# The braid group action on exceptional sequences for weighted projective lines

aa r X i v : . [ m a t h . R T ] F e b THE BRAID GROUP ACTION ON EXCEPTIONAL SEQUENCES FORWEIGHTED PROJECTIVE LINES

EDSON R. ALVARES, EDUARDO N. MARCOS, AND HAGEN MELTZER

We dedicate this work to the memory of Andrzeyj Skowroński

Abstract.

We give a new and intrinsic proof of the transitivity of the braid groupaction on the set of full exceptional sequences of coherent sheaves on a weightedprojective line. We do not use here the corresponding result of Crawley-Boevey formodules over hereditary algebras. As an application we prove that the strongest globaldimension of the category of coherent sheaves on a weighted projective line X does notdepend on the parameters of X . Finally we prove that the determinant of the matrixobtained by taking the values of n Z -linear functions deﬁned on the Grothendieckgroup K ( X ) ≃ Z n of the elements of a full exceptional sequence is an invariant, upto sign. Introduction

Let X be a weighted projective line in the sense of Geigle and Lenzing [GL1]. The braidgroup B n on n strings acts on the set of full exceptional sequences in the category coh( X ) of coherent sheaves on X , where n denotes the rank of the Grothendieck group K ( X ) of coh( X ) . This action is given by mutations in the sense of Gorodentsev and Rudakov [GR].The following result was proved in [M1] Theorem 1.1.

The action of the braid group on the set of full exceptional sequences inthe category of coherent sheaves on a weighted projective line X is transitive. The proof was based on induction on the rank of the Grothendieck group of coh( X ) andon the rather strong result of Crawley-Boevey [CB] which states that the braid group actstransitively on the set of full exceptional sequences in the category of ﬁnitely generatedmodules over a hereditary algebra over an algebraically closed ﬁeld.It is desirable to have in the geometric situation a purely sheaf-theoretical proof forthe transitivity of the braid group operation. In this paper we show that this in factcan be done using perpendicular calculus of exceptional pairs. For this we calculate theleft perpendicular category of the sum of two line bundles L ⊕ L ( ~c ) formed in the sheafcategory, where L is a line bundle and ~c the canonical element of the grading group of X .For the convenience of the reader we also state the unchanged parts of the original proof.Furthermore, we give two applications of the transitivity of the braid group action.First we show that the strongest global dimension of a weighted projective line X , anotion which we deﬁned in this paper, is independent of the parameters of X . This meansthat if X = X ( p , λ ) and X ′ = X ( p , λ ′ ) , are weighted projective lines with the same weightsequence p and diﬀerent parameter sequences λ and λ ′ then the strong global dimensionsfor X and X ′ are the same. Mathematics Subject Classiﬁcation.

Primary 14H05,; Secondary 16G20, 16G99.

Key words and phrases. braid group, exceptional sheaf, exceptional sequence, weighted projective line,tilting sheaf, tilting complex, strong global dimension, Grothendieck group, diophantine equation.The third mentioned author thanks FAPESP, from the grant 2019/08284-4, which made this workpossible, the ﬁrst mentioned author thanks FAPESP, from the grant 2018/08104-3. The second mentionedauthor was supported by the thematic project of FAPESP 2014/09310-5, a research grant from CNPq302003/2018-5 and we all acknowledges support from the "Brazilian-French Network in Mathematics".

Second we prove that the determinant of the matrix obtained by applying n additivefunctions deﬁned on the Grothendieck group of coh( X ) to the sheaves of a full exceptionalsequence on X is independent of the exceptional sequence, up to sign. Finally, we calculatethis invariant for taking the rank function, the degree function and n − Euler forms withrespect to simple exceptional sheaves.2.

Preliminaries k be an algebraically closed ﬁeld. A weight sequence p = ( p , . . . , p t ) is a sequenceof natural numbers p i with p i ≥ . For a weight sequence p denote by L ( p ) the abeliangroup with generators ~x , . . . , ~x t and relations p ~x = · · · = p t ~x t := ~c . The element ~c is called the canonical element. L ( p ) is an ordered group with P ti =1 N ~x i as cone ofnon-negative elements. Furthermore, each element ~x can be written, on a unique way, innormal form ~x = l~c + P ti =1 l i ~x i with l ∈ Z and ≤ l i < p i . Consider further a sequenceof parameters λ = ( λ , . . . , λ t ) , that is the λ i are non-zero and pairwise distinct elementsfrom k . We denote S = S ( p , λ ) = K [ X , . . . , X t ] / ( X p i i + X p + λ i X p , i = 3 , . . . , t ) . Thealgebra S ( p , λ ) is L ( p ) -graded by deﬁning deg ( X i ) = ~x i . Then the weighted projectiveline X = X ( p , λ ) is deﬁned to be the L ( p ) -graded projective scheme Proj L ( p ) ( S ( p , λ )) andthe category coh( X ) of coherent sheaves on X is the quotient of the category of ﬁnitelygenerated L ( p ) -graded S modules modulo the L ( p ) -graded S modules of ﬁnite length. Thecategory coh( X ) is abelian, hereditary, that is Ext i ( A, B ) = 0 for all A and B in coh( X ) and i ≥ , and it has ﬁnite dimensional Hom and

Ext spaces. Moreover, coh( X ) admitsSerre duality in the form Ext ( A, B ) ≃ D Hom(

B, A ( ~ω )) , where ~ω denotes the dualizingelement ( t − ~c − P ti =1 ~x i , and consequently coh( X ) has Auslander-Reiten sequences.We denote the structure sheaf on X by O . It is well known that the isomorphism classof line bundles on X form a group, via the tensor product and this group is isomorphicto the group L ( p ) via the map ~x

