Journal of Mathematical Sciences, Vol. 124, No. 1, 2004

COHOMOLOGY OF ALGEBRAS OF DIHEDRAL TYPE. III : THE FAMILY D(2A) A. I. Generalov and E. A. Osiyuk

UDC 512.5, 512.6

For algebras of dihedral type that form the family D(2A) (from K. Erdmann’s list), the Yoneda algebras are described in terms of quivers with relations. Bibliography: 8 titles.

Introduction In [1–3], the Yoneda algebras were determined for several families of algebras of dihedral type, namely, for algebras of the families D(3A)1 , D(3B)1 , D(3D)1 , and D(3K) (in the notation of [4]). An interest in these families is partially due to the fact that the blocks of group algebras of finite groups that have precisely three simple modules may occur only among the algebras of these four families (see [5]). As an application of pertinent calculations, a description of the cohomology ring of the groups PSL(2, q), where q is a power of an odd prime number, over an algebraically closed field of characteristic two was obtained (see also [6, 7]). In the present paper, we describe the Yoneda algebras for the algebras in the family D(2A). Contrary to the algebras studied in [1–3], the algebras of this family are not all special biserial. For this reason, we cannot immediately apply to these algebras the technique of constructing minimal projective resolutions that is based on the so-called diagrammatic method of Benson and Carlson. In this connection, using some previous empirical observations, we offer at once respective resolutions for simple modules over algebras in the family under consideration and prove their exactness with the help of spectral sequences. Next we obtain a description of the Yoneda algebras of those algebras in terms of quivers with relations, using the techniques developed in [1–3, 8]. Since we compute also the Ext-algebras of simple modules, these results can be immediately applied to a description of the cohomology ring of finite groups for which the principal block (over an algebraically closed field of characteristic two) is Morita equivalent to the algebras in the family D(2A) (see, e.g., tables in [4]). 1. Statement of the main results (A)

Let K be an arbitrary field. The algebras R(A) = Rn,c with n ∈ N and c ∈ {0, 1} that form the family D(2A) are defined by the following quiver Q(A) with relations (compositions are written from the right to the left):

0 βγ = 0,

1

α2 = c(γβα)n ,

(γβα)n = (αγβ)n .

(1)

As was proved in [4], R(A) is a symmetric basic K-algebra of tame representation type. For any module M over a finite-dimensional K-algebra Λ, the direct sum
finite-dimensional m E(M ) = Ext (M, M ) is an (associative) K-algebra with respect to the usual Yoneda product, and Λ m≥0 this algebra is called the Ext-algebra of the module M . If Λ is a basic K-algebra with Jacobson radical J(Λ), then the Ext-algebra E(Λ/J(Λ)) is called the Yoneda algebra of Λ; it will be denoted by Y(Λ). To determine the Yoneda algebra Y(R(A) ) of the algebra R(A) , we consider the quiver R(A) : z1 x3

x1 0

x2

1

z2

y1 Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 289, 2002, pp. 113–133. Original article submitted November 2, 2002. c 2004 Springer Science+Business Media, Inc. 1072-3374/04/1241-4741 

4741

On the path algebra K[R(A) ] of this quiver, we introduce a grading such that deg(xi) = 1 (i = 1, 2, 3),

deg(y1 ) = 2,

deg(zj ) = 3 (j = 1, 2).

(A)

Consider the following relations on the quiver R :   x2 x1 = x3 x2 = x1 x3 = x1 y1 = y1 x2 = 0, x1 z1 = −z2 x1 , x2 z2 = −z1 x2 ,  y1 x3 + x3 y1 = cx33 , 

z1 x3 = x3 z1 = 0,

(2)

y1 z1 = −z1 y1 ,

= −x3 z1 , z1 x3 − x3 z1 = y1 z1 + z1 y1 = −cx3 y12 . y12

c(x23 y1

(3)

− x3 y1 x3 ),

(4)

We define an algebra E = Ec with the aid of the quiver R(A) with relations (2) and (3) and also define an algebra E  = Ec with the aid of the quiver R(A) with relations (2) and (4). Clearly, the algebras Ec and Ec inherit the natural grading of the algebra K[R(A) ]. (A)

Theorem 1.1. Let R = Rn,c with n ∈ N and c ∈ {0, 1}. As a graded algebra, the Yoneda algebra Y(R) is isomorphic to the algebra Ec if n > 1 and to the algebra Ec if n = 1. Remark. If char K = 2, then, by [4], the algebras in the family under consideration can be defined by relations (1), in which it is sufficient to put only c = 0. Consequently, in studying the algebras for which we have c = 1, we may assume that char K = 2. However, we do not use the latter condition in the sequel, because this does not simplify the presentation. The remaining part of the paper is devoted to the proof of Theorem 1.1. 2. Resolutions (A)

