Indecomposables live in all smaller lengths

Indecomp osables liv e in all smaller lengths Klaus Bongartz Univ ersit¨ at W upp ertal German y ∗ Dedicated to A.V.Roiter and P .Gabriel Abstract W e sho w that there are no g aps in the lengths of the indecomp osa ble mod u les of a fin ite dimensional algebra ov er an algebraical ly closed fi eld. This result extends to indecomposables in any k-linear abelian category where the end omor phism algebras of th e simples are k . F or the pro of w e sho w th at any distributive minimal represen tation-infinite algebra is isomorphic to its ra y c ategory and it has an interv al -finite universal co ve r with a free fundamental group. In tro duction The meaning of the title is the following r esult: Theorem 1 L et A b e an asso ciative algebr a of finite dimension over an alge- br ai c al ly close d field k. If ther e is a non-simple inde c omp osable A -mo dule of length n , t her e is also o ne of length n − 1 . F or repres en tation-finite algebras - i.e. algebras having only finitely many isomorphism class es of finite-dimensiona l indecomp osable r epresen tations - one even knows since a lo ng time that any non- s imple mo dule is an ex tension of a simple and a n indecomp osable. This is pr obably not true in gene r al altho ugh Ringel has r ecen tly shown that a n y non-simple indecomp osable is an extension of indecomp osables. The statemen t of the theorem is very naiv e a nd elementary , but the proof given here is not. It dep ends stro ngly on the work of Roiter, Gabr iel and others on algebr as of finite r e presen tation type. This is summar iz e d very well in chapters 13 and 14 of their b ook ab out repr esen tations of finite dimensiona l algebras . T o des c ribe in this in tro duction shortly the s imple strategy of the proof I freely use no tions fro m that b oo k without a n y further explanatio n. Later on, there will be precis e references. ∗ E-mail:b ongartz@math.uni-wuppertal.de 1 Up to Morita-equiv alence one ca n a ssume that all simple modules ha ve di- mension one ther e by replacing the length by the dimensio n. In view of the known representation-finite case it only remains to b e shown that a minimal representation-infinite algebra - i.e. the algebra itself is no t representation- finite, but eac h pro per quo tient is - a dmits indec omposable s in all dimensions . If the a lgebra is distributive and zig zag-free w e app eal to the well-elabo rated cov ering theory a nd we can complete the pro of b y ’standard’ arguments. So all we still ha ve to deal with are non-distributive alge bras or distributive alg e br as with a zig zag. F or non-distributive alg ebras there is a s imple dir ect constr uction that was later on mo dified by Ringel to prove the existence of a so called accessible mo dule in each dimension ( a first version of this ar ticle in the archiv e dates back to april 20 09 ). F or distributive alg ebras with a zigzag I could not find a direct pro of. The reason seems to b e that the technique of cleaving dia grams due to Bautista, Larri´ on and Salmer´ on allows to construct easily infinite families of indecomp os- ables, but their dimensions cannot b e determined precisely . This situatio n is familiar from mo dular r epresen tation theor y: Induction implies that the whole group has the same re pr esen tation type a s a p-Sylow subgroup, but the deco m- po sition of an induced module in to indecomposa bles can b e v ery complica ted. Thu s for gro up alg ebras the tw o well-known Brauer-Thr all conjectures a re easy to prov e whereas theorem 1 is not. This migh t b e the reason wh y it was no t formulated as the ’Br a uer-Thrall 0 conjecture’. Now to handle the distributive algebras with a zigzag I ha d to generalize a little bit a central result ab out co verings whic h migh t be of independent interest: Theorem 2 L et A b e a distributive b asic asso ciative algebr a with asso ciate d r ay c ate gory ~ A . Supp ose that A is minima l r epr esentation-infinite. Then we have: a) ~ A has a n int erval -finite u niversa l c over. b) The fundamental gr oup is fr e e. c) A is isomorphic to the li ne arization k ~ A of ~ A . As I learned from V os siec k the last tw o statements of the theo r em ha ve already b een obta ine d by Geiss in his unpublished diploma-thesis from 19 9 0 by observing that Fisch bac hers a rgumen ts can b e adapted to include the minimal representation-infinite ca se. Ho wev er this do es no t work fo r the fir st statement and o ur inductive pro of based on Fis c h bachers result gives all thr ee statements at once. Theorem 1 has so me nice co r ollaries. The fir st one is a gener alization from A - mo dules to ob jects of finite length in an abelian k -linea r category . The a nologous generaliza tion is no t true for the Brauer-Thra ll conjectur e s as trivial ex a mples show. 2 Corollary 1 L et C b e an ab elian k -line ar c ate gory over an a lgebr aic al ly close d field k . Supp ose that al l simple obje ct s in C have endomorphism algebr a k . If ther e is an inde c omp osabl e non-simple obje ct in C of length n , ther e is also one of length n − 1 . Note that by a well-kno wn co un ting ar gumen t the assumption on the endo- morphism alg ebras is always true fo r mo dules over algebr as whose dimension is strictly s ma ller than the cardinality of the field. So fo r example, the coro lla ry applies to complex repr esen tations o f Ka c-Moo dy-alge br as and their deforma- tions. Corollary 2 ( The naive criterion for finite re pr esentation typ e ) The fol low - ing c onditions ar e e quivalent for an algebr a A of fin ite dimension over an alge- br ai c al ly close d field. a) A is r epr esentation-finite. b) Ther e is a natur al numb er n such that ther e is no inde c omp osa ble A -m o dule of that length. In fact, under these c onditions the numb er n = 2 · dimA + 100 0 wil l a lways do. Back in 1974 I tried to finish m y diploma-thesis by applying this criterio n. Much to m y surpr ise Ga br iel re jected my ’solution’ b ecause that obvious crite- rion was no t proven. Now it is - hop efully . The a rticle is or ganized as follows. In chapter 1 we consider non-distributive algebras and in the cen tral chapter 2 w e study cro wns - i.e. perio dic zigzags - in minimal representation-infinite ray categor ies where a t leas t one comp osition of irreducible morphisms does not v anish. The pro of of the main reduction resembles the pro ofs of some of the ma in results in the a rticle on multiplicative bases. As explained at the b eginning of section 2.2 it r equires first a finite strategy and second enough energ y to ca rry this thro ugh. Finally in chapter 3 the statements made in this introduction are easy consequences of the general theory develop ed for representation-finite a lgebras. I w ant to thank Dieter V ossieck for dr a wing my attention to an error in lemma 9 as stated in the first archive version. F or the sak e of s implicit y w e will a lw ays work o ver an algebra ically closed field k of arbitrar y characteristic, but the o nly thing that we r eally need is that all simple mo dules have endomor phism alg ebra k . W e co nsider le ft mo dules. 3 1 The non-distributiv e case Now A denotes a basic asso ciative alg ebra of finite dimens ion over k with Ja- cobson radical J . Such a n algebr a is given by a uniquely determined quiv er Q and a tw o-sided ideal I inside the path a lgebra k Q , that is genera ted by cer tain linear co m binations of paths of length ≥ 2. There is a commutativ e semi-simple subalgebra B in A that is a vector space supplemen t of J . F or the ne x t pro of we need the following ea sy o bserv a tio n. Lemma 1 L et M b e a non-zer o A -mo dule wi th an endomorph ism φ . Then we have: a) φ is nilp otent iff the induc e d map on the so cle of M is n ilp otent. b) φ is n ilp otent iff t he induc e d map on the top M /J M is nilp oten t. c) M is inde c omp osable iff e ach en do morphism ha s ex actly one eigenvalue. Recall that A is distributive if its ideal lattice is distributive. This is equiv a- lent to the fact that for all pr imitiv e idemp oten ts e, f the algebra eAe is uniserial and that f Ae is cy clic as an f Af left mo dule or as an eAe right mo dule ( [2 0, 13.2] ). Thus, if A is no t distributive, there ar e not neces sarily different primi- tive idemp oten ts e , f and a natural num b er l such that for the ra dical filtration ( R i ) of f Ae as a f Af − eAe -bimo dule we hav e dimR i /R i +1 = 1 for all i < l , but di mR l /R l +1 ≥ 2 . W e cho o se elements v , w in R l whose images in R l /R l +1 are linearly indep enden t, and we lo ok at the tw o-sided idea l K of A gener a ted by R l +1 , J v , v J, J w, w J . In the quo tient A/ K we obtain primitive idemp otent elements e, f a nd linearly indep endent elements v , w that are annihilated on bo th s ides b y the Jacobson radical o f the quotien t. Our aim is to co nstruct in each dimension an indecomp osable A/K -module. T o simplify the no tation a little bit we assume right from the b eginning that the o riginal e, f , v , w hav e the prop erties mentioned b efore. Let d b e the dimension of the indecompo s able pro jective Ae . All non-zero quotients of this lo cal mo dule are ag ain loca l, whence indecompo