Report on locally finite triangulated categories

REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES HENNING KRAUSE Abstract. The basic prop erties of lo cally ﬁnite tri angulated categories are discussed. The focus is on Auslander–Reiten theory and the lattice of thick subcategories. Contents 1. Int ro duction 1 2. Lo cally no etheria n triangulated catego ries 2 3. Auslander–Reiten theory fo r triangulated ca teg ories 7 4. The lattice of thick sub catego ries 10 5. Simply connected triangulated categor ies 12 6. Thick subcategor ies and non-cro ssing partitions 14 Appendix A. Auslander–Re iten theor y 19 References 25 1. Introduction This is a rep or t on a pa rticular class of tr iangulated ca tegories . A triangulated category T is said to b e lo c al ly ﬁnite if every cohomolo gical functor from T or its opposite category T op int o the c a tegory of ab elian gr oups is a direct s um of representable functors. W e present a num ber o f bas ic r esults for such tria ngulated categor ies. Some of these results seem to b e new, but w e include als o results which are v ariations or generalisa tions of known results. Thu s our aim is to provide the foundatio ns for studying the lo cally ﬁnite triangulated ca teg ories. A basic to ol for under standing a tria ngulated category T is the c o llection of representable functors Hom T ( − , X ) : T op → Ab where X r uns throug h the ob j ects of T . W e show that T is locally ﬁnite if and o nly if eac h represen table functor is of ﬁnite length. This pr op erty justiﬁes the ter m ‘lo ca lly ﬁnite’ whic h is due to Xiao and Zhu [51] in the triangulated context and go es back to Gabriel [21]. An imp ortant thread in the study of lo cally ﬁnite tria ngulated categor ies is the use of A uslander–Reiten theory . The principal idea is to analyse for eac h o b ject X in T the r adical ﬁltration . . . ⊆ Rad 2 T ( − , X ) ⊆ Rad 1 T ( − , X ) ⊆ Rad 0 T ( − , X ) = Hom T ( − , X ) which is ﬁnite when T is lo cally ﬁnite. Some of this infor mation is enco ded in the Auslander–Reiten quiver of T which can be describ ed fairly explicitly . 2010 Mathematics Subje ct Classiﬁc ation. 18E30(primary), 16E35, 16G 70. Key wor ds and phr ases. T r iangulated category , derived category , lo cally no etherian, lo cally ﬁnite, thick subcategory , Auslander–Reiten the ory . 1 2 HENNING KRA USE Another in triguing in v a riant of a triangulated category is the lattice of thic k sub c ategories . Ass uming lo cally ﬁniteness, we show t hat the inclus ion of each thic k sub c ategory admits a le ft and a right a djoint . In fact, the lattice has in teresting symmetries and is e ven ﬁn ite if the c a tegory is ﬁnitely g e ne r ated. The r esults pres ented here ar e mo st c o mplete when the catego ry is simply c on- ne cte d , that is, the Auslander– Reiten quiver is co nnected and contains no oriented cycle. F or instance, the lattice o f thick sub categor ie s is in this case isomor phic to the lattice o f non-crossing partitions ass o ciated to some diagra m of Dynkin type. Using co v ering theory , the study o f a gener a l lo cally ﬁnite triangulated c a tegory can often b e reduced to the s imply co nnected ca se. F or this direction we refer to recent w ork of Amiot [1] and K¨ ohler [36]. Ac kno wledgemen ts. It is a pleasure to thank n umerous colle agues for in teresting discussions and helpful comments on the sub ject of this w ork. Let me mention explicitly Ap ostolos B e ligiannis, Otto Kerner , Claudia K¨ ohler, Shiping Liu, Cla us Ringel, Jan ˇ S ˇ tov ´ ıˇ cek, Hugh Tho mas, and Dieter V ossieck. I am also gra teful to an anonymous referee for many helpf ul sugge stions for improving the expositio n. 2. Locall y noetherian triangul a ted ca tegories In this section, lo cally no etherian and lo ca lly ﬁnite triangulated categories are int ro duced. W e provide v arious characterisa tions a nd a host of examples. Then w e establish the existence o f adjoints for inclusions of thick subcateg o ries. Throughout this work let T denote a tria ngulated catego ry with suspensio n Σ . The ab eli anisation of a triangul ated category. F ollowing F reyd [2 0, § 3] and V erdier [48, I I.3], w e consider the ab elianisation A ( T ) of a triangulated categ ory T which is by deﬁnition the ab elian ca tegory consisting of all a dditive functors F : T op → Ab into the categor y of ab elian gr oups that ﬁt into an exact sequence Hom T ( − , X ) − → Hom T ( − , Y ) − → F − → 0 . The fully faithful Y oneda functor H : T → A ( T ) taking an ob ject X to the rep- resentable functor Hom T ( − , X ) is the universal cohomolog ical functor sta r ting in T , that is, each cohomolog ic a l functor T → A to an ab elia n ca tegory A f actors essentially uniquely thr ough H . Observe that the functor ta king Hom T ( − , X ) to Hom T ( X, − ) induces an eq uiv a lence (2.1) A ( T ) op ∼ − → A ( T op ) . This is an immediate consequence of the universal proper ty o f the Y oneda functor. Lo cally no etherian triangulated categories. Given a n es sentially small trian- gulated category T , w e use its abelianisatio n to for mulate a useful ﬁniteness condi- tion; see also [9]. W e say that T is lo c al ly no et herian 1 if the equiv alent conditions of the following theorem are satisﬁed. Theorem 2.1 . F or an essen tial ly smal l triangulate d c ate gory T t he fol lowing c on- ditions ar e e quivalent. (1) Every c ohomolo gic al fu nctor T op → Ab into the c ate gory of ab elian gr oups is a dir e ct sum of r epr esentable functors. 1 The terminology refers to th e equiv alent fact tha t the ab elian category of additiv e f unctors T op → Ab is l ocally noetherian in the sense of [21, I I.4]. REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 3 (2) Every obje ct in T is a ﬁnite c opr o duct of inde c omp osable obje cts with lo c al endomorphi sm rings, and for every se qu enc e X 1 φ 1 − → X 2 φ 2 − → X 3 φ 3 − → . . . of non-isomorphisms b etwe en inde c omp osable ob je cts ther e exists some numb er n s u ch that φ n . . . φ 2 φ 1 = 0 . (3) Idemp otents in T split and every obje ct of the ab elianization A ( T ) is no e- therian, that is, every asc ending chain of su b obje ct s in A ( T ) eventual ly stabilises. Pr o of. W e v iew the additive c a tegory T as a ring with sever al obje cts and think of additive functors T op → Ab as T -mo dules. Note that a T -mo dule is ﬂat if and only if it is a cohomolo g ical functor; s ee [37, Lemma 2.7]. Bass has characterised the rings for which ev ery ﬂat module is pro jective. This can be generalise d to mo dules ov er rings with several ob jects, see [3 0, Theor em B.12], a nd yields the equiv a le nce of conditions (1) and (2). Recall that a mo dule M is fp-inje ctive if Ext 1 ( − , M ) v anishes on a ll ﬁnitely pr e- sented mo dules. Note that T a s a ring with several o b jects is no etherian (that is, each represe n table functor Hom T ( − , X ) satisﬁes the as cending chain condition o n subfunctors) if and o nly if every fp-injectiv e T -mo dule is injectiv e; see [30, The- orem B.1 7]. The fp-injective T -mo dules are precisely the co homologica l functors T op → Ab , by [37, Lemma 2.7]. Suppo se th at (1) holds and ﬁx a n fp-injectiv e T -module M . Cho ose a n injectiv e env elop e φ : M → Q . The snake lemma shows that the cokernel Coker φ is coho- mological. Th us Coker φ is a direct sum of representable functor s and therefo re pro jective; in particular φ splits. It follows that M is injective, and therefore T is no etherian. It re ma ins to show that T is idemp otent co mplete. But this is c lear bec ause a direct s ummand of a r epresentable functor is co homologica l and therefor e representable. Th us (3) holds. Now s uppo se that (3) holds . Thus the ascending chain condition holds for chains of ﬁnitely presented submo dules of mo dules of the form Ho m T ( − , X ). This implies the as cending chain conditions for ar bitr ary submo dules, since each submo dule is a union of ﬁnitely gener ated submodules a nd ea ch ﬁnitely g enerated submo dule of a ﬁnitely prese nted one is aga in ﬁnitely presented. It follows that T is no etherian. Fix a ﬂat T -mo dule M and choose an epimor phism φ : P → M such that P is pro jective. The snake lemma shows that the kernel Ker φ is cohomolog ical. Thus Ker φ is fp-injective a nd therefore injective; in particula r φ splits. It follows that M is pro jective, and therefore a direct sum of ﬁnitely generated pro jective mo dules; see [3 0, Corolla ry B.13]. The ﬁnitely generated pro jective modules a re precisely the representable functors since T has split idempo ten ts. T hus (1) holds.  Example 2.2. Fix a ﬁeld k and denote by A the category of k -line a r r e presentations of the quiver Γ : 1 − → 2 − → 3 − → 4 − → · · · that are ﬁnite dimensional and have ﬁnite suppo rt. Then the bo unded derived category D b ( A ) is lo cally no etherian but its opp os ite category is not. Indeed, each ob ject in D b ( A ) decomp os e s int o a ﬁnite copro duct of indecom- po sable o b jects with lo cal endomorphism r ings. The indecompo sable o b jects are isomorphic to co mplexes c oncentrated in a sing le deg r ee (thus of the form X [ i ] with X ∈ A and i ∈ Z ) since Ext p A ( − , − ) = 0 fo r p > 1. Given a sequence X 1 , . . . , X r of ob jects in A a nd a sequence o f morphism X 1 [ i 1 ] → X 2 [ i 2 ] → · · · → X r [ i r ] in D b ( A ) such tha t their comp osite is non-zero, we hav e 4 HENNING KRAUSE i 1 ≤ . . . ≤ i r ≤ i 1 + 1. Th us it remains to observe that inﬁnite chains of non- isomorphisms b etw een indecomp osa ble ob jects in A exist o nly in one dir ection. More precisely , fo r each n ≥ 1 let A n denote the full sub categor y of repr esenta- tions with supp ort contained in { 1 , . . . , n } . Then any non- z e ro mo r phism X → Y betw een indecomp os able representations ha s the pro p er ty that X ∈ A n implies Y ∈ A n . This is an immediate conseq uence o f the fact that each indeco mp o s able representation of Γ is, up to isomorphism, of the following fo rm: 0 → · · · → 0 → k 1 − → · · · 1 − → k → 0 → · · · On the other ha nd, ther e is an o bvious chain of prop er epimorphisms · · · → X 3 → X 2 → X 1 in A , where X n denotes the unique indecomposable representation with supp ort { 1 , . . . , n } . Lo cally ﬁnite triangulated categories. An essentially small triangulated ca t- egory T is said to be lo c al ly ﬁn ite if T and T op are lo cally no etherian. The ﬁrst condition means that each representable functor Hom T ( − , X ) is a no etheria n T - mo dule. In particular, ea ch s ubo b ject b elong s to A ( T ). Combining the second condition with the equiv alenc e A ( T ) op ∼ → A ( T op ), it fo llows that Hom T ( − , X ) is an a rtinian T -mo dule. Thus lo cally ﬁnite means that each r epresentable functor Hom T ( − , X ) is of ﬁnite length as a T -mo dule. In par ticular, T is lo cally ﬁnite if and only if the categor y of T -mo dules is lo cally ﬁnite in the sense of [21, II.4 ]. Suppo se th at T is lo c ally ﬁnite and ﬁx an ob ject Y . Then there ar e only ﬁnitely many isomorphism cla sses o f indecomp osable ob jects X satisfying Hom T ( X, Y ) 6 = 0, bec ause Hom T ( X, Y ) 6 = 0 implies that Hom T ( − , X ) is the pro jective cov e r of a comp osition factor of Hom T ( − , Y ). This obse rv ation gives rise to the following characterisation whic h can b e deduced from [3, Theorem 2.1 2]. Prop ositi o n 2.3 (Auslander) . An essential ly smal l triangulate d c ate gory T with split id emp otents is lo c al ly ﬁnite i f and only if for e ach obje ct Y t he f ol lowing holds: (1) The obje ct Y de c omp oses into a ﬁnite dir e ct sum of ind e c omp osable obje cts. (2) Ther e ar e only ﬁnitely many isomorphism cla sses of inde c omp osable obje cts X satisfying Hom T ( X, Y ) 6 = 0 . (3) F or e ach inde c omp osable obje ct X , the E nd T ( X ) - m o dule Hom T ( X, Y ) is of ﬁnite length.  