Let R(A) = Rn,c with n ∈ N and c ∈ {0, 1} be the algebras defined in Sec. 1. We introduce abridged notation for the following elements of the algebra K[Q(A) ] (and also for its canonical images in the algebra R): g = (γβα)n−1 ,

a = (αγβ)n−1 ,

α ˜ = α − c(γβα)n−1 γβ

(if n = 1, then as g and a we take appropriate idempotents ei ). The modules P0 and P1 have the following K-bases: P0 =K e0 , β, γβ, αγβ, . . . , (αγβ)n = (γβα)n , βα(γβα)n−1 , . . . , γβα, βα, α, P1 =K e1 , γ, αγ, βαγ, γβαγ, . . . , (βαγ)n . In particular, dimK P0 = 6n and dimK P1 = 3n + 1. Moreover, it is clear that P1 is a uniserial module. We construct the following bicomplex B•• lying in the first quadrant of the plane (i.e., its rows and columns are enumerated by 0, 1, 2, . . . ): ... ... ...       β γβ  γβ  aα

aα

α

−aαγ

−aα

−α

−α ˜

aα

α

α ˜

α

−aαγ

−α

−α ˜

−α

−α ˜

α

α ˜

α

α ˜

α

P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .       γβ  γβ  gγβ  P1 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .         β γβ  gγβ  gγβ  P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .         γβ  gγβ  gγβ  gγβ  P1 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .           β gγβ  gγβ  gγβ  gγβ  P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . . . 4742

(5)

Proposition 2.1. Let n > 1. Then the totalization Tot(B•• ) of bicomplex (5) is the minimal projective resolution of the simple R-module S0 . Proof. We use the spectral sequence of the bicomplex 2 Epq = Hph Hqv (B•• ) ⇒ Hp+q (Tot(B•• )).

We claim that this spectral sequence degenerates at the second level; namely, we prove that its first level objects form exact sequences, except for the position (p, q) = (0, 0). For i > 0, from diagram (5) we obtain

1 Ep,2i

 0     Ker(γβ)/Imβ = Ker(γβ)/Im(γβ)     Ker(gγβ)/Im(γβ) Ker(gγβ)/Im(gγβ)

1 Ep,2i+1

 0     Ker β = Ker(γβ)/Im(γβ)     Ker(gγβ)/Im(γβ) Ker(gγβ)/Im(gγβ)

if if if if in

0 ≤ p < i, p = i, i < p < 2i, p = 2i, the other cases;

if if if if in

0 ≤ p < i, p = i, i < p < 2i + 1, p = 2i + 1, the other cases.

Direct calculations show that Ker(gγβ)/Im(gγβ) =β, γβ, αγβ, β(αγβ), . . . , β · a, βα · g, . . . , γβα, βα, Ker(gγβ)/Im(γβ) =β, βα · g, α · g, . . . , α(γβα), γβα, βα, Cokerβ =e0 , β, γβ, . . . , gγβ, βαg, . . . , βα, α, Coker(gγβ) =e0 , β, γβ, . . . , βa, βαg, . . . , βα, α, Ker(γβ)/Im(γβ) =β, βα · g, Ker(γβ)/Imβ =βα · g, Ker β =(βαγ)n  (here, the bar marks the canonical images of elements in respective quotient modules). Now, it is easily seen that 1 1 and Ep,2i+1 (i > 0) calculated above, together with the first level differentials of the spectral the objects Ep,2i sequence (these latter are induced by horizontal differentials of the bicomplex B•• ), form exact sequences. Now, since  if p = 0,  Ker β 1 Ep,1 = Ker(gγβ)/Im(γβ) if p = 1,  Ker(gγβ)/Im(gγβ) in the other cases, −aαγ it is clear that the homomorphism P1 ← −−−− P0 in the first row of the bicomplex B•• induces an isomorphism  1 1 . Finally, we have E0,1 ←−−−− E1,1

 1 Ep,0 =

Cokerβ if p = 0, Coker(gγβ) in the other cases,

α α 1 1 and, moreover, the homomorphism P0 ←−− with kernel ←−−−− E1,0 −− P0 induces a monomorphism E0,0 Cokerα S0 . Consequently, we have proved that Q• = Tot(B•• ) is a projective resolution of the module S0 . The minimality of this resolution follows from the fact that, by construction, we have Im(dQ m : Qm+1 → Qm ) ⊂ Rad Qm for any m ≥ 0.


4743

 In the case where n = 1, we consider the following bicomplex B•• :

...   β

...   γβ 

...   γβ 

α

α ˜

α

−αγ

−α ˜

−α

−α ˜

α ˜

α

α ˜

α

−αγ

−α

−α ˜

−α

−α ˜

α

α ˜

α

α ˜

α

P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .       γβ  γβ  γβ  P1 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .         β γβ  γβ  γβ 

(6)

P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .         γβ  γβ  γβ  γβ  P1 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . .           β γβ  γβ  γβ  γβ  P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− P0 ←−−−− . . . . Observe that for c = 0, this bicomplex coincides with the bicomplex B•• from (5), in which we take n = 1.  Proposition 2.2. Let n = 1. Then the totalization Tot(B•• ) of bicomplex (6) is the minimal projective resolution of the simple R-module S0 .