sable. Th us we easily find indecomp osables of dimensio n m fo r all m ≤ d . In pa rticular, the family Ae/ h v − xw i , xǫk , consists of pairw is e non- isomorphic indecomp osables. Here a nd la ter on we denote b y h X i the k -s ubspace g enerated by some subset X inside s ome vector spac e . T o construct a t least one indecompo sable in each dimension w e take the Kronecker-modules a s our pro to -t ypes . So let n ≥ 2 be a natura l num b er and take n copies Ax 1 , Ax 2 , . . . , Ax n of the indecompos able pro jective Ae . Define M = ⊕ n i =1 Ax i and in tro duce the tw o subspaces U 0 = h w x i − v x i +1 | 1 ≤ i ≤ n − 1 i and H = h v x 1 , v x 2 , . . . , v x n i . Note that these are actually semi-simple submo dules b ecause v and w ar e annihilated by J . Lemma 2 Using al l the notations fr om ab ove, the fol lowing is true: a) L et U b e a su bmo dule of M s u ch that 4 i) U c ontains U 0 and also w x n for U 6 = U 0 . ii) U is c ontaine d in the r adic al J M of M . iii) U ∩ H = 0 . Then M /U is inde c omp osable. b) U 0 satisfies the c onditions in p art a). L et U b e a fi xe d maximal su bmo dule satisfying t hese c onditions. Then the so cle of N = M / U is iso morphic to H un der the c anonic al pr oje ction π : M − → M / U . c) L et V b e a submo dule of N that c ontains the submo dule V 0 gener ate d by π x 1 , π x 2 , . . . , π x n − 1 and is c ontaine d in V 0 + J N . Then V is inde c omp os- able. d) F or e ach m with nd − ( n − 1 ) ≥ m ≥ ( n − 1) d − ( n − 2) we find an inde c omp osabl e sub quotient of M with di mension m . Pro of: a) Condition i) just says v x i +1 = wx i for 1 ≤ i ≤ n − 1 in M = M /U . Here x denotes as usual the image of an element o f M in M / U . Condi- tion ii) guara ntees that the top of M / U is ⊕ n i =1 k x i . Finally , the element s v x 1 , v x 2 , . . . , v x n are still linearly indepe ndent in M by iii). W e claim that for all i with 0 ≤ i ≤ n − 1 we hav e ( v − 1 w ) i ( M ) = h x i +1 , x i +2 , . . . , x n i + J M . The start of the induction is trivial. In the step from ( i − 1) to i we g et w ( v − 1 w ) i − 1 ( M ) = h w x i , . . . , wx n i b ecause of wJ = 0. Here wx n = 0 for U 6 = U 0 . T aking the in v erse imag e under multiplication with v w e find after a short calc ula tion ( v − 1 w ) i ( M ) = v − 1 h wx i , . . . , wx n i = h x i +1 , x i +2 , . . . , x n i + J M . T ake now any endomor phism φ of M . Then φ r espects the flag M ⊇ ( v − 1 w ) M ⊇ . . . ⊇ ( v − 1 w ) n − 1 M ⊇ 0 . Therefore we have φ ( x i ) = n X j = i φ ij x j + r i for some appropria te scalars φ ij and r i in J M . W e obtain φ ( wx i ) = n X j = i φ ij wx j = φ ( v x i +1 ) = n X j = i +1 φ i +1 j v x j for all 1 ≤ i ≤ n − 1, whence φ ii wx i = φ ii v x i +1 = φ i +1 i +1 v x i +1 . Th us we hav e φ ii = φ 11 =: a for a ll i . Therefore, φ − a · id induces a nilp oten t endo morphism 5 on the top of M and also on M b y the last lemma. So a is the only eig e n v alue of φ and M is indecomp osable. b) It is easy to see that U 0 satisfies all conditions. Let U be a ma x imal submo dule with that pro p erty . Under the pro jectio n π : M − → M /U the semisimple mo dule H is embedded into the so cle of N . W e hav e to prove that any simple submo dule o f N lies in the image of H . If S is a simple submo dule of N its in v erse imag e I = π − 1 S cont ains U prop erly so th at I cannot satisfy all three co nditions of part a ). The first c ondition holds for I . So a ssume I is not co ntained in J M . Then S is not con tained in J N b ecause of π − 1 ( J N ) = J M . Using S ∩ J N = 0 we can choose a B -mo dule supplement N ′ of S in N that contains J N . Then w e get N = S ⊕ N ′ even as an A -mo dule. B ut N is indecomp osable by pa r t a ). W e conclude n = 1 contradicting o ur a s sumption n ≥ 2. Therefore I cannot satisfy the third condition, i.e. T = I ∩ H 6 = 0. This implies U ⊂ π − 1 ( S ) = U + T ⊆ U + H and therefore S = π ( π − 1 ( S )) ⊆ π ( H ). c) Of cour se the socle of V co ntains v x 1 , . . . v x n − 1 and also v x n = w x n − 1 , i.e. π ( H ) whic h in turn is the so cle of N . Since V is a submodule of N , the so cle of V is π ( H ). W rite K for the supspace of V consisting of all elemen ts killed b y multiplication with v and w . So K c o n tains the ra dical J V , but it will be strictly bigger in general. W e claim that ( v − 1 w ) i ( V ) = h x i +1 , x i +2 , . . . , x n − 1 i + K holds for all i = 0 , 1 , . . . n − 2 . This is t rue for i = 0 b ecause V is co n tained in V 0 + J N . The induction-s tep is easy a nd s imilar to that in part a ). Now take an endomor phis m φ of V . It resp ects the filtration by the ( v − 1 w ) i ( V ) and we have φ ( x i ) = n − 1 X j = i φ ij x j + r i for some appropria te scalars φ ij and r i in K . W e obtain φ ( wx i ) = n − 1 X j = i φ ij wx j = φ ( v x i +1 ) = n − 1 X j = i +1 φ i +1 j v x j for a ll i with 1 ≤ i ≤ n − 2, whence φ ii wx i = φ ii v x i +1 = φ i +1 i +1 v x i +1 . Thus we have φ ii = φ 11 =: a for all i ≤ n − 1 and in addition φ ( v x n ) = φ ( wx n − 1 ) = a v x n . There fore, φ − a · id induces a nilp oten t endomorphism on the socle of V and a lso on V by the la st lemma. So a is the only eig en v a lue of φ a nd V is indecomp osable. d) Cho ose a complete flag o f submo dules b e t w een U 0 and the maximal mo d- ule U fixed in par t d). Dividing M b y these mo dules pro duces indecomp os- ables of dimensions m with dimM / U 0 = nd − ( n − 1 ) ≥ m ≥ dimM /U . Similarly a complete flag o f submo dules of N = M /U starting with V 0 and 6 ending with V 0 + J N gives us indecomposa bles with dimensions ranging from dimV 0 to dimM / U − 1. Since V 0 is genera ted by x 1 , . . . x n − 1 we have dimV 0 ≤ ( n − 1 ) d − ( n − 2). The re a der is invited to verify what the prece ding constr uc tio n means a t least in the following tw o simple, but typical examples that are g iv en by quivers with rela tions. ✲ ✲ ✒✑ ✓✏ ❅ ❅ ❅ ❘ ❄    ✠ ❅ ❅ ❅ ❘ ❄    ✠ α β γ β 2 = 0 α 1 α 2 α 3 β 1 β 2 β 3 P 3 i =1 β i α i = 0 Our findings can b e summar ized in the following result: Prop osition 1 L et A b e a b asic non-distributive algebr a. Then ther e is an inde c omp osabl e W of c o untable dimension having for e ach natur al numb er m an inde c omp osable su b quotient V of dimension m . Pro of: W e use the reductions and notatio ns in tro duced b efore. Let M ′ be the direct sum of an infinite sequence Ax i of co pie s of Ae and let U ′ be the s ubmodule gener ated b y all differences wx i − v x i +1 . Cle a rly , there is an endomorphism T of M ′ that maps x i to x i +1 for all i. Since U ′ is T -inv aria n t, we obtain a n induced endomor phism T on the quotient W := M ′ /U ′ . T o see that W is indecomp osable, we determine its a ndo morphism a lgebra B . As befor e one sees that any e ndomorphism φ respec ts the infinite descending chain o f submo dules W i generated by J W and the x j with j ≥ i . This implies inductively tha t a n endomor phism φ with φ ( x 1 ) ǫJ W maps W in to J W . Let I be the set of all endomorphis ms φ with φ ( W ) ⊆ J W . Then I is a nilp oten t ideal and we cla im that B is the direct sum o f I and the s uba lgebra k [ T ] gener ated by T which is iso morphic to the p olynomial alg ebra in o ne indeterminate. Indeed, let ψ be a n endomorphism. Then we ha v e ψ ( x 1 ) = P n i =1 λ i x i + j with so me scalar s λ i and j in J W . Thus φ := ψ − P n i =1 λ i T i maps x 1 int o J W , whence it b elongs to I and B is the sum of I a nd k [ T ] whos e in tersection is trivial. It follows e a sily that 0 and 1 a re the only idemp oten ts in B . Therefore W is indecomp osable. Finally , for a given natural num b e r m there is an n with nd − ( n − 1) ≥ m ≥ ( n − 1) d − ( n − 2). Then the mo dule M /U 0 considered in lemma 2 is a submo dule of W and it con tains a n indecomp osable sub quotient of dimension m . 7 2 On cro wns in minimal represen tation-infinite ra y categories 2.1 Reminder on ra y cat egor ies and clea ving diagrams Unfortunately , we ha ve to recall now a lot of definitions and r esults mainly from the b oo k [2 0]. A lo cally bounded categor y k -categor y A is a k -linea r ca tegory where dif- ferent ob jects are not isomo rphic, w he r e a ll endomorphism alg e bras A ( x, x ) ar e lo cal and where the dir e c t sums L y ǫA A ( x, y ) and L y ǫA A ( y , x ) a re of finite dimension for all xǫA . A finite dimensional A -mo dule M is a cov ariant k -linear functor from A to the category of k - vectorspac es such that the sum of the di- mensions of a ll M ( x ) , xǫA , is finite. A is lo cally representation finite, if for any ob ject xǫA there are up to isomorphism only finitely many indecomp osable mo dules U with U ( x ) 6 = 0. A lo cally b ounded k -categ ory is distributive if a ll endomor phism algebras A ( x, x ) are uniserial and all homomorphism spaces A ( x, y ) are cyclic as an A ( x, x ) right mo dule or as an A ( y , y ) left mo dule. The pro duct A ( x, x ) ∗ × A ( y , y ) ∗ of the tw o automorphism g roups acts on A ( x, y ) a nd the orbit of a morphism is the cor respo nding ray . These rays are the morphisms of the ray category ~ A a ttac hed to A . The proper ties of ~ A a re subs umed in the following axioms that define the abstract notion of a ray categor y P ( [20, sec tion 