The Ext-algebra of the Brauer tree algebra associated to a line
aa r X i v : . [ m a t h . R T ] J a n THE EXT-ALGEBRA OF THE BRAUER TREE ALGEBRAASSOCIATED TO A LINE
OLIVIER DUDAS
Abstract.
We compute the
Ext -algebra of the Brauer tree algebra associatedto a line with no exceptional vertex.
Introduction
This note provides a detailed computation of the
Ext -algebra for a very specificfinite dimensional algebra, namely a Brauer tree algebra associated to a line, with noexceptional vertex. Such algebras appear for example as the principal p -block of thesymmetric group S p , and in a different context, as blocks of the Verlinde categories Ver p studied by Benson–Etingof in [2] (our computation is actually motivated by[2, Conj. 1.3]).Let us emphasise that Ext -algebras for more general biserial algebras were explic-itly computed by Green–Schroll–Snashall–Taillefer in [4], but under some assump-tion on the multiplicity of the the vertices, assumption which is not satisfied for thesimple example treated in this note. Other general results relying on Auslander–Reiten theory were obtained by Antipov–Generalov [1] and Brown [3]. Howeverwe did not manage to use their work to get an explicit description in our case.Nevertheless, the simple structure of the projective indecomposable modules forthe line allows a straightforward approach using explicit projective resolutions ofsimple modules. The Poincar´e series for the
Ext -algebra is given in Proposition 2.2and its structure as a path algebra with relations is given in Proposition 3.2.
Acknowledgments
We thank Rapha¨el Rouquier and Rachel Taillefer for providing helpful references.1.
Notation
Let F be a field, and A be the self-injective finite dimensional F -algebra. All A -modules will be assumed to be finitely generated. Given an A -module M , wedenote by Ω( M ) the kernel of a projective cover P ։ M . Up to isomorphism itdoes not depend on the cover. We then define inductively Ω n ( M ) = Ω(Ω n − ( M ))for n ≥ n ( M ) is indecomposable and non-projective then Ext nA ( M, S ) ≃ Hom A (Ω n ( M ) , S )for all simple A -module S and all n ≥ The author gratefully acknowledges financial support by the ANR, Project No ANR-16-CE40-0010-01.
For computing the algebra structure on the various
Ext -groups it will be conve-nient to work in the homotopy category Ho ( A ) of the complexes of finitely generated A -modules. If S (resp. S ′ ) is a simple A -module, and P • → S (resp. P ′• → S ′ ) is aprojective resolution then Ext nA ( S, S ′ ) ≃ Hom Ho ( A ) ( P • , P ′• [ n ])with the Yoneda product being given by the composition of maps in Ho ( A ).Assume now that A is the F -algebra associated to the following Brauer tree with N + 1 vertices: S S S N Here, unlike in [4] we assume that there are no exceptional vertex. The edgesare labelled by the simple A -modules S , . . . , S N . The head and socle of P i areisomorphic to S i and rad ( P i ) /S i ≃ S i − ⊕ S i +1 with the convention that S = S N +1 = 0. 2. Ext-groups
Given 1 ≤ i ≤ j ≤ N with i − j even, there is, up to isomorphism, a uniquenon-projective indecomposable module i X j such that • rad ( i X j ) = S i +1 ⊕ S i +3 ⊕ · · · ⊕ S j − • hd ( i X j ) = S i ⊕ S i +2 ⊕ · · · ⊕ S j .The structure of i X j can be represented by the following diagram: S i S i +2 S i +4 · · · S j − S ji X j = · · · S i +1 S i +3 · · · S j − Similarly we denote by i X j the unique indecomposable module with the followingstructure: S i +1 S i +3 · · · S j − i X j = · · · S i S i +2 S i +4 · · · S j − S j Finally, in the case where i − j is odd we define the modules i X j and i X j as theindecomposable modules with the following respective structure: S i +1 S i +3 · · · S ji X j = · · · S i S i +2 S i +4 · · · S j − S i S i +2 S i +4 · · · S j − i X j = · · · S i +1 S i +3 · · · S j HE EXT-ALGEBRA OF THE BRAUER TREE ALGEBRA ASSOCIATED TO A LINE 3
For convenience we will extend the notation i X j , i X j , i X j and i X j to any integers i, j ∈ Z (with the suitable parity condition on i − j ) so that the following relationshold:(1) i X = − i X , i X j = j X i , i ± N X = i X . Note that this also implies X j = X − j and X j ± N = X j . Lemma 2.1.
Let i, j ∈ Z with i − j even. Then Ω( i X j ) ≃ i − X j +1 . Proof.