The proof of this proposition is similar to that of Proposition 2.1 and is left to the reader. (A)

Proposition 2.3. Let R = Rn,c with n ∈ N and c ∈ {0, 1}. The minimal projective resolution of the module S1 is of the form β

(βαγ)n

γ

β

γ

· · · −−−−→ P1 −−−−→ P0 −−−−→ P1 −−−−→ P1 −−−−→ P0 −−−−→ P1 → S1 → 0. The proof is a direct verification of the exactness of the above sequence; it is left to the reader. Propositions 2.1–2.3 imply the following statement. (A)

Corollary 2.4. Let R = Rn,c with n ∈ N and c ∈ {0, 1}. Then (a) m dimK Extm R (S0 , S1 ) = dimK ExtR (S1 , S0 ) =

(b)

 dimK Extm R (S1 , S1 )

=



1 if m ≡ 1 (mod 3), 0 otherwise;

0 if m ≡ 1 (mod 3), 1 otherwise;

(c) for any m ≥ 2 we have  dimK Extm R (S0 , S0 )

=

dimK Extm−2 (S0 , S0 ) + 2 if m ≡ 0 (mod 3), R m−2 dimK ExtR (S0 , S0 ) + 1 otherwise. (k)

Remark 2.5. The differentials in the minimal projective resolution of the module Sk will be denoted by dm , m ≥ 0. If B•• is a bicomplex of the form (5) (or a similar bicomplex), then we always arrange
the direct summands in the decomposition of the mth member of the minimal projective resolution Qm (Sk ) = i+j=m Bij for the simple module Sk in increasing order of the second index (cf. [2]). 4744

3. Generators In this section, we indicate (finite) sets of generators for the Yoneda algebras 1

Y(R) = E(R/J(R)) =

Extm R (Si , Sj )

m≥0 i,j=0 m of the algebras in the family D(2A) under consideration. The elements ψ ∈ Extm R (Si , Sj ) HomR (Ω (Si ), Sj ) of the extension groups are given by commutative diagrams of the form π

Qm (Si ) −−−m−→ Ωm (Si ) ⊂ Qm−1 (Si )    

f f ψ 0 1 Pj =Q0 (Sj ) −−−−→

Sj

ρj

⊂

(7)

Pj ,

where ρj is the projective cover. Such a diagram can be briefly represented as (i)

dm−1

Qm (Si ) −−−−→ Qm−1 (Si )   f f 1 0 Q0 (Sj ) −−−−→ ρj

(8)

Pj ,

because commutative square (8) determines the map ψ in (7) uniquely. The Ω-translates of the map ψ are also described by using similar commutative squares. On the other hand, up to isomorphism, diagram (7) can be (i) recovered uniquely, starting with a homomorphism f1 such that ρj · f1 annihilates Kerdm−1 . Therefore, this diagram can be briefly described as follows: f1 ψ = sq(Qm (Si ) −−−−→ Q0 (Sj )).

Let R = R(A) . We introduce several homogeneous elements: of degree 1: x1 ∈ Ext1R (S0 , S1 ), x2 ∈ Ext1R (S1 , S0 ), x3 ∈ Ext1R (S0 , S0 ); of degree 2: y1 ∈ Ext2R (S0 , S0 ); of degree 3: z1 ∈ Ext3R (S0 , S0 ), z2 ∈ Ext3R (S1 , S1 ); namely, the maps corresponding to them are briefly described by the following squares: (0,1)

x

2 = sq(Q1 (S1 ) −−−−→ P0 )

(1,0)

y 1 = sq(Q2 (S0 ) −−−−→ P0 )

(0,0,1)

z 2 = sq(Q3 (S1 ) −−−−→ P1 ).

x

1 = sq(Q1 (S0 ) −−−−→ P1 ), x

3 = sq(Q1 (S0 ) −−−−→ P0 ), z 1 = sq(Q3 (S0 ) −−−−→ P0 ),

1

(0,1) 1

In the statement below, we indicate basis elements (over K) of several groups Extm R (Si , Sj ) with m ≤ 6. Proposition 3.1. Viewed as K-vector spaces, the Ext-groups presented below have the following bases: Ext2R (S0 , S0 ) = x23 , y1 ,

Ext2R (S1 , S1 ) = x1 x2 ,

Ext3R (S0 , S0 ) = x33 , x3 y1 , z1 ,

Ext3R (S1 , S1 ) = z2 ,

Ext4R (S0 , S0 ) = x43 , y1 x23 , y12 ,

Ext4R (S0 , S1 ) = x1 z1 ,

Ext4R (S1 , S0 ) = x2 z2 ,

Ext5R (S1 , S1 ) = x1 x2 z2 , 4745

Ext5R (S0 , S0 ) = x53 , y1 x33 , x3 y12 , y1 z1 , Ext6R (S0 , S0 ) = x63 , x43 y1 , x23 y12 , x3 z1 y1 , z12 . Proof. In the cases where dimK Extm R (Si , Sj ) = 1, it is sufficient to observe that the above elements are nonzero, and this is verified directly. If dimK Extm R (Si , Sj ) > 1, then more detailed calculations are needed. We illustrate this with the proof of the statement concerning the group Ext2R (S0 , S0 ). The element x32 ∈ Ext2R (S0 , S0 ) is 1 represented by the homomorphism x
x3 ) : Ω2 (S0 ) −→ S0 . This composition is described by the 3 x3 = x3 ◦ Ω (

surrounded square of the following diagram: Q2 (S0 ) −−−−→ Q1 (S0 )    1 0 (1,0)  −cgγ gγ  Q1 (S0 ) −−−−→ Q0 (S0 )   gγβ (1,0)   Q0 (S0 ) −−−−→

P0 .