13.4] ): a) The ob jects form a set a nd they are pa irwise not isomor phic. b) There is a family o f zer o-morphisms 0 xy : x → y , x, y ǫP , satisfying µ 0 = 0 = 0 ν whenever the comp osition is defined. c) F or each xǫP , P ( x, y ) = { 0 } and P ( y , x ) = { 0 } for a lmost a ll y ǫP . d) F or eac h x one has P ( x, x ) = { i d x , σ, . . . , σ n − 1 6 = 0 = σ n } . Here n dep ends on x . e) F or each x, y , the set P ( x, y ) is cyclic under the action of P ( x, x ) or of P ( y , y ). f ) If κ, λ, µ, ν are morphisms with λµκ = λν κ 6 = 0 then µ = ν . Starting with such an abstract r a y catego ry P o ne constructs in a natura l way its linea rization k ( P ), w hich is a lo cally bo unded distributive k -categor y having the orig inal ca tegory P as the asso ciated ray category ~ k ( P ) ( [20, 13.5] ). In sharp contrast, a loca lly bounded distributive category A is in general not isomorphic to k ( ~ A ). If it is, A is called standard. W e say that P is ( lo cally ) representation finite o r minimal repr esen tation-infinite if k ( P ) is so, and this is independent of the field by [20, 1 4.7]. T o study a ray categor y P a nd its universal cover ˜ P we lo ok at the quiver Q P of P ( [20, sectio n 13.6] ). Its po in ts are the ob jects o f P a nd its arrows the irreducible mor phis ms in P , i.e. those no n- zero mor phisms that canno t b e 8 written a s a pro duct o f t wo morphisms different from identities. Each non-zer o non-inv ertible µ in P is then a pr oduct of irreducible morphisms, and the depth d ( µ ) of µ is the ma ximal num b er of f actor s o ccuring in these pro ducts. The non- zero morphisms in P are partially o r dered by defining µ ≤ ν iff ν = αµβ for so me morphisms α and β . A mo r phism is long if it is maximal with resp ect to this order and not irreducible. F or xǫP w e also co nsider the finite par tia lly order e d set x/P of all non-zero morphis ms with doma in x . Here w e define φ  ψ iff ψ = χφ . The dual order on the set P / x of the no n-zero morphisms with a fixed co domain is also denoted by  . The path ca tegory P Q P has the p oints of Q P as ob jects and the paths in Q P as non-zero morphisms, to which we add formal zero-mor phisms. There is a canonica l full functor ~ : P Q P − → P fr o m the path category to P whic h is the ’iden tit y’ on o b jects, arrows and zer o -morphisms. Two paths in Q P are in terlaced if they b e long to the transitive closur e of the relation R given by ( v, w ) ǫR iff v = pv ′ q , w = pw ′ q a nd ~ v ′ = ~ w ′ 6 = 0 where p and q are not b o th identities. A contour o f P is a pair ( v , w ) of non- interlaced paths with µ = ~ v = ~ w 6 = 0 ( see [20, section 13.6] ). Then we s a y that µ o ccurs in the contour ( v , w ). Note that these c o n tours a re called essential contours in [3, 16]. A dec o mposition v = v r v r − 1 . . . v 1 of a path is non-trivia l if a ll subpaths v i hav e length 1 a t least. Similarly , a fac torization of a morphis m is no n-trivial if none o f the factors is an identit y . A functor F : D − → P b et ween ray categor ies is clea ving ( [20, 1 3.8] ) iff it satisfies the follo wing tw o conditions and their duals: a ) F µ = 0 iff µ = 0 ; b) If αǫD ( x, y ) is irreducible and F µ : F x → F z factors thro ugh F α then µ factors already thro ugh α . The key fact ab out cleaving functors is that P is not ( lo cally ) representation finite if D is not. In this article D will always b e g iv en by its quiver Q D , that has no o rien ted cycles, and s o me r elations. Tw o paths betw een the s a me p o in ts give alwa ys the same morphism, and zero r elations are wr itten down ex plicitely . As in [20, section 13] the cleaving functor is then defined by drawing the quiver of D with relations and by writing the morphism F α in P close to ea c h ar ro w α . T o av oid confusions by to many letters in o ur figures we include sometimes not all names of morphisms ( see figure 2.1 ) o r we only mention all mor phis ms occurring in a figure in the text. F or instance, let D b e the ray catego ry with the natural nu mbers as ob jects and with ar ro ws 2 n ← 2 n + 1 and 2 n + 1 → 2 n + 2 for all n . Then a cleaving functor from D to P is called a zig zag in [20, section 13 .9] a nd P is said to contain a zig zag. A functor fro m D to P is just an infinite seque nc e of morphisms ( σ 1 , ρ 1 , σ 2 , ρ 2 , . . . ) in P such that ρ i and σ i alwa ys have commo n domain and ρ i and σ i +1 common co domain. The functor is cleaving iff none of the eq uations σ i = ξ ρ i , ξ σ i = ρ i , σ i +1 ξ = ρ i or σ i +1 = ρ i ξ ha s a solution. The situation is usually illustr a ted by the following zigzag : ❅ ❅ ❅ ❘    ✠ σ 1    ✠ ρ 1 ❅ ❅ ❅ ❘ σ 2    ✠ ρ 2 ❅ ❅ ❅ ❘    ✠ ❅ ❅ ❅ ❘ - - - - - - - - - - figure 2.1 9 A c ro wn in P of length 2 n is a zig- zag that be c omes p erio dic a fter n steps, i.e. o ne ha s σ i = σ n + i and ρ i = ρ n + i . If P is finite and contains a zig zag, it contains also a crown. W e denote such a crown b y ( σ 1 , ρ 1 , σ 2 , . . . , ρ n ). By axiom e) o f a r a y ca tegory the length of a cr o wn is at least 4. Later on we need the following represent ation infinite ray categories. The nu mbers r efer to the list in [2 0, section 1 0.7]. ✲ ❄ ❄ ✲ 12 q q q q q q q q ✲ ❄ ❄ ✲ q q q q q q q q 14 ✲ ❄ ❄ ✲ q q q q q q q 11 ✲ ❄ ❄✲ q q q q q q q q q q 20 ✛ ✛ ❄ ❄ ✛ q q q q q q q q q 44 ✛ ✲ ❄ ❄ ❄ ✲ ✛ q q q q q q q q 93 ✛ ✲ ❄ ❄ ❄ ✲ ✛ q q q q q q q q q 96 figure 2.2 Here an unoriented e dg e c a n b e oriented in an a rbitrary w ay . As usual, a branch can even be repla ced b y a r ooted tree with appr opriate zero relations ( see [20, 10.7] ). The same remar k applies to a ll extended D ynkin-diag rams. All ca tegories obtained from a Dynkin-diagra m of t ype T by orienting the edge s and by repla cing c e r tain br anc hes a re then called o f type T . F or later use we collect some simple facts in the following lemma . Lemma 3 L et P and D b e ra y c ate gori es. a) L et σ and ρ b e morphisms with in P with c ommon domain. If ther e ar e morphisms φ and ψ such that σ φ = 0 6 = ρφ and σψ 6 = 0 = ρψ , or if ther e is a morphism η such that ρ η and σ η ar e neigh b ors in a cr own then the fol lowing diagr am is cle aving:   ✠ ❅ ❅ ❘ σ ρ b) The c omp osition of cle a ving functors is cle aving. c) If τ is lo ng in P and F : D → P is cle aving with F µ 6 = τ for al l µ i n D , then the i nduc e d functor D → P /τ is stil l cle aving. 10 d) F : D → P is cle aving iff it satisfies t he c ondi tions: i) F µ = 0 iff µ = 0 . ii) No irr e ducible morphi sm is mapp e d to an identity. iii) F or any two irr e ducible morphisms α : x → y a nd β : x → z in D and e ach  - m aximal morphism µ : x → t with β  µ that do es not factor thr ough α , the i mage F µ do es not factor thr ough F α . iv) The dual of ii i). e) The c ate go ry D given by the quiver in figur e 3.1 without r elations c ontains the cr own ( αγ , β γ , β δ, αδ ) of lengt h 4 . Similarly, the c ate gory D given by the qu ive r in figur e 3.2 with zer o-r elations α 2 α 1 , β 2 β 1 and γ 2 γ 1 c ontains the cr own ( β 2 α 1 , β 2 γ 1 , α 2 γ 1 , α 2 β 1 , γ 2 β 1 , γ 2 α 1 ) of le ngth 6 . ❅ ❅ ❅ ❘    ✠    ✠ ❅ ❅ ❅ ❘ β α γ δ figur e 2.1 ❅ ❅ ❅ ❘ ❄    ✠    ✠ ❄ ❅ ❅ ❅ ❘ γ 2 α 2 α 1 β 1 β 2 γ 1 figur e 2.2 f ) L et ( v = α s α s − 1 . . . α 1 , w = β t β t − 1 , . . . β 1 ) b e a c ontour in P . Then ther e is the cle aving diagr am shown in figur e 3.3.    ✒ ✲ ✲ ...... ✲ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ✲ ✲ ...... ✲   ✒ α 1 α 2 α s − 1 β t − 1 α s β t β 2 β 1 ✲ ✲ ...... ✲   ✒ figur e 2.3 The easy pr o ofs are left to the r eader. The parts a) and its dua l, c) and d) are used again and again in the technical s ections to come. A t the end of this par agraph I ma k e some comments on the fundamental article [3] on multiplicativ e bases . It consists of a lo cal and a glo bal part. The lo cal part dea ls only with small pieces of the given a lgebra A . Here o ne uses quite often the c lea ving technique. Unfortuna tely , one has to dea l with arbitrar y k -catego ries instead o f r a y-categor ies which makes the v erification of the cleaving co nditions muc h more complicated. T his is one rea son why the lo cal part is hard to r ead. How ever, due to [9], it suffices to deal only with ~ A instead o f A which is mu ch easier. This considera ble simplification men tioned already in [3] is explained with many details in chapter 13 o f the b oo k [20]. In a forthcoming pap er [11] w e refine the structure and disjointness theo r ems for no n-deep contours and o bta in shorter pro ofs esp ecially for the so called diamonds. 11 After the rather tech nical lo cal par t there is the global top ological part starting with section 8. This part is alrea dy v ery elegan t and independent of lengthy case- b y-case cons iderations. Nev ertheless, in [16] it was simplified and generalized by Fisc h bacher whos e ma in r esults are presen ted in 13.9 and 14.2 of the b ook [20]. But observe that the crucial reduction lemma in [1 6] is based on a b eautiful lemma ab out ho oks in efficient tackles contained in [3, 8.4] r esp. [28, lemma 2 5]). 