Using the relations (1) it is enough to prove that for 1 ≤ k ≤ l ≤ N we havethe following isomorphismsΩ( k X l ) ≃ k − X l +1 , Ω( k X l ) ≃ k +1 X l +1 , Ω( k X l ) ≃ k − X l − , Ω( k X l ) ≃ k +1 X l − . We only consider the first one, the others are similar. If 1 ≤ k ≤ l ≤ N a projectivecover of k X l is given by P k ⊕ P k +2 ⊕ · · · ⊕ P l ։ k X l , whose kernel equals k − X l +1 .Note that this holds even when k = 1 since X l +1 = X l +1 or when l = N since k − X N +1 = k − X − N +1 = k − X N . (cid:3) We deduce from Lemma 2.1 that for any simple module S i and for all k ≥ k ( S i ) = Ω k ( i X i ) ≃ i − k X i + k as A -modules. Consequently we have Ext kA ( S i , S j ) = (cid:26) F if S j appears in the head of i − k X i + k , Ext -groups.
Proposition 2.2.
Given ≤ i, j ≤ N , the Poincar´e series of Ext • A ( S i , S j ) is givenby X k ≥ dim F Ext kA ( S i , S j ) t k = Q i,j ( t ) + t N − Q i,j ( t − )1 − t N where Q i,j ( t ) = t | j − i | + t | j − i | +2 + · · · + t N − −| N +1 − j − i | .Proof. Without loss of generality we can assume that i ≤ j . Let k ∈ { , . . . , N − } .If i + j ≤ N + 1, the simple module S j appears in the head of i − k X i + k if and onlyif k = j − i, j − i + 2 , . . . , j + i −
2. The limit cases are indeed i − j X j for k = j − i and − j X i + j − = j − X i + j − for k = j + i −
2. Note that if j − i ≤ k ≤ i + j − j ≤ i + k and j ≤ N − i − k so that S j appears in the head of i − k X i + k = i − k X N − i − k +1 whenever k has the suitable parity. If i + j > N + 1 one must ensurethat j ≤ N − i − k and therefore S j appears in the head of i − k X i + k if and only if k = j − i, j − i + 2 , . . . , N − i − j . Consequently we have(2) N − X k =0 dim F Ext kA ( S i , S j ) t k = t j − i + t j − i +2 + · · · + t N − −| N +1 − j − i | = t | j − i | + t | j − i | +2 + · · · + t N − −| N +1 − j − i | = Q i,j ( t ) . Now the relationΩ N ( S i ) = i − N X i + N = N − i X − N − i = N − i X N − i = S N +1 − i OLIVIER DUDAS yields N − X k =0 dim F Ext kA ( S i , S j ) t k = N − X k =0 dim F Ext kA ( S i , S j ) t k + t N N − X k =0 dim F Ext kA ( S N +1 − i , S j ) t k . and the proposition follows from (2) after observing that Q N +1 − i,j ( t ) = t N − Q i,j ( t − ). (cid:3) Algebra structure
Minimal resolution.
Given 1 ≤ i ≤ N − f i : P i −→ P i +1 and f ∗ i : P i +1 −→ P i such that f ∗ i ◦ f i + f i − ◦ f ∗ i − = 0 for all 2 ≤ i ≤ N − ≤ i ≤ j ≤ N with j − i even we denote by i P j the following projective A -module i P j := P i ⊕ P i +2 ⊕ · · · ⊕ P j − ⊕ P j . For 1 ≤ i < j ≤ N with j − i even we let d i,j : i P j −→ i +1 P j − be the morphismof A -modules corresponding to the following matrix: d i,j = f i f ∗ i +1 · · · · · · f i +2 f ∗ i +3 · · · · · · f j − f ∗ j − The definition of i P j extends to any integers i, j ∈ Z with the convention that(3) i P j = j +1 P i − , i P − j = i P j , i P j ± N = i P j . Note that these relations imply − i P j = i P j and i ± N P j = i P j . Furthermore,the definition of d i,j extends naturally to any pair i, j if we set in addition d i,i = ( − i f ∗ i f i = ( − i − f i − f ∗ i − , a map from i P i = P i to i +1 P i − = P i . With this notation one checks that for all k > d i − k,i + k : i − k P i + k −→ i − k +1 P i + k − is isomorphic to i − k X i + k ≃ Ω k ( S i ) so that the bounded above complex R i := · · · d i − k − ,i + k +1 −−−−−−−→ i − k P i + k d i − k,i + k −−−−→ · · · d i − ,i +2 −−−−→ i − P i +1 d i − ,i +1 −−−−→ P i −→ S i .3.2. Generators and relations.