7→ O ( ~x ) where O ( ~x ) is the twisted by ~x structure sheaf.Moreover, the homomorphism space between two line bundles can be calculated by theformula Hom( O ( ~x ) , O ( ~y )) ≃ S ~y − ~x and if ~z = l~c + P tj =1 l i ~x i is in normal form, then dim S ~z = l + 1 provided l ≥ − . For coherent sheaves on X we have the rank and thedegree function, which are deﬁned on the Grothendieck group K ( X ) . The sheaves of rank are those of ﬁnite length. One of the key results in [GL1] is that the sheaf L ≤ ~x ≤ ~c O ( ~x ) is a tilting sheaf such that its endomorphism algebra is a canonical algebra.2.2. Recall that an object in a hereditary k -category H is called exceptional if End( E ) = k and Ext ( E, E ) = 0 . Moreover, a sequence of exceptional objects ǫ = ( E , . . . , E r ) iscalled an exceptional sequence if Hom( E j , E i ) = 0 = Ext ( E j , E i ) for all j > i . If r = 2 then ǫ is called an exceptional pair and if r equals the rank of the Grothendieck group K ( H ) then ǫ is called a full exceptional sequence. This nomenclature is justiﬁed sinceevery exceptional sequence has at most K ( H ) entries and any exceptional sequence canbe extended to at least one full exceptional sequence.Gorodentsev and Rudakov deﬁned mutations of exceptional sequences on P n whichgive rise to an operation of the braid group B r = h σ , . . . , σ r − | σ i σ j = σ j σ i for i − j ≥ σ i σ i +1 σ i = σ i +1 σ i σ i +1 i on the set of (isomorphism classes) of exceptional sequencesof length r [GR]. For a categorical treatment we refer to [B].We will study the action of the braid group B n , where n is the rank of K ( X ) , on theset of full exceptional sequences in coh( X ) . In this case each line bundle is exceptional.Moreover, the simple exceptional sheaves of rank ﬁt in exact sequences −→O ( j~x i ) −→O (( j + 1) ~x i ) −→ S i,j −→ . RAID GROUP ACTION 3

We continue this section with the following lemma, which is probably well known, butfor the sake of completeness we state and give a proof.

Lemma 2.1.

Let A be an abelian k -category and M an object in it whose endomorphismring is a ﬁnite dimension k -algebra, and f an element in End ( M ) . Then the followingare equivalent: (1) f is a monomorphism, (2) f is an epimorphism (3) f is an isomorphism Proof.

We show that (1) implies (3). Let f be a monomorphism. We can assume that f isnon-zero. There is n such that { f, f , · · · , f n } is linearly dependent. So there is a nontrivial linear combination λ t f t + · · · + λ n f n = 0 with λ t and λ n non-zero. So we have f t ( λ t Id + · · · + λ n f n − t ) = 0 . Since f t is a monomorphism, we get that ( λ t Id + λ t +1 f + · · · + λ n f n − t ) = 0 which implies that λ t Id = − ( λ t +1 f + · · · + λ n f n − t ) . Factoring out f we get that f is invertible, which shows that (1) implies (3.) Analogously we have that(2) implies (3). Since clearly (3) implies (1) and (2), we have that the three assertions areequivalent. (cid:3) Corollary 2.2.

Let us assume the same hypotheses as in lemma 2.1. If M and N areobjects in A and there are monomorphisms, f : M → N and g : N → M then f and g are isomorphisms. The analogous statement is valid for epimorphism. (cid:3) Given two sheaves A and B over a weighted projective line, we deﬁne the trace map can : Hom( A, B ) ⊗ A → B in the usual way, i.e. can( f ⊗ a ) = f ( a ) . In the literature, theimage of can is also called the trace of A in B .Furthermore, if the space Hom(

A, B ) is diﬀerent from zero, then the canonical map can : Hom( A, B ) ⊗ k A −→ B is surjective or injective but not bijective, the proof for thisfact is similar to the proof of [HR, Lemma 4.1]. In order to make our text complete, wegive it now. Lemma 2.3.

Let

A, B be an exceptional pair in coh( X ) , then the trace map can :Hom( A, B ) ⊗ A → B is either a monomorphism or an epimorphism. Proof.

We let U be the image of can , by µ the inclusion µ : U → B and can = µδ , where δ is induced by can .So we get the following exact sequence: ( ∗ ) 0 → U → B → B/U → Using that

Ext = 0 , we get an epimorphism Ext ( B/U,

Hom(

A, B ) ⊗ A ) → Ext ( B/U, U ) .This shows that the short exact sequence ( ∗ ) comes from an extension in the group Ext ( B/U,

Hom(

A, B ) ⊗ A ) , i.e. we have the following commutative diagram with ex-act rows:(2.1) / / Hom(

A, B ) ⊗ A δ (cid:15) (cid:15) µ ′ / / V δ ′ (cid:15) (cid:15) / / B/U Id (cid:15) (cid:15) / / / / U µ / / B / / B/U / / where δ and δ ′ are epimorphisms. From this diagram we get the following exact sequence: → Hom(

A, B ) ⊗ A ( δ µ ′ ) tr −→ U ⊕ V ( µ − δ ′ ) −→ B → . Since

Ext ( B, A ) = 0 the exact sequence above splits. We consider now two cases.

E. R. ALVARES, E.N. MARCOS AND H. MELTZER

Case 1: B is a direct summand of U . In this case, there is a monomorphism from B to U , then since there is a monomorphism µ : U → B , we use Corollary 2.2 and get that µ is an epimorphim. So the map can is an epimorphism.Case 2: B is not a direct summand of U . Therefore, U ≃ A t for some t and U ⊕ V ≃ A t ⊕ ( A s ⊕ B ) where t + s = dim Hom( A, B ) . Since U is the image of can , we have Hom(Hom(

A, B ) ⊗ A, B ) = Hom(Hom(

A, B ) ⊗ A, U ) ≃ Hom(Hom(

A, B ) ⊗ A, A t ) and dim Hom( A, B ) = t dim Hom( A, A ) . Therefore, t = dim Hom( A, B ) and s = 0 .The morphism Hom(

A, B ) ⊗ A δ → U , the isomorphism U ≃ Hom(

A, B ) ⊗ A and theCorollary 2.2 give us that δ is a monomorphism. (cid:3) The left mutation of ( A, B ) is the exceptional pair ( L A B, A ) , where L A B is given byone of the following three exact sequences: if Hom(