2.2 Long morphisms in cro wns The next r esult is basic fo r the inductive pro of o f theore m 2. Prop osition 2 L et P b e a minimal r epr esentation-infinite r ay c ate go ry c on- taining at le ast one cr own and one long morph ism. Then ther e is a long mor- phism not o c curing in a c ontour. W e will explain now the gener a l strategy how to prov e this and w e give the deta ils in the following sections . T o each crown C we consider the pa ir of natural num ber s ( n, t ), where 2 n is the length of the crown and t is the sum P n i =1 ( d ( ρ i ) + d ( σ i )) of the depths of all morphisms in C . The lexikographic order on the pairs ( n, t ) induces a partia l order on the s et of crowns. W e cho ose a minimal elemen t C . Of course, each lo ng morphis m τ o ccurs in C , b ecause otherwise C induces a crown in P / τ contradicting th e fact that P is minimal representation-infinite. It is clear that we a re in a self-dual situation: The minimal cr o wn C is a lso a minimal cro wn in the minimal representation-infinite ray-categ o ry P op , tha t contains also a long mor phism. F urther more, the propo sition holds for P iff it holds for P op . So if we hav e pr o ved that C has a certain pro perty , it has also the dual pr operty . Assume that the prop osition is not true i.e. tha t all lo ng mo r phisms b elong to a contour. W e will derive in several steps the contradiction that C as ab o ve do es no t ex ist. W e can a ssume that σ 1 is long, a nd we choose a contour ( v = α s α s − 1 . . . α 1 , w = β t β t − 1 . . . β 1 ) with ~ v = ~ w = σ 1 . W e sa y that ρ 1 factors through v resp. w if we ha ve ρ 1 = ρ ′ 1 ~ α 1 resp. ρ 1 = ρ 1 ~ β 1 . W e show in 2.2.1 that ρ 1 factors through exactly one of the tw o pa ths v or w . Dua lly , one defines when ρ n factors through v resp. w . Of course, by the ab o ve self-duality , ρ n also factors through exactly one of the t w o paths. So there are tw o cases p ossible for a contour ( v , w ) be lo nging to the long mor phism σ 1 . Either b oth neighbours ρ 1 and ρ n factor thro ugh different paths o r b oth factor through the sa me path. In the first case the chosen contour is c alled per meable, in the seco nd reflec ting . Analogous definitions a nd statements hold for all long morphisms o ccurring in C and for all choices of contours. 12 In the very technical section 2.2 .2 we show that tw o long morphisms ar e no t neighbors in C and in 2.2.3 that n = 2 is no t p ossible. In 2 .2.4 we lo ok at the long morphism σ 1 with the chosen contour ( v , w ) and we assume that we have a non-trivial factor ization v = v 2 v 1 such that ρ 1 = ρ ′ 1 ~ v 1 . W e prove that then σ 2 do es no t fac tor through ρ ′ 1 . Finally in 2.2.5 we show that C do es no t ex ist. F or the pro ofs of the firs t and especia lly the last step one has to lo ok at lar ge parts of the category whereas the other pro ofs only require a ca reful a nalysis of some s mall parts. Howev er this lo cal part is more complica ted than in the article on m ultiplicative bas es b ecause we hav e to consider also deep contours. W e end this sectio n with some easy , but useful obser v ations. Lemma 4 We ke ep al l the notations and assum ptio ns ma de in thi s se ction. a) F or xǫP , t he r ay c ate go ry induc e d by the p artial ly or der er e d set S := x/P is zigzag-fr e e. b) ρ 1 do es not factor thr ough v and w . c) Supp ose ρ 1 ≤ σ 1 . If ~ v = δ γ is a non-trivial factorization and ρ 1 = ρ ′ 1 γ , then δ is not irr e du cible. F urthermor e, ther e is a non-t rivi al factorization v = v 3 v 2 v 1 such that ρ 1 = ρ ′ 1 ~ v 1 and such that t he diagr am in figu r e 4.1 is cle a ving. Her e w = w 2 w 1 is any non- t rivi al d e c omp osition of w .    ✠ ❅ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❄   ✠ ρ ′ 1 ~ w 1 ~ v 1 ~ w 2 ~ v 2 ~ v 3 figur e 4.1 Pro of: a) Supp ose not. Since S is finite, there is a crown ( µ 1 , ν 1 , . . . , ν m ) in S . The morphism µ i in S gives us tw o morphisms φ i , ψ i with domain x a nd a morphism µ ′ i satisfying ψ i = µ ′ i φ i . Similar ly , we obtain morphisms ν ′ i with ψ i +1 = ν ′ i φ i , where ψ m +1 = ψ 1 . Here no φ i is the identit y of x , because other- wise µ i factors thr ough ν i − 1 ( ν 0 := ν m ). It follows ea sily that ( µ ′ 1 , ν ′ 1 , . . . , ν ′ m ) is a crown in P that do es not cont ain τ , wher e τ is a no n-zero mo rphism of max- imal depth. This is imp ossible b ecause P is minimal r epresen tation infinite. b) So assume ρ 1 factors through b oth, i.e. we have ρ 1 = ρ ′ ~ α 1 = ρ ′′ ~ β 1 . Set v ′ = α s . . . α 2 and w ′ = β t . . . β 2 . Then C ′ = ( ρ ′ , ~ v ′ , ~ w ′ , ρ ′′ ) is a cr o wn in x /P , where x is the doma in of ρ 1 . c) By as sumption we hav e ρ 1 ≤ σ 1 , i.e. σ 1 = ψ ρ 1 φ for so me morphisms φ and ψ . The definition of a ray c ategory implies that one of the following four equations relating ψ and δ ρ ′ 1 holds for so me appropr iate ξ . If we hav e ξ ψ = δ ρ ′ 1 we get ξ ψ~ v 1 = δ ρ ′ 1 γ = ξ σ 1 6 = 0. Because σ 1 is long , ξ is an identit y . But then we obtain σ 1 = ψ~ v 1 = δ ρ ′ 1 γ = δ ρ 1 contradicting the fact that σ 1 and ρ 1 are neighbors in a crown. The case ψ ξ = δ ρ ′ 1 leads to 0 6 = ψ~ v 1 = δ ρ ′ 1 φγ = ψ ξ φγ . Cancellation s hows that ξ is an identit y and we a re in the imp ossible firs t case. 13 The third p ossibilit y ψ = ξ δ ρ ′ 1 contradicts aga in to the fact that σ 1 and ρ 1 are neighbors. Thus we are in the case ψ = δ ρ ′ 1 ξ where ξ is not a n identit y bec ause other wise we ar e back in the third case . Since ρ ′ 1 is not an ident ity either we a r e done . Now, cho ose i ma x imal with the pro perty that ρ 1 factors thro ugh ~ v 1 where v 1 = α i . . . α 1 . Then w e have i < s a s shown be fo re which gives the wan ted non-trivial factor ization. The diagra m is cleaving by the maxima l choice of i . 2.2.1 ρ 1 factors through v or w Lemma 5 We ke ep al l notations and assumptions. a) ρ 1 factors thr ough v or w . b) A t most one neighb or ρ i of the lo ng morphism σ 1 satisfies ρ i ≤ σ 1 . c) Ther e ar e at le ast two differ ent long morph isms in C . d) If σ 1 and ρ 1 ar e long a nd σ 2 ≤ σ 1 , then σ 2 ≤ ρ 1 . Pro of: a) Supp ose a ) is not true. W e star t with the long mor phism σ 1 and we mo v e ahead in the crown until w e reach the next lo ng morphism τ which might be σ 1 again. W e co nsider first the cas e where τ o ccurs as some ρ i . W e choose a contour ( v ′ = α ′ s ′ . . . α ′ 1 , w ′ = β ′ t ′ , . . . , β ′ 1 ) corres ponding to the long morphism τ . If τ = ρ 1 the first ar rows in v , w , v ′ , w ′ are all different. This is impossible. Thu s we hav e i > 1 . If σ i do es not factor through v ′ or w ′ , we consider the mor - phisms α 1 , β 1 , ρ 1 , . . . , σ i , α ′ 1 , β ′ 1 . They define a quiver of t yp e ˜ D m as a cleaving diagram in P /σ 1 as can b e se en in figure 5.1 .   ✠ ❄ ❅ ❅ ❘   ✠ - - - - ❅ ❅ ❘   ✠ ❅ ❅ ❘ ❄ figure 5.1 figure 5.2   ✠ ❄ ❅ ❅ ❘   ✠ - - - - ❅ ❅ ❅ ❘   ✠   ✠ ❅ ❅ ❘ So suppo se that w e have a non-tr ivial decomp osition v ′ = v ′ 2 v ′ 1 such that σ i = σ ′ i ~ v ′ 1 . Fir st we lo ok at the sub case where ρ i − 1 do es not factor throug h σ ′ i and we consider the cleaving diagram of type ˜ D m defined by the morphisms α 1 , β 1 , ρ 1 , . . . , σ ′ i , ~ v ′ 1 , ~ v ′ 2 and dr a wn in figure 5 .2 . If σ 1 6 = τ , this lies in P /σ 1 which is imp ossible. If σ 1 = τ and if there is a no ther long mor phism φ we o btain the same cleaving diagr am in P /φ . Finally if σ 1 = τ is the o nly long mor phism we hav e σ i ≤ ρ i , whence ther e is by lemma 4 a non-trivia l decomp osition v ′ 2 = uu ′ with an a rrow u . Replacing ~ v ′ 2 by ~ u ′ we find a gain a ˜ D m - quiver in P /τ . W e ar e left with the cas e ρ i − 1 = σ ′ 1 ρ ′ i − 1 . F or i − 1 = 1 o ne gets a ˜ D 5 -quiver in P /σ 1 defined by the morphisms α 1 , β 1 , ρ ′ 1 , σ ′ 1 , ~ v ′ 1 . F or i > 2 we go on with these facto rizations σ i − 1 = σ ′ i − 1 ρ ′ i − 1 , ρ i − 2 = ρ ′ i − 2 σ ′ i − 1 and so on as long as po ssible. If a ll mo rphisms ca n b e facto r ized we e nd up with a similar cleaving diagram as ab o ve with ρ ′ 2 instead of ~ v ′ 1 . 14 If ρ k is the first morphism that do es not factor we obtain in P /σ 1 the cleaving diagram of figure 5.3 defined by the morphisms α 1 , β 1 , ρ 1 , . . . , ρ k , σ ′ k +1 , ρ ′ k +1 , σ ′ k +2 , and if it is some σ k the similar cleaving diag ram of figur e 5.4 .   ✠ ❄ ❅ ❅ ❘   ✠ - - - - figure 5.3 ❅ ❅ ❅ ❘   ✠ ❄ ❄   ✠ ❄ ❅ ❅ ❘   ✠ - - - - figure 5.4    ✠ ❅ ❅ ❘ ❄ ❄ The cas e where τ is some σ i can b e tre a ted with the same arguments. b) Suppo se that σ 1 go es from x to y . By ax iom e) of a ray categor y P ( x, y ) is cyclic ov er P ( x, x ) or o ver P ( y , y ). If P ( x, y ) is generated by χ ov er P ( x, x ) we cannot have ρ n ≤ σ 1 . F or ch o osing a genera tor γ of P ( x, x ) w e obtain from σ 1 = ψ ρ n φ the relatio ns σ 1 = χγ s and ρ n φ = χγ t with s ≥ t , whence σ 1 = ρ n φγ s − t . This contradicts the fact, that σ 1 and ρ n are neighbors in a crown. Dually , if P ( x, y ) is cyclic ov er P ( y , y ), we cannot have ρ 1 ≤ σ 1 . c) Assume that σ 1 is the only long morphism. Then the neig h bor s ρ 1 and ρ n factor through v or w by part a), whenc e they are not irreducible. Therefo re they ar e smaller than the only long morphism σ 1 which co n tradicts pa rt b). d) By assumption, we hav e σ 1 = ψ σ 2 φ 6 = 0 . B y axiom e) of a ray ca tegory we hav e σ 2 φ ≤ ρ 1 and also σ 2 ≤ ρ 1 . 