THE EXT-ALGEBRA OF THE BRAUER TREE ALGEBRA ASSOCIATED TO A LINE

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OLIVIER DUDAS

Abstract. We compute theExt-algebra of the Brauer tree algebra associated to a line with no exceptional vertex.

Introduction

This note provides a detailed computation of theExt-algebra for a very specific finite dimensional algebra, namely a Brauer tree algebra associated to a line, with no exceptional vertex. Such algebras appear for example as the principal p-block of the symmetric group S_p, and in a different context, as blocks of the Verlinde categoriesVer_p² studied by Benson–Etingof in [1].

Let us emphasise that Ext-algebras for more general biserial algebras were explicitly computed by Green–Schroll-Snashall–Taillefer in [3], but under some assumption on the multiplicity of the the vertices, assumption which is not satis- fied for the simple example treated in this note. Other general results relying on Auslander–Reiten theory can be found in the work of Brown [2]. However we did not manage to use his work to get an explicit description in our case. Nevertheless, the simple structure of the projective indecomposable modules for the line allows a straightforward approach using explicit projective resolution of simple modules.

The Poincar´e series for theExt-algebra is given in Proposition2.2and its structure as a path algebra with relations is given in Proposition3.2.

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, we denote by Ω(M) the kernel of a projective cover P ։ M. Up to isomorphism it does not depend on the cover. We then define inductively Ωⁿ(M) = Ω(Ωⁿ⁻¹(M)) forn≥1.

To compute the extension groups between simple modules we will use the prop- erty that if Ωⁿ(M) is indecomposable and non-projective then

Extⁿ_A(M, S)≃Hom_A(Ωⁿ(M), S) for all simpleA-moduleS and alln≥1.

For computing the algebra structure on the variousExt-groups it will be conve- nient to work in the homotopy categoryHo(A) of the complexes of finitely generated

The author gratefully acknowledges financial support by the ANR, Project No ANR-16-CE40- 0010-01.

1

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A-modules. IfS (resp. S^′) is a simple A-module, andP•→S (resp. P_•^′ →S^′) is a projective resolution then

Extⁿ_A(S, S^′)≃HomHo(A)(P•, P_•^′[n])

with the Yoneda product being given by the composition of maps inHo(A).

Assume now thatAis theF-algebra associated to the following Brauer tree with N+ 1 vertices:

S1 S2 SN

Here, unlike in [3] we assume that there are no exceptional vertex. The edges are labelled by the simple A-modules S1, . . . , SN. The head and socle of Pi are isomorphic to Si and rad(Pi)/Si ≃ Si−1⊕Si+1 with the convention that S0 = SN+1= 0.

2. Ext-groups

Given 1 ≤i ≤j ≤ N with i−j even, there is, up to isomorphism, a unique non-projective indecomposable moduleⁱX^j such that

• rad(ⁱX^j) =Si+1⊕Si+3⊕ · · · ⊕Sj−1

• hd(ⁱX^j) =Si⊕Si+2⊕ · · · ⊕Sj.

The structure ofⁱX^j can be represented by the following diagram:

Si Si+2 Si+4 · · · Sj−2 Sj

iX^j= · · ·

Si+1 Si+3 · · · Sj−1

Similarly we denote byiX_j the unique indecomposable module with the following structure:

Si+1 Si+3 · · · Sj−1

iX_j= · · ·

Si Si+2 Si+4 · · · Sj−2 Sj

Finally, in the case where i−j is odd we define the modules iX^j and ⁱX_j as the indecomposable modules with the following respective structure:

Si+1 Si+3 · · · Sj

iX^j= · · ·

Si Si+2 Si+4 · · · Sj−1

iX_j= · · ·

Si+1 Si+3 · · · Sj

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For convenience we will extend the notationⁱX^j, iX_j, iX^j and ⁱX_j to any integers i, j∈Z(with the suitable parity condition oni−j) so that the following relations hold:

(1) ⁱX=1−iX, ⁱX^j =jX_i, ^i±2NX=ⁱX.