Comparing the maps x


1 , we see that they are linearly independent. It remains to observe that 3 x3 and y dimK Ext2R (S0 , S0 ) = 2. Verification in the remaining cases is left to the reader.
Proposition 3.2. Let R = R(A) . Any map f : Ωm (S0 ) −→ Sj (j = 0, 1) can be represented as a linear combination of compositions of the maps x

1 , x

2 , x

3 , y 1 , z 1 , and z 2 and their Ω-translates. Proof. If m ≤ 5, then the desired statement follows from Proposition 3.1 (and also from the definition of the above uniform elements of the algebra Y(R)); hence we may assume that m > 5. Case 1: j = 1. The periodicity of the module S1 implies that multiplication by z2 defines an isomorphism 3   m−3 Extm−3 (S0 , S1 ) −→ Extm (S0 ) −→ S1 , R (S0 , S1 ). Consequently, we have f = z 2 ·Ω (f ) for a certain map f : Ω R and we can apply induction on m. Case 2: j = 0. Put p = fπm , where πm : Qm (S0 ) −→ Ωm (S0 ) is a projective cover. Let Bi,m−i P0 be a direct summand in the canonical decomposition
of the module Qm (S0 ) (see Remark 2.5) such that p(Bi,m−i ) = S0 ; we may assume additionally that Ker p ⊃ l=i Bl,m−l . We should consider all possible positions of such a module P0 in the bicomplex B•• . (a) Let n > 1, and assume that we have the configuration P1   β aα

Bi,m−i =P0 ←−−−− P0 ; (0)

in other words, in the matrix representation of the differential dm : Qm+1 (S0 ) −→ Qm (S0 ) there is a row of the form (. . . , 0, aα, β) (and this is the last row). We note that now we have m ≡ 0 (mod 3) and also n > 1. Construct the following commutative diagram with exact rows: d(0)

π

m Qm+1 (S0 ) −−− −→ Qm (S0 ) −−−m−→ Ωm (S0 ) −−−−→ 0       ϕ1  ϕ0  ϕ (0)

Q4 (S0 ) 

where (0) d3

4746

α ˜ gγβ =  0 −˜ α 0 0

d

π

3 −−− −→ Q3 (S0 ) −−−3−→ Ω3 (S0 ) −−−−→ 0,

0 gγβ aα

 0 0  : Q4 (S0 ) = P03 ⊕ P1 −→ Q3 (S0 ) = P03 , β

(9)

ϕ1 is the projection onto the last four direct summands of the module Qm+1 (S0 ) in its canonical decomposition and ϕ0 is the compositionof an analogous projection of the module Qm (S0 ) onto the respective direct summands  0 0 0 (0) (0) and the homomorphism  0 a 0  : Q3 (S0 ) −→ Q3 (S0 ). Then we have ϕ0 dm = d3 ϕ1 , and therefore there 0 0 1 is a homomorphism ϕ that makes the entire diagram (9) commutative. Moreover, since Ker ϕ0 ⊂ Ker p, there (0) exists a homomorphism p : Q3 (S0 ) −→ S0 such that p ϕ0 = p. Because of Im d3 ⊂ Rad Q3 (S0 ) ⊂ Ker p , there is a homomorphism p : Ω3 (S0 ) −→ S0 , for which we have p π3 = p ; moreover, p is a multiple of z 1 with a scalar coefficient k ∈ K ∗ . Then we have p ϕπm = p = fπm , whence p ϕ = f. Since we have ϕ = Ω3 (ϕ) ˜ for a certain ϕ ˜ : Ωm−3 (S0 ) −→ S0 , we obtain the desired statement by induction. (b) Now we consider the case where the module Bi,m−i P0 is included in a configuration of the following form: Bi,m+1−i =P0   γβ  −α

Bi,m−i = P0 ←−−−− P0 = Bi+1,m−i   gγβ 

(10)

α

Bi+1,m−i−1 = P0 ←−−−− P0 = Bi+2,m−i−1 . Observe that m ≡ 2 (mod 4); moreover, if m = 2l, l ∈ N, and n > 1, we have i = m − i = l. Construct the following commutative diagram with exact rows: d(0)

π

m Qm+1 (S0 ) −−− −→ Qm (S0 ) −−−m−→ Ωm (S0 ) −−−−→ 0       ψ1  ψ0  ψ (0)

Q3 (S0 ) 

Here (0)

d2 =

α 0

d

(11)

π

2 −−− −→ Q2 (S0 ) −−−2−→ Ω2 (S0 ) −−−−→ 0.

