Brauer graph algebras are closed under derived equivalence
aa r X i v : . [ m a t h . R T ] A ug ALGEBRAS DERIVED EQUIVALENT TO BRAUER GRAPHALGEBRAS AND DERIVED INVARIANTS OF BRAUER GRAPHALGEBRAS REVISITED
MIKHAIL ANTIPOV AND ALEXANDRA ZVONAREVA
Abstract.
In this paper the class of Brauer graph algebras is proved to be closedunder derived equivalence. For that we use the rank of the maximal torus of the identitycomponent of the group of outer automorphisms
Out ( A ) of a symmetric stably biserialalgebra A . Introduction
Brauer graph algebras or equivalently symmetric special biserial algebras, originatingfrom modular representation theory, are studied quite extensively. They appear in clas-sifications of various classes of algebras including blocks with cyclic or dihedral defectgroups [14, 15], blocks of Hecke algebras [9, 10] and others. Brauer tree algebras, thesubclass of Brauer graph algebras of finite representation type, contain all blocks withcyclic defect group.In this paper we make a final step in the proof of the fact that Brauer graph algebrasare closed under derived equivalence. This fact was believed to be true, based on the workof Pogorza ly [27]. In [11], counterexamples to some of the statements of [27] were given.In [8], we revised the proof of the fact that the only algebras possibly stably (and thusderived) equivalent to self-injective special biserial algebras (a class containing Brauergraph algebras) are self-injective stably biserial (see Section 2). As a finite-dimensionalalgebra derived equivalent to a symmetric algebra is itself symmetric [31] we can restrictour attention to symmetric stably biserial algebras. It turns out that in odd characteristicthe class of symmetric stably biserial algebras coincides with the class of Brauer graphalgebras, whereas in characteristic 2 this is not the case [8].The general strategy of the proof of the fact that Brauer graph algebras are closedunder derived equivalence follows the classical proof for Brauer tree algebras. The factthat Brauer tree algebras are closed under stable equivalence was proved in [19]. Sinceby [30] derived equivalence for self-injective algebras implies stable equivalence, it followsthat this class is closed under derived equivalence as well. It turns out that the proof forthe whole class of Brauer graph algebras is much more involved and requires an extrastep in characteristic 2, which is provided in this paper.A symmetric stably biserial algebra can be given by the same combinatorial data asthe Brauer graph algebra, that is a graph on a surface and a number attached to eachvertex of this graph, called the multiplicity. Additionally, one needs to fix a distinguishedclass of loops in the quiver, satisfying certain conditions, which we call deformed loops(see Section 2). In case the number of deformed loops is 0 we recover the usual definitionof a Brauer graph algebra. Since for local algebras derived equivalence implies Moritaequivalence [34], we will sometimes assume that A has at least 2 simple modules. Forfurther reference, let us denote by V (Γ) , E (Γ) and F (Γ) the vertices, edges and faces ofthe Brauer graph Γ. he main technique, used in this paper, is the computation of the rank of the maximaltorus D ( A ) of the identity component of the group of outer automorphisms Out ( A ) fora symmetric stably biserial algebra A . The group Out ( A ) is a derived invariant [23, 33]used quite seldom. The only previous systematic application we know of is the proof ofthe fact that the number of arrows in the quiver of a gentle algebra is a derived invariant[1]. Theorem 1.1.
Let k be an algebraically closed field. Let A be a symmetric stably biserialalgebra over k ( char ( k ) = 2 ) or a symmetric special biserial algebra over k ( char ( k ) = 2 )with at least two non-isomorphic simple modules, which is not a caterpillar (see Section3). Let Γ be the Brauer graph of A and let d be the number of deformed loops in A ( d = 0 for the symmetric special biserial case). The rank of D ( A ) is | E (Γ) | − | V (Γ) | − d + 2 . In section 3, we investigate basic properties of symmetric stably biserial algebras. Insection 4, we revisit the known derived invariants for Brauer graph algebras [3, 4, 5, 6] inarbitrary characteristic, providing simpler proofs of their invariance for the larger classof symmetric stably biserial algebras and correcting some inaccuracies in the existingliterature.
Theorem 1.2.
Let A be a symmetric stably biserial algebra with a Brauer graph Γ andwith at least two simple modules. The following are invariants of A under a derivedequivalence of symmetric stably biserial algebras: | V (Γ) | , | E (Γ) | , | F (Γ) | , the multiset ofperimeters of faces, the multiset of multiplicities and bipartivity of Γ . As a corollary of Theorems 1.1 and 1.2 and the fact that Brauer graph algebras can bederived equivalent only to symmetric stably biserial algebras [8] we obtain the following:
Corollary 1.3.
The class of Brauer graph algebras is closed under derived equivalence.Namely, if A is an algebra Morita equivalent to a Brauer graph algebra and B is analgebra such that D b ( A ) ≃ D b ( B ) , then B is Morita equivalent to a Brauer graph algebra. In forthcoming work [21], among other results, W. Gnedin independently obtains Corol-lary 1.3 in characteristic 2 and for bipartite Brauer graphs by different methods. Notethat the list of invariants from Theorem 1.2 is crucial to the forthcoming joint work [26] ofS. Opper and the second named author, where a complete classification of Brauer graphalgebras up to derived equivalence will be provided.
Acknowledgement:
AZ would like to thank Alexey Ananyevskiy for many fruitfuldiscussions. 2.
Preliminaries
Throughout this paper, A is a basic, connected, finite dimensional algebra over analgebraically closed field k and mod- A is the category of right A -modules. The stablecategory of mod- A will be denoted by mod- A and Ω : mod- A → mod- A will denotethe syzygy. The bounded derived category of the category mod- A will be denoted by D b ( A ). A quiver Q consists of a set of vertices Q and a set of arrows Q . The map s : Q → Q will denote the start of an arrow, the map e : Q → Q will denote theend of an arrow. In the path algebra k Q the multiplication of arrows α and β is αβ = 0,provided e ( α ) = s ( β ), by J ( A ) we will denote the Jacobson radical of the algebra A ,which is the ideal generated by the arrows of the quiver Q in case A ≃ k Q/I . By K ( C )we are going to denote the Grothendieck group of an Abelian or a triangulated category C . n this paper we are going to be interested in symmetric special biserial and symmetricstably biserial algebras. Definition 2.1.
Let Q be a quiver, I an admissible ideal of k Q . A self-injective algebra A = k Q/I is called special biserial if the following conditions are satisfied.(1) For each vertex v ∈ Q , the number of outgoing arrows and the number of incomingarrows are less than or equal to 2.(2) For each arrow α ∈ Q , there is at most one arrow β ∈ Q such that αβ / ∈ I .(3) For each arrow α ∈ Q , there is at most one arrow β ∈ Q such that βα / ∈ I . Definition 2.2.