Examples. W e lis t some examples of tria ng ulated categories that ar e loc a lly ﬁnite. Throughout we ﬁx a ﬁeld k . (1) Let T b e an essentially small k -linear triangula ted categor y . Suppose that idempo tent s split and tha t mo r phism spac es are ﬁnite dimensional. Then T is lo cally ﬁnite if and only if for each ob ject Y there are only ﬁnitely many is omorphism classes of indecompo sable ob jects X satisfying Hom T ( X, Y ) 6 = 0 . This f ollows fro m Prop ositio n 2.3 and serves as a deﬁnition in [51]. (2) Let A b e a ﬁnite dimensional k -algebra and supp ose that k is algebraica lly closed. Then the b ounded derived categor y D b (mo d A ) o f the categor y of ﬁnite dimensional A -mo dules is lo cally ﬁnite if a nd o nly if it is tr iangle eq uiv a lent to D b (mo d k Γ ) for some path alge bra k Γ of a ﬁnite quiver Γ such that its underly- ing diagr am is a disjoin t union of diagrams of Dynkin type; s ee [25, § 5] and [10, Theorem 12.2 0]. (3) Let A b e an ess ent ially small he r editary ab elia n categor y . Then the derived category D b ( A ) is lo cally ﬁnite if a nd o nly if A sa tisﬁes the co nditions in Pro po si- tion 2.3. This follows from the fact that each indecomposa ble ob ject is isomorphic to a complex that is concentrated in a single deg ree. If A is the categ ory of ﬁnitely REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 5 generated modules over a n ar tinian r ing, then this conditio n mea ns that the r ing is of ﬁnite r e pr esentation t yp e. (4) Let A be a noetherian ring and suppose that A is Gor enst ein , that is, A ha s ﬁnite injective dimension a s an A -mo dule. Denote b y MCM( A ) the catego ry of ﬁnitely ge nerated A -mo dules X that are maximal Cohen–Mac aulay , which means that Ext i A ( X, A ) = 0 for all i > 0. This is an exact F rob enius categ ory , and the stable categ ory MCM ( A ) mo dulo all morphisms that factor through a pro jective ob ject is a triang ulated categor y [16]. If A is a ﬁnite dimensional and se lf-injective k -algebra, then all A -mo dules ar e maximal Cohen–Ma caulay , and MCM ( A ) is locally ﬁnite if and only if A is of ﬁ nite r epr esentation typ e , that is, there are only ﬁnitely ma ny isomorphism classes of indecomp osable A -mo dules. These alge bras ha ve be e n classiﬁed [4 4, 49 ]. If A is a commut ative co mplete lo cal ring, then A is by deﬁnition of ﬁnite Cohen– Mac aulay typ e if there there a re o nly ﬁnitely many isomo rphism classes of indecom- po sable maximal Cohen– Macaulay mo dules over A . In tha t c a se MCM ( A ) is lo cally ﬁnite. There is a whole theor y describing such r ings and a par a llel theory for gra ded Gorenstein algebr as; see [50]. (5) Le t Γ b e a quiver of Dynkin type. Then the orbit categor y D b (mo d k Γ ) / G of the derived categor y with resp ect to a n appro pr iate group G of a utomorphisms is a loca lly ﬁnite triangulated category [33, 1]. Exa mples are the clus ter categor ies of ﬁnite type [15 ]. (6) The categor y of ﬁnitely generated pro jective modules over the ring Z / 4 Z carries a tria ngulated structure that admits no mo del [4 2]; it is a lo ca lly ﬁnite triangulated categ ory . Orthogonal s u b categories. Let T b e a triangulated categor y and S a triangu- lated sub categor y . Then we deﬁne t wo full sub categor ies S ⊥ = { Y ∈ T | Hom T ( X, Y ) = 0 for all X ∈ S } ⊥ S = { X ∈ T | Hom T ( X, Y ) = 0 for all Y ∈ S } and call them ortho gonal sub c ate gories with resp ect t o S . No te that S ⊥ and ⊥ S a re thic k sub categor ie s of T . The following lemma co llects some basic facts ab o ut o r thogonal subc a tegories which a re well-kno wn. F or a pro of, see [39, Prop o sition 4.9.1]. Lemma 2.4. L et T b e a triangulate d c ate gory and S a thick sub c ate gory. Then the fol lowing ar e e qu ivalent. (1) The inclusion functor S → T admits a right adjoint. (2) The c omp osite S ⊥ inc − − → T can − − → T / S is an e quivalenc e. (3) The inclusion functor S ⊥ → T admits a left adjoint and ⊥ ( S ⊥ ) = S .  There is an interesting conse q uence. If S is a thick sub categ ory o f T such that the inclusion admits a left and a r ight adjoin t, then one has equiv alences ⊥ S ∼ − → T / S ∼ ← − S ⊥ . Existence of adjoin ts. T riangula ted ca teg ories that a re lo cally no etheria n hav e the following remark a ble prop erty . Theorem 2.5. L et T b e an essential ly smal l tria ngulate d c ate gory a nd su pp ose that T is lo c al ly no et herian. Then for e ach thick s u b c ate gory of T the inclusion functor admits a right adjoint. 6 HENNING KRAUSE Pr o of. Fix a thick sub categor y U and an ob ject X in T . W e need to co nstruct a morphism U → X with U in U inducing a bijection Hom T ( U ′ , U ) → Hom T ( U ′ , X ) for all U ′ in U . T ake the c o mma categ ory U /X consisting of a ll morphisms U φ → X with U in U . A mor phism fro m U φ → X to U ′ φ ′ → X is a morphism µ : U → U ′ such t hat φ ′ µ = φ . This ca tegory is closely related to V erdier’s construction of the lo calisation functor T → T / U ; see [48, I I.2]. The arg uments giv en there show that U /X is ﬁltered. Therefore the functor colim U → X Hom T ( − , U ) is c ohomolog ical. Moreover, one obtains an ex act sequence o f co ho mologica l func- tors · · · − → Hom T / U ( − , Σ − 1 X ) − → colim U → X Hom T ( − , U ) − → − → Hom T ( − , X ) − → Hom T / U ( − , X ) − → · · · since one has by deﬁnition Hom T / U ( − , X ) = colim X → V Hom T ( − , V ) where X → V r uns through all morphis ms with cone in U . Now w e use that T is lo cally no ether ian and write colim U → X Hom T ( − , U ) = M i ∈ I Hom T ( − , U i ) as a direct sum o f repres ent able functors. Similarly , we get Hom T / U ( − , X ) = M j ∈ J Hom T ( − , V j ) . W e may a ssume that U i and V j are no n-zero for a ll i , j . Obser ve that U i ∈ U and V j ∈ U ⊥ for all i, j . The morphism Hom T ( − , X ) − → M j ∈ J Hom T ( − , V j ) factors thro ug h a ﬁnite sum L j ∈ J 0 Hom T ( − , V j ). In fac t, the exactness o f the ab ov e sequence implies that V j belo ngs to U for e a ch j ∈ J r J 0 . Thus J = J 0 and therefore Hom T / U ( − , X ) b e longs to A ( T ). It follows that I is also ﬁnite, since colim U → X Hom T ( − , U ) is an ex tens io n of tw o o b jects in A ( T ). This yields a morphism U = ` i U i → X inducing a bijection Hom T ( U ′ , U ) → Hom T ( U ′ , X ) for all U ′ in U .  Corollary 2.6. Le t T b e an essential ly s m al l triangulate d c ate gory and supp ose that T is lo c al ly no etherian. I f U is a t hick sub c ate gory of T , then ⊥ ( U ⊥ ) = U . Pr o of. O ne could deduce th is from Lemma 2.4, but we give the co mplete arg ument bec ause it is short and simple. Clea rly , ⊥ ( U ⊥ ) co n tains U . Now pick an ob ject X in ⊥ ( U ⊥ ). Let U → X b e the universal mo rphism from an ob ject in U to X , a nd complete this t o an exact triangle U → X → V → Σ U . Then V b elongs to U ⊥ and therefore Hom T ( X, V ) = 0 . It follows that X b elo ngs to U .  The following example shows that the identit y ⊥ ( U ⊥ ) = U do es not hold in general. REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 7 Example 2.7. Consider the b ounded de r ived ca teg ory D b (mo d Z ) o f ﬁnitely ge n- erated mo dules over the ring Z o f integers. The complexes with torsio n cohomolog y form a thick subca tegory U = D b tor (mo d Z ) such that ⊥ U = 0. Corollary 2.8. Le t T b e an essential ly s m al l triangulate d c ate gory and supp ose that T is lo c al ly no etherian. If U a t hick sub c ate gory of T , then the c ate gories U and T / U ar e lo c al ly n o etherian. Pr o of. The inclusion U → T induces a fully fa ithful and exa ct functor A ( U ) → A ( T ). Thus every ob ject of A ( U ) is no etheria n. On the o ther hand, there is an equiv alence U ⊥ ∼ − → T / U by Lemma 2.4. Thus T / U is lo cally no etheria n, since U ⊥ is lo cally no etheria n.  3. Auslander–Reiten theor y for trian gula ted ca tegories W e describ e brieﬂy the Auslander– Reiten theory for an es s entially small triangu- lated category T that is lo c ally no ether ian. F or genera l concepts from Auslander – Reiten theory we refer to Appe ndix A. Auslander–Reiten triangles. Recall from [24] that a n exact triangle X α − → Y β − → Z γ − → Σ X is an Auslander–R eiten t riangle sta r ting at X and ending at Z if α is a left al- most s plit morphism and β is a r ight almost split morphism. Observe that these prop erties hold if a nd only if the morphism β is minimal right almo s t split, b y [38, Lemma 2.6]. Prop ositi o n 3.1. Given an essential ly smal l triangulate d c ate gory that is lo c al ly no etherian, t her e ex ists for e ach inde c omp osable obje ct an Auslander– R eiten trian- gle ending at it. Pr o of. Fix an indecomp osa ble ob ject Z in T . Then the simple T -mo dule S Z = Hom T ( − , Z ) / Rad T ( − , Z ) is ﬁnitely presented since T is lo cally no ether ian. Th us we c an choose in A ( T ) a minimal pr o jective pres entation Hom T ( − , Y ) − → Hom T ( − , Z ) − → S Z − → 0; see Pro po sition A.1. It follows from Lemma A.7 that the induced mor phism Y → Z is minimal rig ht a lmost split. Completing this morphis m to a n exact tria ngle y ie lds an Auslander–Reiten tria ng le ending at Z .  