A, B ) = 0 then −→ L A B −→ Hom(

A, B ) ⊗ k A can −→ B −→ , −→ Hom(

A, B ) ⊗ k A can −→ B −→ L A B −→ , and if Ext ( A, B ) = 0 then −→ B −→ L A B −→ Ext ( A, B ) ⊗ k A −→ , where the third sequence is the universal extension. If Hom(

A, B ) = 0 = Ext ( A, B ) then L A B = B and the left mutation of the pair ( A, B ) is called a transposition. Now, thegenerators of B n act on the set of full exceptional sequences in coh( X ) as follows: σ i · ( E , . . . E i − , E i , E i +1 , E i +2 , . . . , E n ) = ( E , . . . E i − , L E i E i +1 , E i , E i +2 , . . . , E n ) . Further the right mutation of an exceptional pair ( A, B ) is the exceptional pair ( B, R B A ) ,where R B A is given by one of the following three exact sequences −→ A cocan −→ DHom(

A, B ) ⊗ k B −→ R B A −→ , −→ R B A −→ A cocan −→ DHom(

A, B ) ⊗ k B −→ , −→ DExt ( A, B ) ⊗ k B −→ R B A −→ A −→ , where D = Hom k ( − , k ) , cocan denotes the co-canonical map and the third sequence isthe universal extension. Then σ − i acts in the following way. σ − i · ( E , . . . E i − , E i , E i +1 , E i +2 , . . . , E n ) = ( E , . . . E i − , E i +1 , R E i +1 E i , E i +2 , . . . , E n ) . The following lemma is a useful tool.

Lemma 2.4.

We have(i) σ . . . σ n − ( E , E , . . . , E n ) = ( E n ( ~ω ) , E , E , . . . , E n − ) (ii) σ n − . . . σ ( E , E , . . . , E n ) = ( E , . . . , E n − , E ( − ~ω )) (iii) In the orbit of an exceptional sequence ( E , . . . E a , E a +1 , . . . ) there is an exceptionalsequence of the form ( E a , E a +1 , . . . ) The proof for (i) and (ii) is given in [M1, Proposition 2.4] and (iii) is a consequence of(i) and (ii).2.3. Recall that for an object X in a hereditary category H the left perpendicular cat-egory with respect to X is deﬁned as the full subcategory of all objects Y satisfying Hom(

Y, X ) = 0 and

Ext ( Y, X ) = 0 (see [GL2]). The right perpendicular category isdeﬁned dually.

RAID GROUP ACTION 5 Proof of Theorem 1.1 X be a weighted projective line ofweight type p = ( p , . . . , p t ) and rank of K ( X ) equals n . We start with the followingobservation. Proposition 3.1. (a) Let ( L, L ′ ) be an exceptional pair of line bundles in coh( X ) with dim k Hom(

L, L ′ ) ≥ . Then L ′ ≃ L ( ~c ) and dim Hom( L, L ′ ) = 2 .(b) The left perpendicular category with respect to L ⊕ L ( ~c ) for a line bundle L , formedin coh( X ) , consists only of ﬁnite length sheaves. Moreover, this perpendicular categoryis equivalent to the category of ﬁnite dimensional modules over the path algebra of thedisjoint union of linearly oriented quivers of type A p i − , i = 1 , . . . , t . Proof. (a) We have L ′ = L ( ~x ) for some ~x . We write ~x in normal form and, afterrenumbering the indices, if necessary, ~x = l~c + P rj =1 l j ~x j , where l = 0 , . . . , l r = 0 forsome r . Since dim Hom k ( L, L ′ ) ≥ we have l ≥ . Using Serre duality and the factthat ( L, L ( ~x )) is an exceptional pair we have ( L ( ~x ) , L ) ≃ Hom(

L, L ( ~x + ~ω )) ≃ Hom( O , O ( ~x + ~ω )) . Now ~x + ~ω = l~c + P rj =1 l j ~x j + ( t − ~c − P ti =1 ~x i = ( l − r ) ~c + P rj =1 ( l j − ~x j + P ti = r +1 ( p i − ~x i . This element is in normal form and it follows that l − r < , hence l = 1 and r = 0 . Consequently ~x = ~c .(b) After renumbering the indices, if necessary, for the simple exceptional sheaves inthe tubes we can assume that Ext ( S i, , L ) = 0 for i = 1 , . . . , t .The Riemann-Roch formula [GL1, 2.9] applied to S i, and L yields p − X j =0 h τ j S i, , L i = p (1 − g ) rk ( S i, ) rk ( L ) + det (cid:18) rk ( S i, ) rk ( L )deg ( S i, ) deg ( L ) (cid:19) where p denotes the least common multiple of the weights p , . . . , p t , g is the genus ofthe weighted projective line and h A, B i = dim Hom( A, B ) − dim Ext ( A, B ) the Eulerform. Since the τ period of S i, is p i , rk ( S i, ) = 0 and deg ( S i, ) = pp i we conclude that pp i P p i − j =0 h τ j S i, , L i = − pp i .Since there are no non-zero homomorphisms from ﬁnite length sheaves to vector bundleswe obtain that P p i − j =0 dim Ext ( τ j S i, , L ) = 1 and therefore Ext ( τ j S i, , L ) = 0 for j =1 , . . . , p i − . Again using that there are no non-zero homomorphisms from ﬁnite lengthsheaves to vector bundles we see that the sheaves S i,j for j = 1 , . . . , p i − , i = 1 , . . . , t belong to the left perpendicular category, formed in coh( X ) , ⊥ ( L ) ,formed in coh( X ) . Since S i,j = S i,j ( ~c ) , the same argument can be applied to the line bundle L ( ~c ) . Therefore theﬁnite length sheaves S i,j for j = 1 , . . . , p i − , i = 1 , . . . , t belong to the left perpendicularcategory H = ⊥ ( L ⊕ L ( ~c )) .The category H can be obtained by forming ﬁrst the left perpendicular category H with respect to L in coh( X ) and then the left perpendicular category H with respect to L ( ~c ) in H . The category H is known to be equivalent to the category of ﬁnitely generatedmodules over a hereditary algebra, in fact the path algebra of the quiver, obtained fromthe quiver of the canonical algebra End( L ≤ ~x ≤ ~c L ( ~x )) by removing the vertex whichcorresponds to L , see ( [LP]). Then by a result of Happel [H], using the fact that L ( ~c ) considered in the module category mod( H ) is exceptional, the category H is equivalentto the category of ﬁnitely generated modules over a hereditary algebra H . Moreover,both results together imply that the rank of the Grothendieck group K ( H ) equals n − .Denote by [ j ] S i, the indecomposable sheaf with socle S i, and quasi-length j . The sheaf T = L ti =1 L p j − j =1 [ j ] S i, satisﬁes Ext ( T, T ) = 0 and consists of n − indecomposabledirect summands. Therefore T is a tilting sheaf in H and consequently H consists of theobjects of the wings for [ j ] S i,p i − , i = 1 , . . . t . This shows that the endomorphism algebraof T is the disjoint union of linear quivers of type A p i − , i = 1 , . . . , t . (cid:3) E. R. ALVARES, E.N. MARCOS AND H. MELTZER