Note that this also impliesX^j =X_1−j andX^j±2N =X^j. Lemma 2.1. Let i, j∈Zwith i−j even. Then

Ω(ⁱX^j)≃ⁱ⁻¹X^j+1.

Proof. Using the relations (1) it is enough to prove that for 1≤k≤l≤N we have the following isomorphisms

Ω(^kX^l)≃^k−1X^l+1, Ω(kX^l)≃k+1X^l+1, Ω(^kX_l)≃^k−1X_l−1, Ω(kX_l)≃k+1X_l−1. We only consider the first one, the others are similar. If 1≤k≤l≤N a projective cover of^kX^l is given byPk⊕Pk+2⊕ · · · ⊕Pl։^kX^l, whose kernel equals^k−1X^l+1. Note that this holds even when k = 1 since ⁰X^l+1 =1X^l+1 or when l = N since

k−1X^N⁺¹=^k−1X^−N+1 =^k−1X_N.

We deduce from Lemma2.1that for any simple moduleSi and for allk≥0 we have

Ω^k(Si) = Ω^k(ⁱXⁱ)≃^i−kX^i+k asA-modules. Consequently we have

Ext^k_A(Si, Sj) =

F ifSj appears in the head of^i−kX^i+k, 0 otherwise.

From this description one can compute explicitly the Poincar´e series of the Ext- groups.

Proposition 2.2. Given1≤i, j≤N, the Poincar´e series ofExt^•_A(Si, Sj)is given by

X

k≥0

dim_FExt^k_A(Si, Sj)t^k =Qi,j(t) +t^2N−1Qi,j(t⁻¹) 1−t^2N

whereQi,j(t) =t^|j−i|+t^|j−i|+2+· · ·+t^N−1−|N^+1−j−i|.

Proof. Without loss of generality we can assume thati≤j. Letk∈ {0, . . . , N−1}.

Ifi+j ≤N+ 1, the simple moduleSj appears in the head of î−kXî+k if and only ifk=j−i, j−i+ 2, . . . , j+i−2. The limit cases are indeed^2i−jX^j fork=j−i and^2−jX^2i+j−2=j−1X^2i+j−2 fork=j+i−2. Note that ifj−i≤k≤i+j−2 then j ≤i+k and j ≤2N−i−k so thatSj appears in the head of î−kXî+k =

i−kX2N−i−k+1wheneverkhas the suitable parity. Ifi+j > N+ 1 one must ensure thatj≤2N−i−k and thereforeSj appears in the head of^i−kX^i+k if and only if k=j−i, j−i+ 2, . . . ,2N−i−j. Consequently we have

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N−1

X

k=0

dim_FExt^k_A(Si, Sj)t^k =t^j−i+t^j−i+2+· · ·+tN−1−|N+1−j−i|

=t^|j−i|+t^|j−i|+2+· · ·+t^N−1−|N+1−j−i|

=Qi,j(t).

Now the relation

Ω^N(Si) =^i−NX^i+N =1+N−iX1−N−i=1+N−iX1+N−i=SN+1−i

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yields

2N−1

X

k=0

dim_FExt^k_A(Si, Sj)t^k =

N−1

X

k=0

dim_FExt^k_A(Si, Sj)t^k+t^N

N−1

X

k=0

dim_FExt^k_A(SN+1−i, Sj)t^k. and the proposition follows from (2) after observing thatQN+1−i,j(t) =t^N−1Qi,j(t⁻¹).

3. Algebra structure

3.1. Minimal resolution. Given 1≤i≤N−1 we fix non-zero mapsfi :Pi−→

Pi+1andf_i^∗:Pi+1 −→Pi such thatf_i^∗◦fi+fi−1◦f_i−1^∗ = 0 for all 2≤i≤N−1.

Given 1 ≤i ≤ j ≤N with j−i even we denote by iPj the following projective A-module

iPj:=Pi⊕Pi+2⊕ · · · ⊕Pj−2⊕Pj.