The deﬁnition of a n Auslander– Reiten triang le is symmetric. Thus there are Auslander–Reiten triang les in T s tarting at ea ch indeco mpo sable ob ject if T op is lo cally noether ian. This gives the existence o f Ausla nder–Reiten triangles for lo cally ﬁnite triangulated categor ies. F or compa c tly generated triangulated categor ies, this result is due to Beligiannis [11, Theor em 10.2]. Corollary 3.2. Given an essential ly sm al l triangulate d c ate gory that is lo c al ly ﬁnite, t her e exist Auslander–R eiten t riangles starting and ending at e ach inde c om- p osable obje ct .  R emark 3.3 . Let X → Y → Z → Σ X b e an Ausla nder–Reiten triangle in an essentially small triangulated catego ry that is lo ca lly no etherian. The relation betw een the end terms can b e e xplained as follows. Let A = End T ( Z ) and denote by E = E ( A/ rad A ) an injective en velop e. Then (3.1) Hom A (Hom T ( Z, − ) , E ) ∼ = Hom T ( − , Σ X ); 8 HENNING KRAUSE see [38, Theor em 2.2]. In particular, End T ( X ) ∼ = End T ( Σ X ) ∼ = Hom A (Hom T ( Z, Σ X ) , E ) ∼ = Hom A (Hom A (Hom T ( Z, Z ) , E ) , E ) ∼ = End A ( E ) . The Auslander–R eiten quiv e r. F or a lo cally ﬁnite triangulated ca tegory the structure o f the Auslander–Reiten quiv er has b een determined in w o r k of Xiao and Zhu [51] a nd Amiot [1]. In fact, these a uthors consider tria ngulated catego ries that are linear ov er a ﬁeld with ﬁnite dimensional morphism spaces. The n they apply the structura l results on v a lued tr a nslation q uivers due to Riedtmann [43] and Happel, Pr eiser, and Ring el [26]. The same argumen ts work in a slightly more general setting, thanks to the following result; see also [41, P rop ositio n 2.1]. F or the deﬁnition of a value d tr anslation qu iver , see [41, § 2]. The orig ina l deﬁni- tion [27, § 2] excludes lo ops, but they are p ossible in our setting. Prop ositi o n 3.4. L et k b e a c ommutative ring and T a n essential ly s m al l k -line ar triangulate d c ate gory such t hat al l morphism sp ac es ar e of ﬁn ite length over k . Supp ose t hat T is lo c al ly ﬁnite. Then the Auslander– R eiten quiver of T is a value d tr anslation quiver. Assigning to a vertex X the length ℓ ( X ) of Hom T ( − , X ) in the ab elianisation A ( T ) yields a sub additive function on t he set of vertic es such that for e ach vertex Z 2 ℓ ( Z ) = ℓ ( Z ) + ℓ ( τ Z ) = 2 + X Y → Z δ Y , Z ℓ ( Y ) . Pr o of. The existence of Auslander– Reiten triangles has alr eady b een establis hed, and this gives the tra nslation τ . The identities for the v a luation ar e pr ecisely the statements of Lemmas A.11 and A.12. F or the seco nd par t o ne us e s the fact that each Ausla nder–Reiten triangle τ Z → ¯ Y → Z → Σ ( τ Z ) induces an exact s equence 0 → S Σ − 1 Z → Hom T ( − , τ Z ) → Hom T ( − , ¯ Y ) → Hom T ( − , Z ) → S Z → 0 in A ( T ) by Lemma A.7. Then one applies Lemma A.8 which g ives a deco mpo sition ¯ Y = a Y → Z Y δ Y ,Z where Y → Z r uns through all a r rows ending at Z . It remains to o bserve that ℓ ( Z ) = ℓ ( τ Z ). This follo ws fro m the iso morphism (3.1) and the alter na tive descrip- tion of ℓ via ℓ ( X ) = X C ℓ End T ( C ) Hom T ( C, X ) , where C runs throug h a representativ e set of indecomp osable o b jects; see [3, P rop o- sition 2.11]. Note that Hom T ( − , X ) a nd Hom T ( X, − ) have the same length, thanks to the duality (2.1).  Theorem 3.5 (Xiao– Zhu) . L et k b e a c ommutative ri ng and T an essential ly smal l k -line ar triangulate d c ate gory such that al l morphism sp ac es ar e of ﬁnite length over k . Su pp ose that T is lo c al ly ﬁnite. Then e ach c onne cte d c omp onen t of the Auslander– R eiten quiver of T is of the form Z ∆/G for some tr e e ∆ of Dynkin typ e and some gr oup G of au t omorphisms of Z ∆ . Pr o of. Adapt the proo fs o f [51, Theo r em 2.3.4] or [1, Theorem 4.0 .4], using P rop o- sition 3.4.  REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 9 Example 3 .6. Let k b e a ﬁeld and Γ be a quiver o f Dynkin type. Then the Auslander–Reiten quiver of D b (mo d k Γ ) is of the form Z Γ ; see [24, § 4]. A de c omp osition C = C 1 × C 2 of an additive category C is a pair of f ull additive sub c ategories C 1 and C 2 such that ea ch ob ject in C is a direct sum of tw o ob jects from C 1 and C 2 , and Hom C ( X 1 , X 2 ) = 0 = Hom C ( X 2 , X 1 ) for all X 1 ∈ C 1 and X 2 ∈ C 2 . An additive catego ry C is c onne cte d if any decomp osition C = C 1 × C 2 implies C = C 1 or C = C 2 . Prop ositi o n 3 .7. L et T b e an essential ly smal l triangulate d c ate gory that is lo c al ly ﬁnite. Then any non-zer o morphism X → Y b etwe en t wo inde c omp osable obje cts in T g ives a p ath X → · · · → Y in the Auslander–R eiten quiver of T . In p articular, the c ate gory T is c onne cte d if and only if its Auslander–R eiten quiver is c onne cte d. Pr o of. W e prove the ﬁrst statement; the second statemen t is then an immediate consequence. Let φ : X → Y b e a non-z e ro mo r phism. If φ is in vertible, then the path b etw een X and Y has le ngth zero . O therwise, φ fa ctors through the right almost split morphism ending at Y 0 = Y , which exists by Prop ositio n 3.1. It follows fro m Lemma A.8 th at there is an arrow Y 1 → Y 0 and a non-zero morphism X → Y 0 that factors throug h an irre ducible morphism Y 1 → Y 0 . W e contin ue with the co rresp onding mo rphism X → Y 1 , a nd the pro cess terminates since T is locally ﬁnite.  It is in teresting to note that the Auslander –Reiten quiver of T can b e ident iﬁed with the Ex t-quiver [22, 7.1 ] o f the ab elia n leng th category A ( T ), using the bijection from Lemma A.5 b etw een the indecomp osable ob jects of T a nd the simple ob jects of A ( T ). F or the bijection b etw e e n arrows o ne uses Lemma A.6. The Nak a ya ma functor. Let k b e a ﬁeld and T an essentially small k -linear triangulated categ ory with ﬁnite dimensional mo r phism spaces. Supp o se that T is lo cally ﬁnite. Then there is for each ob ject X in T an ob ject N X representing the k -dual of Hom T ( X, − ), that is, Hom k (Hom T ( X, − ) , k ) ∼ = Hom T ( − , N X ) . More precisely , we hav e an isomorphism Hom k (Hom T ( X, − ) , k ) ∼ = M i ∈ I Hom T ( − , Y i ) for some collection of indecomp osable ob jects Y i since T is loca lly no etherian. It follows from Prop ositio n 2.3 that I is ﬁnite. Th us N X = ` i Y i is a r epresenting ob ject. This gives a functor N : T → T which is known as Nakayama functor in representation theory [23], or as Serr e functor in algebraic geometry [13]. It is easily chec ked that this functor is an equiv a lence; a qua s i-inv ers e is g iven by the Nak ay ama functor for T op which sends a n ob ject X to the ob ject repr esenting Hom k (Hom T ( − , X ) , k ). The exactness then follows fr om [13, Prop os itio n 3.3] or [47, Theorem A.4.4 ]. Note that for each indecomp osable ob ject Z in T , one obtains an Ausla nder– Reiten tria ngle X → Y → Z → Σ X b y ﬁrst c ho osing a no n- zero k - linear map End T ( Z ) → k a nnihilating the unique ma ximal ideal of End T ( Z ) and then com- pleting the corresp onding mo rphism Z → N Z to an exact triang le Σ − 1 ( N Z ) → Y → Z → N Z ; s e e [38] for details. 10 HENNING KRAUSE 4. The la ttice of thick subca tegories Let T be an essentially small triangulated category . W e deno te b y T ( T ) the set of a ll thick s ub ca tegories o f T . This set is partially order ed by inclusion. Observe that for a n y co lle ction of thick sub ca tegories U i its in tersection T i U i is thick. Thus T ( T ) is a lattic e , that is, we can for m for ea ch pair of thick sub categ ories U , V its inﬁm um U ∧ V = U ∩ V and its supremum U ∨ V = T S S , where S runs thro ug h all thic k sub categor ie s con taining U and V . In fact, the lattice T ( T ) is c omplete since for any collection o f thic k sub categor ies U i its inﬁmum V i U i and its supremum W i U i exist. If X is an ob ject or co llection of o b jects in T , we write Thic k( X ) for the thick sub c ategory gener a ted b y X , tha t is, the smallest thick subcateg ory containing T . The triangulated categ ory T is ﬁnitely gener ate d if ther e is some ob ject X such that T = Thick( X ). Compactness. Let L be a la ttice. An elemen t x o f L is c omp act if x ≤ W i ∈ I y i implies x ≤ W i ∈ J y i for some ﬁnite subset J ⊆ I . The lattice L is c omp act if it has a greates t element that is compac t. Lemma 4.1. The lattic e T ( T ) is c omp act if and only if T is ﬁnitely gener ate d.  No etherianes s. A lattice L is no etherian if there is no inﬁnite strictly increasing chain x 0 < x 1 < x 2 < . . . in L . Prop ositi o n 4.2 . L et T b e an essential ly smal l triangulate d c ate gory that is ﬁn itely gener ate d and lo c al ly n o etherian. Then the lattic e T ( T ) is no et herian. Pr o of. Let T = Thick( X ) and write H = Hom T ( − , X ). F or eac h t hick sub categ ory U let φ U : X U → X b e the universal morphism from an ob ject in U end ing at X , which exists b y Theorem 2.5. Note that U = Thick( X U ) since T = Thick( X ). Denote by H U the image of the induced morphism Hom T ( − , X U ) → H in A ( T ), and observe that the ind uced morphism π U : Hom T ( − , X U ) → H U is a pro jective cov er . This follows fr o m the uniqueness of φ U with Lemma A.2, since any endomorphism Hom T ( − , X U ) → Hom T ( − , X U ) commuting with π U is an isomo rphism. Let V b e another thick sub catego ry of T . Then H U = H V implies U = V . Indeed, H U = H V implies that their pr o jective covers are isomorphic. Thus X U ∼ = X V , and therefore U = Thick( X U ) = Thic k( X V ) = V . Clearly , U ⊆ V implies H U ⊆ H V . It follows that T ( T ) is no etherian, since the lattice of sub ob jects of H in A ( T ) is no etheria n.  Example 4.3. Recall from Example 2.2 that re presentations of the quiv e r 1 − → 2 − → 3 − → 4 − → · · · give rise to a tria ngulated ca tegory D b ( A ) that is lo cally no etherian. The full sub c ategories A n ⊆ A consisting of all repre s entations with s uppo rt con tained in { 1 , . . . , n } yield an inﬁnite str ic tly increasing chain D b ( A 1 ) ⊆ D b ( A 2 ) ⊆ . . . of thick sub c ategories in D b ( A ). Compleme n ts. Assigning to a thick sub categ ory U its orthog onal sub ca teg ories U ⊥ and ⊥ U yields tw o o rder reversing maps T ( T ) → T ( T ). The s e maps ar e of int erest b ecause we hav e for tw o ob jects X , Y in T Hom ∗ T ( X, Y ) = 0 ⇐ ⇒ Thick( X ) ⊥ ≥ Thick( Y ) REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 11 where Hom ∗ T ( X, Y ) = M n ∈ Z Hom T ( X, Σ n Y ) . Let us collect the basic pro pe r ties of b oth maps when T is lo cally ﬁnite. Prop ositi o n 4 . 