
T T′ S (F,φ)
Let and be two triangulated
ategories. An -fun
tor is given by
k F : T → T′ φ
a -linear fun
tor and a fun
tor isomorphism between the fun
tors
F ◦S S′◦F S T S′ T′
and , where is the suspension of and the suspension of . The
ν τ S
notion of -fun
tor, or -fun
tor is then
lear. A triangle fun
tor is an -fun
tor
u v w
(F,φ) U //V //W // SU T
su
hthatforea
htriangle of ,thesequen
e
FU Fu //FV Fv //FW φU◦Fw //S′FU T′
is a triangle of .
T d>0
The
ategory is Calabi-Yau if there exists an integer su
h that we have
Sd ν T
a triangle fun
tor isomorphism between and . We say that is maximal d-
T d T′ → T T′
Calabi-Yau if is -Calabi-Yau and if for ea
h
overing fun
tor with
d k T T′
-Calabi-Yau, we have a -linear equivalen
e between and .
k E modE
For an additive -
ategory , we write for the
ategory of
ontravariant
E modk
(cid:28)nitely presented fun
tors from to (se
tion 8), and if the proje
tives of
modE modE

oin
ide with the inje
tives, will be the stable
ategory.
1. Serre duality and Auslander-Reiten triangles
T
1.1. Serre duality. Re
all from [23℄ that a Serre fun
tor for is an autoequiva-
ν : T → T DHom (X,?) ≃ Hom (?,νX)
T T
len
e together with an isomorphism for
X ∈T D Hom (?,k)
k
ea
h , where is the duality .
T
Theorem 1.1.1. Let be a Krull-S
hmidt, lo
ally (cid:28)nite triangulated
ategory.
T ν
Then has a Serre fun
tor .
X T X∧ Hom (?,X) F
T
Proof. Let be an obje
t of . We write for the fun
tor and
DHom (X,?)
T
for the fun
tor . Using the lemma [23, I.1.6℄ we just have to show
F Top
that isrepresentable. Indeed,the
ategory islo
ally(cid:28)niteaswell. Theproof
is in two steps.
F
Step 1: The fun
tor is (cid:28)nitely presented.
Y ,...,Y
1 r
Let be representatives of the iso
lasses of inde
omposable obje
ts of
T FY Hom(Y∧,F)
su
h that i is not zero. The spa
e i is (cid:28)nite-dimensional over
k FY
i
. Indeed it is isomorphi
to by the Yoneda lemma. Therefore, the fun
tor
Hom(Y∧,F)⊗ Y∧
i k i is representable. We get an epimorphism from a representable
4 CLAIREAMIOT
F
fun
tor to :
r
MHom(Yi∧,F)⊗kYi∧ −→F.
i=1
F
By applying the same argumentto its kernel we get a proje
tive presentationof
U∧ −→V∧ −→F −→0 U V T
of the form , with and in .
H :Top →modk
Step 2: A
ohomologi
al fun
tor is representable if and only if
it is (cid:28)nitely presented.
u∧ φ
U∧ //V∧ // H //0 H
Let be a presentation of . We form a triangle
u v w
U //V // W //SU.
We get an exa
t sequen
e
U∧ u∧ // V∧ v∧ //W∧ w∧ //(SU)∧ .
φ u∧ H
Sin
e the
omposition of with is zero and is
ohomologi
al, the morphism
φ v∧ H u∧ v∧ φ
fa
tors through . But is the
okernel of , so fa
tors through . We
obtain a
ommutative diagram:
u∧ v∧ w∧
U∧ //φV(cid:15)(cid:15)∧yy||yyyiyyyyyyyφyy′y//<<yW∧ // SU.
H
φ′◦i◦φ=φ′◦v∧ =φ φ′◦i H
The equality implies that is the identity of be
ause
φ H W∧
isan epimorphism. Wededu
ethat isadire
tfa
torof . The
omposition
i◦φ′ = e∧ e ∈ End(W) H = W′∧
is an idempotent. Then splits and we get for a
W′ W
dire
t fa
tor of . (cid:3)
1.2. Auslander-Reiten triangles.
u v w
X //Y // Z // SX T
De(cid:28)nition1.2.1. [10℄Atriangle of is
alledan
Auslander-Reiten triangle or AR-triangle if the following
onditions are satis(cid:28)ed:
X Z
(AR1) and are inde
omposable obje
ts;
w 6=0
(AR2) ;
f :W → Z f′ :W →Y
(AR3) if is not a retra
tion, there exists su
h that
vf′ =f
;
g : X → V g′ : Y → V
(AR3') if is not a se
tion, there exists su
h that
g′u=g
.