2.2.2 Long morphisms are not neighbors This lengthy se c tio n is only devoted to prov e: Lemma 6 Two long morp hisms ar e not neighb ors in C . Pro of: Supp ose on the contrary that σ 1 and ρ 1 are long . F or σ 1 we tak e the already chosen contour ( v , w ) a nd for ρ 1 we choo se an arbitr ary contour ( v ′ , w ′ ). Let x b e the domain of σ 1 . A t most three arrows start at x . Th us we can assume that α 1 is the fir st arr o w of v and v ′ . Ins ide the partia lly ordered set x/P we lo ok at the set S of all mo rphisms φ sa tisfying ~ α 1  φ , φ  σ 1 and φ  ρ 1 . This set co n tains a g reatest element ψ be c ause otherwis e S contains a crown of length 2 con tradicting par t a) of lemma 4. Now there are paths u = γ r . . . γ 1 , u ′ = δ p δ p − 1 . . . δ 1 γ r ′ . . . γ 1 and r ′ < r suc h that α 1 = γ 1 , ψ = ~ γ r ′ . . . ~ γ 1 , ~ u = σ 1 and ~ u ′ = ρ 1 . Since u and v are interlaced b y constr uction, ( u, w ) is a co n tour with ~ u = σ 1 . Similar ly , ( u ′ , w ′ ) is a co n tour with ~ u ′ = ρ 1 . Now define u 2 = γ r . . . γ r ′ +1 , u ′ 2 = δ p δ p − 1 . . . δ 1 . Then we obta in for an y non- trivial deco mp ositions w = w 2 w 1 , w ′ = w ′ 2 w ′ 1 the following cle aving diag ram in P : 15 ❅ ❅ ❅ ❘ ~ w 1 ψ ~ w ′ 1 ~ w 2 ~ w ′ 2 ~ u 2 ~ u ′ 2    ✠ ❄ ❄    ✠ ❅ ❅ ❅ ❘ ❄ figure 6.0 Namely , σ 1 do es no t factor through ~ w ′ 1 by part b) of lemma 4. By symmetry , ρ 1 do es not facto r through ~ w 1 . Mor eo ver ~ u 2 = ξ~ u ′ 2 implies σ 1 = ξ ρ 1 . This is impo ssible since σ 1 and ρ 1 are neighbors in a crown. F or the same r eason, ~ u ′ = ξ ~ u lea ds to a co n tradiction. Observe that ρ n cannot facto r through ~ w 2 and ~ u 2 . The analogo us sta temen t holds fo r σ 2 and we hav e the tedious tas k to analyze the different p ossibilities. This is no t difficult, but very leng th y . W e give alwa ys a repres en tation-infinite ray categ ory t hat is cleaving by its num ber in figure 2.2 or b y the type of an extended Dynkin diagr am, but we do not chec k in detail a ll the conditions impo sed o n a cleaving functor . F or instance, pa rt a) of lemma 3 and its dual will b e used very often without mentioning it explicitely . First let σ 2 be long too . Then in one of the t wo paths of a co ntour to σ 2 there is an arr o w η with the s ame co domain a s ρ 1 such that ρ 1 do es no t fac tor through ~ η . If ρ n is a ls o long one ge ts an a rrow θ having the co rrespo nding prop erties with r espect to σ 1 . This gives in P /σ 1 the cle a ving diagr a m from figure 6.1 inv olving from the left to the r igh t the morphisms θ , ~ w 2 , ~ u 2 , ~ u ′ 2 , ~ w ′ 2 , η .   ✠ ~ η ❅ ❅ ❘ ~ θ ❄    ✠ ❅ ❅ ❅ ❘ ❄ figure 6.1   ✠ ❄ ❄    ✠ ❅ ❅ ❅ ❘ ❄ ρ ′ n    ✠ ❄ figure 6.2 So ρ n is no t long. If it do es not factor thr ough ~ w 2 or ~ v 2 we o btain a similar cleaving diag ram a s a bov e with ρ n instead of ~ θ . So supp ose ρ n = ~ w 2 ρ ′ n . If σ n = σ ′ n ρ ′ n we hav e in P / σ 1 the cleaving dia g ram drawn in figur e 6.2 given by the morphis ms ρ ′ n , σ ′ n , ~ w 2 , ~ u 2 , ~ u ′ 2 , ~ w ′ 2 , η . If σ n do es not factor thro ugh ρ ′ n we get in P /σ 2 the cleaving diagr a m shown in fig ure 6.3. It involv es the morphisms φ, ρ ′ n , ~ w 2 , ~ u 2 , ~ u ′ 2 , ~ w ′ 2 , ~ w 1 , ψ , ~ w ′ 1 , η . Here φ is σ n for σ n 6 = σ 2 or else an ir reducible mo rphism o ccuring in the paths chosen to a contour corr esponding to the long mor phism σ 2 that ρ n do es not factor through. 16 ❅ ❅ ❅ ❘    ✠ ❄ ❄ φ    ✠ ❅ ❅ ❅ ❘ ❄   ✠ ✲ ❄ figure 6.3 ❅ ❅ ❅ ❘    ✠ ❄ ❄    ✠    ✠ figure 6.4 ❅ ❅ ❅ ❘    ✠ Next w e co nsider the case wher e ρ n factors through u . W e will always find an appropria te ˜ D m - quiver that admits a cle aving functor into a pro per quotient of P . If ρ n do es not fac to r through ~ u 2 we ha ve a no n trivial decomp osition u 2 = ab with ρ n = ~ bρ ′ n . By the definition of ψ , ~ u ′ 2 do es not factor through ~ b and the morphisms ρ ′ n , ~ b, ~ a, ~ u ′ 2 , ~ w ′ 2 , η define our wan ted cleaving functor into P /ρ 1 . So we hav e ρ n = ~ u 2 ρ ′ n . Suppose first, that 0 6 = ~ u ′ 2 ρ ′ n . Then P co n tains the crown ( ρ n , σ 1 , ρ 1 , ~ u ′ 2 ρ ′ n ). Therefore we hav e n = 2 and σ 2 = ~ u ′ 2 ρ ′ n by the minimal c hoice of C . By part d) of lemma 5 ρ n is compa rable to σ 1 or t o σ 2 . F rom ρ n ≤ σ 1 we see using lemma 4 part c) that ψ can b e non-trivia lly factored as ψ 2 ψ 1 such that the morphisms ψ 2 , ~ u 2 , ~ u ′ 2 , ~ w ′ 2 , η g iv e r ise to a cleaving functor from a ˜ D 5 -quiver to P /ρ 1 . Similarly , ρ n ≤ σ 2 implies that u ′ 2 admits a non- trivial factoriza tio n u ′ 2 = ab . Then we find to P /ρ 1 the cleaving functor from a ˜ D 4 - quiver given by the morphis ms ρ ′ 2 , ψ 2 , ~ u 2 , ~ b . W e a r e reduced to the cas e 0 = ~ u ′ 2 ρ ′ n . If σ n = σ ′ n ρ ′ n then we ha v e in P / σ 2 the ˜ D 5 -diagra m supp orted by the mor phisms σ ′ n , ~ u 2 , ~ u ′ 2 , ~ w ′ 2 , η . If this is not cleaving, we ha v e ξ~ u ′ 2 = σ ′ n or else ~ u ′ 2 = ξ σ ′ n . The first case implies the contradiction 0 = ξ ~ u ′ 2 ρ ′ n = σ ′ n ρ ′ n = σ n 6 = 0 . In the seco nd ca se we can as s ume that ξ is not an identit y . F rom 0 6 = ~ u ′ 2 ψ = ξ σ ′ n ψ we obtain 0 6 = σ ′ n ψ . Then we hav e in P the crown ( σ ′ n ψ , σ n , ρ n , σ 1 ) which is strictly smaller than the given chain C because the depth of ρ 1 is strictly greater than the depth o f σ ′ n ψ . So from now on σ n do es no t factor through ρ ′ n . If σ 2 factors through w ′ we can cho ose the decompo sition so that it factors alr eady through ~ w ′ 2 . Then we obtain in P /ρ 1 or for σ n = ρ 1 in P /σ 2 the cleaving diagr a m from figure 6.4 that shows an algebra with num b er 11 from the list. The o ccurring morphisms a re σ n , ρ ′ n , ~ w 2 , ~ w 1 , ψ , ~ u 2 , ~ w ′ 1 , σ ′ 2 . So σ 2 factors thro ugh u ′ . If we even hav e σ 2 = ~ u ′ 2 σ ′ 2 we obtain in fig ur e 6.5 the next cleaving diagra m in P / σ 1 containing the mo rphisms ρ ′ n , ~ w 1 , ψ , ~ w ′ 1 , σ ′ 2 .: 17 ❅ ❅ ❅ ❘    ✠ ❄ ❅ ❅ ❅ ❘    ✠ figure 6.5    ✠    ✠ ❄ ❄    ✠ ❅ ❅ ❘   ✠ ❅ ❅ ❘ ρ ′ n ~ b ❅ ❅ ❅ ❘   ✠ figure 6.6 ~ bρ ′ n = 0 Here σ ′ 2 = ρ ′ n ξ leads to 0 6 = ~ u ′ 2 σ ′ 2 = ~ u ′ 2 ρ ′ n ξ = 0, whereas ρ ′ n = σ ′ 2 ξ with a non-identit y ξ implies 0 6 = ~ u 2 σ ′ 2 . Then w e find in P a ˜ D 4 -quiver a s a cleaving diagram inducing the cr o wn ( σ 1 , ~ u 2 σ ′ 2 , σ 2 , ρ 1 ) which is strictly smaller than C . In the last case remaining with a lo ng σ 2 there is a non-trivia l deco mposition u ′ 2 = ab suc h that σ 2 = ~ aσ ′ 2 . Then w e get ~ bρ ′ n = 0 because otherwise we hav e an obvious cleaving diagram of trype ˜ D 4 in P / ρ 1 . Now figur e 6.6 shows a cleaving functor from the categor y with num ber 20 fro m our list in to P /ρ 1 . The o ccurring morphisms are σ n , ρ ′ n , ~ w 2 , ~ w 1 , ψ , ~ u 2 , ~ w ′ 1 , ~ b, σ ′ 2 . Recall here that σ n do es not factor thr ough ρ ′ n . W e hav e trea ted a ll case s where σ 2 is long. By duality , w e are r educed to the situation that neither ρ n nor σ 2 are long. Fir st, let ρ n factor through w . Cho osing an appro priate decompo sition for w w e hav e ρ n = ~ w 2 ρ ′ n . Then u ′ 2 is an a rrow, b ecause for u ′ 2 = ab one has the cleaving functor sho wn in figur e 6.7 from the ca tegory with n umber 11 to P /ρ 1 . The occurr ing mor phisms are ρ ′ n , ~ w 2 , ~ w 1 , ψ , ~ u 2 , ~ w ′ 1 , ~ b. If σ 2 factors thro ugh w ′ we find a decomp osition s uc h that σ 2 = ~ w ′ 2 σ ′ 2 . This g ives in P the category 93 as a cleaving diagr am dr awn in figure 6.8 with morphisms ρ ′ n , ~ w 2 , ~ w 1 , ψ , ~ u 2 , ~ w ′ 1 , ~ u ′ 2 , ~ w ′ 2 , σ ′ 2 .    ✠ ❄ ❄    ✠ ❅ ❅ ❘ ✲ ❅ ❅ ❘ figure 6.7    ✠ ❄ ❄    ✠ ❅ ❅ ❅ ❘ ❄ ❅ ❅ ❅ ❘ ✛ ✲ figure 6.8 If there is a third long morphism τ different from σ 1 and ρ 1 this diagra m is in P / τ . In the other case we ha v e ρ n ≤ σ 1 and σ 2 ≤ ρ 1 by lemma 5 part d). Thu s part c) of le mma 4 implies that w 1 = ab a nd w ′ 1 = a ′ b ′ with non- trivial de- comp ositions and arr o ws b, b ′ . and we find in figure 6.9 a cleaving diagra m from a ˜ D 8 -quiver to P / σ 1 . The occur ring morphisms are ρ ′ n , ~ w 2 , ~ a, ~ u 2 , ~ u ′ 2 , ~ w ′ 2 , ~ a ′ , σ ′ 2 . Observe that ρ ′ n resp. σ ′ n do es no t factor throug h ~ a r esp. ~ a ′ by lemma 4 ag ain. 18 ❄    ✠ ❅ ❅ ❅ ❘ ❄ ❄ ❄ ✛ ✲ figure 6.9    ✠ ❄    ✠ ❄    ✠ ❅ ❅ ❅ ❘ ✲ figure 6.1 0 ❅ ❅ ❅ ❘ ❄ ❅ ❅ ❅ ❘    ✠ ❄ ❅ ❅ ❅ ❘    ✠ φ ✲ figure 6.1 1 If σ 2 factors thr ough u ′ it factors alrea dy through the irr educible morphism ~ u ′ 2 , i.e. we ha v e σ 2 = ~ u ′ 2 σ ′ 2 . Then we hav e ~ u 2 σ ′ 2 = 0 because otherwise P /ρ 1 contains a catego ry of type 1 1 a s a cleaving diagram. This is indica ted in figure 6.10. The morphisms inv olved are ρ ′ n , ~ w 2 , ~ w 1 , ψ , ~ u 2 , ~ w ′ 1 , σ ′ 2 . If ρ 2 do es not factor through σ ′ 2 we obtain the cleaving functor of figur e 6.11 from a category of t ype 12 to P /σ 1 . Here φ is ρ 2 for ρ 2 6 = σ 1 or else an irreducible morphism that belong s to the path u or w that σ 2 do es not factor through. T he second co nstruction alwa ys works for ρ 2 = σ 1 . The inv ov ed morphisms ar e ρ ′ n , ~ w 1 , ψ , ~ u ′ 2 , ~ w ′ 1 , ~ w ′ 2 , σ ′ 2 , φ . So we ha v e ρ 2 = ρ ′ 2 σ ′ 2 and ρ 2 6 = σ 1 . If ρ ′ 2 ψ 6 = 0 we find a ˜ D 4 quiver co nsisting of the morphisms ψ, σ ′ 2 , ρ ′ 2 , ~ u ′ 2 in P /σ 1 . Th us we hav e ρ ′ 2 ψ = 0 and we find in figure 6.12 a n ˜ E 8 -quiver or in figure 6 .13 a ˜ D 6 -quiver in P /σ 1 depe nding o n the fact whether σ n factors through ρ ′ n or not. The inv olv ed morphisms are obvious. ❅ ❅ ❅ ❘ r ❄ σ ′ 2    ✠ ❄ ρ ′ 2    ✠ ❄ ✲ ✛ ~ u 2 σ ′ 2 = 0 figure 6.1 2 ❅ ❅ ❅ ❘ ❄    ✠ ❄    ✠ ✲ figure 6.1 3 By the symmetry of our situation we ar e left with the case where ρ n factors through u and σ 2 through u ′ . One of the tw o paths u 2 or u ′ 2 is an ar ro w because otherwise w e hav e an o b vious ˜ D 5 - quiv er in P /σ 1 . Say u 2 is an arrow. Then we hav e σ 2 = ~ u ′ 2 σ ′ 2 . If we hav e also ρ n = ~ u 2 ρ ′ n we lo ok at the diagra m of fig ur e 6.14 in P /σ 1 . It contains the morphisms ρ ′ n , ~ w 1 , ψ , ~ w ′ 1 , σ ′ 2 . If this diagra m is not cleaving we hav e up to symmetry σ ′ 2 = ρ ′ n ξ . This implies ~ u ′ 2 ρ ′ n 6 = 0 and we obtain the cr o wn ( ρ n , σ 1 , ρ 1 , ~ u ′ 2 ρ ′ n ). The minimal choice of C implies n = 2. F rom parts c) resp. d) of lemmata 4 resp. 5 we get that ρ n ≤ σ 1 and that ψ has a non-tr ivial facto rization ψ = ψ 2 ψ 1 with irreducible ψ 1 . Then fig ure 6 .1 5 shows a ˜ D 4 -quiver in P /σ 1 consisting of the morphis ms ρ ′ n , ~ u 2 , ψ 2 , ~ u ′ 2 . 