For 1≤i < j≤N withj−ieven we let di,j :iPj −→i+1Pj−1 be the morphism ofA-modules corresponding to the following matrix:

di,j =







fi f_i+1^∗ 0 · · · 0 0 fi+2 f_i+3^∗ 0 ... ... . .. ... ... ... ... ... . .. ... ... 0 0 · · · 0 fj−2 f_j−1^∗







The definition ofiPj extends to any integersi, j∈Zwith the convention that (3) iPj=j+1Pi−1, iP−j=iPj, iPj±2N =iPj.

Note that these relations imply 1−iPj = 1+iPj and i±2NPj =iPj. Furthermore, the definition ofdi,j extends naturally to any pairi, j if we set in addition

di,i= (−1)ⁱf_i^∗fi= (−1)ⁱ⁻¹fi−1f_i−1^∗ ,

a map from iPi =Pi to i+1Pi−1 =Pi. With this notation one checks that for all k > 0 the image of the map di−k,i+k : i−kPi+k −→ i−k+1Pi+k−1 is isomorphic to

i−kX^i+k ≃Ω^k(Si) so that the bounded above complex Ri:=· · ·−−−−−−−→^dⁱ⁻^k⁻^1,i+^k⁺¹ i−kPi+k

di−k,i+k

−−−−→ · · ·−−−−→^dⁱ⁻^2,i+2 i−1Pi+1 di−1,i+1

−−−−→Pi−→0 forms a minimal projective resolution ofSi.

3.2. Generators and relations. We will consider two kinds of generators for the Ext-algebra, of respective degrees 1 andN. We start by defining a mapxi ∈ HomHo(A)(Ri, Ri+1[1]) for any 1 ≤ i ≤ N −1. Let k be a positive integer. If k /∈ NZ, the projective modules i−kPi+k and i+1−(k−1)Pi+1+(k−1) = i−k+2Pi+k have at least one indecomposable summand in common and we can consider the map Xi,k : i−kPi+k −→ i−k+2Pi+k given by the identity map on the common factors. If k∈N+ 2NZthen from the relations (3) we have

i−kPi+k=i−NPi+N =i+N+1Pi−N−1=−i−N+1P−i+N+1=PN+1−i

and

i−k+2Pi+k =i−N+2Pi+N =N−iP−N−i=PN−i.

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In that case we setXi,k:= (−1)^N−if_N^∗_−i. Ifk∈2NZtheni−kPi+k=Pi,i−k+2Pi+k=

i+2Pi=Pi+1and we setXi,k := (−1)ⁱfi. Ifk≥0 we setXi,k:= 0. Then the family of morphisms of A-modules Xi := (Xi,k)k∈Z defines a morphism of complexes of A-modules fromRi to Ri+1[1] and we denote byxi its image in Ho(A).

Similarly we define a mapX_i^∗:Ri+1−→Ri[1] by exchanging the role off and f^∗. More precisely we consider in that caseX_i,−N^∗ := (−1)^N⁻ⁱfN−i andX_i,−2N^∗ :=

(−1)ⁱf_i^∗. We denote byx^∗_i the image ofX_i^∗ inHo(A).

Assume now that 1≤i≤N. The modulesi−kPi+kand(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 complexes Ri and RN+1−i[N] coincide. We denote by Yi : Ri −→ RN+1−i[N] the natural projection betweenRiand its obvious truncation at degrees≤ −N, and by yi its image in Ho(A).

Lemma 3.1. The following relations hold inEnd^•Ho(A)(L Ri):

(a) x^∗₁◦x1=xN−1◦x^∗_N−1= 0;

(b) xi◦x^∗_i =x^∗_i+1◦xi+1 for alli= 1, . . . , N−2;

(c) yi+1◦xi=x^∗_N_−i◦yi for all i= 1, . . . , N−1;

(d) yi◦x^∗_i =xN−i◦yi+1 for alli= 1, . . . , N−1.