Lemma 3.2. [M1, Lemma 2.7]

Two distinct complete exceptional sequences, diﬀers in atleast two places.

An exceptional sequence ( E , ..., E n ) in coh( X ) is called orthogonal if Hom( E i , E j ) =0 for all i = j . Proposition 3.3. [M1, Proposition 2.8]

There are no orthogonal complete exceptionalsequences in coh( X ) . Lemma 3.4. [M1, Lemma 3.1]

Let E , . . . , E n be an exceptional sequence in coh( X ) suchthat dim k Hom( E , E ) ≥ .(i) Suppose that LE = L E E is deﬁned by an exact sequence → LE → Hom( E , E ) ⊗ E → E → . Then morphisms = h ∈ Hom( LE , E ) and = f ∈ Hom( E , E ) are either bothmonomorphisms or both epimorphisms.(ii) Suppose that RE = R E E is deﬁned by an exact sequence → E → D Hom X ( E , E ) ⊗ E → RE → . Then morphisms = h ∈ Hom( E , RE ) and = f ∈ Hom( E , E ) are either bothmonomorphisms or both epimorphisms. ǫ = ( E , . . . , E n ) we deﬁne k ǫ k = (rk ( E π (1) ) , ..., rk ( E π ( n ) )) , where π is a permutation of , ..., n such that rk ( E π (1) ) ≥ ... ≥ rk ( E π ( n ) ) . Proposition 3.5.

Let X be a weighted projective line with at least one weight, i.e. X = P .Then in each orbit, under the braid group action, there is a complete exceptional sequencecontaining a simple sheaf of rank . Proof.

We show ﬁrst the following claim: if ǫ = E , . . . , E n ) is a complete exceptionalsequence in coh( X ) with rk ( E i ) ≥ for all i then there exists σ ∈ B n such that k σ · ǫ k < k ǫ k .Let ǫ = ( E , . . . , E n ) be a complete exceptional sequence in coh( X ) with rk ( E i ) ≥ forall i . We know from 3.3 that ǫ is not orthogonal. Choose a < b such that Hom( E a , E b ) = 0 ,but Hom( E i , E j ) = 0 for the remaining a ≤ i < j ≤ b .Let f : E a → E b a nonzero morphism. We know that f is a monomorphism or anepimorphism, thus we distinguish two cases.Case 1: f is a monomorphism.Then f induces epimorphisms Ext ( E b , E i ) ։ Ext ( E a , E i ) for all i . Since the ﬁrst Ext -group is zero for i ≤ b the second Ext -group also vanishes for these i . We see that both Hom( E a , E i ) = 0 , and Ext ( E a , E i ) = 0 for all a < i < b , therefore applying transpositionswe obtain that σ − b − ...σ − a +1 σ − a ǫ = ( E , ..., E a − , E a +1 , ..., E b − , E a , E b , ..., E n ) . Moreover, using Lemma 2.4, we can assume that a = 1 and b = 2 .Now, the left mutation LE = L E E is deﬁned by an exact sequence being of the form (i) 0 → Hom( E , E ) ⊗ E → E → LE → or (ii) 0 → LE → Hom( E , E ) ⊗ E → E → . In the case (i) we have rk ( LE ) < rk ( E ) , hence k σ ǫ k < k ǫ k and we are done. RAID GROUP ACTION 7

In the case (ii) there exists a nonzero morphism h : LE → E . Again, h is a monomor-phism or an epimorphism. Because f is a monomorphism we infer from the sequence (ii)that dim k Hom( E , E ) ≥ . But then, in view of Lemma ?? , h is a monomorphism. Thus rk ( LE ) ≤ rk ( E ) ≤ rk ( E ) . If rk ( LE ) < rk ( E ) we apply σ − as before and obtain k σ − ǫ k < k ǫ k .Assume otherwise that rk ( LE ) = rk ( E ) . Then also rk ( E ) = rk ( E ) and therefore dim k Hom( E , E ) = 2 .Consider an exact sequence → E f → E → C → where C = coker(f) . Clearly rk ( C ) = 0 . Furthermore, applying the functor Hom( E i , − ) we conclude that dim k Hom( E i , C ) = 1 , for i = 1 , . Finally, applying the functor Hom( − , C ) we obtain Hom(

C, C ) = k and Ext ( C, C ) = k , in particular C is inde-composable.We have to consider two cases. First assume that C is a ﬁnite length sheaf concentratedat an ordinary point. Now End( C ) = k which implies that C is a simple sheaf. TheRiemann-Roch formula yields Hom(

L, C ) = k for each line bundle L . Thus using a linebundle ﬁltration for E we obtain dim k Hom( E , C ) = rk ( E ) . We have shown beforethat dim k Hom( E , C ) = 1 . Thus we obtain rk ( E ) = rk ( E ) = 1 and we have also dim Hom( E , E ) = 2 . But then we have rk ( E i ) = 0 for i > by Proposition 3.1.Now, assume that C is a sheaf of ﬁnite length concentrated at an exceptional point, say λ i of weight p i . It follows from Hom(