19 ❅ ❅ ❅ ❘ ❄    ✠ ❅ ❅ ❅ ❘    ✠ figure 6.1 4 ❄ ❅ ❅ ❘   ✠ ❅ ❅ ❘ figure 6.1 5 ❅ ❅ ❅ ❘ ❄   ✠   ✠ ✲ figure 6.1 8 Finally , ρ n do es no t factor through ~ u 2 . Then u 2 has a non-trivial decomp o- sition u 2 = ba and ρ n = ~ bρ ′ n . One has ~ aσ ′ 2 = 0 because other wise a ˜ D 4 -quiver with mo rphisms ψ , σ ′ 2 , ~ a, ~ u ′ 2 is cleaving in P / σ 1 . If σ n do es not facto r through ρ ′ n we hav e in P / σ 1 an ˜ E 6 -quiver as shown in figure 6.1 6. figure 6.1 6    ✠    ✠ ❅ ❅ ❅ ❘ ❄ ❄ ✲ ❅ ❅ ❅ ❘ ❄ ❅ ❅ ❅ ❘ ❄    ✠    ✠ ❅ ❅ ❅ ❘    ✠ ❅ ❅ ❅ ❘ σ ′ 2 ~ a ~ aσ ′ 2 = 0 figure 6.1 7 If ρ 2 do es not fac to r through σ ′ 2 we find in P /σ 1 a catego ry o f t yp e 20 as a cleaving diagram. This is illustrated in figur e 6.17. So supp ose we ha ve ρ 2 = ρ ′ 2 σ ′ 2 . Then P /σ 1 contains the ˜ D 5 -quiver shown in figur e 6.18. as a diagram. The morphisms inv olved are ρ ′ n , ~ b, ~ a, ρ ′ 2 , ~ u ′ 2 . If this diag r am is not cleaving, we ha ve ~ b ~ a = ξ ρ ′ 2 with a non-iden tit y ξ . But then P /σ 1 contains as cleaving diagr am the ˜ D 4 -quiver with morphisms ψ , σ ′ 2 , ρ ′ 2 , ~ u ′ 2 . 2.2.3 The length of C is at least 6 Lemma 7 The length of C is n ot 4 . Pro of: Suppo se n = 2. By part c) of lemma 5, there are at least tw o diff erent long mor phisms which cannot b e neighbor s b y the last section. So up to duality we hav e b y part b) of lemma 5 that σ 1 and σ 2 are long and that ρ i ≤ σ i holds for i = 1 , 2. So by pa r t c) of lemma 4 we obtain from σ 1 and ρ 1 a cleaving diagram as in figur e 2.3 . Cho osing a co n tour ( v ′ , w ′ ) for σ 2 we obtain a similar dia gram from σ 2 and ρ 2 thereby ass uming that ρ 2 factors thro ugh v ′ . Then there is a n arrow η in v ′ or in w ′ that ρ 1 do es not factor through. So w e obtain in P /σ 2 the cle a ving diagr a m shown in figure 7.1. If one of the v i ’s is no t an arrow or if the length of w is ≥ 5 , then we obtain o bviously an ˜ E 6 - o r an ˜ E 8 -quiver a s a cleaving diagr am in P /σ 2 . This is indicated in fig ure 7 .2. 20    ✠ ~ η   ✠ ❅ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❄   ✠ ρ ′ 1 ~ w 1 ~ v 1 ~ w 2 ~ v 2 ~ v 3 figure 7.1 ❅ ❅ ❘ ❄ ❄   ✠ ❅ ❅ ❘   ✠ ❅ ❅ ❘   ✠ ❄ ❅ ❅ ❘ ❄   ✠ ❅ ❅ ❘   ✠ ❄ ❅ ❅ ❘ ❄ ❄ ❄   ✠ figure 7.2 Thu s we ca n as sume that all v i and all v ′ i are arrows and also that ( radP ) 5 = 0. O therwise there is a path u of leng th ≥ 5 suc h that ~ u is a long morphism say σ 1 and we can r eplace the contour ( v , w ) b y ( v , u ) or by ( u, w ) whic h is impo ssible. First we co nsider the case wher e the contour ( v ′ , w ′ ) chosen to σ 2 is p erme- able. F or w 1 we take an a rrow and we factor ize ρ 1 = ρ ′ 1 ~ v 1 . If ρ ′ 1 factors through w ′ we find the ray ca tegory 44 from our list as a cleaving diagra m in P /σ 2 as drawn in figure 7 .3. The morphisms in volv ed are ~ w 1 , ~ w 2 , ~ v 1 , ~ v 2 , ~ v 3 , ρ ′ 1 , ~ v ′ 3 , ~ v ′ 2 , ρ ′ 2 . Here ρ 1 do es no t factor through ~ v ′ 3 , b ecause it fac tors already thro ugh w ′ . Also ~ v ′ 3 ~ v ′ 2 do es not factor through ρ ′ 1 , b ecause v ′ and w ′ are not interlaced. ❅ ❅ ❘ ✁ ✁ ✁ ☛ ❆ ❆ ❆ ❯ ❄   ✠ ❏ ❏ ❏ ❏ ❏ ❫ ρ ′ 1   ✠ ❄ ❅ ❅ ❘ figure 7.3 ❅ ❅ ❘ ✁ ✁ ✁ ☛ ❅ ❅ ❘ ❍ ❍ ❍ ❥ γ   ✠ ❄ ❅ ❅ ❘ figure 7.4 If ρ ′ 1 do es not fac to r throug h w ′ we hav e a nother factoriza tion ρ 1 = ~ w ′ 2 γ giving ris e in P /σ 1 to the cleaving diag ram shown in figure 7.4 . It inv olves the morphisms ~ w 1 , ~ v 1 , γ , ~ w ′ 1 , ~ v ′ 1 , ~ v ′ 2 , ρ ′ 2 . Here γ do es no t fac to r thro ugh ~ w 1 , b ecause ρ 1 do es not factor throug h w , and it do es no t factor throug h ~ v 1 , b ecause ρ ′ 1 do es no t fac tor through w ′ . By sy mmetr y , w e can a ssume now that the c o n tours chosen to σ 1 and σ 2 are b oth reflecting. Then ρ 1 factors alre a dy through ~ v ′ 3 ~ v ′ 2 , b ecause for ρ 1 = ~ v ′ 3 φ we find a ˜ D 5 -quiver as a cleaving diagram in P /σ 1 consisting of the morphisms φ, ~ v ′ 1 , ~ v ′ 2 , ~ v ′ 3 , ρ ′ 2 . By symmetry we are in the situatio n of figur e 7.5 . 21    ✠ ❅ ❅ ❅ ❘ ❅ ❅ ❘ ❄ ˜ ρ 1 ~ w 1 ~ v 1 ~ v 2   ✠ ❅ ❅ ❅ ❘ ❳ ❳ ❳ ❳ ❳ ❳ ③ ✘ ✘ ✘ ✘ ✘ ✘ ✾    ✠   ✠ ❄ ˜ ρ 2 ~ w ′ 1 ~ v ′ 1 ~ v ′ 2 ❅ ❅ ❘ figure 7.5 Here we hav e chosen w 1 and w ′ 1 as a rrows. W e have ρ 1 = ~ v ′ 3 ~ v ′ 2 ˜ ρ 1 and ρ 2 = ~ v 3 ~ v 2 ˜ ρ 2 . Because ( r adP ) 5 = 0 the ˜ ρ i ’s are irreducible and ( ˜ ρ 1 , ~ v ′ 1 , ˜ ρ 2 , ~ v 1 ) cannot b e a cr o wn in P /σ 1 . This implies ~ v 1 = ˜ ρ 1 and ~ v ′ 1 = ˜ ρ 2 . F or v 1 = v ′ 1 we get v 2 6 = v ′ 2 bec ause other wise ρ 1 = ρ ′ 1 ~ v 2 ~ v 1 . Since ρ 1 ≤ σ 1 and v 3 is an arr o w, this co n tradicts part c) of lemma 4. F urthermor e w 1 = w ′ 1 implies ρ 1 = ~ v ′ 3 ~ v ′ 2 ~ v 1 = ~ w ′ 2 ~ w ′ 1 = ~ w ′ 2 ~ w 1 , i.e. ρ 1 factors through w . Th us w e find a ˜ D 5 -quiver as a cleaving diagr am in P /σ 1 that inv olves the mor phisms ~ w 1 , ~ v 1 , ~ w ′ 1 , ~ v 2 , ~ v ′ 2 . F or v 1 6 = v ′ 1 we get as ab o ve that v 2 6 = v ′ 2 bec ause v 3 is ir reducible. But now one has a ˜ D 4 - quiver in P / σ 1 consisting o f the morphisms ~ v 1 , ~ v 2 , ~ v ′ 1 , ~ v ′ 2 . 2.2.4 The factorization of neig h bors of long morphi sms stops after one step W e keep all the no tations and use all the reductio ns alr eady obtained. In pa r- ticular, long morphisms are not neig h bor s and n > 2. So there is no cleaving diagram as in figur e 2.1 in P . Lemma 8 L et ( v , w ) b e a c ontour with ~ v = σ 1 . L et v = v 2 v 1 b e any n on-trivial de c omp ositio n with ρ 1 = ρ ′ 1 ~ v 1 . Then σ 2 do es not factor thr ough ρ ′ 1 . Pro of: Supp ose on the contrary that σ 2 = ρ ′ 1 σ ′ 2 . Then we hav e ~ v 2 σ ′ 2 = 0 bec ause otherwise ~ v 1 , ~ v 2 , σ ′ 2 , ρ ′ 1 define a cleaving diag ram as in figure 2.1. Firs t, we treat the case where the co n tour is p ermeable. If ρ 2 do es no t factor through σ ′ 2 we o btain an obvious ˜ E 7 -quiver drawn in fig ure 8.1 as a cleaving diagra m in P /τ . Here we take τ = σ 1 for σ 1 6 = ρ 2 or else a long morphism different from σ 1 . ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘    ✠ ❅ ❅ ❅ ❘    ✠ ❅ ❅ ❅ ❘    ✠ ρ ′ n ~ w 1 ~ v 1 ρ ′ 1 σ ′ 2 ρ 2 σ 3 figure 8.1 F or ρ 2 = ρ ′ 2 σ ′ 2 we get ρ ′ 2 ~ v 1 = 0 because figur e 2.1 is not cleaving in P . If σ n = σ ′ n ρ ′ n , figur e 8.2 shows a cleaving diagra m in P /σ 1 . 22    ✠ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘    ✠ ❅ ❅ ❅ ❘ P P P P P P P P P q figure 8.2 ρ ′ 1 ~ w 2 ~ v 2 ρ ′ 2 ρ ′ n σ ′ n If σ 3 do es not factor throug h ρ ′ 2 we lo ok a t the cleaving diag ram of type ˜ E 8 in P that is given in figur e 8.3. Here φ is σ n if this is not long or else an irreducible morphism that ρ ′ n do es no t factor through. F or σ 1 6 = ρ 2 this diagr am is a lready in P /σ 1 , while for σ 1 = ρ 2 there is ano ther lo ng morphism τ a nd the diagram is in P /τ . If σ 3 = ρ ′ 2 σ ′ 3 , it follows ρ ′ 1 σ ′ 3 = 0 since figure 2.1 is not cleaving. Then the diagram of figure 8.4 is cle aving in P where φ is defined as in the case b efore. Again this diagram is in P /σ 1 for σ 1 6 = ρ 2 or else in P /τ for a long morphism τ 6 = σ 1 . ❅ ❅ ❘ ❅ ❅ ❘   ✠ ❍ ❍ ❍ ❍ ❥   ✠   ✠   ✠ ❅ ❅ ❘ ρ ′ 1 σ ′ 2 σ 3 ρ ′ 2 ~ v 2 σ ′ 2 = 0 figure 8.3 φ ρ ′ n ~ v 2 ❅ ❅ ❘ ❅ ❅ ❘   ✠ ❍ ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✟ ✙   ✠   ✠ ❅ ❅ ❘ figure 8.4 ρ ′ 1 σ ′ 2 ~ v 1 ρ ′ 2 ~ v 1 = ρ ′ 1 σ ′ 3 = 0 ρ ′ 2 σ ′ 3 φ ρ ′ n Now we consider the case wher e the contour ( v , w ) is refle c ting . Let τ 6 = σ 1 be ano ther long morphis m. W e cla im that ρ n = ~ v 2 ρ ′ n . Suppose this is false. Then we get a prop er dec o mposition v 2 = ba with ρ n = ~ bρ ′ n . If ρ ′ 1 do es no t factor through ~ a , we hav e a ˜ D 5 -quiver in P / τ in volving the morphisms ~ v 1 , ~ a, ~ b, ρ ′ n , ρ ′ 1 . Thu s we ha ve ρ ′ 1 = ρ ′′ 1 ~ a with ρ ′′ 1 ρ ′ n = 0, b ecause otherwis e we obtain a ˜ D 4 -quiver defined b y the morphisms ~ a, ~ b, ρ ′ n , ρ ′′ 1 . If ρ 2 is long or if it do es not factor through σ ′ 2 , then P /σ 1 contains a cleaving diagra m of t yp e ˜ E 6 . It involv es the morphisms ~ w 1 , ~ v 1 , ~ a, ρ ′′ 1 , σ ′ 2 , φ , where φ is an a rrow that σ ′ 2 do es not fac to r through if ρ 2 is long or else φ = ρ 2 . Therefo re, ρ 2 = ρ ′ 2 