Letusre
allthat,if(AR1)and(AR2)hold,the
onditions(AR3)and(AR3')are
T
equivalent. We say that a triangulated
ategory has Auslander-Reiten triangles
Z T
if, for any inde
omposable obje
t of , there exists an AR-triangle ending at
u v w
Z X // Y //Z // SX
: . In this
ase, the AR-triangle is unique up to
Z
triangle isomorphism indu
ing the identity of .
The following proposition is proved in [23, Proposition I.2.3℄
T
Proposition 1.2.1. The
ategory has Auslander-Reiten triangles.
τ = S−1ν
The
omposition is
alled the Auslander-Reiten translation. An AR-
T Z
triangle of ending at has the form:
u v w
τZ // Y //Z //νZ.
ON THE STRUCTURE OF TRIANGULATED CATEGORIES 5
2. Valued translation quivers and automorphism groups
2.1. Translationquivers. Inthisse
tion,were
allsomede(cid:28)nitionsandnotations
Q = (Q ,Q ,s,t) Q
0 1 0

on
erning quivers [8℄. A quiver is given by the set of its
Q s t x ∈ Q
1 0
verti
es, the set of its arrows, a sour
e map and a tail map . If is a
x+ x x−
vertex, we denote by the set of dire
t su

essors of , and by the set of its
Q x∈Q
0
dire
t prede
essors. We say that is lo
ally (cid:28)nite if for ea
h vertex , there
x x x+ x−
are (cid:28)nitely many arrows ending at and starting at (in this
ase, and
Q
are (cid:28)nite sets). The quiver is said to be without double arrows, if two di(cid:27)erent
arrows
annot have the same tail and sour
e.
(Q,τ)
De(cid:28)nition 2.1.1. A stable translationquiver is a lo
ally (cid:28)nite quiver with-
τ : Q → Q (τx)+ = x−
0 0
out double arrows with a bije
tion su
h that for ea
h
x α:x→y σα τy →x
vertex . For ea
h arrow , let the unique arrow .
Note that a stable translation quiver
an have loops.
(Q,τ,a)
D(Qe,(cid:28)τn)ition2.1.2.aA:vQalu→edNtranslationqau(iαv)er=a(σα) isastabletransαlationαquiver
1
with a map su
h that for ea
h arrow . If is an
x y a a(α)
xy
arrow from to , we write instead of .
∆ ∆
DZ∆e(cid:28)nition 2.1.3. Let be an oriented tree. The repetition of is the quiver
de(cid:28)ned as follows:
• (Z∆) =Z×∆
0 0
• (Z∆) = Z × ∆ ∪ σ(Z × ∆ ) (n,α) : (n,x) → (n,y)
1 1 1
with arrows and
σ(n,α):(n−1,y)→(n,x) α:x→y ∆
for ea
h arrow of .
Z∆ τ(n,x)=(n−1,x)
The quiver with the translation is
learlyastable trans-
∆
lation quiver whi
h does not depend (up to isomorphism) on the orientation of
(see [24℄).
2.2. Groups of weakly admissible automorphisms.
G
De(cid:28)nition 2.2.1. An automorphism group of a quiver is said to be admissible
G {x}∪x+ {x}∪x−
[24℄ if no orbit of interse
ts a set of the form or in more than
g ∈G−{1}
one point. It said to be weaklyadmissible [8℄ if, for ea
h and for ea
h
x∈Q x+∩(gx)+ =∅
0
, we have .
Note that an admissible automorphism group is a weakly admissible automor-
phismgroup. Letus(cid:28)xanumberingandanorientationofthesimply-la
edDynkin
trees.
A : 1 //2 // ··· // n−1 //n
n
n−1
zzvvvvvvvvv
D : 1 // 2··· ////n−2
n ddIIIIIIIIII
n
4
OO
E : 1oo 2 oo 3 //5 // ··· // n
n
∆ S Z∆
Let be a Dynkin tree. We de(cid:28)ne an automorphism of as follows:
• ∆=A S(p,q)=(p+q,n+1−q)
n
if , then ;
6 CLAIREAMIOT
• ∆=D n S =τ−n+1
n
• if ∆ = D with neven, thenS = τ−n+1;φ φ
n
iDf with odnd, thenn−1 where is the automorphism of
n
• ∆wh=i
hEex
hangeSs =aφndτ−6 ; φ E
6 6
if , then where is the automorphism of whi
h
2 5 1 6
• ex∆
ha=ngEes andS,=anτd−9 and ;
7
• if ∆=, tEhen S =;τ−15
8
and if , then .