C, C ) = k , Ext ( C, C ) = k and the tube structure ofthe Auslander-Reiten quiver that the length of C is p i , and therefore for the classes in theGrothendieck group K ( X ) group we have [ C ] = P p i − j =0 [ S i,j ] where S i,j are the objectson the mouth of the tube. From the exact sequences stated in subsection 2.2 we inferthat [ S i,j ] = [ O ( j + 1) ~x i )] − [ O ( j ) ~x i ] for i = 1 , . . . , t , j = 1 , . . . p i . Hence P p i − j =0 [ S i,j ] =[ O ( ~c )] − [ O ] . On the other hand there is an exact sequence −→O−→O ( ~c ) −→ S −→ where S is a simple ﬁnite length sheaf concentrated in an ordinary point and consequently [ C ] = [ S ] . We conclude that k Hom( E , C ) = χ ([ E ] , [ C ]) = χ ([ E ] , [ S ]) =dim k Hom( E , S ) = rk ( E ) , where χ is the Euler form. Then we have rk ( E ) = rk ( E ) =1 and dim Hom( E , E ) = 2 , and consequently rk ( E i ) = 0 for i > by Lemma 3.1,contrary to our assumption.Case 2: f is an epimorphism.Then f induces epimorphisms Ext ( E i , E a ) ։ Ext ( E i , E b ) , for all i . The ﬁrst Ext -group is zero for i ≥ a , thus also the second Ext -group vanishes for these i . We seethat both Hom( E i , E b ) = 0 and Ext ( E i , E b ) = 0 for all a < i < b , and again applyingtranspositions we have σ − a +1 ...σ − b − ǫ = ( E , ..., E a − , E a , E b , E a +1 , ..., E n ) . As before we can assume a = 1 and b = 2 . Then RE = R E E is deﬁned by an exactsequence (i) 0 → RE → E → D Hom X ( E , E ) ⊗ E → or (ii) 0 → E → D Hom X ( E , E ) ⊗ E → RE → In the ﬁrst case we have rk ( RE ) < rk ( E ) , and consequently k σ − ǫ k < k ǫ k . In thesecond case there is a nonzero map h : E → RE , which again is a monomorphism or anepimorphism. Since f is an epimorphism we conclude that Hom( E , E ) ≥ and therefore h is an epimorphism by Lemma 3.4.Now, in this case, rk ( E ) > rk ( E ) > rk ( RE ) and therefore again k σ − ǫ k < k ǫ k . E. R. ALVARES, E.N. MARCOS AND H. MELTZER

So the claim is proved. We see that after applying successively the norm reductionabove, if necessary, we can shift by a braid group element any full exceptional sequenceto a sequence containing an exceptional sheaf of rank .We will show now that in the same orbit there is an exceptional sequence containing asimple sheaf.Now let s be the minimal number with the property that the orbit of ǫ contains anexceptional sequence with a rank sheaf F of length s . By Lemma 2.4 we can assumethat this exceptional sequence is of the form ( E , ..., E n − , F ) .We have to show that s = 1 . Assume contrary that F is not simple and denote by S the socle of F . We claim that ( E , ..., E n − , S ) is an exceptional sequence, too. Indeed,we have Ext ( S, E i ) = 0 for ≤ i ≤ n − , because the embedding S ֒ → F inducesepimorphisms Ext ( F, E i ) ։ Ext ( S, E i ) and the ﬁrst Ext -group vanishes by assumption.On the other hand,

Hom(

S, E i ) = 0 for ≤ i ≤ n − , because the existence of a nonzeromorphism from S to some E i implies that E i also has ﬁnite length, and equals therefore [ r ] S , for some r , the unique indecomposable ﬁnite length sheaf with socle S and length r . Then r ≥ s by minimality of s . But this implies Hom(

F, E i ) = 0 , contrary to the factthat ( E , ..., E n − , F ) is an exceptional sequence. Thus we have two exceptional sequenceswhich coincide in the ﬁrst n − terms but are diﬀerent in the last one. By Lemma 3.2this is impossible. (cid:3) n of K ( X ) that the group B n acts transitively on the set of complete exceptional sequences in coh( X ) .If n = 2 then X = P . In this case an exceptional sequence is of the form ( O ( i ) , O ( i +1)) for some i ∈ Z and the braid group B ∼ = Z obviously acts transitively on the set of theseexceptional sequences.Now, suppose n > and assume that ǫ = ( E , . . . , E n ) is a full exceptional sequencein coh( X ) . By Proposition 3.5 and applying, if necessary, Lemma 2.4 we have g.ǫ =( E ′ , ...E ′ n − , S ) for some g ∈ B n and some simple exceptional ﬁnite length sheaf S . Denoteby κ = ( O , O ( ~x ) , O (2 ~x ) , . . . , O (( p − ~x ) , O ( ~x ) , . . . O (( p t − ~x t ) , O ( ~c )) the exceptionalsequence corresponding to the canonical tilting sheaf. Since S is exceptional simple, S = S i,j for some i, j . From the exact sequence → O ( j~x i ) → O (( j + 1) ~x i ) → S i,j → we see that the right mutation of the pair ( O ( j~x i ) , O (( j + 1) ~x i ) equals ( O (( j + 1) ~x i ) , S i,j ) .Thus, for some g ∈ B n we get g .κ = ( O , ..., O (( j + 1) ~x i ) , S i,j , ... ) . Observe that incase j = p i − , i = t , we ﬁrst can apply transpositions in order to arrange that O ( j~x i ) and O (( j + 1) ~x i ) are neighbours. Applying if necessary, Lemma 2.2, we obtain g .κ =( F , ..., F n − , S ) for some g ∈ B n and line bundles F , ..., F n − . Now, by [GL2] theright perpendicular category S ⊥ is equivalent to a sheaf category coh( X ′ ) for a weightedprojective line X ′ = X ( p ′ , λ ) with weight sequence p ′ = ( p , ...p i − , p i − , p i +1 , ..., p n ) . Byinduction ( E ′ , ..., E ′ n − ) and ( F , ..., F n − ) , considered as complete exceptional sequencesin S ⊥ , are in the same orbit under the action of the braid group B n − on the set ofcomplete exceptional sequences in coh( X ′ ) . We conclude that ǫ and κ are in the sameorbit, which ﬁnishes the proof. (cid:3) The strongest global dimension of coh( X ) Deﬁnition 4.1.