Let Q be a quiver, I an admissible ideal of k Q . A self-injective algebra A = k Q/I is called stably biserial if the following conditions are satisfied.(1) For each vertex v ∈ Q , the number of outgoing arrows and the number of incomingarrows are less than or equal to 2.(2) For each arrow α ∈ Q , there is at most one arrow β ∈ Q such that αβ α rad( A ) β + soc( A ) .(3) For each arrow α ∈ Q , there is at most one arrow β ∈ Q such that βα β rad( A ) α + soc( A ) . The following description of stably biserial algebras was provided in [11]:
Proposition 2.3 (Proposition 7.5 [11]) . If A = k Q/I is stably biserial then we canchoose the presentation of A in such a way that the following conditions hold.(1) If αβ = 0 , αγ = 0 , β = γ , for arrows α, β, γ then either αβ ∈ soc( A ) or αγ ∈ soc( A ) .(2) If βα = 0 , γα = 0 , β = γ , for arrows α, β, γ then either βα ∈ soc( A ) or γα ∈ soc( A ) . Self-injective special biserial algebras are a subclass of stably biserial algebras. Wewill call an algebra symmetric special biserial (SSB for short) or symmetric stablybiserial , if in addition to being special biserial or stably biserial it is symmetric.Consider the following data:(1) A quiver Q such that every vertex has two incoming and two outgoing arrows.(2) A permutation π on Q with e ( α ) = s ( π ( α )) for all α ∈ Q .(3) A function m : C ( π ) → N , where C ( π ) is the set of cycles of π . We will denote by C ( α ) := απ ( α ) π ( α ) . . . π | C ( α ) |− the cycle, containing α ∈ Q and call m ( C ( α ))the multiplicity of the cycle C ( α ).(4) A set of loops L = { α i , . . . , α i d } , such that π ( α i j ) = α i j and a set of elements { t α i , . . . , t α id } , with t α ij ∈ k ∗ .In [8] the following description of symmetric stably biserial algebras in terms of gener-ators and relations was obtained: Theorem 2.4.
Any symmetric stably biserial algebras can be given as A = k Q/I , wherethe ideal of relations is generated by(1) αβ for all α, β ∈ Q , β = π ( α ) , α does not belong to the set of loops L ,(2) (cid:18) απ ( α ) π ( α ) . . . π | C ( α ) |− ( α ) (cid:19) m ( C ( α )) − (cid:18) βπ ( β ) π ( β ) . . . π | C ( β ) |− ( β ) (cid:19) m ( C ( β )) forall α, β ∈ Q with s ( α ) = s ( β ) ,(3) α − t α (cid:18) απ ( α ) π ( α ) . . . π | C ( α ) |− ( α ) (cid:19) m ( C ( α )) for each α ∈ L , (cid:18) απ ( α ) π ( α ) . . . π | C ( α ) |− ( α ) (cid:19) m ( C ( α )) β for all α, β ∈ Q .Moreover, any symmetric stably biserial algebra over an algebraically closed field k with char ( k ) = 2 is isomorphic to an algebra k Q/I as above with the empty set of loops L . Remark 2.5. In [8] we considered quivers Q such that every vertex has either two incom-ing and two outgoing arrows or one incoming and one outgoing arrow, and an admissibleideal of relations I . To pass to this equivalent description from the description in Theo-rem 2.4 one needs to delete loops α such that π ( α ) = α, m ( α ) = 1 and modify the idealof relations accordingly. In the case where the algebra has only two loops α i such that π ( α i ) = α i , m ( α i ) = 1 , one needs to delete only one loop to get the algebra isomorphic to k [ x ] / ( x ) . The loops from the set L will be called deformed loops . It is well known that anySSB-algebra can be given in the above form with the empty set of deformed loops.Note also that for this description of stably biserial algebras | Q | = 2 | Q | is invariantunder derived equivalence, since | Q | is the rank of K ( D b ( A )).It is well known [32, 7, 35] that the class of SSB-algebras coincide with the classof Brauer graph algebras. Brauer graph is a graph with a cyclic ordering of (half-)edges around each vertex and a number assigned to each vertex. This graph Γ can beconstructed using the data (
Q, π, m ) as follows: the vertices of Γ correspond to the cyclesof π , the edges of Γ correspond to the vertices of Q , an edge connects two vertices of Γif the corresponding π -cycles have the corresponding vertex of Q in common. The cyclicordering of edges around a vertex comes from the order in which vertices of Q appear inthe π -cycle, the multiplicities come from the function m . Along the same lines, to eachBrauer graph one can assign the data ( Q, π, m ) and the corresponding SSB-algebra.In [4] each Brauer graph was considered together with a minimal compact orientedsurface S , into which it is embedded, in such a way that its complement is a union ofdisks (see also [24]). The orientation of the edges around the vertices of Γ comes from theorientation of the surface, now it makes sense to consider not only vertices and edges ofΓ but also faces of Γ. The set L corresponds to a subset of faces of perimeter 1 of Γ. Wewill use the terms SSB-algebra and Brauer graph algebra interchangeably.3. Stably biserial algebras
In this section we are going to investigate basic properties of symmetric stably biserialalgebras. As stated in Theorem 2.4 any symmetric stably biserial algebra A can be givenas a certain deformation of a Brauer graph algebra with a Brauer graph Γ. We are goingto show that with one exception the Brauer graph Γ does not depend on the presentationof A and that any deformation from Theorem 2.4 is indeed symmetric.Let us introduce a special class of algebras, called caterpillar in this paper. This classof algebras behaves differently form other symmetric special biserial algebras and has tobe excluded from some considerations.The algebra k Q/I will be called a caterpillar of length n > Q is of the form1 2 3 n-1 n. α α ααβ β ββ In this case, the ideal of relations has the form I or I . Where I is generated by relations αe i β = 0 = βe i α, i = 1, αe α = 0 = βe β , ( α k e β n α n − k ) m α = ( β k e α n β n − k ) m α , thus it as one π -cycle, the multiplicity of which is m α . The ideal I is generated by relations αβ = 0 = βα, α nm α = β nm β , thus it has two π -cycles, the multiplicity of one π -cycle is m α , the multiplicity of the other π -cycle is m β . The Brauer graphs of these algebras are: • · · · • •· · · The following Lemma is most likely know for Brauer graph algebras, but we could notfind the proof, so we include it for the larger class of symmetric stably biserial algebras.
Lemma 3.1.