gγβ −α

0 γβ

 : Q3 (S0 ) = P03 −→ Q2 (S0 ) = P02 ,

ψ0 is the projection onto the direct summands Bi+1,m−i−1 ⊕ Bi,m−i P0 ⊕ P0 and ψ1 is the projection onto the direct summands Bi+2,m−i−1 ⊕ Bi+1,m−i ⊕ Bi,m−i+1 P0 ⊕ P0 ⊕ P0 that are included in configuration (10). Now, as in case 2(a), we construct a homomorphism p : Ω2 (S0 ) −→ S0 such that p = p ψ and p = k · y 1 ˜ for a certain map ψ˜ : Ωm−2 (S0 ) −→ S0 , the proof can be completed again by with k ∈ K ∗ . Since ψ = Ω2 (ψ) induction on m. (c) Consider the configuration P0   γβ  (12) α

Bi,m−i =P0 ←−−−− P0 . Observe that m ≡ 0 (mod 4) in this case. Next we construct a diagram similar to (11), but as ψ0 we take the composition   of the projection onto the direct summands Bi+1,m−i−1 ⊕ Bi,m−i P0 ⊕ P0 and the homomorphism −1 0 : P02 −→ Q2 (S0 ) and as ψ1 we take the composition of the projection onto the direct summands 0 1   1 0 0 Bi+2,m−i−1 ⊕ Bi+1,m−i ⊕ Bi,m−i+1 P0 ⊕ P0 ⊕ P0 and the homomorphism  0 −1 0  : P03 −→ Q3 (S0 ). 0 0 1 Now, the proof is completed as in case 2(a). (d) Consider the configuration P0   gγβ  (13) α

Bi,m−i =P0 ←−−−− P0 . 4747

(i) We assume additionally that the homomorphisms occurring in configuration (13) are included in the (0) following submatrix of the matrix representation of the differential dm :   . . . α gγβ 0 ... . . . . 0 −α gγβ . . . In this case, we construct a diagram of the form (11): as ψ0 we take the projection onto Bi,m−i ⊕Bi−1,m−i+1 , and 3 as ψ1 we take the  composition  of the projection onto the direct summands Bi+1,m−i ⊕Bi,m−i+1 ⊕Bi−1,m−i+2 P0 1 0 0 and the map  0 1 0  : P03 −→ Q3 (S0 ). 0 0 g (ii) Now we assume that the homomorphisms occurring in configuration (13) are included in the following (0) submatrix of the matrix representation of the differential dm :   . . . α gγβ 0 . . . . . . . 0 −α γβ . . . Observe that if n > 1, then m ≡ 2 (mod 4) in this case. Now we construct a diagram of the form (11) in which ψ0 is the projection onto Bi,m−i ⊕ Bi−1,m−i+1 and ψ1 is the projection onto the direct summands Bi+1,m−i ⊕ Bi,m−i+1 ⊕ Bi−1,m−i+2 ; then the proof is completed as in case 2(a). (e) Consider the configuration P0   gγβ  (14) −α

Bi,m−i =P0 ←−−−− P0 . (i) Assume that the homomorphisms occurring in configuration (14) are included in the following submatrix (0) of the matrix of the differential dm :   . . . −α gγβ 0 ... . ... 0 α gγβ . . . We construct again a diagram of the form  (11): as  ψ0 we take the composition of the projection onto Bi,m−i ⊕ −1 0 : P02 −→ Q2 (S0 ) and as ψ1 we take the composition of the Bi−1,m−i+1 P02 and the homomorphism 0 1   1 0 0 projection onto the direct summands Bi+1,m−i ⊕ Bi,m−i+1 ⊕ Bi−1,m−i+2 P03 and the map  0 −1 0  : 0 0 g P03 −→ Q3 (S0 ). (ii) Assume that the homomorphisms occurring in configuration (14) are included in the following submatrix (0) of the matrix of the differential dm :   . . . −α gγβ 0 . . . . ... 0 α γβ . . . Observe that if n > 1, then m ≡ 0 (mod 4). We again construct a diagram of the form (11): as ψ  0 we take −1 0 the composition of the projection onto Bi,m−i ⊕ Bi−1,m−i+1 P02 and the homomorphism : P02 −→ 0 1 Q2 (S0 ) and as ψ1 we take the composition of   the projection onto the direct summands Bi+1,m−i ⊕ Bi,m−i+1 ⊕ 1 0 0 Bi−1,m−i+2 P03 and the map  0 −1 0  : P03 −→ Q3 (S0 ). 0 0 1 (f) Consider the configuration P0   gγβ  (15) ±α ˜

Bi,m−i =P0 ←−−−− P0 . 4748

(i) Assume that the homomorphisms occurring in configuration (15) are included in the following submatrix (0) that consists of the (m − i)th and (m − i + 1)th rows of the matrix of the differential dm : 

... ...

α ˜ gγβ 0 −˜ α

0 gγβ

... ...



(note that henceforth the rows of the matrix of a differential are enumerated by the numbers of the corresponding rows of the bicomplex B•• ). In this case, we construct a diagram similar to (11) in which the bottom row has the form (0)

d

π

1 Q2 (S0 ) −−− −→ Q1 (S0 ) −−−1−→ Ω1 (S0 ) −−−−→ 0 :



 1 0 and the map : as ψ0 we take the composition of the projection onto Bi,m−i ⊕ Bi−1,m−i+1 0 gγ 2 P0 −→ Q1 (S0 ) and as ψ1 we take the projection onto Bi+1,m−i ⊕ Bi,m−i+1 . Now, as in case 2 (a), we build a homomorphism p : Ω1 (S0 ) −→ S0 such that p = p ψ, and, moreover, we have p = k · x

3 with k ∈ K ∗ . Since ˜ for a certain map ψ˜ : Ωm−2 (S0 ) −→ S0 , the proof can be completed by induction on m. ψ = Ω1 (ψ) (ii) Assume that the homomorphisms occurring in configuration (15) are included in the following submatrix (0) of the matrix representation of the differential dm : P02



... ...