The strongest global dimension is the maximum of the strong globaldimension of all algebras which are derived equivalent do coh( X ) We have the following characterization of the strongest global dimension.

RAID GROUP ACTION 9

Proposition 4.2.

The strongest global dimension of a weighted projective line X is oneif X = P or is the maximal number m + 2 such that there exists a tilting complex T in the derived category of coh( X ) of the form L mi =0 T i [ i ] with T i ∈ coh( X ) , i ∈ Z and T = 0 = T m . Proof.

See Theorem 1 in [ALM]. (cid:3)

The strongest global dimension of X will be denoted by st . gl . dim X . It follows from thedeﬁnition that if the bounded derived category of an algebra A is triangular equivalent tothe bounded derived category of coh( X ) , then the s . gl . dim A ≤ st . gl . dim X .In [M2] it was shown that if X has weight type (2 , , . . . , ( t entries) then st . gl . dim X =2 . For tubular weighted projective lines X bounds of the strongest global dimension weregiven in [S].Before the main theorem in this section we have the following remarks and facts. Recallthat, if ( A, B ) is an exceptional pair in coh( X ) , then Hom X ( A, B ) = 0 or Ext ( A, B ) = 0 .First we assume that

Hom(

A, B ) = 0 . We have then two cases for the left mutation toconsider: ( α ) : 0 −→ L A B −→ Hom(

A, B ) ⊗ k A can −→ B −→ , ( β ) : 0 −→ Hom(

A, B ) ⊗ k A can −→ B −→ L A B −→ . It is important to note that the surjectivity of the canonical map depends only on rk ( A ) , rk ( B ) and on the dimension of the spaces Hom X ( A, B ) . We have that • if rk ( A ) = 0 then can is surjective ⇐⇒ dim k Hom(

A, B ) · rank A > rank B . • if rk ( A ) = 0can is surjective ⇐⇒ dim k Hom(

A, B ) · dim k A > dim k B .Applying Hom X ( , A ) in ( α ) we have Hom X ( α, A ) :0 −→ Hom(

B, A ) −→ Hom(Hom(

A, B ) ⊗ k A, A ) −→ Hom( L A B, A ) −→−→ Ext ( B, A ) −→ Ext (Hom( A, B ) ⊗ k A, A ) −→ Ext ( L A B, A ) −→ Applying

Hom X ( , A ) in ( β ) we have Hom X ( β, A ) :0 −→ Hom( L A B, A ) −→ Hom(

B, A ) −→ Hom(Hom(

A, B ) ⊗ k A, A ) −→−→ Ext ( L A B, A ) −→ Ext ( B, A ) −→ Ext (Hom( A, B ) ⊗ k A, A ) −→ Now, the following remarks follows from both long exact sequences:

Remark 4.3.

The mutation of ( A, B ) is the exceptional pair ( L A B, A ) and: • In the case ( α ) , the conditions Hom X ( B, A ) = 0 = Ext ( B, A ) imply that dim k Hom( L A B, A ) = dim k Hom(

A, B ) and dim k Ext ( L A B, A ) = 0 . • In the case ( β ) , the conditions Hom X ( B, A ) = 0 = Ext ( B, A ) imply that dim k Hom( L A B, A ) = 0 and dim k Ext ( L A B, A ) = dim k Hom(

A, B ) . Now assume that

Ext(

A, B ) = 0 . Then have the universal extension ( γ ) : 0 −→ B −→ L A B −→ Ext(

A, B ) ⊗ k A −→ . Applying

Hom X ( , A ) in ( γ ) we have Hom X ( γ, A ) :0 −→ Hom(Ext(

A, B ) ⊗ k A, A ) −→ Hom( L A B, A ) −→ Hom(

B, A ) −→−→ Ext (Ext( A, B ) ⊗ k A, A ) −→ Ext ( L A B, A ) −→ Ext ( B, A ) −→ . The mutation of ( A, B ) is the exceptional pair ( L A B, A ) and: • In the case ( γ ) , the conditions Hom X ( B, A ) = 0 = Ext ( B, A ) imply that dim k Hom( L A B, A ) = dim k Ext X ( A, B ) and dim k Ext ( L A B, A ) = 0 . ( A.B ) is an exceptional pair, then on the mutationpair ( L A B, A ) we can compute the dimensions dim k Hom(( L A B, A ) , dim k Ext ( L A B, A ) ,and rk ( L A B ) without using the parameters λ . Lemma 4.4.

Let ǫ = ( E , . . . , E n ) be a complete exceptional sequences in coh( X ) and σ bethe generator of the braid group B n such that σ.ǫ = ( E , . . . , E k − , LE k +1 , E k , E k +2 , . . . , E n ) ,where we write shortly LE k +1 instead of L E k E k +1 . Then the respective dimensions dim k Hom( E i , LE k +1 ) , dim k Ext ( E i , LE k +1 ) for ≤ i ≤ k − , dim k Hom X ( LE k +1 , E i ) , dim k (Ext ( LE k +1 , E i ) for i ∈ { k, k + 2 , . . . , n } and rank ( LE k +1 ) depend only on thedimensions of the Hom , Ext and the ranks of the elements in ǫ . Proof.