Proof. IfN = 1 there are no relation to check. Therefore we assumeN ≥2. The relations in (a) follow from the fact thatExt²_A(S1, S1) =Ext²_A(SN, SN) = 0, which is for example a consequence of Proposition2.2whenN ≥2.

To show (c), we observe that the morphism of complexes Xi : Ri −→ Ri+1[1]

defined above coincide with X_N−i^∗ [N] : RN+1−i[N] −→ RN−i[N + 1] in degrees less than −N. Since Yi and Yi+1 are just obvious truncations we actually have Yi+1◦Xi=X_N−i^∗ ◦Yi. The relation (d) is obtained by a similar argument.

We now consider (b). The morphism of complexes Xi◦X_i^∗ and X_i+1^∗ ◦Xi+1

coincide at every degreekexcept whenkis congruent to 0 or−1 moduloN. Let us first look in details at the degrees−N and−N−1. The mapXi◦X_i^∗is as follows:

· · · PN−1−i⊕PN+1−i PN−i PN−i

PN−i⊕PN+2−i PN+1−i PN+1−i PN−i⊕PN+2−i

PN−i PN−i PN−1−i⊕PN+1−i · · ·

h

fN−1−i f_N−^∗ _i i

h 0 1i

(−1)^N−ⁱf_N−^∗ _i◦fN−i

(−1)^N−ⁱfN−i



 1 0



 h

fN−i f_N+1^∗ ₋_i i

h 1 0i

(−1)^N−ⁱfN−i◦f_N−^∗ _i

(−1)^N−ⁱf_N−^∗ _i



 f_N−i^∗ fN+1−i







 0 1





(−1)^N−ⁱf_N−i^∗ ◦fN−i



 f_N−^∗ 1−i

fN−i





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whereas the mapX_i+1^∗ ◦Xi+1 corresponds to the following composition:

· · · PN−1−i⊕PN+1−i PN−i PN−i

PN−2−i⊕PN−i PN−1−i PN−1−i PN−2−i⊕PN−i

PN−i PN−i PN−1−i⊕PN+1−i · · ·

h

fN−1−i f_N−^∗ _ii

h 1 0i

(−1)^N−if_N−^∗ _i◦fN−i

(−1)^N−¹⁻ⁱf_N−^∗ ₁_−i



 0 1



 h

fN−2−i f_N−^∗ 1−i

i

h 0 1i

(−1)^N−¹⁻ⁱf_N−^∗ ₁₋_i◦fN−1−i

(−1)^N−¹⁻ⁱfN−1−i



 f_N−^∗ ₂_−i fN−1−i







 1 0





(−1)^N−ⁱf_N−^∗ _i◦fN−i



 f_N−^∗ 1−i

fN−i





We deduce that at the degrees−N and−N−1 the mapXi◦X_i^∗−X_i+1^∗ ◦Xi+1 is given by

PN−1−i⊕PN+1−i PN−i

PN−i PN−1−i⊕PN+1−i h

(−1)^N−ⁱh

(−1)^N−i



 f_N−^∗ ₁₋_i

fN−i







 f_N−^∗ ₁₋_i

fN−i





A similar picture holds at the degrees−2N and−2N−1:

Pi⊕Pi+2 Pi+1

Pi+1 Pi⊕Pi+2 h

fi f_i^∗+1

i

(−1)ⁱh fi f_i^∗+1

i (−1)ⁱ



 f_i^∗ fi+1







 f_i^∗ fi+1





Using the maps:Xi+1→Xi+1[1] defined by sk :=







(−1)^N⁻ⁱIdPN−i if−k∈N+ 2NN, (−1)ⁱIdPi+1 if−k∈2N+ 2NN,

0 otherwise,

we see thatXi◦X_i^∗−X_i+1^∗ ◦Xi+1 is null-homotopic, which proves thatxi◦x^∗_i − x^∗_i+1◦xi+1 is zero inHomHo(A)(Pi+1, Pi+1[2]).

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The next proposition shows that the relations given in Lemma 3.1 are actually enough to describe the Ext-algebra. We use here the concatenation of paths as opposed to the composition of arrows, which explains the discrepancy in the relations.