In [24, Anhang 2℄, Riedtmann des
ribes all admissible automorphism groups of
Dynkin diagrams. Here is a more pre
ise result:
∆ G
Theorem 2.2.1. Let be aZ∆Dynkin tGree and a non tZrivial group of weakly
admissible automorphisms of . Then is isomorphi
to , and here is a list of
its possible generators:
• ∆=A n τr φτr r ≥1
n
iφf=τn+21Swith odd, possible geneZra∆tors are and with , where
• ∆ = A is annautomorphism of of order 2; ρr r ≥ 1
n
if ρ=τwn2iSth even,ρt2h=enτp−o1ssiτbrle generators are , where and
• wh∆ere=D .n(≥Sin5
e , is a possible gτerneratoτrr.φ) r ≥1
n
if wφith= (n −,1t,hnen) possible generatorsare Dand , wheren
n
and where is the automorphism of ex
hanging and
n−1
• ∆ .= D φτr r ≥ 1 φ
4
if ,Sthen possible generators are , where and where
3
belongs to theDpermutation group on 3 elements seen as subgroup of
4
• au∆tom=oErphisms of . τr φτr r≥1
6
iφf , then possible geEnerators are 2and 5 , whe1re 6 and where
6
• i∆s t=heEautomonrp=his7m,8of ex
hanging and τ,rand anrd≥.1
n
if with , possible generators are , where .
TheAuniqnue weakly admissible automρorphism group whi
h is not admissible exists
n
for , even, and is generated by .
3. Property of the Auslander-Reiten translation
Γ T
T
We de(cid:28)ne the Auslander-Reiten quiver of the
ategory as a valued quiver
(Γ,a)
. The verti
es are the iso
lasses of inde
omposable obje
ts. Given two inde-
X Y T x = [X] y = [Y]

omposable obje
ts and of , we draw one arrow from to
R(X,Y)/R2(X,Y) R(?,?)
if the ve
tor spa
e is not zero, where is the radi
al of
Hom (?,?) R(X,Y)
T
the bifun
tor . A morphism of whi
h does not vanish in the
R(X,Y)/R2(X,Y)
quotient will be
alled irredu
ible. Then we put
a =dim R(X,Y)/R2(X,Y).
xy k
T
Remarkthatthefa
tthat islo
ally(cid:28)nite implies thatitsAR-quiverislo
ally
Γ τ
T
(cid:28)nite. The aim of this se
tion is to show that with the translation de(cid:28)ned in
the (cid:28)rst part is a valued translation quiver. In other words, we want to show the
proposition:
X Y T
Proposition 3.0.2. If and are inde
omposable obje
ts of , we have the
equality
dim R(X,Y)/R2(X,Y)=dim R(τY,X)/R2(τY,X).
k k
Let us re
all some de(cid:28)nitions [11℄.
g :Y →Z
De(cid:28)nition 3.0.2. A morphism is
alled sinkmorphism if the following
hold
g
(1) is not a retra
tion;
h:M →Z h g
(2) if is not a retra
tion, then fa
tors through ;
ON THE STRUCTURE OF TRIANGULATED CATEGORIES 7
u Y gu=u u
(3) if is an endomorphism of whi
h satis(cid:28)es , then is an automor-
phism.
f :X →Y
Dually, a morphism is
alled sour
e morphism if the following hold:
f
(1) is not a se
tion;
h:X →M h f
(2) if is not a se
tion, then fa
tors through ;
u Y uf =f u
(3) if is an endomorphism of whi
h satis(cid:28)es , then is an automor-
phism.
X Z
These
onditions imply that and are inde
omposable. Obviously, if
u v w
X //Y // Z //SX u
is an AR-triangle, then is a sour
e morphism
v v ∈ Hom (Y,Z)
T
and is a sink morphism. Conversely, if is a sink morphism
u ∈ Hom (X,Y)
T
(or if is a sour
e morphism), then there exists an AR-triangle
u v w
X //Y // Z //SX
(see [11, I 4.5℄).