In remarks 4.3 and 4.1 we have seen that the dimensions dim k Hom( LE k +1 , E k ) , dim k Ext ( LE k +1 , E k ) depend only of the dimension of Hom X ( X k , X k +1 ) or Ext ( X k , X k +1 ) .Now we will prove the claim for dim k Hom( E j , LE k +1 ) , dim k Ext ( E j , LE k +1 ) for ≤ j ≤ k − .Suppose that the mutation is given by type ( α ) , then we have the exact sequence −→ LE k +1 −→ Hom( E k , E k +1 ) ⊗ k E k can −→ E k +1 −→ , which induces a long exact sequence −→ Hom( E j , LE k +1 ) −→ Hom( E j , Hom( E k , E k +1 ) ⊗ k E k ) −→ Hom( E j , E k +1 ) −→−→ Ext ( E j , LE k +1 ) −→ Ext ( E j , Hom( E k , E k +1 ) ⊗ k E k ) −→ Ext ( E j , E k +1 ) −→ for ≤ j ≤ k − .Since by [M1, Lemma 3.2.4] Hom X ( E j , E k +1 ) = 0 or Ext ( E j , E k +1 ) = 0 we have either dim k Hom( E j , LE k +1 ) = dim k Hom( E j , E k ) · dim k Hom( E k , E k +1 ) and dim k Ext ( E j , LE k +1 ) = dim k Ext ( E j , E k ) · dim k Hom( E k , E k +1 ) − dim k Ext ( E j , E k +1 ) or dim k Hom( E j , E k ) · dim k Hom( E k , E k +1 ) + dim k Ext ( E j , LE k +1 ) =dim k Hom( E j , LE k +1 ) + dim k Hom( E j , E k +1 ) . Since ( E j , LE k +1 ) is an exceptional pair, we have Hom X ( E j , LE k +1 ) = 0 or Ext ( E j , LE k +1 ) =0 . Each one gives us that dim k Hom( E j , LE k +1 ) and Ext ( E j , LE k +1 ) depend only of thedimensions of the Hom , Ext spaces of ǫ .In the cases that the left mutation is given by type ( β ) or type ( γ ) the proof is similar. (cid:3) We have as a consequence of the previous discussion the following:

Corollary 4.5.

Suppose that ǫ = ( E , . . . , E n ) and ǫ ′ = ( E ′ , . . . , E ′ n ) are complete excep-tional sequences in coh( X ) such that the following formulas are valid dim k Hom X ( E j , E l ) =dim k Hom X ( E ′ j , E ′ l ) , dim k Ext X ( E j , E l ) = dim k Ext X ( E ′ j , E ′ l ) and rk ( E j ) = rk ( E ′ j ) forall ≤ j, l ≤ n . Let σ ∈ B n and σǫ = ( F , · · · , F n ) , σǫ ′ = ( F ′ , · · · , F ′ n ) . Then dim k Hom X ( F j , F l ) = dim k Hom X ( F ′ j , F ′ l ) , dim k Ext X ( F j , F l ) = dim k Ext X ( F ′ j , F ′ l ) and rk ( F j ) = rk ( F ′ j ) for all ≤ j, l ≤ n . (cid:3) Theorem 4.6.

Let X = ( p , λ ) and X ′ = ( p , λ ′ ) be weighted projective lines with the sameweight type. Then st . gl . dim X = st . gl . dim X ′ . Proof.

Let m be maximal such that there exists a tilting complex T of the form L mi =0 T i [ i ] with T i ∈ coh( X ) and T = 0 = T m . Write T = L E j [ n j ] with indecomposable sheaves E j and n j ∈ Z . The E j can be ordered in such a way that they form a full exceptional sequence ǫ in coh( X ) . By Theorem 1.1. there exists a braid group element σ ∈ B n such that ǫ = σ · κ where κ = ( O X , O X ( ~x ) , . . . , O X (( p − ~x ) , . . . , O X ( ~x t ) , . . . , O X (( p t − ~x t ) , O X ( ~c )) is theexceptional sequence obtained from the canonical tilting sheaf L ≤ ~x ≤ ~c O X ( ~x ) on X . RAID GROUP ACTION 11

Now the application of the same braid group element σ to the exceptional sequence κ ′ = ( O X ′ , O X ′ ( ~x ) , . . . , O X ′ (( p − ~x ) , . . . , O X ′ ( ~x t ) , . . . , O X ′ (( p t − ~x t ) , O X ′ ( ~c )) obtainedfrom the canonical tilting sheaf L ≤ ~x ≤ ~c O X ′ ( ~x ) on X ′ yields a full exceptional sequence ǫ ′ for the weighted projective line X ′ .The exceptional sheaves O X ( u~x i ) and O X ′ ( s~x i ) of κ and κ ′ , respectively, satisfy thesame dimension for the Hom and

Ext spaces that is, dim Ext k X ( O X ( m~x i ) , O X ( n~x j )) = dim Ext k X ′ ( O X ′ ( m~x i ) , O X ′ ( n~x j )) for i, j ∈ { , · · · , t } , m, n ∈ { , · · · , max { p , · · · , p t }} and k ∈ { , } . Moreover, the ranksof all sheaves of κ and κ ′ equal .The sequence ǫ ′ is constructed from κ ′ using successively the same kind of mutations asin the construction of ǫ from κ . Therefore, following Corolary 4.5 the exceptional sheaves E ′ j of ǫ ′ satisfy the same dimension formulas for the Hom , Ext and rank spaces as theexceptional sheaves E j of ǫ .Therefore the exceptional sheaves E ′ j of ǫ ′ satisfy the same dimension formulas for the Hom and

Ext spaces as the exceptional sheaves E j of ǫ . Hence the E ′ j can be shifted inthe derived category of coh( X ′ ) as the E j which yields a tilting complex L mi =0 T ′ i [ i ] with T ′ i ∈ coh( X ′ ) for X ′ and with T ′ = 0 = T ′ m . Consequently st . gl . dim X = m ≤ st . gl . dim X ′ .By symmetry, st . gl . dim X ′ ≤ st . gl . dim X and consequently st . gl . dim X = st . gl . dim X ′ . (cid:3) Note that from Corollary 4.5 we obtain that the ordinary quivers of the algebras

End T and End T ′ in the former theorem are the same which was already stated in [M2].The former proof also suggests the following: Conjecture:

Let T be a tilting complex of the form L mi =0 T i [ i ] with T i ∈ coh( X ) and T = 0 = T m and A = End T . The strong global dimension of A does not depend on theparameters.The validity of this conjecture implies the statement of the Theorem (4.6).5. Determinants for exceptional sequences

Let f , . . . f n be group homomorphisms deﬁned on the Grothendieck group K ( X ) of aweighted projective line with values in Z . For a full exceptional sequence ǫ = ( E , . . . , E n ) on X we form the n × n matrix M ( ǫ ) whose coeﬃcient at the place ( i, j ) equals f i ( E j ) and we consider the determinant of that matrix det( M ( ǫ )) . Theorem 5.1.