Proposition 3.2. TheExt-algebra ofAis isomorphic to the path algebra associated with the following quiver

S1 S2 S3 · · · SN−2 SN−1 SN x1

y1

x^∗1

x2

y2

x^∗2

y3

yN−2

xN−2

x^∗_N−2

yN−1

xN−1

x^∗_N−1

yN

withxi’s of degree1 andyi’s of degreeN, subject to the relations (a) x1x^∗₁=x^∗_N₋₁xN−1= 0;

(b) x^∗_ixi=xi+1x^∗_i+1 for alli= 1, . . . , N−2;

(c) xiyi+1=yix^∗_N−i for alli= 1, . . . , N−1;

(d) x^∗_iyi=yi+1xN−i for alli= 1, . . . , N−1.

Proof. Let Q (resp. I) be the quiver (resp. the set of relations) given in the proposition. Let Γ =FQ/hIibe the corresponding path algebra. By Lemma 3.1, theExt-algebra ofA is a quotient of Γ. To show that A≃Γ it is enough to show that the graded dimension of Γ is smaller than that ofA.

Let 1≤i, j≤N and γbe a path between Si andSj in Qcontaining onlyxl’s.

Letk be the length ofγ. We havek≥ |i−j|, which is the length of the minimal path fromSi toSj. Using the relations, there exist loopsγ1andγ2 aroundSi and Sj respectively such that

γ=

γ1xixi+1· · ·xj−1=xixi+1· · ·xj−1γ2 ifi≤j;

γ1x^∗_i−1x^∗_i−2· · ·x^∗_j =x^∗_i−1x^∗_i−2· · ·x^∗_jγ2 otherwise.

Maximal non-zero loops starting and ending atSiare eitherx^∗_i−1x^∗_i−2· · ·x^∗₁x1x2· · ·xi−1

or xixi+1· · ·xN−1x^∗_N₋₁· · ·x^∗_i+1· · ·x^∗_i depending on whether Si is closer to S1 or SN. Indeed, any longer loop will involvex1x^∗₁orx^∗_N₋₁xN−1, which are zero by (a).

Therefore ifdeg(γ1)>2(i−1) ordeg(γ1)>2(N−i) thenγ1= 0. Using a similar argument for loops aroundSj we deduce thatγis zero whenever

k=deg(γ)≥ |i−j|+ 2min(i−1, j−1, N−j, N−j)

which is equivalent tok=deg(γ)> N−1− |N+ 1−j−i|. This proves thatγ is zero unless|i−j| ≤k≤N−1− |N+ 1−j−i|in which case it equals to

γ=xixi+1· · ·xr−1x^∗_r−1x^∗_r−2· · ·x^∗_j wherek= 2r−i−j.

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Assume now that γ is any path between Si and Sj in Q. Using the relations one can write γ as γ=y_i^aγ1γ2 where γ2 is a loop aroundSj containing only yl’s, γ1 is a product of xl’s and a ∈ {0,1}. Note that deg(γ2) is a multiple of 2N and γ1 is either a path from Si to Sj if a = 0 or a path from SN+1−i to Sj if a = 1. From the previous discussion and Proposition 2.2 we conclude that γ is zero if dim_FExt^k_A(Si, Sj) = 0 or unique moduloI otherwise. This shows that the

projection Γ։Amust be an isomorphism.

References

[1] D. Benson, P. Etingof. On cohomology in symmetric tensor categories in prime character- istic. PreprintarXiv:2008.13149, 2020.

[2] P. Brown. The Ext-algebra of a representation-finite biserial algebra.J. Algebra221(1999), 611–629.

[3] E. Green, S. Schroll, N. Snashall, R. Taillefer. The Ext algebra of a Brauer graph algebra.J. Noncommut. Geom.11(2017), 537–579.

Universit´e de Paris and Sorbonne Universit´e, CNRS, IMJ-PRG, F-75006 Paris, France.

E-mail address: olivier.dudas@imj-prg.fr