σ ′ 2 . In cas e ρ ′ 2 = ρ ′′ 2 ~ a one has ρ ′′ 2 ~ a ~ v 1 = 0 since otherwis e P co n tains fig ure 2 .1 as a cleaving diag ram with the morphisms ~ a ~ v 1 , ~ aσ ′ 2 , ρ ′′ 1 , ρ ′′ 2 . But now the ˜ D 4 -quiver defined b y ~ a, ~ b, ρ ′′ 1 , ρ ′′ 2 is cleaving in P . Thu s ρ ′ 2 do es not factor throug h ~ a and we also get ρ ′ 2 ~ v 1 = 0 by the ’figure 2.1 ’ argument. So we finally o bta in in P / σ 1 the ˜ E 7 -quiver inv olving the mo rphisms ~ w 1 , ~ v 1 , ~ a, ρ ′ n , ρ ′′ 1 , σ ′ 2 , ρ ′ 2 with the relatio ns ρ ′′ 1 ρ ′ n = ρ ′ 2 ~ v 1 = 0 . W e have shown our claim ρ n = ~ v 2 ρ ′ n . Lo oking a t the morphisms ~ v 1 , ρ ′ n , ρ ′ 1 , ~ v 2 we infer ρ ′ 1 ρ ′ n = 0. Then the diag ram of figure 8 .5 is cleaving in P . ❄ ❅ ❅ ❘ ❅ ❅ ❘   ✠ ❅ ❅ ❘ ❄ ❅ ❅ ❘ ❅ ❅ ❘   ✠ ❅ ❅ ❘   ✠ ~ w 2 ~ v 2   ✠ ρ ′ n ~ w 1 ~ v 1 ρ ′ 1 σ ′ 2 ρ ′ 1 ρ ′ n = ~ v 2 σ ′ 2 = 0 figure 8.5 ❄ ❅ ❅ ❘   ✠ ❅ ❅ ❘ figure 8.6   ✠ ❅ ❅ ❘ ρ ′ n ~ v 1 ρ ′ 1 σ ′ 2 φ ρ ′ 1 ρ ′ n = 0 23 Define φ = ρ 2 if ρ 2 is not long and do es not factor thr ough σ ′ 2 or take for φ an irreducible morphism that σ 2 do es not fac to r throug h if ρ 2 is long. Then the ˜ E 6 -diagra m of figure 8.6 is c lea ving in P / σ 1 . Thus we hav e that ρ 2 is not lo ng and ρ 2 = ρ ′ 2 σ ′ 2 . This implies ρ ′ 2 ~ v 1 = 0. Suppo se no w that σ n do es not factor through ρ ′ n . Then P admits an ˜ E 8 - quiver as the c le aving diagra m shown in fig ur e 8.7. This lies a lready in P /σ 1 if σ 1 6 = ρ n − 1 . F or σ 1 = ρ n − 1 there is another long mor phism τ . F or τ 6 = σ 2 the diagram lies in P /τ . F or τ = σ 2 one gets a n ˜ E 6 -quiver dr a wn in fig ure 8 .8 as a cleaving dia gram in P /σ 2 . Here φ is an ir reducible morphism that σ ′ 2 do es not factor thro ugh.   ✠ ❅ ❅ ❘ ❄ ❅ ❅ ❘   ✠ ❅ ❅ ❘   ✠ ❅ ❅ ❘ ρ ′ 1 σ ′ 2 ~ v 1 ρ ′ n ρ ′ 1 ρ ′ n = 0 σ n ρ n − 1 σ n − 1 figure 8.7   ✠ ❅ ❅ ❘ ❄ ✲   ✠ ❅ ❅ ❘ φ σ ′ 2 ~ v 1 ρ ′ n σ n figure 8.8 Thu s w e can a ssume that σ n = σ ′ n ρ ′ n which implies σ ′ n ~ v 1 = 0. Then the three morphisms ~ v 2 , ρ ′ 1 , ρ ′ 2 induce a cle a ving diagr am in P . If we add σ ′ n this cannot stay a cleaving diagr a m b ecause no ne of the inv olved mo rphisms is long. Thus we get ξ ρ ′ 2 = σ ′ n or ρ ′ 2 = ξ σ ′ n . In both case s we obtain an ˜ E 6 -quiver as a cleaving diagram in P . In the firs t case it inv olves the morphisms ~ v 1 , ρ ′ n , σ ′ 2 , ~ v 2 , ρ ′ 1 , ρ ′ 2 and in the seco nd case ~ v 1 , ρ ′ n , σ ′ 2 , ~ v 2 , σ ′ n , ρ ′ 1 . These induce by part e) of lemma 3 the t wo c r o wns ( σ 1 , ρ 1 , σ 2 , ρ 2 , ρ ′ 2 ρ ′ n , ρ n ) and ( σ 1 , ρ 1 , σ 2 , σ ′ n σ ′ 2 , σ n , ρ n ). B y minimalit y we get n = 3 and ξ = id in both c a ses. Since ρ 2 is not long, only the σ i ’s can be long. If all of them are lo ng , we can a lw ays choo se a n irr educible morphism where ~ v 1 resp. ρ ′ 3 resp. σ ′ 2 do es no t fac tor through. This gives an ˜ E 6 -quiver in a prop er quo tien t P /τ as indicated in figure 8.9. Th us, using duality , w e can assume tha t σ 1 and σ 2 are the only lo ng morphisms. Then w e get ρ 1 ≤ σ 1 or ρ 1 ≤ σ 2 . In the first ca se v 2 = b a by par t c) o f lemma 4. W e find a ˜ D 4 -quiver with morphisms ~ v 1 , ρ ′ 3 , σ ′ 2 , ~ a or an ˜ E 6 -quiver in P /σ 1 as shown in figure 8.10 . The case ρ 1 ≤ σ 2 is dual be c a use there is a lso a reflecting contour to σ 2 .   ✠ ❅ ❅ ❘ ❄ ❅ ❅ ❘   ✠ ❅ ❅ ❘ ρ ′ 3 ~ v 1 σ ′ 2 figure 8.9   ✠ ❅ ❅ ❘ ❄ ❄   ✠ ❅ ❅ ❘ ρ ′ 1 ~ w 1 ~ v 1 σ ′ 2 ρ ′ 3 ~ a ρ ′ 1 ρ ′ 3 = ~ aσ ′ 2 = 0 figure 8.10 24 2.2.5 C do es not exist Assume tha t the co n tour ( v , w ) chosen for the long morphism σ 1 in C is p erme- able. Since the fa ctorization of the neig h bo r s stops after one step, we obtain up to permutation of v and w the following clea ving diagram drawn in figure 9.1 in P . ❅ ❅ ❅ ❘    ✠    ✠ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘    ✠ ❅ ❅ ❅ ❘    ✠ ρ ′ n σ n σ 2 ~ w 1 ~ v 1 ρ ′ 1 ~ w 2 ~ v 2 figure 9.1 Replacing σ 1 by v 2 and ρ 1 by ρ ′ 1 we o btain a smalle r crown than C b ecause the sum of the depths of the inv olved morphisms has strictly decreased. W e are left with the case where a ll the contours c hosen for the long mor- phisms in the crown C a re reflecting. W e start with σ 1 and go o n unt il we reach the next long morphism τ . Bec a use long mor phis ms are no t neighbors the first p ossible case is τ = σ 2 . W e can assume that ρ 1 factors through v and v ′ where ( v ′ , w ′ ) is a contour with ~ v ′ = σ 2 . Th us we ha ve non-trivial factor - izations v = v 2 v 1 and ρ 1 = ρ ′ 1 ~ v 1 . If ρ ′ 1 factors through v ′ , w e ha ve ρ ′ 1 = ~ v ′ 2 ρ ′′ 1 for some non-trivial fa c torization v ′ = v ′ 2 v ′ 1 . Here ρ ′′ 1 is not an identit y b ecause otherwise σ 2 factors thr ough ρ ′ 1 contradicting the last sectio n. Th us we ge t the following repres en tation infinite cleaving diagram shown in figur e 9.2 in P :   ✠ ❅ ❅ ❘ ❆ ❆ ❆ ❆ ❯ ✁ ✁ ✁ ✁ ☛ ❅ ❅ ❘ ❅ ❅ ❘   ✠ ✁ ✁ ✁ ✁ ☛ ❆ ❆ ❆ ❆ ❯ figure 9.2   ✠ ❅ ❅ ❘ ❄   ✠ ❄   ✠ ✁ ✁ ✁ ✁ ☛ ✲ figure 9.3 Because P is minimal repres en tation-infinite, σ 1 and σ 2 are the only long morphisms. But then ρ 1 ≤ σ 1 up to duality . By part c) of lemma 4 v 2 = ba with a n arrow b , and there is or a ˜ D 5 -quiver as a cleaving diagr am in P /σ 1 . The morphisms inv olved are ~ v 1 , ~ a, ρ ′′ 1 , ~ v ′ 1 , ~ v ′ 2 . in the first cas e or in P /σ 2 in the second. If ρ ′ 1 do es not factor thr ough v ′ , we obtain in P /σ 1 the cle a ving dia g ram 25 shown in figur e 9.3 . It is a catego ry with nu mber 12 from our list. Therefore we ca n ass ume that the minimal distance of tw o lo ng morphisms in the cr o wn is 3 at lea st. Next w e co nsider the case where τ is some ρ i . W e hav e 1 < i < n . Using the last subsection w e find the following cle aving diagram in P : ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ✁ ✁ ✁ ✁ ✁ ✁ ☛ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ❅ ❅ ❅ ❘    ✠    ✠ ❅ ❅ ❅ ❘    ✠ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✁ ✁ ✁ ✁ ✁ ✁ ☛ ❅ ❅ ❅ ❘ σ 2 σ 1 ρ i ~ w 1 ~ v 1 ~ w ′ 1 ~ w 2 ρ i − 1 ~ w ′ 2 ρ ′ n ~ v 2 ............. ............. σ ′ i +1 ~ v ′ 1 ρ i +1 ~ v ′ 2 σ ′ i ρ 2 σ n ρ ′ 1 ρ ′ 1 ρ ′ n = 0 = σ ′ i σ ′ i +1 ρ n − 1 figure 9.4 Here we have denoted the contour c hosen to the long morphism ρ i by ( v ′ , w ′ ) and we have assumed that bo th neighbors factor through v ′ . The only zero - relations are 0 = ρ ′ 1 ρ ′ n and 0 = σ ′ i σ ′ i +1 . The tw o outer ’diamonds’ connected by the lo w er finite zigzag form a representation-infinite cle aving diag ram D in P that contains by cons truction only the tw o long mor phisms σ 1 and ρ i . Since P is minimal repr esen tation-infinite, we conclude that these tw o morphisms are different a nd ar e the only long morphisms . If v 1 = ab is a no n- trivial factorization, one g ets an ˜ E 8 -quiver as a cleaving diagra m in P /σ 1 that involv es the morphis ms ~ a, ~ w 2 , ~ v 2 , ρ ′ 1 , σ 2 , ρ 2 , σ 3 , ρ 3 . The same argument s ho ws that v 2 , v ′ 1 and v ′ 2 are also a rrows. If we had ρ 1 ≤ σ 1 we would obtain fro m part c) of lemma 4, that ~ v 2 is not irreducible. Beca use ρ 1 = ρ ′ 1 ~ v 1 is neither irreducible nor long it is comparable to a long morphism. In our situation we obtain ρ i = δρ ′ 1 ~ v 1 γ for some appropria te morphisms γ and δ . W e cla im that ( ~ v 1 γ , ρ ′ n , σ n , . . . , ρ i +1 , σ ′ i +1 , ~ v ′ 1 ) is a crown. Since it is strictly smaller than C this is impossible. Only four factorisations b etw een neigh bor s in the c hain remain to b e excluded. The factor isation ~ v 1 γ = ρ ′ n η implies 0 6 = δ ρ ′ 1 ~ v 1 γ = δ ρ ′ 1 ρ ′ n η = 0. F rom ~ v 1 γ η = ρ ′ n we o btain ρ n = ~ v 2 ρ ′ n = ~ v 2 ~ v 1 γ η = σ 1 γ η which is omp ossible for the neighbo r s σ 1 and ρ n . Next assume η~ v ′ 1 = ~ v 1 γ . W e get 0 6 = δ ρ ′ 1 ~ v 1 γ = δ ρ ′ 1 η~ v ′ 1 = ρ i = ~ v ′ 2 ~ v ′ 1 , whence δ ρ ′ 1 η = ~ v ′ 2 . Since all v i and v ′ i are arrows we see that η , δ and γ ar e identities and that ~ v ′ 2 = ρ ′ 1 , ~ v ′ 1 = ~ v 1 . This implies the co n tradiction ρ 1 = ρ i . Finally , lo ok at ~ v ′ 1 = η~ v 1 γ . This implies 26 v 1 = v ′ 1 . In particular , the domains of the tw o long mor phisms in P coincide. Now we prov e the sa me for the cas e where the next long morphism after σ 1 is some σ i . W e can assume in addition 2 < i < n . Then w e obtain a similar cleaving diagr am as b efore: ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ✁ ✁ ✁ ✁ ✁ ✁ ☛ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ❅ ❅ ❅ ❘    ✠    ✠ ❅ ❅ ❅ ❘ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✎ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✁ ✁ ✁ ✁ ✁ ✁ ☛ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❲ σ 2 σ 1 σ i ~ w 1 ~ v 1 ~ w ′ 1 ~ w 2 σ i − 1 ~ w ′ 2 ρ ′ n ~ v 2 ............. ............. ρ ′ i − 1 ~ v ′ 1 σ i +1 ~ v ′ 2 ρ ′ i ρ 2 σ n ρ ′ 1 ρ n − 1 ρ ′ 1 ρ ′ n = 0 = ρ ′ i ρ ′ i − 1 figure 9.5 Observe that now the tw o zig z a gs connecting the outer diamo nds cr oss ea c h other. The o nly zer o-relations are 0 = ρ ′ 1 ρ ′ n and 0 = ρ ′ i ρ ′ i − 1 . Argueing as b efore one reduces to the situatio n whe r e σ 1 and σ i are the only long morphisms, ~ v 1 , ~ v 2 , ~ v ′ 1 and ~ v ′ 2 are irreducible, ρ 1 is smaller than σ i and ρ i smaller than σ 1 . So we hav e for so me a ppr opriate morphisms the equations δ ρ ′ 1 ~ v 1 γ = σ i and δ ′ ρ ′ i ~ v ′ 1 γ ′ = σ 1 . W e c la im that ( ~ v 1 γ , ρ ′ n , σ n , . . . σ i +1 , ρ i ) is a cr o wn in P . Again only four factorizations b et ween neighbors are not yet excluded by o b vious rea sons. ρ ′ n = ~ v 1 γ η implies ρ n = ~ v 2 ρ ′ n = σ 1 γ η which is imp ossible for neigh bor s in a crown. Applying δ ρ ′ 1 to the equation ρ ′ n η = ~ v 1 γ leads to 0 = δ ρ ′ 1 ρ ′ n η = δ ρ ′ 1 ~ v 1 γ 6 = 0 . The third fac to rization to b e ex c luded is η ρ ′ i ~ v ′ 1 = ~ v 1 γ . This gives δ ρ ′ 1 ~ v 1 γ = σ i = δ ρ ′ 1 η ρ ′ i ~ v ′ 1 = δ ρ ′ 1 η ρ i contradicting the fact that ρ i and σ i are neighbors in a crown. Finally supp ose ρ ′ i ~ v ′ 1 = η~ v 1 γ . W e g e t σ 1 = δ ′ ρ ′ i ~ v ′ 1 γ ′ = δ ′ η~ v 1 γ γ ′ so that δ ′ η 6 = 0. So we hav e δ ′ η = ~ v 2 ξ or δ ′ η = ξ~ v 2 . In the seco nd case ξ is an identit y bec ause we have 0 6 = δ ′ η~ v 1 = ξ~ v 2 ~ v 1 = ξ σ 1 . So we only hav e to consider the first case. F rom 0 6 = ~ v 2 ~ v 1 = σ 1 = δ ′ η~ v 1 γ γ ′ = ~ v 2 ξ~ v 1 γ γ ′ we s ee that γ , γ ′ and ξ are a ll ident ities. In particular , the domains o f the tw o lo ng morphisms coincide again. W e ha ve s hown that P contains only tw o long mo rphisms that hav e the same domain x . By the self-dualit y of our situation they also ha ve the sa me co domain y . This contradicts the fact that in any ray-catego ry P ( x, y ) is linear ly order ed. 27 3 The pro ofs of the main results 3.1 The pr o of of theorem 2 W e need the following e a sy le mma . Lemma 9 L et P b e a r ay c ate gory c ontainig a long morphism µ t hat do es not o c cur in a c ontour. Then we have: a) P and P /µ have the s ame quivers and the same c ontours. b) The quivers of the u niversal c overs and t he fundamental gr oups c oincide. c) F or al l ab elian gr oups Z the c ohomolo gy gr oup ( [3, se ction 8] ) H 2 ( P, Z ) emb e ds int o H 2 ( P /µ, Z ) . Pro of: a) The quivers co incide b ecause µ is long, and the contours, b ecause µ do es not b elong to a contour. b) The quivers of the universal cov ers and the fundamen tal groups are defined by the homotopy re la tion on the universal co vers o f the common quiver of P and P /µ . ( see [20, section 14.1] or [3, section 10]). Since the definition of homotopic walks de p ends o nly on the contours, part b) follows from par t a). c) As shown in [3, sectio n 8.2], H 2 ( P, Z ) is isomorphic to the q uotien t of the space C ( P, Z ) of Z -v a lued contour functions b y the space E ( P, Z ) o f e xact contour fun ctions. The point is that here one ha s to take the ’old’ definition of contour as given in [3, section 2 .7], where our cont ours are called essential contours. Now in the old lang ua ge part a ) says that P and P /µ have the same quivers and the same essential contours, but P has in g e neral more contours than P /µ . Restriction is a homomo rphism from C ( P , Z ) to C ( P /µ, Z ) which induces an isomorphism on the spaces of exa ct co n tour functions beca use the quivers co incide. Since a contour function is uniquely determined by its v alues on the e s sen tial cont ours which are the same for P and P /µ , the r estriction induces the wanted embedding. Now, let A b e a distr ibutiv e minimal representation-infinite alg ebra. Then there is the asso ciated ray c a tegory P ([3, section 1.7] or [20, section 13.4 ]). By theo rem 1 3.17 in [20], tha t is based on [3] and [9], P is a lso minimal representation-infinite. T o prov e the theorem we distinguis h three c a ses. If there is no long mo rphism in P , the ray category and the algebra are given by zero- relations o f le ngth 2, whence there a re no co n tours. Then theorem 2 is a well-known fac t from ele mentary alg ebraic top ology . If there is no cr o wn in P , then P is zig zag-free in the ter minology of [20] and we ar e done by [2 0, theorems 14.2,13 .17a]. Finally , in the las t case there is a crown and a long morphism. By pro position 2, there is a long morphis m µ as in the lemma ab o ve. Since P /µ con tains no crown, the univ ersal cover of P /µ is in terv a l-finite and the fu ndamental group is free. By the lemma, the same holds for P . Theorem 13.17 a) of [20] says that A is isomorphic to k f ( P ) fo r some cohomolo gy class in H 2 ( P, k ∗ ). But this 28 cohomolog y group v anishes for P /µ and by the lemma also for P . Therefor e, A is isomor phic to k ( P ). 3.2 The pr o of of theorem 1 W e have to sho w that a bas ic algebr a of infinite re presen tation type has inde- comp osable r epresen tations in a ll dimensions. If A is not distributive, w e obtain this from pro position 1 . If A is dis- tributive, we can a s sume that it is minimal representation-infinite and therefore by theorem 2 iso morphic to the lineariza tio n k ( P ) of its ray-category . More- ov er, the fundamental group is free. Th us the dimension pr eserving push-down functor as socia ted to the universal cov er pres erv es indecomp osabilit y by ( [20, section14.4 ] or [1 9] ). So it is enough to find indecomp osables of all dimensions for the universal cover ˜ P . Section 2.3 in [13] shows that the first homology group H 1 ( k ( ˜ P )) of the Sch urian ca tegory k ( ˜ P ) v anis he s . Beca use the par tia lly orde r ed sets x/P a nd P /y are zigzag-free by part a) of lemma 3, it f ollows from [6 , section 2.3 ] that each finite conv ex sub category B of ˜ P satisfies also H 1 B = 0 . By the separation- criterion [4] of Ba utista-Larri´ on as sligh tly generalized in [6, section2.5] an y suc h B has a prepro jective comp onent in its Auslander - Reiten quiver. W e hav e to distinguis h tw o ca ses. If ther e is such a subcatego ry B that is not r epresen tation-finite, w e apply some results o f Ringel in [2 4, section 4.3 ]. Namely there is a quotient C of B that is tame concealed. In pa rticular, C has indecomp osable representations in all dimens io ns. No te that in this case b y [10, s ection 6] each non-simple indecomp osable is aga in an extension of an indecomp osable a nd a simple as in the repr esen tation-finite case. If all these finite sub categor ie s a re r epresen tation-finite there ca n b e no common b ound for the dimensions of all the indecompo sable representations of ˜ P . F or then the push-down functor would pro duce a b ounded comp onent of the Ausla nder-Reiten quiver of P b y [20 , theorem 14 .4]. But then the category P , which is connected being minimal repr esen tation-infinite, would be representation-finite by a basic result in [1, c hapter 6]. So ther e a re indec o mpos- ables of arbitrar ily large dimensions, whence indeco mposables in all dimensions by the k nown repre s en tation-finite ca se. Observe that the pro of of theorem 1 uses no cla ssification lists a t all. In contrast, all pro ofs of the seco nd Brauer -Thrall conjecture via coverings depend on the list o f the la rge faithful simply connected alg ebras in [5] ( but not o n the lists in [2 2] o r [7] as indicated in [1, chapter 6 ] ). 3.3 The pr o of of corollary 1 Let U b e an indecomp osable in C of leng th n and height h . Let C ( h ) b e the full sub c ategory o f C c o nsisting of ob jects of heigh t at most h , that hav e only the comp osition factors of U as simple subquotients and that a re of finite leng th. Then C ( h ) is a n abelian sub category containing the indecompo s able U . It is 29 well-kno wn that C ( h ) is a mo dule catego ry ([17], [18, se ction 8]) over some algebra whic h is finite-dimensional if all extensio n- groups b et w een simples ar e finite-dimensional. F or the convenience of the reader we give so me details. If one of the extension groups E xt ( S, T ) betw een simples in C ( h ) is no t finite- dimensional, one constr ucts easily lo cal mo dules o f arbitrary le ng th n > 2 having top S . Indeed one ta k es n − 1 linear ly indep enden t elements E 1 , E 2 , . . . E n − 1 in E xt ( S, T ) and lo oks at the exact se q uence E : 0 → T n − 1 → X → S → 0 suc h that the push-out under the pro jection π i : T n − 1 → T is E i . Then X is the wan ted lo cal mo dule. If a ll extension groups betw een the simples a re o f finite dimension one c on- structs finitely many pro jectiv e indeco mposables P i ( h ) who se direct s um is a progener ator P ( h ) of finite length inside C ( h ). W e pro ceed by induction o n h . F or h = 1 we set P i (1) = S i for a repr esen tative system S 1 , S 2 , . . . , S r of the comp osition factors of U . In the inductive step we set dim k E xt ( P i ( h − 1) , S j ) = n ij . Note that by the ha lf-exactness of E xt these extension-groups are finite- dimensional. W e define P i ( h ) as the ( uniquely deter mined ) middle term of the universal extensio n 0 − → r M j =1 S n ij j − → P i ( h ) − → P i ( h − 1) − → 0 . W e leav e it as an exercise to show that P i ( h ) is the pro jective cov er of S i in C ( h ). The functor H om ( P ( h ) , ) iden tifies C ( h ) with the finite dimens io nal right mo dules ov er the finite dimensional endomor phism algebra of P ( h ). Thus the corolla r y fo llo ws fro m theorem 1. 3.4 The pr o of of corollary 2 The implication fr o m a) to b) is trivial. Reversely , theor em 1 implies that a ll indecomp osables hav e length at most n . By Roiters theo rem in [27] the algebra is repres en tation-finite. 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