There exists a constant c ∈ k such that det( M ( ǫ )) = c or − c for all fullexceptional sequences ǫ in coh( X ) . Proof.

We are going to show that the determinant of the matrix does not change if weapply to the exceptional sequence in coh( X ) the left mutation σ i . For right mutationsthe proof is analogous.For a full exceptional sequence ǫ = ( E , E , . . . E n ) we denote dim k Hom( E i , E i +1 ) = h and dim k Ext ( E i , E i +1 ) = e . Now, σ i · ǫ equals ( E , . . . , E i − , LE i +1 , E i , E i +2 , . . . E n ) and we have [ L E i E i +1 ] = h [ E i ] − [ E i +1 ] , [ L E i E i +1 ] = [ E i +1 ] − h [ E i ] or [ L E i E i +1 ] = e [ E i ] + [ E i +1 ] depending on the type of the left mutation of the pair ( E i , E i +1 ) (seesection 2). The matrix for the exceptional sequence σ i · ǫ is obtained from that of ǫ byreplacing the values in the i -th column by f j ( E i +1 ) − hf j ( E i ) , − f j ( E i +1 ) + hf j ( E i ) or f j ( E i +1 ) + ef j ( E i ) , j = 1 , . . . , n and by replacing the values in the i + 1 -th column by f j ( E i ) , j = 1 , . . . n . Then the statement follows from the known rules for determinants. (cid:3) In particular we can apply the method above to the rank function, the degree functionand the n − Euler forms h− , S i,j i , j = 1 , . . . p i − , i = 1 , . . . t . Corollary 5.2.

For each full exceptional sequence ǫ = ( E , E , . . . , E n ) in coh( X ) thedeterminant of the matrix M ( ǫ ) = rk E rk E rk E . . . rk E p +1 . . . rk E n deg E deg E deg E . . . deg E p +1 . . . deg E n h E , S , i h E , S , i h E , S , i . . . h E p +1 , S , i . . . h E n , S p , i ... h E , S ,p − i h E , S ,p − i h E , S ,p − i . . . h E p +1 , S ,p − i . . . h E n , S ,p − i ... h E , S t, i h E , S t, i h E , S t, i . . . h E p +1 , S t, i . . . h E n , S t, i ... h E , S t,p t − i h E , S t,p t − i h E , S t,p t − i . . . h E p +1 , S t,p t − i . . . h E n , S t,p t − i equals p or − p . Recall that p denotes the least common multiple of the weights p , . . . , p t . Proof.

The determinant is easily calculated to be p or − p for the exceptional sequence ( O , O ( ~c ) , S , , . . . , S ,p − , . . . S t, , . . . , S t,p − ) using the block structure of the matrix andthe fact that rk O = rk O ( ~c ) = 1 , deg O = 0 and deg O ( ~c ) = p . Then the statement followsfrom Theorem 5.1. (cid:3) Remark 5.3.

The determinantal equation obtained in the way above can be interpretedas a diophantine equation for the weighted projective line X . Diophantine equationsexpressed for data in terms of exceptional sequences seems to be typical. So Rudakovshowed that the ranks of the vector bundles of an exceptional triple on the projectiveplane satisfy the Markov equation X + Y + Z = 3 XY Z [Ru]. Diophantine equationsfor partial tilting sequences on weighted projective lines were given in [M2, Chapter 10.2].

References [ALM] E. R. Alvares, P. Le Meur, and E. N. Marcos.

The strong global dimension of piecewise hereditaryalgebras . J. Algebra, 481:36-67, 2017.[S] S. C. Schmidt,

Complexos Tilting e Dimensão Global Forte em Álgebras Hereditárias por partes.

PhDThesis. Universidade Federal do Paraná 2017.[B] A. I. Bondal,

Representation of associative algebras and coherent sheaves , Math. USSR, Izv. 34, No.1, 23-42 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 1, 25-44 (1989).[CB] W. Crawley-Boevey,

Exceptional sequences of representations of quivers , Dlab, Vlastimil (ed.) etal., Representations of algebras. Proceedings of the sixth international conference on representations ofalgebras, Carleton University, Ottawa, Ontario, Canada, August 19-22, 1992. Providence, RI: AmericanMathematical Society. CMS Conf. Proc. 14, 117-124 (1993).[GL1] W. Geigle and H. Lenzing,

A class of weighted projective curves arising in representation theoryof ﬁnite-dimensional algebras , Singularities, representation of algebras, and vector bundles (Lambrecht,1985), Lecture Notes in Math., vol. 1273, 265-297 Springer, Berlin, (1987).[GL2] W. Geigle and H. Lenzing,

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Perpendicular categories to exceptional modules , An. Stiint. Univ. “Ovidius” Constanta,Ser. Mat. 4, No. 2, 66-75 (1996).[HR] D. Happel and C. M. Ringel

Tilted algebras , Trans. Am. Math. Soc. 274, 399-443 (1982).[LP] H. Lenzing and J. A. de la Peña,

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Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projectivelines , Mem. Am. Math. Soc. 808, 138 p. (2004).[Ri] C. M. Ringel,

Tame algebras and integral quadratic forms , Lecture Notes in Mathematics. 1099.Berlin etc.: Springer-Verlag. XIII, 376 p. (1984).[Ru] A. N. Rudakov,

The Markov numbers and exceptional bundles on P , Math. USSR, Izv. 32, No. 1,99-112 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 1, 100-112 (1988). RAID GROUP ACTION 13

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