4. L et T b e an essent ial ly smal l triangulate d c ate gory and supp ose that T is lo c al ly ﬁnite. (1) The maps T ( T ) → T ( T ) taking U to U ⊥ and ⊥ U ar e m u tual ly inverse. Thus the lattic e T ( T ) is s elf-dual. (2) L et U b e a t hick su b c ate gory. Then U ∧ U ⊥ = 0 and U ∨ U ⊥ = T . (3) L et V ⊆ U ⊆ T b e thick sub c ate gories. Then the qu otient U / V is a lo c al ly ﬁnite triangulate d c ate gory, and t her e is a latt ic e isomorphism [ V , U ] = { X ∈ T ( T ) | V ⊆ X ⊆ U } ∼ − → T ( U / V ) . The map takes X ⊆ U to X / V . Pr o of. (1) W e hav e ⊥ ( U ⊥ ) = U = ( ⊥ U ) ⊥ by C o rollar y 2.6. (2) Clearly , U ∩ U ⊥ = 0. On the other hand, each ob ject X in T ﬁts into a n exact triangle U → X → V → Σ U with U ∈ U a nd V ∈ U ⊥ , by Theorem 2.5. Thu s U ∨ U ⊥ = T . (3) The c a tegory U / V is lo cally ﬁnite by Cor ollary 2 .8, a nd the inv erse map T ( U / V ) → [ V , U ] takes X to its inv e r se imag e under the lo calisa tio n functor U → U / V .  This pr op osition s ays that the lattice is r elatively c omplemente d , that is, each int erv al is complemen ted. A lattice L is c omplemente d if for ea ch x ∈ L there exists y ∈ L such that x ∨ y = 1 and x ∧ y = 0. If T is lo c a lly ﬁnite and admits a Nak ay ama functor N : T ∼ − → T , then this induces a lattice automorphism o f T ( T ) b y taking a thic k subc a tegory U to N U . It follows from the deﬁnition o f N that ⊥ N U = U ⊥ . Thus N U = ( U ⊥ ) ⊥ by C o rollar y 2.6. Finiteness . Ther e are further ﬁniteness results for the lattice T ( T ) if T is lo cally ﬁnite. Prop ositi o n 4.5. L et T b e an essential ly smal l t riangulate d c ate gory. If T is ﬁnitely gener ate d and lo c al ly ﬁnite, then T has only ﬁnitely many thick sub c ate- gories. Pr o of. Let T = Thick( X ). F o r each thic k sub catego ry U let X U → X b e the univer- sal mor phism f rom an ob ject in U ending at X , whic h exists by Theorem 2.5. Note that U = Thick( X U ) since T = Thick( X ). Each indecomp os a ble dir ect s umma nd X ′ of X U satisﬁes Ho m T ( X ′ , X ) 6 = 0. There a re only ﬁnitely many isomorphism classes of such ob jects b y P rop ositio n 2.3. It follows that T ( T ) is ﬁnite.  Auslander–Reiten theory is us ed in an es sential way f or proving the following. Theorem 4.6. L et T b e an essential ly smal l and lo c al ly ﬁn it e triangulate d c ate gory. If T is c onne ct e d then T is ﬁnitely gener ate d and has ther efor e only ﬁ nitely many thick s u b c ate gories. W e need the following lemma. Lemma 4.7 (Amiot) . L et T b e an essential ly smal l triangulate d c ate gory that is lo c al ly ﬁ n ite. Then any c onne cte d c omp onent of the Auslande r–R eiten quiver of T is – after r emoving p ossible lo ops – of the form Z ∆/G for some ﬁnite tre e ∆ and some gr oup G of aut omorphisms of Z ∆ . 12 HENNING KRAUSE Pr o of. If X → X is an ir reducible morphism in T then X = τ X , by [1, Prop osi- tion 4.1.1]. After removing such arrows X → X , the Auslander–Reiten quiver of T is a stable tra nslation quiver and therefo re of the form Z ∆/G by Riedtmann’s Struktursatz [43]. The tree is ﬁnite by [1, Lemma 4.2.2].  Pr o of of The or em 4.6. The Auslander– Reiten quiver of T is connected by Prop o- sition 3.7 and therefor e – mo dulo lo ops – of the form Z ∆/G for s ome ﬁnite tree ∆ and some group G of a utomorphisms by Lemma 4 .7. Note tha t Z ∆ do es not depe nd on the orientation of ∆ . Thus w e may a ssume that ∆ ha s no path of leng th > 1. Then there a re t wo types of v ertices . Call a vertex x ∈ ∆ 0 minimal if there is no arrow from ∆ ending at x , and maximal if there is no arrow fro m ∆ star ting at x . Let π : Z ∆ → Z ∆/G b e the ca no nical map and write each vertex of Z ∆ a s τ r x with r ∈ Z and x ∈ ∆ 0 . W e claim that T = Thick( T ) for T = ` x ∈ ∆ 0 π ( x ). Thus we ﬁx an indecomp osable ob ject X = π ( y ) in T a nd need to show that X belo ngs to Thick( T ). Deﬁne for y = τ r x d ( y ) = ( 2 r, if x is ma ximal , 2 r − 1 , if x is minimal , and use induction o n d = d ( y ) as fo llows. The cases d = 0 and d = − 1 a re clear . Suppo se now d > 0 and co nsider the Auslander–Reiten triangle X → Y → τ − 1 X → Σ X starting at X . Then eac h indecompos able direct summand of Y is of the form π ( y ′ ) with an ar row y → y ′ in Z ∆ , by Lemma A.8 . W e have d ( y ′ ) = d − 1 and d ( τ − 1 y ) = d − 2. T hus Y and τ − 1 X be lo ng to Thick( T ), a nd it fo llows that X is in Thick( T ). The case d < 0 is similar . The ﬁniteness of T ( T ) follows f rom Pro po sition 4.5.  5. Simpl y connected triangula ted ca tegories Let T b e a triangula ted category tha t is ess ent ially small and lo cally ﬁnite. W e say that T is simply c onne cte d if the Auslander– Reiten quiver of T is connected and has no o r iented cycle. Here , a n oriente d cycle is a path of leng th > 0 starting and ending a t the s ame vertex. Note tha t there exists an oriented cy cle if and only if there is a chain of non-in vertible non-zero morphisms X 0 → X 1 → · · · → X n = X 0 betw een indecomp osable ob jects; see Pro po sition 3.7. Rings of ﬁnite represe ntation t yp e . F or any ring A , we denote by mod A the category of ﬁnitely presented A -mo dules. Reca ll that A ha s ﬁnite r epr esentation typ e if A is artinian and there are o nly ﬁnitely man y is omorphism class es of ﬁnitely presented indecomposa ble A -mo dules. Theorem 5.1. F or a triangulate d c ate gory T the fol lowing ar e e quivalent. (1) The triangulate d c ate gory T is essential ly smal l, algebr aic, lo c al ly ﬁnite, and simply c onne cte d. (2) Ther e exists a c onne ct e d her e ditary artinian ring A of ﬁnite re pr esentation typ e such that T is e qu ivalent to t he b ounde d derive d c ate gory D b (mo d A ) . In this c ase t he Auslander–R eiten qu iver of T is of the form Z ∆ for some t re e ∆ of Dynkin typ e. Let us explain how the ring A is obtaine d from T . A sp e cies ( K i , i E j ) i,j ∈ I consists of a family o f division rings K i and a family of K i − K j -bimo dules i E j . REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 13 There is an a sso ciated tensor alg e bra L n ≥ 0 M n , where M 0 = Y i ∈ I K i , M 1 = M i,j ∈ I i E j , M n = M 1 ⊗ M 0 . . . ⊗ M 0 M 1 for n > 1 , and mu ltiplication is induced by the tensor pro duct. Let T b e a t riangulated category a nd suppose that T is lo c ally ﬁnite and simply connected. T hen the Auslander –Reiten quiver is of the form Z ∆ for some ﬁnite tree b y Lemma 4.7. In fact, we may assume that ∆ has no path of length > 1, since Z ∆ does not dep end on the orien tatio n of ∆ . Cho o se indecomp os able ob jects T i for ea ch vertex i ∈ ∆ 0 . Deﬁne K i = End T ( T i ) a nd i E j = Hom T ( T j , T i ) for i, j ∈ ∆ 0 . Obse r ve that each K i is a division ring by Prop ositio n 3.7, and that each bimo dule i E j is of ﬁnite length on either side by Pr op osition 2 .3. Then the tensor alg e bra corresp o nding to the sp ecies ( K i , i E j ) i,j ∈ ∆ 0 is isomo rphic to the endomorphism ring of T = ` i ∈ ∆ 0 T i . The pr o of is straightforward, using the fact that there ar e no paths o f length > 1 in ∆ . W e denote this ring by A . Observe that A is hereditary and artinian, since K = Q i ∈ ∆ 0 K i is s e misimple and L i,j ∈ ∆ 0 i E j is ﬁnitely gener a ted ov er K o n either side. Lemma 5. 2. Th e obje ct T is a t ilting obje ct, that is, T is gener ate d by T and Hom T ( T , Σ n T ) = 0 for al l n 6 = 0 . Pr o of. W e use P rop osition 3.7 which gives a pa th in the Auslander–Reiten quiver for each non-z e ro morphism betw een indeco mp o sable ob jects. In particular, there is for eac h indecompo sable ob ject X a path X → Y → τ − 1 X → · · · → Σ X , pr ovided there is a non- zero morphism starting in X whic h is not a section. Suppo se that Hom T ( T i , Σ n T j ) 6 = 0 and consider the following t wo cases: n > 0 . F rom Remark 3.3 one has Hom T ( Σ n − 1 T j , τ T i ) 6 = 0. Th us ther e is in Z ∆ a path T j → · · · → Σ n − 1 T j → · · · → τ T i → T ′ i → T i . n < 0 . There is in Z ∆ a path T i → · · · → Σ n T j → · · · → τ T j → T ′ j → T j . In both cases, this contradicts the c hoice of T i , T j ∈ ∆ 0 and the fact that ∆ ha s no path of length > 1. It follows that n = 0. The fact that T = Thick( T ) follows from the pro o f of Theorem 4.6.  Pr o of of The or em 5.1. (1) ⇒ (2 ): The assumptions o n T yield a tilting ob ject T = ` i ∈ ∆ 0 T i , a nd its endomo r phism ring A = E nd T ( T ) is hereditar y artinian. It follows that there are equiv a lences T ∼ ← − K b (add T ) ∼ − → D b (mo d A ) . F or the ﬁr st equiv alence, see [32, 2 .1], while the seco nd is clea r fro m the fact that Hom T ( T , − ) iden tiﬁes add T with the ca tegory of ﬁnitely g enerated pr o jective A - mo dules. Loc a lly ﬁniteness of T implies that A is o f ﬁnite repr esentation type. (2) ⇒ (1): The ab elia n ca tegory mo d A is hereditary , that is , Ext p A ( − , − ) v an- ishes for p > 1 . It follows that ea ch indecomp osable complex is concentrated in a single degr ee. Finite re presentation type of A implies that T is loc ally ﬁnite, by Prop ositio n 2.3. The Auslander– Reiten triangle s in D b (mo d A ) are obtained from almost split seqences in mo d A . T hus the Auslander–Reiten quiv er is of the form Z ∆ for so me tree ∆ ; see [24, § 4] for details . In pa rticular, the quiver has no o riented cycles. F or the sha pe of ∆ , see [18, § 4 ], wher e hereditar y rings o f ﬁnite representation t yp e are discussed.  