The following lemma (and the dual statement) is proved in [28, 2.2.5℄.
g Y Z Z
YLe=mmar 3.Y0.n3i. Let be a morphism fYrom to , where is inde
omposable and
Li=1 i isthede
omposition of intoinde
omposables. Thenthemorphism
g
is a sink morphism if and only if the following hold:
i = 1,...,r j = 1,...,n g
i i,j
(1) For ea
h and , the morphism belongs to the
R(Y ,Z)
i
radi
al .
i = 1,...,r (g ) k
(2) For ea
h , the family i,j j=1,...,ni forms a -basis of the spa
e
R(Y ,Z)/R2(Y ,Z)
i i
.
h ∈ Hom (Y′,Z) Y′ h
T
(3) If is irredu
ible and inde
omposable, then fa
tors
g Y′ Y i
i
through and is isomorphi
to for some .
Using this lemma, it is easy to see that proposition 3.0.2 holds. Thus, the
Γ = (Γ,τ,a) T
T
Auslander-Reiten quiver of the
ategory is a valued translation
quiver.
4. Stru
ture of the Auslander-Reiten quiver
This se
tion is dedi
ated to an other proof of a theorem due to J. Xiao and B.
Zhu ([33℄):
T
Theorem4.0.4. [33℄Let beaKrull-S
hmidt, lo
ally (cid:28)nitetriangulated
ategory.
Γ T
Let ∆be a
onnAe
teDd
omEponent of the AR-quiver of . Then there exGists aZD∆ynkin
tree of type , or , a weakly admissible automorphism group of and
an isomorphism of valued translation quiver
θ :Γ ∼ //Z∆/G.
∆
The undΓerlying graph of tGhe tree is unique up to isomorph(Zis∆m)(it is
alled the
type of ), and the group is unique up to
onjuga
y in Aut .
T
InGparti
ular, if hΓas an in(cid:28)nitenumberofisZo∆
lasses ofinde
omposable obje
ts,
then is trivial, and is the repetition quiver .
4.1. Auslander-Reiten quivers with a loop. In this se
tion, we suppose that
T
the Auslander-Reiten quiver of
ontains a loop, i.e. there exists an arrow with
X
same tail and sour
e. Thus, we suppose that there exists an inde
omposable of
T
su
h that
dim R(X,X)/R2(X,X)≥1.
k
X T
Proposition4.1.1. Let beaninde
omposable obje
tof . Supposethatwe have
dim R(X,X)/R2(X,X)≥1. τX X
k
Then is isomorphi
to .
To prove this, we need a lemma.
8 CLAIREAMIOT
X f1 //X f2 // ··· fn // X
1 2 n+1
Lemma4.1.2. Let beasequen
eofirredu
ible
n ≥ 2 f ◦
n
morphisms between inde
omposable obje
ts with . If the
omposition
f ···f i τ−1X X
n−1 1 i i+2
is zero, then there exists an su
h that is isomorphi
to .
n n=2
Proof. The proof pro
eeds by indu
tion on . Let us show the assertionfor .
X f1 //X f2 //X f ◦f =0
1 2 3 2 1
Suppose is a sequen
e su
h that . We
an then

onstru
t an AR-triangle:
X1 (f1,f)//TX2⊕X(g1,g2)//τ−1X1 // SX1
(f2,0)
β
(cid:15)(cid:15) yy
X
3
f ◦f f g
2 1 2 1
The
omposition is zero, thus the morphism fa
tors through . As the
g f β X
1 2 3
morphisms and are irredu
ible, we
on
lude that is a retra
tion, and
τ−1X X β
1 1
a dire
t summand of . But is inde
omposable, so is an isomorphism
X τ−1X
3 1
between and .
n− 1
Now suppose that the property holds for an integer and that we have
f f ···f = 0 f ···f
n n−1 1 n−1 1
. If the
omposition is zero, the proposition holds by
i ≤ n−2 τ−1X X
i i+2
indu
tion. So we
an suppose that for , the obje
ts and are
i i≤n−1
notisomorphi
. Weshownowbyindu
tionon thatforea
h ,thereexists
β :τ−1X →X f ···f =β g g :X →τ−1X
i i n+1 n i+1 i i i i+1 i
a map su
h that where is
i=1
an irredu
ible morphism. For , we
onstru
t an AR-triangle:
X1 (f1,f1′//)XT 2⊕X1′(g1,g1′)//τ−1X1 // SX1
(fn···f2,0)
(cid:15)(cid:15) yy β1
X
n+1
f ···f f ···f =β g
n 1 n 2 1 1
As the
omposition is zero, we have the fa
torization .
i τ−1X X
i−1 i+1
Nowfor , as is not isomorphi
to , there exists an AR-triangle of
the form:
X (gi−1,fi,fi′)T//τ−1X ⊕X ⊕X′ (gi′′,gi,gi′) //τ−1X //SX
i i−1 i+1 i i i
(−βi−1,fn···fi+1,0)
(cid:15)(cid:15) tt βi
X
n+1
−β g +f ···f f f ···f g
i−1 i−1 n i+1 i n i+1 i
By indu
tion, is zero, thus fa
tors through .
i = n−1 β : τ−1X → X
n−1 n−1 n+1
This property is true for , so we have a map
β g = f g f β
n−1 n−1 n n−1 n n−1
su
h that . AXs anτd−1Xare irredu
ible, we
on
lude that (cid:3)
n+1 n−1
is an isomorphism between and .
Now we are able to prove proposition 4.1.1. There exists an irredu
ible map
f :X →X X τX
. Suppose that and are not isomorphi
. Then from the previous
fn n T
lemma, the endomorphism is non zero for ea
h . But sin
e is a Krull-
R(X,X)
S
hmidt, lo
ally (cid:28)nite
ategory,apowerofthe radi
al vanishes. This isa

ontradi
tion.
Γ˜ =(Γ˜ ,Γ˜ ,a˜)
0 1
4.2. ProofoftΓheorem4.0.4. Let beΓt˜he=vaΓluedΓ˜tra=ns{lαat∈ionΓquiver
0 0 1 1
obtained from by removing the loops, i.e. we have , su
h
s(α)6=t(α)} a˜=a
that , and |Γ˜1.
Γ˜ =(Γ˜ ,Γ˜ ,a˜) τ
0 1
Lemma 4.2.1. The quiver with the translation is a valued trans-
lation quiver without loop.
ON THE STRUCTURE OF TRIANGULATED CATEGORIES 9
σ
Proof. We have to
he
k that the map is well-de(cid:28)ned. But from proposition
α x σ(α) τx = x x
σ4.(1α.1),=ifα is a looΓ˜p on a vertex , Γ is the unique arroσw from to , i.e.
. Thus is obtained from by removing some -orbits and it keeps th(cid:3)e
stru
ture of stable valued translation quiver.
Now, we
an apply Riedtmann's Struktursatz [24℄ and the result of Happel-
∆
GPreiser-Ringel [13℄. There exZis∆t a tree anΓ˜d an admissible auZto∆m/oGrphism group
(whi
h may be trivial) of su
h that is isomorphi
to as a valued
∆
translationquiver. ThGeunderlyinggraphofthetree isth(Zen∆u)niquexuptoisomor-
phism and the group is unique up to
onjuga
y in Aut . Let be a vertex
∆ x x
of . We write for the image of by the map:
∆ //Z∆ π //Z∆/G≃Γ˜(cid:31)(cid:127) //Γ.
C :∆ ×∆ →Z
0 0
Let be the matrix de(cid:28)ned as follows:
• C(x,y) = −a −a x y
xy yx
(resp. ) if there exists an arrow from to (resp.
y x ∆
from to ) in ,
• C(x,x)=2−a
xx
,
• C(x,y)=0
otherwise.
C
The matrix is symmetri
; it is a `generalizedCartan matrix' in the sense of [12℄.
C
If we remove the loops from the `underlying graph of ' (in the sense of [12℄), we
∆
get the underlying graph of .
In order to apply the result of Happel-Preiser-Ringel [12, se
tion 2℄, we have to
show:
∆ ∆
0
Lemma 4.2.2. The set of verti
es of is (cid:28)nite.
∆ x Γ˜
0 0
Proof. Riedtmann's∆
onstru
tion of Γ˜is the followingx. We (cid:28)x a vertex in .
0
Thentheverti
esof αaσr(eαt)hepathsαof beΓ˜ginningon andwhi
∆hdonot
ontain
1 0
subpaths of the form , where is in . Now suppose that is an in(cid:28)nite
n
set. Then for ea
h , there exists a sequen
e:
x0 α1 //x1 α2 //··· αn−1// xn−1 αn //xn
τx 6= x X ,...,X
i+2 i 0 n
su
h that . Then there exist some inde
omposables su
h
R(X ,X )/R2(X ,X )
i−1 i i−1 i
thattheve
torspa
e isnotzero. Thusfromthelemma
f :X →X
i i−1 i
4.1.2, thereexistsirredu
iblemorphisms su
hthatthe
omposition
f f ···f Hom (X ,?)
n n−1 1 T 0
does not vanish. But the fun
tor has (cid:28)nite support.