−˜ α gγβ 0 α ˜

0 gγβ

... ...



(again, these are the (m − i)th and (m − i + 1)th rows of it). In this case, we construct a diagram that is similar 2 to the diagram  considered  in case (i): ψ0 is the composition of the projection onto Bi,m−i ⊕ Bi−1,m−i+1 P0 −1 0 and the map : P02 −→ Q2 (S0 ) and ψ1 is the composition of the projection onto the direct summands 0 gγ   1 0 Bi+1,m−i ⊕ Bi,m−i+1 and the map : Q1 (S0 ) −→ Q1 (S0 ). 0 −1 (iii) Assume that the homomorphisms occurring in configuration (15) are included in the following submatrix (0) of the matrix representation of the differential dm : 

... ...

α ˜ 0

gγβ 0 −aα γβ

... ...

 .

Clearly, now we have n > 1. In this case, as ψ0 we take the composition of the projection onto Bi,m−i ⊕  1 0 Bi−1,m−i+1 P02 and the map : P02 −→ Q1 (S0 ) and as ψ1 we take the projection onto the direct 0 γ summands Bi+1,m−i ⊕ Bi,m−i+1 . (iv) Assume that the homomorphisms occurring in configuration (15) are included in the following submatrix (0) of the matrix representation of the differential dm : 

... ...

−˜ α 0

gγβ aα

0 γβ

... ...

 ;

again we  have n >1. In this case, ψ0 is the composition of the projection onto Bi,m−i ⊕ Bi−1,m−i+1 P02 and −1 0 the map : P02 −→ Q1 (S0 ) and ψ1 is the composition of the projection onto the direct summands 0 γ   1 0 Bi+1,m−i ⊕ Bi,m−i+1 and the map : Q1 (S0 ) −→ Q1 (S0 ). 0 −1 (g) Consider the configuration P0   γβ  (16) ±aα

Bi,m−i =P0 ←−−−− P0 ; 4749

(0)

in other words, in the matrix representation of the differential dm : Qm+1 (S0 ) −→ Qm (S0 ) there is a row of the form (. . . , 0, ±aα, γβ, . . . ). We may assume additionally that n > 1. (i) Assume that the homomorphisms occurring in configuration (16) are included in the following submatrix (0) that consists of the (m − i − 1)th and (m − i)th rows of the matrix of the differential dm and has the form 

... ...

∓aα γβ 0 ±aα

0 γβ

... ...



 or

... ...

∓α γβ 0 ±aα

0 γβ

... ...

 .

In this case, we construct a diagram similar  0 Bi+1,m−i−1 ⊕ Bi,m−i P02 and the map  ±a 0

to (9), in which ϕ0 is the composition of the projection onto 0 0  : P02 −→ Q3 (S0 ) and ϕ1 is the composition of the projection 1   0 0  0 0 onto the direct summands Bi+1,m−i ⊕ Bi,m+1−i P02 and the map   : P02 −→ Q4 (S0 ) P03 ⊕ P1 . ±1 0 0 γ Further the proof is completed as in case 2(a). (ii) Assume that a row of the form (. . . , 0, −aα, γβ, . . . ) is included in the following submatrix of the matrix (0) of dm :   . . . −˜ α gγβ 0 0 0 ... ... 0 α ˜ gγβ 0 0 ... . ... 0 0 −aα γβ 0 . . . Again we construct a diagram of the form (9): take the composition of the projection onto Bi+2,m−i−2 ⊕  as ϕ0 we  1 0 0 Bi+1,m−i−1 ⊕ Bi,m−i P03 and the map  0 −1 0  : P03 −→ Q3 (S0 ) and as ϕ1 we take the composition 0 0 1 of the projection onto the direct summands Bi+3,m−i−2 ⊕ Bi+2,m−i−1 ⊕ Bi+1,m−i ⊕ Bi,m−i+1 P04 and the   −1 0 0 0  0 1 0 0 homomorphism   : P04 −→ Q4 (S0 ). 0 0 −1 0 0 0 0 γ (iii) Assume that a row of the form (. . . , 0, aα, γβ, . . . ) is included in the following submatrix of the matrix of (0) dm :   ... α ˜ gγβ 0 0 0 ...  . . . 0 −˜ α gγβ 0 0 . . .  . ... 0 0 aα γβ 0 . . . In this case, ϕ0 is the projection onto Bi+2,m−i−2 ⊕ Bi+1,m−i−1 ⊕ Bi,m−i and ϕ1 is the composition of the projection onto the direct summands Bi+3,m−i−2 ⊕ Bi+2,m−i−1 ⊕ Bi+1,m−i ⊕ Bi,m−i+1 and the homomorphism   1 0 0 0 0 1 0 0   : P04 −→ Q4 (S0 ). 0 0 1 0 0 0 0 γ (h) If n = 1, we must consider two additional configurations. (i) If m ≡ 3 (mod 6), the following configuration may occur: P1   β α ˜

Bi,m−i =P0 ←−−−− P0 . In this case, we construct a diagram of the form (9), in which ϕ1 (respectively, ϕ0 ) is the projection onto the last four (respectively, three) direct summands of the module Qm+1 (S0 ) (respectively, Qm (S0 )) in its canonical decomposition. 4750

(ii) If m ≡ 0 (mod 6) and m > 6, then the following configuration may hold: P0   β α

Bi,m−i =P0 ←−−−− P0 . In this case, we construct a diagram, similar to (11), in which the bottom row has the form (0)

d

π

6 Q7 (S0 ) −−− −→ Q6 (S0 ) −−−6−→ Ω6 (S0 ) −−−−→ 0 :

as ψ1 (respectively, ψ0 ) we take the projection onto the last six (respectively, five) direct summands of the module Qm+1 (S0 ) (respectively, Qm (S0 )) in its canonical decomposition. Next we use the last relation in Proposition 3.1 to complete the proof of the desired statement. Corollary 3.3. The set χA = {x1 , x2 , x3 , y1 , z1 , z2 } generates the Yoneda algebra Y(R(A) ) as a K-algebra. 4. Relations (A)

Set R = Rn,c , where n ∈ N and c ∈ {0, 1}. Proposition 4.1. The generating set χA of the algebra Y(R) satisfies relations (2) and, additionally, (3) if n > 1 or (4) if n = 1. Proof. We use the same technique as in [2]. For this reason, we restrict our attention to only one relation in (2)–(4), namely, to x1 z1 = −z2 x1 in the case where n > 1. Verification of the remaining relations in (2) and (3) (or (4)) is left to the reader. Since x

1 · Ω1 (

z1 ) and z

2 · Ω3 (

x1 ), first we need to compute the 1 z1 = x 2 x1 = z required Ω-translates of the maps x

1 and z 1 . The results of these computations are presented in the following commutatives diagrams: (0)

(0)

d

d

(0)

(0)

d

d

3 2 1 0 Q4 (S0 ) −−− −→ Q3 (S0 ) −−− −→ Q2 (S0 ) −−− −→ Q1 (S0 ) −−− −→ Q0 (S0 )      (0,0,0,−1) (0,0,aα) (0,−aα) (0,1) aαγ     

Q3 (S1 ) −−−−→ Q2 (S1 ) −−−−→ Q1 (S1 ) −−−−→ Q0 (S1 ) −−−−→ (1)

d2

(1)

d0

(0)

d

d1 d

(βαγ)n

(1)

P1 ;

(0)

3 2 Q4 (S0 ) −−− −→ Q3 (S0 ) −−− −→ Q2 (S0 )     0 0 a 0 (0,0,1) (0,aα)  0 0 0 1  

Q1 (S0 ) −−−−→ Q0 (S0 ) −−−−→ (α,β)

(γβα)n

P0 .

z1 ) and z 2 · Ω3 (

x1 ) are represented by the surrounding squares of the commutative Thus, the maps x

1 · Ω1 (

diagrams (0)

(0)

d

0

0 0 0

3 Q4 (S0 ) −−− −→ Q3 (S0 )    (0,0,1) a 0   0 1

(α,β)

Q1 (S0 ) −−−−→ Q0 (S0 )    aαγ (0,1)  Q0 (S1 ) −−−−→ (βαγ)n

P1 ,

d

3 Q4 (S0 ) −−− −→ Q3 (S0 )   (0,0,0,−1) (0,0,−aαγ)  

(βαγ)n

Q3 (S1 ) −−−−→ Q2 (S1 )     1 1 Q0 (S1 ) −−−−→ (βαγ)n

P1 .

Now, comparing the maps in these surrounding squares, we see that x1 z1 = −z2 x1 (for n > 1).


4751

Let E = K[R(A) ]/I (respectively, E  = K[R(A)]/I  ) be a graded K-algebra; here, I (respectively, I  ) is the ideal of the path algebra of the quiver R(A) that corresponds to relations (2) and (3) (respectively, (2) and (4)). If n > 1, then, by Corollary 3.3 and Proposition 4.1, there exists a surjective homomorphism of graded K-algebras E −→ Y(R) that takes the canonical generators of the algebra E (i.e., those corresponding to the arrows in R(A) ) to respective generators of the algebra Y(R), which were introduced at the beginning of Sec. 3. If n = 1, a similar surjective homomorphism E  −→ Y(R) exists. In the sequel, we consider only the case where n > 1. For n = 1, similar arguments can be applied, and we leave this to the reader. Let E = ⊕m≥0 E m be the direct decomposition of the algebra E into homogeneous direct summands. Let εi , i = 0, 1, denote the idempotents of the algebra K[R(A) ] that correspond to the vertices of the quiver R(A) ; the same notation will be used for the images of these idempotents in E. Now, Theorem 1.1 is a consequence of the following statement. Proposition 4.2. For any i, j ∈ {0, 1} and any m ≥ 0, we have dimK (εi E m εj ) = dimK Extm R (Sj , Si ). Proof. Since the above relation is obvious for m = 0, 1, we assume that m > 1. (a) First we consider the case where i = j = 0. Put dm = dimK (ε0 E m ε0 ). We observe that the K-algebra ε0 Eε0 is generated by the elements x3 , y1 , and z1 . Since these elements satisfy the relations y1 z1 = −z1 y1 ,