14 HENNING KRAUSE 6. Thick subca tegories and non-crossing p ar titions Let A b e a her e ditary Artin algebr a . Thus A is an artinia n ring tha t is ﬁnitely generated o ver its centre and Ex t p A ( − , − ) = 0 for all p > 1. Note that the centre of a heredita ry Artin alge bra is se misimple. The alg ebra A is said to b e c onne ct e d if the cen tre of A is a ﬁeld. F ro m now on as sume that A is connected and w e denote the centre b y k . In this sectio n, we asso ciate to A a po set of no n-cross ing par titions and es- tablish a cor resp ondence b etw een this p o set and the lattice of thick subca tgories of D b (mo d A ). This cor r esp ondence was ﬁrst observed for path algebra s of ﬁnite and aﬃne type by Ingalls and Tho mas [29]. F o r an in tro duction to non-crossing partitions, see [2]. The W eyl group. Fix a r epresentativ e s e t S 1 , . . . , S n of simple A -mo dules . W e asso ciate to A a gener alised Cartan matrix C ( A ) = ( C ij ) 1 ≤ i,j ≤ n as follows. Given t wo simple modules S i , S j , we ha ve Ext 1 A ( S i , S j ) = 0 or E xt 1 A ( S j , S i ) = 0. Assume i 6 = j and Ext 1 A ( S j , S i ) = 0. Then deﬁne C ij = − ℓ End A ( S i ) (Ext 1 A ( S i , S j )) and C j i = − ℓ End A ( S j ) (Ext 1 A ( S i , S j )) . In addition, deﬁne C ii = 2 and d i = ℓ k (End A ( S i )) for each i . Then o ne has d i C ij = d j C j i . Thus C ( A ) is a symmetrisable gener a lised Cartan matrix in the sense of [31]. Next w e consider the W eyl group corr esp onding to C ( A ). Let R n denote the n - dimensional real spa ce with standar d basis ε 1 , . . . , ε n . Deﬁne a symmetr ic bilinea r form by ( ε i , ε j ) = d i C ij and for ea ch α ∈ R n with ( α, α ) 6 = 0 the r eﬂection s α : R n − → R n , ξ 7→ ξ − 2 ( ξ , α ) ( α, α ) α. W e write s i for the simple r eﬂe ct ion s ε i and observe that each s i maps Z n int o itself. The Weyl gr oup is the g roup W generated by the simple reﬂections s 1 , . . . , s n . The r e al r o ots are by deﬁnition the elements of Z n of the form w ( ε i ) for some w ∈ W and some i ∈ { 1 , . . . , n } . No te that for a ny real root α t he corresp onding re ﬂec tion s α belo ngs to W since s w ( α ) = ws α w − 1 . W e deﬁne the absolute or der on W with resp ect to the absolute length ℓ as follows. Consider the set of reﬂections W 1 = { ws i w − 1 | w ∈ W , 1 ≤ i ≤ n } and for each w ∈ W let ℓ ( w ) denote the minimal r ≥ 0 such that w can b e written as pro duct w = x 1 . . . x r of reﬂections x j ∈ W 1 . Given u, v ∈ W deﬁne u ≤ v ⇐ ⇒ ℓ ( u ) + ℓ ( u − 1 v ) = ℓ ( v ) . Note that the leng th function ℓ a nd the absolute order are inv a riant under conju- gation with a ﬁxed element of W . A Coxeter element in W is a ny elemen t of W that is conjugate to one o f the form s σ (1) s σ (2) . . . s σ ( n ) for so me p ermutation σ . Note that ℓ ( c ) = n for each Coxeter element c b y [19, Theorem 1.1]. This has the following immediate consequence. Lemma 6.1. L et c b e a Coxeter element and x 1 , . . . , x n a se qu enc e of r eﬂe ct ions in W 1 such that c = x 1 · · · x n . If 1 ≤ r ≤ s ≤ n , then ℓ ( x 1 . . . x r ) = r a nd x 1 . . . x r ≤ x 1 . . . x s .  REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 15 Relative to a Coxeter element c one deﬁnes the po set of non-cr ossing p artitions NC( W , c ) = { w ∈ W | id ≤ w ≤ c } . Given t wo C oxeter elements c, c ′ in W , we hav e NC( W , c ) ∼ = NC( W , c ′ ) provided that c and c ′ are conjugate. The Grothendieck gro up K 0 (mo d A ) is is omorphic to Z n via the map sending each simple A -mo dule S i to the standard base v ector ε i . The image of a n A - mo dule X under this map is called dimension ve ct or a nd deno ted by dim X ; we write s X = s dim X for the cor resp onding reﬂection. Exceptional mo dules and sequences. An A -mo dule X is called exc eptional if X is indeco mpo sable and Ext 1 A ( X, X ) = 0. Note that dim X is a real r o ot if X is exceptional [45, C o rollar y 2]. A seque nc e ( X 1 , . . . , X r ) of A -modules is called exc eptional if ea ch X i is exceptiona l and Hom A ( X j , X i ) = 0 = Ext 1 A ( X j , X i ) for a ll i < j. Such a sequence is c omplete if r = n . Lemma 6.2 . Sending an A -mo dule X to the r eﬂe ct ion s X = s dim X gives an in- je ctive map fr om the set of isomorphism classes of exc eptional A -mo dules into W . Pr o of. An exceptional A -mo dule is uniquely determined by its dimensio n vector; see [35, Lemma 8.2 ]. On the other hand, given a re ﬂe c tion s ∈ W 1 , w e hav e s = s α where α ∈ Z n is the unique vector with non-negative e n tries satisfying s ( α ) = − α . Thu s s dim X = s dim Y implies X ∼ = Y .  Theorem 6.3 (Crawley-Bo evey , Ringel, Igus a –Schiﬄer) . L et A b e a c onne cte d her e ditary A rt in algebr a with simple m o dules S 1 , . . . , S n satisfying Ext 1 A ( S j , S i ) = 0 for al l i < j . Denote by W the asso ciate d Weyl gr oup and ﬁx t he Coxeter element c = s 1 · · · s n . Then the br aid gr oup B n on n strings acts t r ansitively on — the isomorphism classes of c omplete exc eptional se qu enc es ( X 1 , . . . , X n ) in mo d A , and — the se quenc es ( x 1 , . . . , x n ) of r eﬂe ctions in W 1 such that c = x 1 · · · x n . Mor e over, σ ( X 1 , . . . , X n ) = ( Y 1 , . . . , Y n ) implies σ ( s X 1 , . . . , s X n ) = ( s Y 1 , . . . , s Y n ) for al l σ ∈ B n . Pr o of. F or the action of the braid g roup on complete ex ceptional sequence s , see [17, 45]. F or the action on factorisations o f the Co xeter element , see [28, Theorem 1.4]. The co mpatibilit y o f b oth actions follows from the computation of the dimensio n vectors of the mo dules in a n exceptional sequence under the braid gr o up action; see the pro o f of [28, Cor ollary 2.4] and also the Cor o llary in [17].  Corollary 6.4 . Le t ( x 1 , . . . , x n ) b e a se quenc e of r eﬂe ctions in W 1 such that c = x 1 · · · x n . Then ther e exists up t o isomorphism a unique c omplete ex c eptional se- quenc e ( X 1 , . . . , X n ) such that x i = s X i for al l i . Pr o of. Theo rem 6.3 gives σ ∈ B n such that ( x 1 , . . . , x n ) = σ ( s 1 , . . . , s n ). Let ( X 1 , . . . , X n ) = σ ( S 1 , . . . , S n ). Then x i = s X i for all i . Uniqueness follows from Lemma 6.2.  16 HENNING KRAUSE Thic k sub categories. W e consider the b ounded der ived categor y D b (mo d A ) and ident ify A -mo dules with co mplexes co ncentrated in degree zer o. Reca ll the follo wing corres p o ndence. Prop ositi o n 6 .5 (Br ¨ uning) . L et A b e a her e ditary ab elian c ate gory. The c anonic al inclusion A → D b ( A ) induc es a bije ction b etwe en – the set of ex act ab elian and ex t ension close d su b c ate gories of A , and – the set of t hick su b c ate gories of D b ( A ) . The bije ction sends C ⊆ A to { X ∈ D b ( A ) | H i X ∈ C for al l i ∈ Z } . Its inverse sends U ⊆ D b ( A ) to H 0 U = { Y ∈ A | Y ∼ = H 0 X for some X ∈ U } . Pr o of. See [14, Theorem 5.1].  The L o ewy length of an ob ject X in some abelia n category is the smallest p ≥ 0 such that there e x ists a chain 0 = X 0 ⊆ X 1 ⊆ . . . ⊆ X p = X so that X i +1 /X i is s e misimple for all i . The height of an abelia n ca tegory is the supremum of the Lo ewy lengths of its ob jects. Next we c haracter is e the thic k sub catego ries of D b (mo d A ) such that the inclu- sion admits an a djoint. The connection with exceptional s equences is due to Bondal [12]. Prop ositi o n 6.6. L et A b e a her e ditary Artin algebr a. F or a thick sub c ate gory U of D b (mo d A ) ar e e quivalent. (1) The inclusion U → D b (mo d A ) admits a left adjoint. (2) The inclusion H 0 U → mo d A admits a left adjoint. (3) The ab elian c ate gory H 0 U is of ﬁnite height and has only ﬁnitely many isomorphi sm classes of simple obje cts. (4) Ther e exists an exc eptional se quenc e ( X 1 , . . . , X r ) in mo d A such t hat U = Thick ( X 1 , . . . , X r ) . (5) Ther e exists a c omplete exc eptional se quenc e ( X 1 , . . . , X n ) in mo d A such that U = Thick( X 1 , . . . , X r ) and ⊥ U = Thick ( X r +1 , . . . , X n ) R emark 6.7 . The k -duality (mod A ) op ∼ − → mo d( A op ) induces a dua lit y D b (mo d A ) op ∼ − → D b (mo d A op ) which pr eserves the prop erty (3). Th us the exis tence of left adjoin ts in (1) and (2) is equiv alent to the existence of rig ht a djoints. R emark 6.8 . The equiv alent conditions in Prop o s ition 6.6 are a utomatically s atis- ﬁed if the a lgebra A is of ﬁnite representation t yp e; see Theorem 2.5. Pr o of of Pr op osition 6.6. (1) ⇔ (2): See [4 0, § 2]. (2) ⇒ (3): A le ft a djoint F : mo d A → H 0 U sends a pr o jective generato r to a pro jective g enerator . Thus one takes B = End A ( F A ) and gets a n eq uiv a lence Hom A ( F A, − ) : H 0 U ∼ − → mo d B . Clearly , mo d B has ﬁnite height and only ﬁnitely ma n y simple ob jects. (3) ⇒ (4): It follows fro m [22, 8.2 ] that the ca tegory H 0 U is eq uiv a lent to mo d B for some ﬁnite dimensio nal k - algebra B . The a lgebra B is hereditary s ince A is hereditary . Thus the simple B -mo dules form a complete exceptional sequence ( X 1 , . . . , X r ) in mo d B . This gives an exce ptional seq uenc e in mo d A satisfying U = Thick ( X 1 , . . . , X r ). REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 17 (4) ⇒ (1): See [12, Theorem 3.2 ]. (1)–(4) ⇒ (5): The duality pro vides a right adjoint for the inc lus ion of U . Thus the inclusio n o f ⊥ U a dmits a left a djo int by Lemma 2.4. It follo ws that there is an e x- ceptional sequence ( X r +1 , . . . , X s ) in mo d A with ⊥ U = Thick( X r +1 , . . . , X s ). Then ( X 1 , . . . , X s ) is an exceptional sequence satisfying Thick ( X 1 , . . . , X s ) = D b (mo d A ) which is therefore co mplete. (5) ⇒ (4): Clear.  Example 6. 9. F or a tame hereditary algebra A , the regula r A -mo dules form an exact ab elian and extension closed sub categ ory of mo d A that is of inﬁnite height. A classiﬁcation. The following result is a co ns equence of The o rem 6.3 a nd pro- vides a co mbin atorial classiﬁca tion of the thick sub c a tegories of D b (mo d A ) s at- isfying the equiv alent co nditions in Prop ositio n 6.6. Using the corresp o ndence in Prop ositio n 6.5, this tra nslates into a classiﬁcatio n of certain abelian subcateg ories of mo d A . F or the pa th algebra of a q uiver ov er an algebra ically closed ﬁeld, this result is due to Igusa , Sc hiﬄer, and Thoma s [28], Pr e v ious work o f Ingalls and Thomas [29] establishes the r esult in ﬁnite and aﬃne type. Theorem 6.10. L et A b e a c onne cte d her e ditary A rtin algebr a with simple mo dules S 1 , . . . , S n satisfying E x t 1 A ( S j , S i ) = 0 for al l i < j . Denote by W t he asso ciate d Weyl gr oup and ﬁx t he Coxeter element c = s 1 · · · s n . Then t her e exists an or der pr eserving bije ction b etwe en — the set of thick sub c ate gories of D b (mo d A ) such that the inclusion admits a left adjoint, and — the set of non-cr ossing p artitions NC( W, c ) . The map sends a thick sub c ate gory which is gener ate d by an exc eptional se quenc e ( X 1 , . . . , X r ) to s X 1 · · · s X r . Let us formulate an immediate consequence. Corollary 6.11. L et ( X 1 , . . . , X r ) and ( Y 1 , . . . , Y s ) b e exc eptional se quen c es in mo d A . Then Thick ( X 1 , . . . , X r ) = Thic k( Y 1 , . . . , Y s ) ⇐ ⇒ s X 1 · · · s X r = s Y 1 · · · s Y s .  