Let A be a symmetric stably biserial algebra with a presentation k Q/I as in Theorem 2.4 and the associated Brauer graph Γ . If Γ is not a loop with 1 as themultiplicity of the unique vertex, or an edge with 2 as the multiplicity of both vertices,then Γ does not depend on the choice of the presentation k Q/I .Proof.
Assume that the algebra A has two presentations k Q/I ≃ k Q ′ /I ′ as in Theo-rem 2.4. Let us delete loops α such that π ( α ) = α, m ( α ) = 1 and modify the ideal ofrelations accordingly in both presentations. These deleted loops correspond to the leafswith multiplicity 1 in the Brauer graph and can always be reconstructed from the va-lency of the vertices in the quiver. Since the ideals of relations I and I ′ are admissibleafter the deletion of extra loops, we can assume that Q and Q ′ coincide, thus there is abijection between primitive idempotents for these two presentations and between simplemodules over k Q/I and k Q ′ /I ′ , simple modules will be denoted by S i . This extends toa bijection between the edges of the Brauer graphs Γ and Γ ′ , constructed from these twopresentations, since the edges of the Brauer graph correspond to simple modules. Forthe projective cover P i of S i we can consider the module rad P i / soc P i which has eitherone or two indecomposable summands M i and N i . These modules are uniserial and eachof them gives a unique sequence of simple modules, corresponding to it’s radical series( S i , · · · , S i n ), where S i is the top of M i or N i respectively. Adding S i to this sequence( S i = S i , S i , · · · , S i n ) and numbering the sequence by the elements of Z / ( n + 1) Z weget a collection of cycles of simple modules (coming from each P i for all i ’s), which weidentify up to a cyclic permutation of Z / ( n + 1) Z . If for some P i the modules M i and N i are both zero, then A ≃ k [ x ] / ( x ). The radical series of the modules M i , N i do notdepend on the presentation of the algebra, so in this case the Brauer graph is determineduniquely and is an edge with both vertices of multiplicity 1.Note that by construction of the permutation π the cyclic ordering of the simples inthe sequences constructed above coincides with the cyclic ordering of edges in the Brauergraph.If the module S i appears in two different cyclic sequences, then the edge, correspondingto S i is not a loop and we can reconstruct the cyclic ordering around the ends of theedge, corresponding to S i from the subsequence of the form ( S i , S i , · · · , S i l , S i ), where( S i , · · · , S i l ) does not contain S i . The multiplicities of the vertices is the number of timesthe subsequences ( S i , S i , · · · , S i l ) has to be repeated to get the whole sequences.If S i appears in only one cyclic sequence, but this cyclic sequence has a subsequence ofthe form ( S i , S i , · · · , S i l , S i , S i l +2 , · · · , S i m , S i ), where the subsequences ( S i , · · · , S i l ) and( S i l +2 , · · · , S i m ) do not contain S i , are different and at least one of them is not empty,then the edge corresponding to S i is a loop and we can reconstruct the cyclic orderingof the edges around the vertex adjacent to this loop and the multiplicity is the numberof times the subsequence ( S i , S i , · · · , S i l , S i , S i l +2 , · · · , S i m ) has to be repeated to get thewhole sequence. f S i appears in only one cyclic sequence and this sequence does not have a subsequenceas before, but the projective module P i is uniserial then we can reconstruct the cyclicordering of the edges around one vertex incident to the edge corresponding to S i and itsmultiplicity as before, the other end of this edge has no other edges incident to it andhas multiplicity 1.The only case left to consider is when S i appears in only one cyclic sequence and thissequence does not have a subsequence as before, but the projective module P i is notuniserial. In this case the modules M i and N i have the same radical series but are bothnonzero. If the cyclic sequence containing S i is of the form ( S i , S i , · · · , S i l , S i ), where( S i , · · · , S i l ) does not contain S i , then the edge, corresponding to S i is not a loop and wecan reconstruct the cyclic ordering around each end of this edge, the multiplicities of theends are 1. Assume that ( S i , S i , · · · , S i l , S i ), where ( S i , · · · , S i l ) does not contain S i is asubsequence of the cyclic sequence and it has to be repeated m > S i , · · · , S i l ) is empty, then | Q | = 1, this situation will be considered later.If the edge corresponding to S i is a loop, then all edges corresponding to ( S i , · · · , S i l )are loops and we get a caterpillar with one vertex in the Brauer graph with multiplicity m/ m ). If the edge corresponding to S i is not a loop,then all edges corresponding to ( S i , · · · , S i l ) are not loops and we get a caterpillar withtwo vertices in the Brauer graph, both with multiplicity m . The two algebras we get foreven m are not isomorphic, since they are not even derived equivalent by Proposition 4.4.Note that the proof of Proposition 4.4 does not relay on the results of this section.Let us consider the case | Q | = 1. The Brauer graph is either an edge or a loop. If itis an edge, there are no deformed loops and A ≃ A k,l = k [ x, y ] / h xy, x k − y l i , k, l ≥ A is non-commutative. So it is sufficient to consider algebra B t x ,t y = k [ x, y ] / h x y, y x, x − t x xy, y − t y xy i , which is 4-dimensional. If it is isomorphic to A k,l , then either k = 1 , l = 3,which is not possible, or k = l = 2. In the last case the algebras can be indeed isomorphic,even when t x = t y = 0, char ( k ) = 2. (cid:3) Remark 3.2.
The cyclic ordering of edges in the Brauer graph played an important rolein the proof of Theorem 2.4. Namely, for a symmetric stably biserial algebra A with anarbitrary presentation as in Proposition 2.3, with an admissible ideal of relations, we firstfixed the permutation π and then using the change of basis produced a presentation asin Theorem 2.4. We would like to note here that the change of basis from [8, Lemma10] does not work for the algebras A t and B t,s (see below), which was not noted in theproof of Lemma 10. This does not effect the result, since these algebras turn out not tobe symmetric. For the algebra A t the element α − tβ belongs to the socle of A t , for thealgebra B t,s the element γ − sγ belongs to the socle of B t,s , which is a contradiction.Here β β γ γ Q : I = h J ( k Q ) , γ β = γ β ,β γ = β γ , β γ = tβ γ ,γ β = tγ β , β γ = sβ γ ,γ β = sγ β i , st = 1 B t,s = k Q/I, π ( γ i ) = β i , π ( β i ) = γ i Q : 1 I = h J ( k Q ) , α = β ,αβ = tα , βα = tβ i , t = 1 A t = k Q/I, π ( α ) = β, π ( β ) = αα β et us also denote by A ∞ an algebra isomorphic to any symmetric stably biserialalgebras with one vertex, two loops and one π -cycle of multiplicity 1. Proposition 3.3.