We will consider two kinds of generators forthe
Ext -algebra, of respective degrees 1 and N . We start by defining a map x i ∈ Hom Ho ( A ) ( R i , R i +1 [1]) for any 1 ≤ i ≤ N −
1. Let k be a positive integer. If k / ∈ N Z , the projective modules i − k P i + k and i +1 − ( k − P i +1+( k − = i − k +2 P i + k haveat least one indecomposable summand in common and we can consider the map X i,k : i − k P i + k −→ i − k +2 P i + k given by the identity map on the common factors. If k ∈ N + 2 N Z then from the relations (3) we have i − k P i + k = i − N P i + N = i + N +1 P i − N − = − i − N +1 P − i + N +1 = P N +1 − i and i − k +2 P i + k = i − N +2 P i + N = N − i P − N − i = P N − i . HE EXT-ALGEBRA OF THE BRAUER TREE ALGEBRA ASSOCIATED TO A LINE 5
In that case we set X i,k := ( − N − i f ∗ N − i . If k ∈ N Z then i − k P i + k = P i , i − k +2 P i + k = i +2 P i = P i +1 and we set X i,k := ( − i f i . If k ≥ X i,k := 0. Then the familyof morphisms of A -modules X i := ( X i,k ) k ∈ Z defines a morphism of complexes of A -modules from R i to R i +1 [1] and we denote by x i its image in Ho ( A ).Similarly we define a map X ∗ i : R i +1 −→ R i [1] by exchanging the role of f and f ∗ . More precisely we consider in that case X ∗ i, − N := ( − N − i f N − i and X ∗ i, − N :=( − i f ∗ i . We denote by x ∗ i the image of X ∗ i in Ho ( A ).Assume now that 1 ≤ i ≤ N . The modules i − k P i + k and ( N +1 − i ) − ( k − N ) P ( N +1 − i )+( k − N ) are equal, which means that starting from the degree − N , the terms of the com-plexes R i and R N +1 − i [ N ] coincide. We denote by Y i : R i −→ R N +1 − i [ N ] thenatural projection between R i and its obvious truncation at degrees ≤ − N , and by y i its image in Ho ( A ). Lemma 3.1.
The following relations hold in
End • Ho ( A ) ( L R i ) : (a) x ∗ ◦ x = x N − ◦ x ∗ N − = 0 ; (b) x i ◦ x ∗ i = x ∗ i +1 ◦ x i +1 for all i = 1 , . . . , N − ; (c) y i +1 ◦ x i = x ∗ N − i ◦ y i for all i = 1 , . . . , N − ; (d) y i ◦ x ∗ i = x N − i ◦ y i +1 for all i = 1 , . . . , N − .Proof. If N = 1 there are no relation to check. Therefore we assume N ≥
2. Therelations in (a) follow from the fact that
Ext A ( S , S ) = Ext A ( S N , S N ) = 0, whichis for example a consequence of Proposition 2.2 when N ≥ X i : R i −→ R i +1 [1]defined above coincide with X ∗ N − i [ N ] : R N +1 − i [ N ] −→ R N − i [ N + 1] in degreesless than − N . Since Y i and Y i +1 are just obvious truncations we actually have Y i +1 ◦ X i = X ∗ N − i ◦ Y i . The relation (d) is obtained by a similar argument.We now consider (b). The morphism of complexes X i ◦ X ∗ i and X ∗ i +1 ◦ X i +1 coincide at every degree k except when k is congruent to 0 or − N . Let usfirst look in details at the degrees − N and − N −
1. The map X i ◦ X ∗ i is as follows: · · · P N − − i ⊕ P N +1 − i P N − i P N − i P N − i ⊕ P N +2 − i P N +1 − i P N +1 − i P N − i ⊕ P N +2 − i P N − i P N − i P N − − i ⊕ P N +1 − i · · · h f N − − i f ∗ N − i ih i ( − N − i f ∗ N − i ◦ f N − i ( − N − i f N − i h f N − i f ∗ N +1 − i ih i ( − N − i f N − i ◦ f ∗ N − i ( − N − i f ∗ N − i f ∗ N − i f N +1 − i ( − N − i f ∗ N − i ◦ f N − i f ∗ N − − i f N − i OLIVIER DUDAS whereas the map X ∗ i +1 ◦ X i +1 corresponds to the following composition: · · · P N − − i ⊕ P N +1 − i P N − i P N − i P N − − i ⊕ P N − i P N − − i P N − − i P N − − i ⊕ P N − i P N − i P N − i P N − − i ⊕ P N +1 − i · · · h f N − − i f ∗ N − i ih i ( − N − i f ∗ N − i ◦ f N − i ( − N − − i f ∗ N − − i h f N − − i f ∗ N − − i ih i ( − N − − i f ∗ N − − i ◦ f N − − i ( − N − − i f N − − i f ∗ N − − i f N − − i ( − N − i f ∗ N − i ◦ f N − i f ∗ N − − i f N − i We deduce that at the degrees − N and − N − X i ◦ X ∗ i − X ∗ i +1 ◦ X i +1 isgiven by P N − − i ⊕ P N +1 − i P N − i P N − i P N − − i ⊕ P N +1 − i h f N − − i f ∗ N − i i ( − N − i h f N − − i f ∗ N − i i ( − N − i f ∗ N − − i f N − i f ∗ N − − i f N − i A similar picture holds at the degrees − N and − N − P i ⊕ P i +2 P i +1 P i +1 P i ⊕ P i +2 h f i f ∗ i +1 i ( − i h f i f ∗ i +1 i ( − i f ∗ i f i +1 f ∗ i f i +1 Using the map s : X i +1 → X i +1 [1] defined by s k := ( − N − i Id P N − i if − k ∈ N + 2 N N , ( − i Id P i +1 if − k ∈ N + 2 N N , , we see that X i ◦ X ∗ i − X ∗ i +1 ◦ X i +1 is null-homotopic, which proves that x i ◦ x ∗ i − x ∗ i +1 ◦ x i +1 is zero in Hom Ho ( A ) ( P i +1 , P i +1 [2]). (cid:3) HE EXT-ALGEBRA OF THE BRAUER TREE ALGEBRA ASSOCIATED TO A LINE 7
The next proposition shows that the relations given in Lemma 3.1 are actu-ally enough to describe the
Ext -algebra. We use here the concatenation of pathsas opposed to the composition of arrows, which explains the discrepancy in therelations.
Proposition 3.2.
The
Ext -algebra of A is isomorphic to the path algebra associatedwith the following quiver S S S · · · S N − S N − S Nx y x ∗ x y x ∗ y y N − x N − x ∗ N − y N − x N − x ∗ N − y N with x i ’s of degree and y i ’s of degree N , subject to the relations (a) x x ∗ = x ∗ N − x N − = 0 ; (b) x ∗ i x i = x i +1 x ∗ i +1 for all i = 1 , . . . , N − ; (c) x i y i +1 = y i x ∗ N − i for all i = 1 , . . . , N − ; (d) x ∗ i y i = y i +1 x N − i for all i = 1 , . . . , N − .Proof. Let Q (resp. I ) be the quiver (resp. the set of relations) given in theproposition. Let Γ = F Q/ h I i be the corresponding path algebra. By Lemma 3.1,the Ext -algebra of A is a quotient of Γ. To show that A ≃ Γ it is enough to showthat the graded dimension of Γ is smaller than that of A .Let 1 ≤ i, j ≤ N and γ be a path between S i and S j in Q containing only x l ’s.Let k be the length of γ . We have k ≥ | i − j | , which is the length of the minimalpath from S i to S j . Using the relations, there exist loops γ and γ around S i and S j respectively such that γ = (cid:26) γ x i x i +1 · · · x j − = x i x i +1 · · · x j − γ if i ≤ j ; γ x ∗ i − x ∗ i − · · · x ∗ j = x ∗ i − x ∗ i − · · · x ∗ j γ otherwise . Maximal non-zero loops starting and ending at S i are either x ∗ i − x ∗ i − · · · x ∗ x x · · · x i − or x i x i +1 · · · x N − x ∗ N − · · · x ∗ i +1 · · · x ∗ i depending on whether S i is closer to S or S N . Indeed, any longer loop will involve x x ∗ or x ∗ N − x N − , which are zero by (a).Therefore if deg ( γ ) > i −
1) or deg ( γ ) > N − i ) then γ = 0. Using a similarargument for loops around S j we deduce that γ is zero whenever k = deg ( γ ) > | i − j | + 2 min ( i − , j − , N − j, N − j )which is equivalent to k = deg ( γ ) > N − − | N + 1 − j − i | . This proves that γ iszero unless | i − j | ≤ k ≤ N − − | N + 1 − j − i | in which case it equals to γ = x i x i +1 · · · x r − x ∗ r − x ∗ r − · · · x ∗ j where k = 2 r − i − j . OLIVIER DUDAS
Assume now that γ is any path between S i and S j in Q . Using the relationsone can write γ as γ = y ai γ γ where γ is a loop around S j containing only y l ’s, γ is a product of x l ’s and a ∈ { , } . Note that deg ( γ ) is a multiple of 2 N and γ is either a path from S i to S j if a = 0 or a path from S N +1 − i to S j if a = 1. From the previous discussion and Proposition 2.2 we conclude that γ iszero if dim F Ext kA ( S i , S j ) = 0 or unique modulo I otherwise. This shows that theprojection Γ ։ A must be an isomorphism. (cid:3) References [1]
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