Pr o of of The or em 6.10. Fix a thick sub categ ory U ⊆ D b (mo d A ) such that the inclusion a dmits a left adjoint. There exists a co mplete exceptio nal sequence ( X 1 , . . . , X n ) in mo d A such that U = Thick ( X 1 , . . . , X r ) fo r so me r ≤ n , b y Prop o- sition 6.6. W e ass ign to U the elemen t cox ( U ) = s X 1 · · · s X r in W . Observe tha t cox ( U ) ≤ c by Lemma 6.1, since s X 1 · · · s X n = c by Theorem 6.3. Thus co x gives a map int o NC( W, c ). The map cox is wel l-deﬁne d: Cho ose a second exceptio nal sequence ( Y 1 , . . . , Y s ) in mo d A such that U = Thick( Y 1 , . . . , Y s ). Then ( Y 1 , . . . , Y s , X r +1 , . . . , X n ) is a complete exceptional sequence , and it follows from Theorem 6.3 that s Y 1 · · · s Y s s X r +1 · · · s X n = c = s X 1 · · · s X r s X r +1 · · · s X n . Thu s s Y 1 · · · s Y s = s X 1 · · · s X r . The map c ox is inje ctive: Let U and V b e thick subc a tegories such that cox( U ) = cox ( V ). Th us there a re t wo complete exceptional sequences ( X 1 , . . . , X n ) and ( Y 1 , . . . , Y n ) such that U = Thick( X 1 , . . . , X r ) and V = Thick( Y 1 , . . . , Y s ) for some pair of integers r , s ≤ n . Moreover, s X 1 · · · s X r = s Y 1 · · · s Y s . It follows that r = ℓ ( s X 1 · · · s X r ) = ℓ ( s Y 1 · · · s Y s ) = s 18 HENNING KRAUSE and therefore c = s Y 1 · · · s Y s s X r +1 · · · s X n . F rom Co rollary 6 .4 and Lemma 6.2 one gets that ( Y 1 , . . . , Y s , X r +1 , . . . , X n ) is a complete exceptional sequence. Th us U = Thick( X 1 , . . . , X r ) = Thick( X r +1 , . . . , X n ) ⊥ = Thick( Y 1 , . . . , Y s ) = V . The map cox is su rje ctive: Le t w ∈ N C ( W, c ). Thus w e can write c = x 1 · · · x r x r +1 · · · x n as pro ducts of reﬂections x i ∈ W 1 such that w = x 1 · · · x r . F rom Corolla ry 6.4 one gets a complete exceptio na l sequence ( X 1 , . . . , X n ) with x i = s X i for all i . Se t U = Thick ( X 1 , . . . , X r ). Then cox( U ) = w . The m ap cox is or der pr eserving: Let V ⊆ U be thic k subca tegories. F rom Prop ositio n 6.6 one gets a complete ex ceptional sequence ( X 1 , . . . , X n ) such that V = Thick( X 1 , . . . , X s ) and U = Thick( X 1 , . . . , X r ) for some s ≤ r ≤ n . It follows from Lemma 6.1 that cox ( V ) = s X 1 · · · s X s ≤ s X 1 · · · s X r = cox( U ) .  R emark 6.12 . Let U ⊆ D b (mo d A ) b e a thick sub ca tegory such that the inclusion admits a left a djoint . Then cox( U ) cox( ⊥ U ) = c . Example: The Kronec k er al g ebra. Let k be a ﬁeld and consider the Kr one cker algebr a , that is, the path algebra of the quiver ◦ / / / / ◦ . This is a ta me her editary Artin algebra; w e denote it by K a nd compute the lattice of thick subcategor ies of D b (mo d K ). F or a descr iption of mo d K , we refer to [6, VI I I.7]. Each ﬁnite dimensiona l indecomp osable K -mo dule is either exceptiona l or reg - ular. The dimension vectors of the exceptional K -mo dules are ( p, q ) ∈ Z 2 with p, q ≥ 0 and | p − q | = 1. Thus the non-cros s ing partitions with resp ect to the Coxeter elemen t c = s 1 s 2 form the lattice NC( W , c ) = { s ( p,q ) | p, q ≥ 0 and | p − q | = 1 } ∪ { id , c } with the following Hasse diagram: c m m m m m m m m | | | | B B B B Q Q Q Q Q Q Q Q · · · P P P P P P P • @ @ @ • • ~ ~ ~ · · · n n n n n n n id The regular K -mo dules for m an extens io n closed exact ab elian sub categor y o f mo d K that is uniserial. The simple ob jects of this a be lian c a tegory are par ame- terised by the closed po int s o f the pro jective line over k , w hich we identify with non-maximal and non-zero ho mogeneous prime ideals p ⊆ k [ x, y ]. W e denote the set of closed p oints by P 1 ( k ) and write 2 P 1 ( k ) for its power set. Adding an extra greatest elemen t (the se t of all non-ma ximal homog eneous prime ideals) to 2 P 1 ( k ) yields a new lattice whic h w e deno te by b 2 P 1 ( k ) . The s imple ob ject corr esp onding to p is denoted by S p . W e obtain an injective map b 2 P 1 ( k ) − → T ( D b (mo d K )) by s ending U ⊆ P 1 ( k ) to Thick( { S p | p ∈ U } ) and P 1 ( k ) ∪ { 0 } to D b (mo d K ). Let L ′ , L ′′ be a pair o f la ttices with smallest elements 0 ′ , 0 ′′ and grea tes t elemen ts 1 ′ , 1 ′′ . Denote b y L ′ ∐ L ′′ the new la ttice which is obtained f rom the disjoin t union L ′ ∪ L ′′ (viewed as sum of pos e ts ) by iden tifying 0 ′ = 0 ′′ and 1 ′ = 1 ′′ . REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 19 Prop ositi o n 6 . 13. The lattic e of t hick su b c ate gories of D b (mo d K ) is isomorphic to the lattic e NC( W, c ) ∐ b 2 P 1 ( k ) . Pr o of. W e wr ite T ( K ) = T ( D b (mo d K )) and have injective maps NC( W , c ) − → T ( K ) and b 2 P 1 ( k ) − → T ( K ) which induce an o rder preserving map NC( W, c ) ∐ b 2 P 1 ( k ) − → T ( K ) . In order to prov e bijectivity , ﬁx a pair o f indeco mpo sable K -mo dules X, Y such that X is exceptiona l and Y is reg ula r. W e hav e Thick( X ) ∩ Thick( Y ) = 0 and this g ives injectivit y; surjectivity follows from the fact that Thick( X, Y ) = D b (mo d K ).  The categ ory of cohere n t sheav es on the pro jective line P 1 k admits a tilting o b ject T = O (0) ⊕ O (1) such that End( T ) ∼ = K ; see [8]. Thus R Hom( T , − ) induces a triangle equiv alence D b (coh P 1 k ) ∼ − → D b (mo d K ) and this yields a description of the lattice of thick sub catego r ies of D b (coh P 1 k ). Note tha t the categor y of coherent sheav es car ries a tensor pro duct. The thick tensor subcateg o ries ha ve been classiﬁed by Thomas o n [46, Theo rem 3.5]; they are precisely the o nes parameter is ed b y subsets o f P 1 ( k ). Appendix A. Auslander–Reiten theor y In this appendix we collect some basic facts from Ausla nder–Reiten theory , as initiated b y Ausla nder and Reiten in [5]. K rull–Remak– Schmid t catego r ies form the appropriate setting for this theory , w hile ex act or triangulated structures are irrelev ant for mos t pa rts; see also [7 , 4 1]. This material is w ell- known, a t lea st for categor ies that ar e linear ov er a ﬁeld with ﬁnite dimensional morphism spaces. W e pro v ide full proofs for mo st statemen ts, including references whenev er they are av ailable. Let C b e an essentially small additive category . Then C is called Kru l l–R emak– Schmidt c ate gory if every ob ject in C is a ﬁnite copro duct of indecomp osable ob jects with lo cal endomor phism rings. It is c o nv enient to view C as a ring with several o b jects. Thus we use the categ ory Mo d C of C -mo dules , whic h ar e by deﬁnitio n the additive functors C op → Ab into the catego r y of abelia n groups. There is the following us e ful characterisation in ter ms o f pro jective cov ers . Recall that a morphism φ : P → M is a pr oje ctive c over , if P is a pro jective ob ject and φ is a n ess en tial epimorphi sm , that is, a mor phism α : P ′ → P is a n epimo rphism if and only if φα is an epimor phism. Prop ositi o n A.1. F or an essent ial ly sm al l additiv e c ate gory C with split idemp o- tents the fol lowing ar e e quivalent. (1) The c ate gory C is a Krul l–R emak–Schmidt c ate gory. (2) Every ﬁnitely gener ate d C -mo dule admits a pr oje ctive c over. The pro of require s some pre parations, and we beg in with tw o lemmas. Lemma A.2. L et P b e a pr oje ctive obje ct. Then t he fol lowing ar e e qu ivalent for an epimorphism φ : P → M . (1) The morphism φ is a pr oje ctive c over of M . (2) Every endomorphism α : P → P satisfying φα = φ is an isomorph ism. 20 HENNING KRAUSE Pr o of. (1) ⇒ (2): Let α : P → P b e a n endo mo rphism satisfying φα = φ . Then α is an epimor phism s inc e φ is essential. T hus ther e ex ists α ′ : P → P sa tisfying αα ′ = id P since P is pro jective. It follows that φα ′ = φ and therefore α ′ is an epimorphism. On the other hand, α ′ is a mo no morphism. Thus α ′ and α are isomorphisms. (2) ⇒ (1): Let α : P ′ → P b e a morphism such that φα is an epimorphism. Then φ factor s through φα via a morphism α ′ : P → P ′ since P is pr o jective. The comp osite αα ′ is a n is o morphism and therefore α is an epimorphism. Thus φ is essential.  Lemma A.3. L et φ : P → S b e an epimorphi sm such that P is pr oje ctive and S is simple. Then the fol lowing ar e e quivalent. (1) The morphism φ is a pr oje ctive c over of S . (2) The obje ct P has a unique maximal sub obje ct. (3) The endomorphism ring of P is lo c al. Pr o of. (1) ⇒ (2): Let U ⊆ P be a subo b ject and suppos e U 6⊆ Ke r φ . Then U + K er φ = P , and therefor e U = P since φ is ess ential. Thus Ke r φ co nt ains e very prop er sub ob ject of P . (2) ⇒ (3): First obser ve that P is a n indeco mpo sable ob ject. It follows that every endomorphism o f P is inv ertible if and only if it is an epimor phism. Given t wo non-units α, β in End( P ), we ha ve therefor e Im( α + β ) ⊆ Im α + Im β ( P . Here we use that P has a unique ma ximal sub ob ject. Th us α + β is a non-unit and End( P ) is lo cal. (3) ⇒ (1): Consider the E nd( P )-submo dule H of Hom( P, S ) which is generated by φ . Supp o se φ = φα for so me α in End ( P ). If α b elongs to the Jacobson r adical, then H = H J (End( P )), whic h is not pos sible b y Nak ay ama’s lemma. Thus α is a n isomorphism since End( P ) is lo cal. It follo ws from Lemma A.2 that φ is a pro jective cov er .  Given any C - mo dule M , w e deno te by r ad M its r adic al , that is, the in ter section of all maxima l s ubmo dules o f M . Pr o of of Pr op osition A.1. First obser ve that the Y oneda functor C → Mo d C taking an ob ject X to Hom C ( − , X ) identiﬁes C with the catego ry of ﬁnitely g enerated pro jective C -mo dules . The o ther tools a re Lemmas A.2 and A.3, which a re used without further refere nce. (1) ⇒ (2): Let M b e a simple C -mo dule and choose an indecompo sable ob ject X with M ( X ) 6 = 0 . Then there exists a non-zero morphism Ho m C ( − , X ) → M which is a pro jective cov er since End C ( X ) is loc al. Thus ev ery ﬁnite sum of simple C -mo dules admits a pr o jective cover. Now le t M b e a ﬁnitely gener ated C -mo dule and choo se an epimorphism φ : P → M with P ﬁnitely genera ted pro jective. Let P = L i P i be a decomp osition into indecomp osable mo dules. Then P / r ad P = M i P i / rad P i is a ﬁnite sum of simple C -mo dules since each P i has a lo cal endomorphism r ing. The epimorphism φ induces an epimor phism P / rad P → M / rad M and therefo r e M / r ad M decomp oses into ﬁnitely many simple mo dules. Ther e exists a pro jective cov er Q → M / rad M and this factors thro ugh the canonica l morphism π : M → M / r ad M via a mo r phism ψ : Q → M . The morphism π is essential since M is REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 21 ﬁnitely generated. It follows that ψ is an epimorphism. The morphism ψ is essential since π ψ is essential. Thus ψ is a pro jective co ver. (2) ⇒ (1): Let X b e a n ob ject in C . W e show that X admits a decompo sition int o ﬁnitely many indecomp osa ble ob jects. Set P = Hom C ( − , X ). W e claim that P / r ad P is semisimple. Choo se a quotien t P /U , where r ad P ⊆ U ⊆ P , and let Q → P / U b e a pro jective cov er . This gives maps P → Q and Q → P . The c o mpo site Q → P → Q is inv ertible and induces a n iso morphism P /U → P / rad P → P /U . Thu s the canonica l morphism P / r ad P → P /U has a right inv erse, and we conclude that P / rad P is s e misimple. Let P / ra d P = L i S i be a decomp osition into ﬁnitely many simple ob jects and c ho ose a pro jective cov er P i → S i for each i . Then P ∼ = L i P i , since P → P / ra d P and L i P i → L i S i are b oth pro jective covers. It remains to observe that ea ch P i is indecomp osable with a lo ca l endomorphism ring.  