Let us consider any data of the form ( Q, π, m, L , { t α } α ∈L ) , and A ≃ k Q/I , where I is the ideal of relations described in Theorem 2.4. If A is not isomorphicto A ∞ with both loops deformed, then the algebra A is symmetric.Proof. Recall that an algebra A is symmetric if and only if there exists a non-degeneratesymmetric k -bilinear form h a, b i : A × A → k such that h ab, c i = h a, bc i for all a, b, c ∈ A .Let us define the standard bilinear form h a, b i := φ ( ab ), where the value of φ on thepath basis of A is defined as follows: φ ( C ( α ) m ( C ( α )) ) = 1 for any arrow α , thus φ ( γ ) = l γ for any deformed loop γ , and φ ( p ) = 0 for any path p soc A . The values of φ on otherelements of A is defined by linearity.The defined form is bilinear, symmetric and satisfies the property h ab, c i = h a, bc i forall a, b, c ∈ A . Let us check that it is non-degenerate.Let us assume that φ is degenerate, that is φ (( P c i p i ) a ) = 0 for some P c i p i = 0 andfor all a ∈ A , where c i ∈ k ∗ and p i are paths, by the symmetry of φ , for all a ∈ A wehave φ ( a ( P c i p i )) = 0. We can assume that all p i start at the same vertex i and endat the same vertex j (multiplying by two idempotents and keeping P c i p i non-zero). All p i ’s are subpaths of the standard socle paths of the form C ( α ) m ( C ( α )) . Since there are atmost two such standard socle paths starting at i , all p i ’s can be divided into two groupsdepending on the socle path. Let us chose the shortest path from one of these two groups p . Let ¯ p p be the standard socle path containing p (that is not γ for a deformed loop γ ), then ¯ p ( P c i p i ) = c ¯ p p + c ¯ p p , where ¯ p p appears only in case when the shortestpath from the second group is an arrow p and ¯ p is also an arrow. In this case p mustbe a deformed loop p = ¯ p . If ¯ p p = 0, then φ ( ¯ p ( P c i p i )) = 0 iff c = 0 and we aredone.Let us do the same exchanging p and p . Then ¯ p ( P c i p i ) = c ¯ p p + c ¯ p p , where¯ p p appears only in case when the shortest path from the first group is an arrow p and¯ p is also an arrow. In this case p must be a deformed loop p = ¯ p . And we get exactlythe excluded case of 2 deformed loops at one vertex. (cid:3) Combinatorial derived invariants
The aim of this section is to show that the following combinatorial data are invariantunder derived equivalences of stably biserial algebras: number of vertices, edges and facesof the Brauer graph, multisets of perimeters of faces, multisets of multiplicities of vertices,bipartivity. Note that the corresponding results were shown to be true for Brauer graphalgebras with some minor inaccuracies in [2, 3, 4, 6], the proofs are identical or relayon the corresponding results for Brauer graph algebras, except for some simplifications.From here on we are going to exclude the case | Q | = 1 from some considerations, sinceby [34] a local algebra can be derived equivalent only to itself.4.1. The centre of a symmetric stably biserial algebra.
In this subsection wecompute the centre Z ( A ) of a symmetric stably biserial algebra A , which is known to beinvariant under derived equivalence, see [29]. We will use this to establish, that the numberof π -cycles, or the vertices of the Brauer graph, is invariant under derived equivalence.This will also gives us an opportunity to correct the above-mentioned inaccuracies inthe description of the centre of an SSB-algebra made in [4]. Let { C , C , . . . , C r } be theset of π -cycles. For each i = 1 , . . . , r consider a cyclic sequence ( α i, , α i, , . . . , α i,l i ) ofarrows of the cycle C i , where π ( α i,j ) = α i,j +1 , l i denotes the length of the cycle C i . Let ( C ) , m ( C ) , . . . , m ( C r ) denote the multiplicities of the π -cycles and let r ′ ≤ r be aninteger such that m ( C i ) > , i = 0 , . . . , r ′ and m ( C i ) = 1 , i = r ′ + 1 , . . . , r . For each loop γ such that π ( γ ) = γ there are i and j such that γ = α i,j . For each such loop γ set q γ = q α i,j = ( α i,j +1 α i,j +2 . . . α i,l i α i, . . . α i,j ) m ( C i ) − α i,j +1 α i,j +2 . . . α i,l i α i, . . . α i,j − . Proposition 4.1.
Let A be a symmetric stably biserial algebra with the correspondingdata ( Q, π, m, L ) . As a vector space over k the centre Z ( A ) is generated by and by theelements of the following form:a) Elements m i,t = ( α i, α i, . . . α i,l i ) t + ( α i, α i, . . . α i, ) t + · · · + ( α i,l i α i, . . . α i,l i − ) t , for i = 1 , , . . . , r ′ and t = 1 , . . . , m ( C i ) − .b) Elements q γ for each loop γ such that π ( γ ) = γ .c) Elements s v for each vertex v ∈ Q , where s v is the socle element corresponding to v . Moreover, if A is not isomorphic to some A ∞ , then considered as an al-gebra, Z ( A ) / (soc( Z ( A )) ≃ k [ x , x , . . . , x r ′ ] / h x m ( C i ) i , ( x i x j ) i = j i . So the multiset { m ( C ) , m ( C ) , . . . , m ( C r ′ ) } is invariant under derived equivalence. The number of loops γ such that π ( γ ) = γ , or equivalently, the number of faces of Γ of perimeter 1 is a derivedinvariant as well.Proof. It is clear that all the listed elements belong to the centre of A . Let us prove thatany element of the centre is a linear combination of elements of the form (a)-(c).Each z ∈ Z ( A ) has a form z = P Ni =1 a i p i + z ′ , where p i are the elements of the path basisof A which do not belong to the socle of A and z ′ ∈ soc( A ). Without loss of generality,we can assume z ′ = 0. All elements p i with a i = 0 are necessarily closed paths, that is p i = e v p i e v for some idempotent e v , corresponding to a vertex v . Fix p i = β β . . . β m for some β j ∈ Q , let β m +1 = π ( β m ), then p i β m +1 = 0. Assume that p i β m +1 does notbelong to the socle of A , then β β . . . β m β m +1 has coefficient a i in the sum β m +1 z , hence β m +1 = β and the coefficient of β . . . β m β m +1 in z is a i , so p i = ( α j,s α j,s +1 . . . α j,s − ) t forsome π -cycle, and z contains a i m j,t as a summand, z − a i m j,t contains less summands,then z . If β β . . . β m β m +1 belongs to the socle of A , then β m +1 is a loop, since p i is aclosed path. Then p i is either m j,m ( C j ) − for a cycle C j , consisting of a single loop (if π ( β m +1 ) = β m +1 ) or q β m +1 (if π ( β m +1 ) = β m +1 ). Either z − a i q β m +1 or z − a i m j,m ( C j ) − has less summands then z and we can proceed by induction on the number of nonzerocoefficients a i in the sum z = P Ni =1 a i p i . By induction we get that z is a linear combinationof elements of the form (a)-(c).In case A A ∞ , soc( Z ( A )) is clearly generated by the elements of type (b) and (c).Moreover, m i,t m j,t = δ i,j m i,t + t and m m ( C i ) i, ∈ soc Z ( A ). Hence Z ( A ) / (soc( Z ( A ))) ≃ k [ x , x , . . . , x r ] / h x m ( C i ) i , ( x i x j ) i = j i . Since Z ( A ) is invariant under derived equivalence asan algebra, the multiset { m ( C ) , m ( C ) , ..., m ( C r ′ ) } is invariant under derived equiva-lence. The socle of Z ( A ) is spanned by the elements of the form s v , v ∈ Q and q γ , γ isa loop, π ( γ ) = γ . Since the number of the elements s v is a derived invariant, so is thenumber of loops γ such that π ( γ ) = γ . (cid:3) Number and perimeters of faces.
Let A be a symmetric stably biserial algebrawith the corresponding Brauer graph Γ, let p , p , . . . p m be the perimeters of faces of Γ.Namely, using the graph Γ the surface S can be cut into polygons, by a perimeter ofa face F we mean the number of edges in the corresponding polygon, thus, for example,the perimeter of a self-folded triangle is 3. Note that the perimeter of a face F coincideswith the length of the corresponding Green walk (see [32, 16]). The aim of this sectionis to prove that the multiset { p , . . . , p m } (and, in particular, the number of faces m ) s an invariant of the derived category of A . For this we are going to use the structureof the Auslander-Reiten quiver of the stable category of mod- A . Note that by [30] forself-injective and in particular for symmetric algebras derived equivalence implies stableequivalence, so any invariant of stable equivalence is automatically a derived invariant.Indecomposable modules over special biserial algebras are classified in terms of stringsand bands, the description of the Auslander-Reiten sequences and of the Auslander-Reiten quiver for such algebras is well understood [20, 13, 36, 17, 18]. Let us consider the AR -quiver Γ mod- A of mod- A . If A is SSB, then each periodic component of Γ mod- A is atube. Moreover, all tubes are either tubes of rank 1, consisting of band modules or tubesconsisting of string modules, called exceptional tubes. Exceptional tubes correspond tofaces of Γ: if a face has an even perimeter p , then it produces two tubes of rank p/ p , which is stable under the action of Ω. For a detailed exposition see [16, Section4].In case A is symmetric stably biserial and not necessarily special biserial its AR -quiverΓ mod- A of mod- A coincides with the same quiver for the SSB-algebra A ′ constructedfrom the same data ( Q, π, m ). Indeed, A/ soc( A ) ≃ A ′ / soc( A ′ ) is a string algebra, so theclassification of indecomposable non-projective modules is the same. The AR-sequencesnot ending at the module P/ soc( P ) for a projective module P coincide for A and A/ soc( A )by [12, Proposition 4.5], the fact that the sequences 0 → rad P → rad P/ soc P ⊕ P → P/ soc P → mod- A and Γ mod- A ′ can be checked by hand.Consequently, we get that the number of faces of a given perimeter p > p , stable under Ω, in case p is odd and the number of tubes of rank p/ p is even. In both cases the perimeter can bealso reconstructed from the stable category. By Proposition 4.1 the number of faces ofperimeter 1 is a derived invariant. The number of faces of perimeter 2 can be reconstructedas follows: (2 | E (Γ) | − P p i =2 p i ) /
2. Thus, the following holds:
Proposition 4.2.
Let A , A be two symmetric stably biserial algebras with Brauer graphs Γ and Γ , such that neither Γ nor Γ is a loop with multiplicity 1 or an edge withmultiplicity of both vertices 2. If D b ( A ) ≃ D b ( A ) , then the number of faces and themultisets of perimeters of faces of Γ and Γ coincide. Remark 4.3.
The proof of the fact that the number of faces and the multisets of perime-ters of faces of Γ is invariant under an equivalence of stable categories of SSB-algebraswas provided in [3] with a mistake, which was corrected in [5] . Note that the proof is muchmore involved, since one can not use the centre of the algebra Z ( A ) (as in Proposition4.1), so one has to deal with the tubes coming from the faces of perimeter 1 and 2 andwith the tubes containing band modules. Number of vertices and their multiplicities.
Let Z | Q | be the Grothendieckgroup of a self-injective algebra A with the Cartan matrix C ( A ). Then C ( A ) defines agroup homomorphism φ A from Z | Q | to itself and K (mod- A ) ≃ Z | Q | / Im( φ A ). To obtainthe standard description of this Abelian group one can use Smith’s normal form of C ( A ),which can be obtained by computing the greatest common divisors of all t × t minorsof C ( A ), this was done for SSB-algebras in [2, 6]. We are going to use only the rank of C ( A ), which is equal to | Q /π | − A is bipartite and to | Q /π | ,otherwise. Note that by construction | Q /π | is the number of vertices of Γ. Proposition 4.4.
Let A , A be stably biserial algebras with Brauer graphs Γ and Γ ,such that neither Γ nor Γ is a loop with multiplicity 1 or an edge with multiplicity of oth vertices 2. If D b ( A ) ≃ D b ( A ) , then | V (Γ ) | = | V (Γ ) | . Moreover, the multisets ofmultiplicities of the vertices and the bipartivity of Γ and Γ coincide.Proof. Let A ′ i be the special biserial algebra corresponding to the data given by the Brauergraph Γ i . As A i and A ′ i have the same Cartan matrices, we can use the description ofthe structure of the Grothendieck group of A ′ i for A i . Since derived equivalences of self-injective algebras imply stable equivalences, K (mod- A ) ≃ K (mod- A ), thus the ranksof C ( A ) and C ( A ) coincide.By [6], rk( C ( A i )) is equal to | V (Γ i ) | − A i ) of A i is bipartiteand to | V (Γ i ) | otherwise. By Proposition 4.2 | F (Γ ) | = | F (Γ ) | , the same holds for | E (Γ ) | = | E (Γ ) | , since | V (Γ i ) | − | E (Γ i ) | + | F (Γ i ) | is even as the Euler characteristic ofthe surface S i , we see that | V (Γ i ) | can not differ by 1, hence, | V (Γ ) | = | V (Γ ) | . Since theranks of the Cartan matrices of A and A coincide, Γ i are either simultaneously bipartiteor simultaneously not bipartite.The multiplicities of the vertices > (cid:3) Lemma 4.5.
Let A , A be derived equivalent symmetric stably biserial algebras, where A is a caterpillar. Then A is special biserial.Proof. The algebra A has no loops γ such that π ( γ ) = γ , so by Proposition 4.1 A hasno such loops as well, hence, A is symmetric special biserial. (cid:3) Proof of Theorem 1.2.
Combining the results of Proposition 4.2 and 4.4 we get thatthe following are derived invariants of a symmetric stably biserial algebra A withat least two non-isomorphic simple modules and the corresponding Brauer graph Γ: | V (Γ) | , | E (Γ) | , | F (Γ) | , the multiset of perimeters of faces, the multiset of multiplicitiesof vertices, bipartivity of Γ. (cid:3) The group of outer automorphisms
Throughout this section we are going to assume that either char ( k ) = 2 or that char ( k ) = 2 and the number of deformed loops d = 0 (that is A is symmetric spe-cial biserial); that A is not a caterpillar and that A has at least two non-isomorphicsimple modules. We are going to show that derived equivalent symmetric stably biserialalgebras have the same number of deformed loops using the identity component of thegroup of outer automorphisms. By [23, 33] the identity component of the group of outerautomorphisms Out ( A ) of an algebra A is invariant under derived equivalence as analgebraic group. We are going to use the necessary notions and facts about algebraicgroups freely, for more details see [25].5.1. H ′ is trigonalizable. Let A = k Q/I be a stably biserial algebra in the standardform given in Theorem 2.4, i.e. the ideal of relations is not necessarily admissible. Let
L ⊂ Q be the set of deformed loops. Let A = B ⊕ J ( A ) be a Wedderburn-Maltsevdecomposition (i.e. B is semisimple subalgebra). Then it is known that Out ( A ) = H/H ∩ Inn ( A ), where H = { f ∈ Aut ( A ) | f ( B ) ⊂ B } [22, 28]. If { e v } v ∈ Q is a set of primitiveidempotents and B = h{ e v } v ∈ Q i , then obviously for any v ∈ Q and f ∈ H , f ( e v ) = e v ′ for some v ′ ∈ Q . Therefore H ′ = { f ∈ H | f | B = Id } is a closed subgroup of finite indexin H , i.e. it is a union of connected components, since H ∩ Inn ( A ) = H ′ ∩ Inn ( A ) = Inn ( A ) acts on each component, to understand Out ( A ) we can consider only H ′ withoutloss of generality. emma 5.1. If A is not a caterpillar and rk K ( A ) ≥ , then there is an embedding i : H ′ → T ( l, k ) of algebraic groups over k , where T ( l, k ) is the group of lower triangularmatrices and l = dim A .Proof. Let P be a set of paths in Q which forms a basis for k Q/I , such that α / ∈ P for α ∈ L and all primitive idempotents { e v } v ∈ Q and all arrows of Q are in P . Foreach p ∈ P let l p = max { k : p ∈ J ( A ) k } , with the convention that l e v = 0 , v ∈ Q . Apair ( β, β ′ ) ∈ Q × Q with s ( β ) = s ( β ′ ) , e ( β ) = e ( β ′ ) ( β, β ′ are parallel arrows) and π ( β ) = β, π ( β ′ ) = β ′ will be called an exceptional pair.Let us consider some linear extension of the following partial order on P :1)If l p < l q , then p < q .2)If ( β, β ′ ) is an exceptional pair, then β < β ′ . Note that since | Q | > l β = l β ′ = 1.We are going to express the matrix of an automorphisms of A in the basis P withrespect to this linear order and show that, it is lower-triangular, i.e, for f ∈ H ′ and p ∈ P we have f ( p ) = k p p + P p ′ >p k p,p ′ p ′ .Let us consider p = β ∈ Q , such that l β = 1. If β has no parallel arrows, then f ( β ) = k β β + r where r ∈ J ( A ) and we are done. Now suppose that β has a parallelarrow β ′ , in this case we can have f ( β ) = k β β + k β,β ′ β ′ + r with k β,β ′ = 0. Note that since | Q | 6 = 1, β is not a loop. There are three possible cases:1) π ( β ) , π ( β ′ ) are not parallel. In this case f ( π ( β ′ )) = k π ( β ′ ) π ( β ′ ) + r , r ∈ J ( A ) .Then 0 = f ( βπ ( β ′ )) = f ( β ) f ( π ( β ′ )) = k β,β ′ k π ( β ′ ) β ′ π ( β ′ ) + r ′ , r ′ ∈ J ( A ) . A path oflength two β ′ π ( β ′ ) belongs to J ( A ) , hence β ′ π ( β ′ ) ∈ soc(A). Since π ( β ) , π ( β ′ ) are notparallel, ( β, β ′ ) is an exceptional pair and β < β ′ . The same argument for ( β ′ , β ) gives k β ′ ,β k π ( β ) βπ ( β ) ∈ soc(A), hence k β ′ ,β k π ( β ) = 0, thus k β ′ ,β = 0, so f ( β ′ ) = k β ′ β ′ + r ′′ , r ′′ ∈ J ( A ) .2) π ( β ) , π ( β ′ ) are parallel arrows but π l ( β ) , π l ( β ′ ) are not parallel for some l (wetake the minimal l ). In this case ( π l − ( β ) , π l − ( β ′ )) is not an exceptional pair (other-wise s ( π l − ( β )) has 3 incoming arrows) and f ( π l ( β ′ )) = k π l ( β ′ ) π l ( β ′ ) + r ′ , r ′ ∈ J ( A ) .So 0 = f ( π l − ( β )) f ( π l ( β ′ )) implies f ( π l − ( β )) = k π l − ( β ) π l − ( β ) + r, r ∈ J ( A ) and thesame holds for β ′ . Then by decreasing induction on i we obtain in the same way that f ( π i ( β ′ )) = k π i ( β ′ ) π i ( β ′ ) + r ′ , r ′ ∈ J ( A ) , f ( π i ( β )) = k π i ( β ) π i ( β ) + r, r ∈ J ( A ) for all0 ≤ i ≤ l , in particular, for i = 0.3) π l ( β ) , π l ( β ′ ) are parallel for all l . In this case A = k Q/I is a caterpillar.For an arbitrary p i ∈ P , chose a presentation p i = β . . . β n with n maximal. Then f ( p i ) = f ( β ) . . . f ( β n ) = Q i k β i β . . . β n + P k j p j , where p j > β . . . β n . Indeed, β . . . β n is of the form β π ( β ) . . . π n ( β ) and since for an exceptional pair ( β, β ′ ) we have β ′ π ( β ) , π − ( β ) β ′ ∈ J ( A ) the sum P k j p j belongs to J ( A ) n +1 . (cid:3) Decomposition with the unipotent subgroup.
We have seen in the previoussubsection that H ′ is a subgroup of the group of lower-triangular matrices. In order not tocompute the groups Out ( A ) for all symmetric stably biserial algebras, which might turnout to be quite technical, we want to deduce some easier invariant of Out ( A ) preservedby isomorphisms of algebraic groups.Let us consider maximal unipotent subgroups in H ′ and Inn ( A ), denoted respectivelyby U H ′ and U I . These groups are given by the intersection of H ′ (respectively Inn ( A ))with U ( l, k ) the group of (lower) unitriangular matrices. We can consider the followingdiagram of algebraic groups: / / U I / / (cid:15) (cid:15) Inn ( A ) / / (cid:15) (cid:15) D I / / (cid:15) (cid:15) / / U H ′ / / (cid:15) (cid:15) H ′ / / (cid:15) (cid:15) D H ′ / / (cid:15) (cid:15) / / U H ′ /U I / / Out ( A ) / / D H ′ /D I / / D I → D H ′ is an embedding. As a quotientof a trigonalizable group Out ( A ) is trigonalizable, D H ′ /D I is diagonalizable and U H ′ /U I is the maximal unipotent subgroup of Out ( A ), it contains all unipotent subgroups of Out ( A ) [25, Theorem 16.6]. Thus we can consider another diagram:1 (cid:15) (cid:15) / / / / (cid:15) (cid:15) X (cid:15) (cid:15) / / ( U H ′ /U I ) / / (cid:15) (cid:15) Out ( A ) / / (cid:15) (cid:15) D ( A ) / / (cid:15) (cid:15) / / U H ′ /U I / / (cid:15) (cid:15) Out ( A ) / / (cid:15) (cid:15) D H ′ /D I / / (cid:15) (cid:15) Y / / Z / / W / / Out ( A ) is connected and solvable its maximal unipotent subgroup is connectedand thus coincides with ( U H ′ /U I ) . Note also that all maximal tori of Out ( A ) are con-jugate. The groups D I , D H ′ , D H ′ /D I are diagonalizable. We also get the following exactsequence 1 → X → Y → Z → W →
1, where
Y, Z are finite, then
X, W are finite as well.Hence the rank of D ( A ) and D H ′ /D I coincide. We have proved the following lemma. Lemma 5.2.
In the notation of the previous construction, the rank of D H ′ /D I coincideswith the rank of the maximal torus D ( A ) of Out ( A ) and so is a derived invariant of A . Computation of the rank of D H ′ /D I . Both D H ′ and D I are induced by theprojection map from the group of lower triangular matrices to the group of diagonalmatrices. So we are going to find out what elements can appear on the diagonal of thematrices from H ′ and Inn ( A ). Clearly the diagonal entries corresponding to the arrowsof the quiver determine all other diagonal entries, so we are going to restrict our attentionto them. Lemma 5.3.
There is an isomorphism of affine algebraic groups D I ≃ ( k ∗ ) | Q |− .Proof. Recall that
Inn ( A ) = { f a | a ∈ A ∗ } , where f a ( x ) = axa − . Each a ∈ A ∗ canbe uniquely written as a = P i ∈ Q a i e i + r , where a i ∈ k ∗ , r ∈ J ( A ). Then a − = P i ∈ Q a − i e i + r ′ , r ′ ∈ J ( A ) and the action of f a on J ( A ) /J ( A ) depends only on a i ’s. For c ∈ k ∗ , f a clearly coincides with f ca .Let f a = f a (mod U I ). Consider any spanning tree of Q (ignoring the orientationof Q ), let { α i } ≤ i ≤ n − be the corresponding set of arrows. For each arrow α i we have f a ( α i ) = aα i a − = a s ( α i ) a − e ( α i ) α i (mod J ( A ) ). Let us define the map D I η −→ ( k ∗ ) | Q |− as f a → ( a s ( α ) a − e ( α ) , . . . , a s ( α | Q |− ) a − e ( α | Q |− ) ). The equality f a ( α i ) = α i (mod J ( A ) ) for theelements of U I guarantees that the map is well defined. ince α i form a spanning tree, for any element ( k , . . . , k | Q |− ) ∈ ( k ∗ ) | Q |− onecan uniquely determine { a i } i ∈ Q up to multiplication of all a i by a common con-stant c ∈ k ∗ . Setting a = P i ∈ Q a i e i one can define a map D I θ ←− ( k ∗ ) | Q |− , clearly θη (( k , . . . , k | Q |− )) = ( k , . . . , k | Q |− ).For a − a ′ ∈ J ( A ) and any path p = β β . . . β k we have f a ( p ) − f a ′ ( p ) ∈ J ( A ) k +1 . Asthe order on the basis P agrees with the path length f a = f a ′ . Since f a = f ca for c ∈ k ∗ , ηθ ( f a ) = f a and we get the desired bijection. (cid:3) Let us consider the following algebraic group D Γ , which can be constructed from thedata ( Q, π, m, L ) or equivalently from the data of a Brauer graph and a fixed number of1-perimeter faces of Γ (the set L is assumed to be empty in case char ( k ) = 2). The group D Γ is a subgroup of ( k ∗ ) | E (Γ) | +1 . The first 2 | E (Γ) | entries k α are labelled by the arrowsof Q , the last entry is denoted k . The subgroup is given by the following equations:( k α ) = t α k for each deformed loop α and Q α ∈ C k m ( C ) α = k for each C ∈ Q /π withmultiplicity m ( C ). Proposition 5.4.
Let A be a symmetric stably biserial algebra corresponding to the data ( Q, π, m, L ) , then D H ′ ≃ D Γ .Proof. As before any f ∈ H ′ has the form f ( α ) = k α α + P p>α k α,p p . We need to checkthat the set of elements of ( k ∗ ) | E (Γ) | appears as the set ( k α ) for some f ∈ H ′ if and onlyif there exists k ∈ k ∗ such that the equations from the description of the group D Γ aresatisfied.If for a set of elements ( k α ) in ( k ∗ ) | E (Γ) | there exists k ∈ k ∗ such that ( k α , k ) ∈ D Γ ,then we can define f ∈ H ′ by f ( α ) = k α α , which clearly gives an automorphism of A .Let us prove that for any f ∈ H ′ the set ( k α ) is an element of D Γ with some k ∈ k ∗ . Forthat we need to better understand which coefficients k α,p can be non-zero. For an arrow α , let us denote by ¯ C ( α ) := C ( α ) m ( C ( α )) = ( απ ( α ) · · · π | π ( α ) |− ( α )) m ( h π i α ) the maximalpower of the cycle passing through α . Let us show that if k α,p = 0 and p is not a subpathof ¯ C ( α ), then p = β − ¯ C ( β ) for some arrow β with s ( β ) = e ( α ) and e ( β ) = s ( α ). Assumethis is not the case and let us take p = β · · · β t with t minimal, since s ( p ) = s ( α ), e ( p ) = e ( α ) and p is not a subpath of ¯ C ( α ), p / ∈ soc( A ), then p is a subpath of some¯ C ( δ ) and such p is unique. Let β be the arrow such that βp ∈ ¯ C ( β ), such β exists and e ( β ) = s ( α ) but α = π ( β ). Note that β is not a loop with π ( β ) = β , otherwise α = β and p is a subpath of ¯ C ( α ). The relation f ( β ) f ( α ) = 0 implies that the coefficient before βp , which contains k β k α,p should be 0. Assume βp / ∈ soc A , then by the minimality of thelength of p , k α,p = 0. So βp ∈ soc A as desired.Let us now check that for any f ∈ H ′ the set ( k α ) satisfies the equations Q α ∈ C k m ( C ) α = k for some k ∈ k . Let us compute f ( ¯ C ( α )) for some α ∈ Q . It has a summand Q α ′ ∈ C ( α ) k m ( C ( α )) α ′ ¯ C ( α ), if it has any other summand, then this summand can only ap-pear in one of the following 3 situations:1) as a product of the elements of the form k π i ( α ) ,p , where p is not a subpath of ¯ C ( π i ( α )).This situation is possible only in the case of a caterpillar with two simple modules, whichwe do not consider.2) as a product of subpaths of ¯ C ( α ) and paths, which are not subpaths of ¯ C ( α ), thisis possible only in the situation | Q | = 1 and A has a deformed loop, which we also donot consider.3) as a product of subpaths of ¯ C ( α ), at least one of which comes from f ( π i ( α )) and isnot π i ( α ). Note that all these subpaths are arrows, otherwise the product is zero. Sinceat least one of the subpaths comes from f ( π i ( α )) and is not π i ( α ) all of them come from ( π j ( α )) but are not π j ( α ), otherwise we are in the situation | Q | = 1 and A has adeformed loop again. Hence every π i ( α ) has a parallel arrow and A is a caterpillar, whichwe do not consider.So f ( ¯ C ( α )) = Q α ′ ∈ C ( α ) k m ( C ( α )) α ′ ¯ C ( α ). Since the relation f ( ¯ C ( α )) = f ( ¯ C ( β )) holds forany α, β ∈ Q with s ( α ) = s ( β ) and the graph Γ is connected, we can denote by k theproduct Q α ′ ∈ C ( α ) k m ( C ( α )) α ′ for some fixed α and get that Q α ∈ C k m ( C ) α = k for any π -cycle C .From here on we assume char ( k ) = 2. Let us deal with the equations ( k α ) = t α k , foreach deformed loop α . For a deformed loop α let w α be the path that makes αw α intoa π -cycle. Let m be the multiplicity of this cycle. Then f ( α ) = k α α + P p k p p (we willuse this simplification of the notation for the rest of the proof), where p can have thefollowing form: ( w α α ) i , i = 1 , . . . m − α ( w α α ) i , i = 0 , . . . m −
1, ( w α α ) i w α , i = 0 , . . . m − α ( w α α ) i w α , i = 0 , . . . m −
1. Since α is a deformed loop it appears only in socle relationsand there are no restrictions on k p so far.Let us consider f ( α ) : the coefficient before αw α α should be zero, this gives k α k w α α + k α k αw α = 0, so since k α ∈ k ∗ , k w α α + k αw α = 0. Let us assume k ( w α α ) i + k ( αw α ) i = 0, for i The rank of D H ′ is | Q | − | Q /π | − d + 1 , where d is the number of deformedloops in A .Proof. Let us construct an epimorphism j : D H ′ → ( k ∗ ) | Q |−| Q /π |− d +1 such that the kernel ker ( j ) is finite.Each cycle of π contains an arrow, which is not a deformed loop. Let us fix one sucharrow in each cycle and denote the collection of these arrows by F . Let us label the ele-ments of ( k ∗ ) | Q |−| Q /π |− d +1 by x α , α ∈ Q , α / ∈ F ∪ L and by an additional indeterminant x . Define the map j as follows: j (( k α , k )) := (( x α , x )), where x α = k α , α / ∈ F ∪ L , x = k .The map j is surjective, since for any ( x α , x ) we can define k α = √ t α x for α ∈ L and k α = m ( C ( α )) q x/ Q α ′ ∈ C ( α ) ,α ′ = α,α ′ / ∈L x m ( C ( α )) α ′ Q α ′ ∈ C ( α ) ,α ′ ∈L ( √ t α ′ x ) m ( C ( α )) for α ∈ F .Let us compute the kernel of j . ( k α , k ) ∈ ker ( j ) if and only if k = 1, k α = 1 for α / ∈ F ∪ L , k α = t α for α ∈ L , k m ( C ( α )) α = 1 / Q α ′ ∈ C ( α ) ,α ′ ∈L t m ( C ( α )) α ′ for α ∈ F . This clearlydefines a finite group.Passing to the groups of characters, if necessary, and using the equivalence betweenthe category of diagonalizable groups and finitely generated commutative groups [25,Theorem 12.9], we see that the rank of D H ′ is | Q | − | Q /π | − d + 1. (cid:3) Proof of Theorem 1.1. Since D I is connected, its image belongs to the maximal torusin D H ′ and passing to the groups of characters again, the exact sequence 1 → D I → D H ′ → D H ′ /D I → 1, gives that the rank of D H ′ /D I is | Q | − | Q /π | − d + 1 − | Q | + 1 = | Q | − | Q /π | − d + 2 = | E (Γ) | − | V (Γ) | − d + 2. (cid:3) sing the fact that Brauer graph algebras can be stably (and hence derived) equivalentonly to symmetric stably biserial algebras [8, Theorem 1 and 3], Corollary 1.3 can bededuced from Theorems 1.1 and 1.2 and Lemma 4.5, as well as from the fact that forlocal algebras derived equivalence implies Morita equivalence. 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