Y
Thusther(eXis)aninde
ompRosNab(Yle,Y)whi
happearsanNin(cid:28)nitenumberoftimesinth(cid:3)e
i i
sequen
e . Butsin
e vanishesforan ,wehavea
ontradi
tion.
S T
Let a system of representatives of iso
lasses of inde
omposables of . For an
Y T
inde
omposable of , we put
l(Y)= X dimkHomT(M,Y).
M∈S
T
This sum is (cid:28)nite sin
e is lo
ally (cid:28)nite.
x ∆ d =l(x) x∈∆
0 x 0
Lemma 4.2.3. For in , we write . Then for ea
h , we have:
X dyCxy =2.
y∈∆0
X U T
Proof. Let and be inde
omposables of . Let
u v w
X //Y // Z // SX
10 CLAIREAMIOT
(U,?) Hom (U,?)
T
be an AR-triangle. We write for the
ohomologi
al fun
tor .
Thus, we have a long exa
t sequen
e:
(U,S−1Z)S−1w∗//(U,X) u∗ //(U,Y) v∗ //(U,Z) w∗ //(U,SX).
S (U) w
Z ∗
Let be the image of the map . We have the exa
t sequen
e:
0 // SS−1Z(U) // (U,X) u∗ //(U,Y) v∗ // (U,Z) w∗ //SZ(U) //0.
Thus we have the following equality:
dimkSZ(U)+dimkSS−1Z(U)+dimk(U,Y)=dimk(U,X)+dimk(U,Z).
U Z U Z S (U)
Z
If isnotisomorphi
to ,ea
hmapfrom to isradi
al,thus iszero. If
U Z w End(Z) S (Z)
∗ Z
is isomorphi
to , the map fa
tors through the radi
al of , so
k U S
is isomorphi
to . Then summing the previous equality when runs over , we
get:
l(X)+l(Z)=l(Y)+2.
l τ l(Z) l(X) Y
Clearly r isY-niinvariant, thus equals . If the de
omposition of is of the
form Li=1 i , we get:
l(Y)=Xnil(Yi)= X aXYil(Yi)+aXXl(X).
i i,X→Yi∈Γ˜
We dedu
e the formula:
2=(2−aXX)l(X)− X aXYil(Yi).
i,X→Yi∈Γ˜
x ∆ x Γ˜ x → Y
LΓ˜et be a vertex of the(xt,r0e)e→(ayn,d0) itsZ∆image in . Then an(xa,r0r)o→w (y,−1) in
Z∆
omes from an arrow (y,0) → (x,0)in or from an arrow Z∆ → Z∆/G in
, i.e. from an arrow . Indeed the proje
tion is a

overing. From this we dedu
e the following equality:
2=(2−axx)dx− X axydy − X ayxdy = X dyCxy.
y,x→y∈∆ y,y→x∈∆ y∈∆0
(cid:3)
C
Nowwe
anprovetheorem4.0.4. Thematrix isa`generalizedCartanmatrix'.
Thepreviouslemmagivesusasubadditivefun
tionwhi
hisnotadditive. Thusby
C C
[12℄, theunderlyinggraphof iAsofD`gEeneralLizedDynkin type'. As issym∆metri
,
the graph is nae
essarily of type , , , orA .DButEthis graph is the graph with
the valuationL . We are done in the
ases , , or .
n
L TheA
ase o

urs when the AR-quiver
ontains at least one loop. We
an see
n n
as with valuationsZoLn the verti
es with aτlorop. Thenr,≥it1is obvious that the
n
automorphism groups of are generated by for an . But proposition
x τx=x G τ
4.1.1 tell us that a vertex with a loop satis(cid:28)es . Thus is generatedby
and the AR-quiver has the following form:
1hh (( 2hh ((3 ff ((n ee
ZA /G G
2n
Thisquiverisisomorphi
to thequiver where isthegroupgeneratedby
τnS =ρ
the automorphism .
S
Thesuspensionfun
tor sendstheinde
omposablesoninde
omposables,thusit

an be seen asan automorphismof the AR-quiver. It is exa
tly the automorphism
S
de(cid:28)ned in se
tion 2.2.
∆
As shown in [33℄, it follows from the results of [19℄ thatGfor eaZ
∆h Dynkin tree
and for ea
h weakly admissible group of automorphisms of , there exists a