z1 x3 = x3 z1 = 0,

y1 x3 + x3 y1 = cx33 ,

it follows easily that the monomials y1s z1t (2s + 3t = m) and xt3 y1s (t + 2s = m) constitute a K-basis of the vector space ε0 E m ε0 . We claim that  dm−2 + 2 if m ≡ 0 (mod 3), dm = (17) dm−2 + 1 otherwise. Indeed, to the monomials y1s z1t and xt3 y1s with s > 0 correspond bijectively elements of a similar K-basis of the vector space ε0 E m−2 ε0 (to see this, it suffices to omit one factor y1 ). Moreover, there is a monomial of the form z1t just in the case m = 3t for t ∈ N, and there is a monomial xm 3 for any m. Finally, note that, by Corollary 2.4(c), the sequence {dimK Extm R (S0 , S0 )}m≥1 satisfies a recurrence relation similar to (17). (b) The case i = j = 1. Denote dm = dimK (ε1 E m ε1 ). The K-algebra ε1 Eε1 is generated by the elements y = x1 x2 and z2 . Since these elements satisfy the relations y2 = 0,

yz2 = z2 y,

the monomials and with 2 + 3t = m (t ≥ 0) and 3s = m (s > 0) constitute a K-basis of the vector space ε1 E m ε1 . Clearly, we have  0 if m ≡ 1 (mod 3), dm = 1 otherwise. yz2t

z2s

By Corollary 2.4(b), we obtain dimK (ε1 E m ε1 ) = dimK Extm R (S1 , S1 ). (c) The case i = 1 and j = 0. Put dm = dimK (ε1 E m ε0 ). Using relations (2) and (3), we easily see that the monomials x1 z1t (t > 0) with 1 + 3t = m form a K-basis of the vector space ε1 E m ε0 . Consequently, we have  1 if m ≡ 1 (mod 3), dm = 0 otherwise, and then, by Corollary 2.4(a), we conclude that dimK (ε1 E m ε0 ) = dimK Extm R (S0 , S1 ). (d) The case i = 0 and j = 1. Put dm = dimK (ε0 E m ε1 ). As in the previous cases, it is easily seen that the monomials x2 z2t (t > 0) with 1 + 3t = m form a K-basis of the vector space ε0 E m ε1 ; consequently, we have  1 if m ≡ 1 (mod 3), dm = 0 otherwise, whence, by Corollary 2.4(a), we obtain dimK (ε0 E m ε1 ) = dimK Extm R (S1 , S0 ).




The proof of Theorem 1.1 implies also a description of the Ext-algebras of simple R(A) -modules. To this end, we consider the K-algebras Hc = Kx, y, z/(yz + zy, zx, xz, yx + xy − cx3 ), Hc = Kx, y, z/(yx + xy − cx3 , xz + y2 , zx + y2 − c(x2 y − xyx), yz + zy + cxy2 ), where deg x = 1, deg y = 2, and deg z = 3. 4752

(A)

Corollary 4.3. Let  R = Rn,c . Then we have Hc if n > 1, (a) E(S0 ) Hc if n = 1. (b)

E(S1 ) K[y, z]/(y2 ), where deg y = 2 and deg z = 3.

Proof. If n > 1, the above statements follow from the proof of Proposition 4.2. If n = 1, the proof is similar and is left to the reader. Translated by A. I. Generalov. REFERENCES 1. O. I. Balashov and A. I. Generalov, “Yoneda algebras of a class of dihedral algebras,” Vestn. Peterburg. Univ., Ser. 1, 3, No. 15, 3–10 (1999). 2. A. I. Generalov, “Cohomology of algebras of dihedral type. I,” Zap. Nauchn. Semin. POMI, 265, 139–162 (1999). 3. O. I. Balashov and A. I. Generalov, “Cohomology of algebras of dihedral type. II,” Algebra Analiz, 13, No. 1, 3–25 (2001). 4. K. Erdmann, “Blocks of tame representation type and related algebras,” Lect. Notes Math., 1428 (1990). 5. K. Erdmann, “Algebras and dihedral defect groups,” Proc. London Math. Soc., 54, No. 1, 88–114 (1987). 6. J. Martino and S. Priddy, “Classification of BG for groups with dihedral or quaternion Sylow 2-subgroups,” J. Pure Appl. Algebra, 73, 13–21 (1991). 7. T. Asai and H. Sasaki, “The mod 2 cohomology algebras of finite groups with dihedral Sylow 2-subgroups,” Commun. Algebra, 21, 2771–2790 (1993). 8. A. I. Generalov, “Cohomology of algebras of semidihedral type. I,” Algebra Analiz, 13, No. 4, 54–85 (2001).

4753