R emark A.4 . Fix a ring A and le t C be the categ o ry of ﬁnitely genera ted pro jective A -mo dules. Then the functor Mo d C → Mo d A taking X to X ( A ) is an equiv ale nce. The catego ry C is Krull–Remak– Schmidt if and only if the ring A is semip er fect. F rom now on suppose that C is a K rull–Remak– Schmid t category . The radical. The r adic al of C is by deﬁnition the collec tio n of subgro ups Rad C ( X, Y ) ⊆ Ho m C ( X, Y ) for each pair X, Y of ob jects in C , where Rad C ( X, Y ) = { φ ∈ Hom C ( X, Y ) | id X − φ ′ φ is inv ertible for all φ ′ : Y → X } = { φ ∈ Hom C ( X, Y ) | id Y − φφ ′ is inv ertible for all φ ′ : Y → X } . The radical is a two-side d ide al o f C , that is, a subfunctor of Hom C ( − , − ) : C op × C → Ab . It is actually the unique t wo-sided ideal J of C such that J ( X , X ) equals the Jacobso n radical of End C ( X ) for each ob ject X ; se e [34]. Given two decompositions X = ` i X i and Y = ` j Y j in C , we ha ve Rad C ( X, Y ) = M i,j Rad C ( X i , Y j ) . A morphism b etw een indeco mpo sable ob jects b elongs to the radical if and only if it is not inv ertible, since indecomp osable o b jects hav e lo cal endomo rphism rings. Simple functors. Given an ob ject X in C , the functor Rad C ( − , X ) equa ls the in- tersection of all maximal subfunctors o f Hom C ( − , X ) : C op → Ab . This observ ation has the following consequence, where S X = Hom C ( − , X ) / Rad C ( − , X ) . Lemma A.5 ([3, P rop osition 2.3]) . The map sending an obje ct X of C to the functor S X induc es (up to isomorphism) a bije ction b etwe en t he inde c omp osable obje cts of C and the simple obje cts of Mo d C . Pr o of. If an o b ject X has a local endomorphism ring, then Rad C ( − , X ) is the unique maximal s ubo b ject of Hom C ( − , X ). Th us S X is simple in that case , and the in verse map s ends a simple ob ject S in Mo d C to the uniq ue indecomp osable o b ject X in C such that S ( X ) 6 = 0.  22 HENNING KRAUSE Irreducible morphisms. The deﬁnition of the radica l is e xtended recursively as follows. F or each n > 1 and each pa ir of ob jects X , Y let Rad n C ( X, Y ) be the set of morphisms φ ∈ Hom C ( X, Y ) that admit a facto risation φ = φ ′′ φ ′ with φ ′ ∈ Ra d C ( X, Z ) and φ ′′ ∈ Rad n − 1 C ( Z, Y ) fo r so me o b ject Z . Then we set Irr C ( X, Y ) = Rad C ( X, Y ) / Ra d 2 C ( X, Y ) . This is a bimo dule ov er the ring s ∆ ( X ) and ∆ ( Y ), where ∆ ( X ) = E nd C ( X ) / Ra d C ( X, X ) . Note that a morphism X → Y be t ween indecompo sable o b jects be lo ngs to Rad C ( X, Y ) r Ra d 2 C ( X, Y ) if a nd only it is irr educible. A mo rphism φ is called irr e ducible if φ is neither a section nor a retr action, and if φ = φ ′′ φ ′ is a factor isa- tion then φ ′ is a sectio n or φ ′′ is a retr action. Lemma A. 6. F or e ach p air of inde c omp osable obje cts X , Y in C , we have Ext 1 C ( S Y , S X ) ∼ = Hom ∆ ( X ) (Irr C ( X, Y ) , ∆ ( X )) as bimo dules over ∆ ( X ) and ∆ ( Y ) . Pr o of. Applying Hom C ( − , S X ) to the exact sequence 0 − → Rad C ( − , Y ) − → Hom C ( − , Y ) − → S Y − → 0 gives Ext 1 C ( S Y , S X ) ∼ = Hom C (Rad C ( − , Y ) , S X ) . Then applying Hom C ( − , S X ) to the exa c t sequence 0 − → Rad 2 C ( − , Y ) − → Rad C ( − , Y ) − → Irr C ( − , Y ) − → 0 gives Ext 1 C ( S Y , S X ) ∼ = Hom C (Irr C ( − , Y ) , S X ) ∼ = Hom ∆ ( X ) (Irr C ( X, Y ) , ∆ ( X )) since S X ( X ) = ∆ ( X ).  Almost spli t morphisms . A mor phis m φ : X → Y is called right almost split if φ is not a r etraction and if ev ery morphism X ′ → Y that is not a retraction factors through φ . The morphism φ is right minimal if ev ery endomo rphism α : X → X with φα = φ is inv er tible. Note that Y is indecomp osa ble if φ is righ t almost split. L eft almost split morphisms and left minimal mo r phisms are deﬁned dually . The term minimal right almost split means right minimal and right almost split . Recall that a pro jective prese ntation P n δ n − → P n − 1 δ n − 1 − → · · · δ 2 − → P 1 δ 1 − → P 0 δ 0 − → M − → 0 is minimal if each morphism P i → Im δ i is a pro jective cov er. Lemma A.7 ([4, Cha p. I I, P rop ositio n 2.7]) . A morphism X → Y in C i s minimal right almost split if and only if it induc es in Mo d C a minimal pr oje ctive pr esentation Hom C ( − , X ) − → Hom C ( − , Y ) − → S − → 0 of a simple obje ct. Pr o of. A no n-zero morphism Hom C ( − , Y ) → S to a simple ob ject is a pro jective cov er if and only if End C ( Y ) is lo cal. In that case the exa ctness of the se q uence means that the image of Hom C ( − , X ) → Hom C ( − , Y ) equals Rad C ( − , Y ); it is therefore equiv alent to the fact that X → Y is right almos t split. The canonical morphism fro m Hom C ( − , X ) to the image of Hom C ( − , X ) → Hom C ( − , Y ) is a pro jective cover if and only if X → Y is right minimal.  REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 23 Almost split morphisms a nd irreducible mor phisms are related as follows. Lemma A.8 ([7, Co rollar y 3.4]) . L et X → Y b e a minimal right almost split morphism in C and let X = X n 1 1 ∐ . . . ∐ X n r r b e a de c omp osition into inde c omp osable obje cts such that the X i ar e p airwise non-isomorphic. Given an inde c omp osable obje ct X ′ , one has Irr C ( X ′ , Y ) 6 = 0 if and only if X ′ ∼ = X i for s ome i . In that c ase n i e quals the length of Irr C ( X ′ , Y ) over ∆ ( X ′ ) . Pr o of. The morphism X → Y induces a minimal pro jective presen tation Hom C ( − , X ) − → Hom C ( − , Y ) − → S Y − → 0 by L e mma A.7, and ther efore a pro jective co ver π : Hom C ( − , X ) − → Rad C ( − , Y ) . This morphism induces a n isomor phism Hom C ( − , X ) / Rad C ( − , X ) ∼ − → Rad C ( − , Y ) / Rad 2 C ( − , Y ) since Ker π ⊆ Rad C ( − , X ). O n the other hand, the decomp osition of X implies Hom C ( − , X ) / Rad C ( − , X ) ∼ = S n 1 X 1 ∐ . . . ∐ S n r X r . It remains to o bs erve tha t S X i ( X ′ ) 6 = 0 if a nd only if X ′ ∼ = X i .  Almost spli t sequences. The following deﬁnition of an almost split sequence is taken from Liu [41]; it cov ers the original deﬁnition of Ausla nder and Reiten for ab elian categories [5], but also Happel’s deﬁnition of an Auslander–Reiten triangle in a triang ulated category [24]. A sequence of morphis ms X α − → Y β − → Z in C is called almost split if (1) α is minimal left almo s t split and a weak k ernel of β , (2) β is minimal right almost split and a weak cok ernel of α , and (3) Y 6 = 0. The end terms X and Z determine each other up to isomo phism, a nd we write X = τ Z and Z = τ − 1 X . O ne calls τ Z the Auslander–R eiten tr anslate of Z . Lemma A.9. A se quenc e of morphisms X → Y → Z in C is almost split if and only if it induc es two minimal pr oje ctive pr esentations Hom C ( − , X ) → Hom C ( − , Y ) → Hom C ( − , Z ) → S Z → 0 Hom C ( Z, − ) → Hom C ( Y , − ) → Hom C ( X, − ) → S X → 0 wher e we u se the notation S X = Hom C ( X, − ) / Rad C ( X, − ) . Pr o of. Apply Lemma A.7.  The Auslander–Reiten transla te is functorial in the following sense. Lemma A.10. L et X , Y b e inde c omp osable obje cts in C and supp ose their Auslander– R eiten tr anslates ar e deﬁne d. Then ∆ ( X ) ∼ = ∆ ( τ X ) and ∆ ( Y ) ∼ = ∆ ( τ Y ) as division rings. Using these isomorphisms, we have Hom ∆ ( X ) (Irr C ( X, Y ) , ∆ ( X )) ∼ = Hom ∆ ( τ Y ) (Irr C ( τ X , τ Y ) , ∆ ( τ Y )) as bimo dules over ∆ ( X ) and ∆ ( Y ) . 24 HENNING KRAUSE Pr o of. W e use the stable c ate gory of ﬁnitely presented C -mo dules which w e denote by mo d C . The ob jects are the C -mo dules M that admit a pres entation ( ∗ ) Hom C ( − , X ) ( − ,φ ) − → Hom C ( − , Y ) − → M − → 0 and the morphisms are all C - linear morphisms modulo the subgro up o f morphisms that factor thro ugh a pro jective C -module. There are tw o functors Ω : mo d C − → mo d C a nd T r : mo d C − → mo d C op taking a mo dule to its syzygy and its t r ansp ose , resp ectively . Both functor s are deﬁned for a mo dule M with pr esentation ( ∗ ) via exact sequences a s follows: 0 − → Ω M − → Hom C ( − , Y ) − → M − → 0 Hom C ( Y , − ) ( φ, − ) − → Hom C ( X, − ) − → T r M − → 0 Note that the tra ns po se yields an e q uiv a lence (mo d C ) op ∼ − → mo d C op . Using the minimal pro jective presentations of S X and S τ X from Lemma A.9, one gets isomor phisms Ω S X ∼ = T r S τ X and Ω S τ X ∼ = T r S X . These yield mutually in verse maps ∆ ( X ) ∼ − → End C ( S X ) → E nd C ( Ω S X ) ∼ − → End C (T r S τ X ) ∼ − → End C op ( S τ X ) op ∼ − → ∆ ( τ X ) and ∆ ( τ X ) ∼ − → End C op ( S τ X ) op → End C op ( Ω S τ X ) op ∼ − → End C op (T r S X ) op ∼ − → End C ( S X ) ∼ − → ∆ ( X ) . Next w e apply Lemma A.6 and obtain Hom ∆ ( X ) (Irr C ( X, Y ) , ∆ ( X )) ∼ = Ext 1 C ( S Y , S X ) ∼ = Hom C ( Ω S Y , S X ) ∼ = Hom C (T r S τ Y , S X ) ∼ = Hom C op (T r S X , S τ Y ) ∼ = Hom C op ( Ω S τ X , S τ Y ) ∼ = Ext 1 C op ( S τ X , S τ Y ) ∼ = Hom ∆ ( τ X ) (Irr C op ( τ Y , τ X ) , ∆ ( τ X )) ∼ = Hom ∆ ( τ X ) (Irr C ( τ X , τ Y ) , ∆ ( τ X )) . The second iso morphism requires an extr a argument, and the same is used for the sixth. Let Ω S Y = Rad C ( − , Y ), and obse r ve that this has no non-zero pro jective direct summand by the minimality of the presentations in Lemma A.9. Then w e hav e Ext 1 C ( S Y , S X ) ∼ = Hom C ( Ω S Y , S X ) ∼ = Hom C ( Ω S Y , S X ) . The ﬁrst is omorphism is fr o m the pro of of Lemma A.6. The second fo llows from the fact that any non-zer o mor phism Ω S Y → S X factoring through a pro jective factors through the pro jective co ver Hom C ( − , X ) → S X . This means that Hom C ( − , X ) is a direct summand of Ω S Y , which has be e n excluded b efore.  REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 25 The Auslander–Rei ten quiv er. The A uslander–R eiten quiver of C is deﬁned as follows. The set of isomorphis m cla sses of indecomp osable o b jects in C form the vertic es , a nd there is a n arr ow X → Y if Irr C ( X, Y ) 6 = 0 . It is often conv enient to ident ify an indecomp osa ble ob ject with its isomo rphism class. The Auslander– Reiten quiver carries a valuation which assig ns to each arrow X → Y the pair ( δ X,Y , δ ′ X,Y ), where δ X,Y = ℓ ∆ ( X ) (Irr C ( X, Y )) and δ ′ X,Y = ℓ ∆ ( Y ) (Irr C ( X, Y )) . Here, ℓ A ( M ) denotes the length of an A -mo dule M . Lemma A.11. L et X , Y b e inde c omp osable obje cts in C and supp ose that τ Y is deﬁne d. Then δ ′ τ Y , X = δ X,Y . Pr o of. W e hav e an almost split sequence τ Y → ¯ X → Y a nd δ X,Y counts the m ultiplicit y o f X in a decompo s ition of ¯ X b y Lemma A.8, which equa ls δ ′ τ Y , X by the dual of Lemma A.8 .  Lemma A.12. Le t k b e a c ommutative ring and supp ose that C is k -line ar with al l morphism sp ac es of ﬁ nite length over k . L et X , Y b e inde c omp osable obje cts in C and supp ose that τ Y is deﬁne d. Then δ τ Y , X = δ ′ X,Y . Pr o of. Using the identit y δ ′ τ Y , X = δ X,Y from Lemma A.11 and the isomor phism ∆ ( Y ) ∼ = ∆ ( τ Y ) from Lemma A.10, o ne computes ℓ ∆ ( Y ) (Irr C ( X, Y )) = ℓ k (Irr C ( X, Y )) · ℓ k ( ∆ ( Y )) − 1 = ℓ ∆ ( X ) (Irr C ( X, Y )) · ℓ k ( ∆ ( X )) · ℓ k ( ∆ ( Y )) − 1 = ℓ ∆ ( X ) (Irr C ( τ Y , X )) · ℓ k ( ∆ ( X )) · ℓ k ( ∆ ( Y )) − 1 = ℓ k (Irr C ( τ Y , X )) · ℓ k ( ∆ ( τ Y )) − 1 = ℓ ∆ ( τ Y ) (Irr C ( τ Y , X )) .  The rep etition. Let Γ b e a quiver without lo ops . Then a new quiver Z Γ is deﬁned as follows. The set o f vertices is Z × Γ 0 . F or each a r row x → y in Γ and each n ∈ Z ther e is an arrow ( n, x ) → ( n, y ) and an arrow ( n − 1 , y ) → ( n, x ) in Z Γ . The quiver Z Γ is a translation quiver [43] with tra nslation τ deﬁned by τ ( n, x ) = ( n − 1 , x ) for e a ch vertex ( n, x ). References [1] C. Amiot, On the structure of triangulated categories with ﬁnitely many indecomposables, Bull. Soc. Math . F rance 135 (2007), no. 3 , 435–474. [2] D. Armstrong, Generalized noncrossing partitions and combinat orics of Coxe ter groups, Mem. Amer. Math. So c. 202 (2009), n o. 949, x+159 pp. [3] M. Auslander, Represen tation the ory of Ar tin algebras. I I, Comm. Algebra 1 (1 974), 269 –310. [4] M. Auslander, F unct ors and morphisms determined b y ob jects, in R e pr esentation the ory of algebr as (Pr o c. Conf., T emple Univ., Philadelphia , Pa., 1976) , 1–244. Lecture Notes in Pure Appl. Math., 37, Dekker, New Y ork, 1 978. [5] M. A uslander and I. Reiten, Representat ion theory of Artin algebras. I II. Almost split se- quences, Comm. Algebra 3 ( 1975), 239–294. [6] M. A uslander, I. Reiten and S. O. Smalø, R epr ese nt ation the ory of Artin algebr as , Cam bri dge Studies in Adv anced M athematics, 36, Cambridge Uni v. Press, Cambridge, 1995. [7] R. Bautista, I rreducible morphisms and t he radical o f a category , An. Inst. Mat. Univ. Nac. Aut´ onoma M´ exico 22 (1982), 83–135 ( 1983). [8] A. A. Be ˘ ıli nson, C oherent s heav es on P n and problems in linear algebra, F unktsional. Anal. i Prilozhen. 12 (1978), no. 3 , 68–69. [9] A. Beli giannis, On t he F r eyd categories of an additive category , Homology Homotop y Appl. 2 (2000), 147–185. 26 HENNING KRAUSE [10] A. Beligiannis, Relativ e homological algebra and puri ty in triangulated categories, J. Algebra 227 (2000), no. 1, 268–361. [11] A. Beligiannis, Auslander-Reiten triangles, Ziegler spectra and Gorenste in rings, K -Theory 32 (2004), no. 1, 1–82. [12] A. I. Bondal, Represent ations of asso ciative algebras and coheren t sheav es, (Russian) Izv. Ak ad. Nauk SSSR Ser. Mat. 53 (19 89), no. 1, 25–44; translation in Math. USSR-Izv. 34 (1990), no. 1, 23–42. [13] A. I. Bondal and M. M. Kap rano v, Representa ble functors, Serre functors, and reconstruc- tions, (Russian) Izv. Ak ad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337; transla- tion in Math. USSR-Izv. 3 5 (1990), no. 3, 519–541. [14] K. Br ¨ uning, Thic k subcategories of the de rived category o f a hereditary alge bra, Homology , Homotop y Appl. 9 ( 2007), no. 2, 165–176. [15] A. B. Buan et al., Tilting theory and cluster combinatorics, Adv. Mat h. 204 ( 2006), no. 2, 572–618. [16] R.-O. Buch weitz, Maximal Cohen-Macaula y modules and T ate-cohomology o v er Gorenst ein rings, unpublished manuscript (1987), 1 55 pp. [17] W. Cr a wley-Bo ev ey , Exceptional sequences of represen tations of quivers, i n Pr o ce e dings of the Sixth International Confer e nce on R epr esentations of Algebr as (O ttawa, ON, 1992) , 7 pp, Carleton-Otta wa M ath. Lecture Note Ser. , 14 Car l eton Univ., Otta wa , ON, 1992. [18] P . Dowbor, C. M. Ringel and D. Simson, Hereditary Artinian rings of ﬁnite represen tation t ype, in R epr esentation the ory, II (Pr o c . Se c ond Internat. Conf., Carleton Univ., O ttawa, Ont., 197 9) , 232–241, Lecture Note s in Math., 83 2 Springer, Berli n, 1980. [19] M. J. Dyer, On minimal l engths of expressions of Co xeter group elements as pro ducts of reﬂections, Pro c. Amer. Math. So c. 129 (2001), no. 9, 25 91–2595. [20] P . F reyd, Stable homotop y , in Pr o c. Conf. Cate goric al Algebr a (L a Jol la, Cal if., 1965) , 121– 172, Springer, New Y ork, 1966. [21] P . Gabriel, Des ca t ´ e gories ab´ eliennes, Bull. So c. Math. F rance 90 (1962), 323–448. [22] P . Gabriel, Inde composable represen tations. II, in Sy mp osia Mathematic a, V ol. XI (Conve gno di A lgebr a Com mutativa, IND AM, R ome, 1971) , 81–104, Academic Press, London, 1 973. [23] P . Gabriel, Ausl ander-R eiten se q uenc es and r epr esentation ﬁnite algebr as , Lect. Notes Math. 831 , Springer-V er l ag (1980), 1 –71. [24] D. Happel, On the derive d categ ory of a ﬁnite-dimensional algebra, C omm en t. Math. Helv. 62 (1987), no. 3, 339–389. [25] D. Happ el, T riangulate d ca te gories in the r epr esentati on the ory of ﬁnit e-dimensional alge- br as , Lond on Mathematical Societ y Lecture Note Seri es, 119, Cambridge Univ. Pr ess, Cam- bridge, 1988. [26] D. Happel , U. Pr eiser and C. M. Ri ngel, Binary p olyhedral groups a nd Euclidean diagrams, Manu scripta Math. 31 ( 1980), no. 1-3, 317–329. [27] D. Happel, U. Preiser and C. M. Ringel, Vin b erg’s c haracterization of Dynkin diagrams using subadditiv e f unctions with application to D T r-p eri odic modules, in R e pr esentation the ory, II (Pr o c. Se c ond Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) , 280–294, Lecture Notes in Math., 832 Springer, Berl in, 1980. [28] K. Igusa and R. Sc hiﬄer, Excep tional sequence s and clusters, J. Algebra 323 (2010), no. 8, 2183–220 2. [29] C. Ingalls and H. Thomas, Noncrossing partitions and represent ations of quiv ers, Comp os. Math. 145 (2009), no. 6, 1533–1562. [30] C. U. Jensen and H . Lenzing, Mo del-t he or etic algebr a with p articular emphasis on ﬁelds, rings, mo dules , Gordon and Breach, New Y ork, 1989. [31] V. G. Kac, Inﬁnite-dimensional Lie algebr as , second e dition, Cambridge Univ. Press, Cam- bridge, 1985. [32] B. Keller, Chain complexes and stable categories, M an uscripta Math. 67 (1990), no. 4, 379– 417. [33] B. Keller, On triangulated or bit categories, Do c. Math. 10 (2005), 551–581. [34] G. M. Kelly , On the radical o f a category , J. Austral. Math. S oc. 4 (1964), 299–307. [35] O. Kerner, Representa tions of wild quiv ers, in R e pr esentation the ory of al gebr as and r elate d topics (Mexic o Cit y, 1994) , 65–107, CMS Conf. Pro c., 19 Amer. Math. So c., Providence, RI, 1996. [36] C. K¨ ohler, Thick sub categories of ﬁnite algebraic tri angulated categories, arXi v:1010.0146v1 [math.CT]. [37] H. Kr ause, Smashing subcategories and the telescope conjecture—an al gebraic approac h, In ven t. Math. 139 (2000), no. 1, 99 –133. REPOR T ON LOCALL Y FINITE TRIANGULA TED CA TEGORIES 27 [38] H. Krause, A usl ander-Reiten theory via B r own representabilit y , K -Theory 20 ( 2000), no. 4, 331–344. [39] H. Krause, Localization theory for triangulated cate gories, in T riangulate d c ate g ories , 161– 235, London Math. So c. Lecture N ote Ser., 375, Cambridge Univ. Press, Cambridge, 2010. [40] H. Krause, J. Sto vicek, The telescope conjecture for hereditary rings via Ext-orthogonal pair s, Adv. Math. 225 (2010), n o. 5, 2341–2364. [41] S. Liu, Auslander-Reiten theory in a Krull -Schmidt category , prepri n t. [42] F. Muro, S. Sc hw ede and N. Strickland, T riangulated categories without m odels, Inv ent. Math. 170 (2007), no. 2, 231–241. [43] C. Ri edtmann, Algebren, Darstellungsk¨ ocher, ¨ Uber lagerungen und zur ¨ uc k, Comment. Math. Helv. 55 (1980), no. 2, 199–224. [44] C. Riedtmann , Representation -ﬁnite self- injectiv e algebras of class D n , Compositio Math. 49 (1983), no. 2, 231–282. [45] C. M. R i ngel, T he braid group action on the set of exc eptional s equences of a hereditary Artin algebra, in Ab elian gr oup the ory and r elate d top ics (Ob erwolfach, 1993) , 339–352, Con temp. Math., 171 Amer. Math. So c., Pro vidence, RI, 1994. [46] R. W. Thomason, The classiﬁcation of triangulated subcategories, Comp ositio Math. 105 (1997), no. 1, 1–27. [47] M. V an den Bergh, The signs of Serr e duality , App endix A to R. Bo cklandt , Graded Calabi Y au algebras of di mension 3, J. Pur e Appl. A lgebra 212 (2008), no. 1, 14– 32. [48] J.-L. V erdi er, Des cat ´ egories d ´ eriv´ ees des cat´ egories ab´ eliennes, A st ´ erisque No. 239 (1996), xii+253 pp. (1997). [49] J. W asch b ¨ usc h, Symmetrisc he Algebren v om endliche n Modultyp, J. Reine A ngew. Math. 321 (1981), 78–98. [50] Y. Y oshino, Cohen-Mac aulay mo dules over Cohen-Mac aulay rings , London Mathematica l Society Lecture Note Series, 146, C ambridge Univ. Press, Ca mbridge, 1990. [51] J. Xiao and B. Zh u, Locally ﬁnite triangulated catego ries, J. Algebra 29 0 (2 005), no. 2, 473–490. Henning Krause, F akul t ¨ at f ¨ ur Ma themat ik, Universit ¨ at Bielefeld, D-335 01 Biele- feld, G ermany. E-mail a ddr e ss : hkrause@math .uni-bie lefeld.de

Report on locally finite triangulated categories

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment