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Universal deformation rings for the symmetric group S_4

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is isomorphic to either k, or W[t]/(t^2,2t), or the group ring W[Z/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.


Introduction
Let k be an algebraically closed field of characteristic p > 0 and let W = W (k) be the ring of infinite Witt vectors over k. Let G be a finite group, and suppose V is a finitely generated kG-module. If the stable endomorphism ring End kG (V ) is one-dimensional over k, it was shown in [6] that V has a universal deformation ring R(G, V ). The ring R(G, V ) is universal with respect to deformations of V over complete local commutative Noetherian rings with residue field k (see §2). In [6,3,4,5], the isomorphism types of R(G, V ) have been determined for V belonging to cyclic blocks, respectively to various tame blocks with dihedral defect groups with one or three isomorphism classes of simple modules. In the present paper, we will consider the case when V belongs to a particular tame block with two isomorphism classes of simple modules. The key tools used to determine the universal deformation rings in all these cases have been results from modular and ordinary representation theory due to Brauer, Erdmann [14], Linckelmann [16,17], Carlson-Thévenaz [11], and others. The main motivation for studying universal deformation rings for finite groups is that this case helps understand ring theoretic properties of universal deformation rings for profinite groups Γ. The latter have become an important tool in number theory, in particular if Γ is a profinite Galois group (see e.g. [12], [21,20], [9] and their references). In [13], de Smit and Lenstra showed that if Γ is an arbitrary profinite group and V is a finite dimensional vector space over k with a continuous Γ-action which has a universal deformation ring R(Γ, V ), then R(Γ, V ) is the inverse limit of the universal deformation rings R(G, V ) when G runs over all finite discrete quotients of Γ through which the Γ-action on V factors. Thus to answer questions about the ring structure of R(Γ, V ), it is natural to first consider the case when Γ = G is finite.
Suppose now that k has characteristic 2 and that S 4 denotes the symmetric group on 4 letters. In the present paper, we consider the group ring kS 4 which is its own (principal) block and has two isomorphism classes of simple modules. These are represented by the trivial simple module T 0 and a 2-dimensional simple module T 1 which is inflated from the symmetric group S 3 . Since the Sylow 2-subgroups of S 4 are dihedral groups of order 8, kS 4 is of tame representation type. It was shown in [7,8] that R(S 4 , T 1 ) ∼ = W [t]/(t 2 , 2t) which provides an example of a universal deformation ring which is not a complete intersection. The results in [7,8] additionally imply that for every finitely generated kS 4 -module V with stable endomorphism ring k, the universal deformation ring R(S 4 , V ) arises from arithmetic in the following sense. By [8,Lemma 3.3 and Thm. 3.7], there are infinitely many real quadratic fields L such that the Galois group G L,∅ of the maximal totally unramified extension of L surjects onto S 4 and R(G L,∅ , V ) ∼ = R(S 4 , V ), where V is viewed as a module for G L,∅ via inflation. In particular, our main result, Theorem 1.1, determines all the universal deformation rings R(G L,∅ , V ).
Our main result is as follows, were Ω denotes the syzygy, or Heller, operator (see for example [1, §20]). Theorem 1.1. Let V be a finitely generated indecomposable kS 4 -module with stable endomorphism ring k, and let C be the component of the stable Auslander-Reiten quiver of kS 4 containing V .
i. Suppose C or Ω(C) contains T 0 . Then C is of type ZA ∞ ∞ , and all modules in C ∪ Ω(C) have stable endomorphism ring equal to k. In this case, ii. Suppose C contains T 1 . Then C is of type ZA ∞ ∞ , and C = Ω(C). In this case, V is isomorphic to Ω j (T 1 ) for some integer j, and R( iii. Suppose C contains a uniserial module Y of length 3 with non-isomorphic top and socle. Then C is a 3-tube with Y belonging to its boundary, and C = Ω(C). In this case, V is isomorphic to Y , Ω 2 (Y ) or Ω 4 (Y ), and R(S 4 , V ) is isomorphic to k. The only components of the stable Auslander-Reiten quiver of kS 4 containing modules with stable endomorphism ring k are the ones in (i) − (iii).
Note that in parts (i) and (iii), R(S 4 , V ) is a complete intersection, whereas in part (ii) this is not the case. In all three parts, R(S 4 , V ) is isomorphic to a subquotient ring of W D 8 when D 8 is a dihedral group of order 8. In particular, this gives a positive answer in case of the unique 2-modular block of S 4 to a question raised by the first author and Chinburg in [6, Question 1.1] whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.
The paper is organized as follows. In §2, we give some background on universal deformation rings. In §3, we state the properties of kS 4 we need to prove Theorem 1.1. In particular, we provide the necessary results concerning the indecomposable modules and the stable Auslander-Reiten quiver of kS 4 . In §4- §7, we prove Theorem 1.1.
Part of this paper constitutes the Ph.D. thesis of the second author under the supervision of the first author [18].

Preliminaries
Let k be an algebraically closed field of characteristic p > 0, let W be the ring of infinite Witt vectors over k and let F be the fraction field of W . Let C be the category of all complete local commutative Noetherian rings with residue field k. The morphisms in C are continuous W -algebra homomorphisms which induce the identity map on k.
Suppose G is a finite group and V is a finitely generated kG-module. A lift of V over an object R in C is a finitely generated RG-module M which is free over R together with a kG-module The isomorphism class of a lift of V over R is called a deformation of V over R, and the set of such deformations is denoted by Def G (V, R). The deformation functor F V : C → Sets is defined to be the covariant functor which sends an object R in C to Def G (V, R).
Suppose there exists an object R(G, V ) in C and a lift (U (G, V ), φ U ) of V over R(G, V ) satisfying the following. For each R in C and for each lift (M, φ) of V over R there is a unique morphism α : . Then R(G, V ) is called the universal deformation ring of V and the isomorphism class of the lift (U (G, V ), φ U ) is called the universal deformation of V . In other words, R(G, V ) represents the functor F V in the sense that F V is naturally isomorphic to Hom C (R(G, V ), −). For more information on deformation rings see [13] and [19].
The following four results were proved in [6] and in [4], respectively. As before, Ω denotes the syzygy, or Heller, operator for kG (see for example [1, §20] . Let V be a finitely generated kG-module with stable endomorphism ring End kG (V ) = k.
ii. There is a non-projective indecomposable kG-module V 0 (unique up to isomorphism) such that End kG (V 0 ) = k, V is isomorphic to V 0 ⊕ P for some projective kG-module P , and R(G, V ) and R(G, V 0 ) are isomorphic.
. Let L be a subgroup of G and let U be a finitely generated indecomposable kL-module with End kL (U ) = k. Suppose there exists an indecomposable kG-module V with End kG (V ) = k and a projective kG-module P such that Ind G L U = V ⊕ P . Assume further that . Let V be a finitely generated kG-module such that there is a non-split short exact sequence of kG-modules Then there exists a W G-module X which defines a lift of V over W .

The group ring kS 4
Let k be an algebraically closed field of characteristic 2, let W be the ring of infinite Witt vectors over k and let F be the fraction field of W . Then kS 4 is its own principal block, and the defect groups are the Sylow 2-subgroups of S 4 which are dihedral groups of order 8. By [14, §V.2.5.1], kS 4 is Morita equivalent to the special biserial algebra Λ = kQ/I where We denote the irreducible Λ-modules by S 0 and S 1 , or, using short-hand, by 0 and 1. Then S 0 corresponds to the trivial simple kS 4 -module T 0 , and S 1 corresponds to the two-dimensional simple kS 4 -module T 1 . The radical series of the projective indecomposable Λ-modules (and hence of the projective indecomposable kS 4 -modules) can be described by the following pictures: The field F is a splitting field for S 4 , and the decomposition matrix of kS 4 has the form Because Λ is a special biserial algebra, all indecomposable non-projective Λ-modules are either string or band modules. Since we will use this description extensively, we now give a brief introduction. For more background material, we refer to [10].
Then strings are representatives w ∈ W of the equivalence classes under ∼ s with the following property: Either w = 1 u for u ∈ {0, 1}, or w = w 1 · · · w n where w i = w −1 i+1 for 1 ≤ i ≤ n−1 and no subword of w or its formal inverse belongs to J.
Let C = w 1 · · · w n be a string of length n. Then there exists an indecomposable Λ-module M (C), called the string module M (C) corresponding to the string C, which can be described as follows. There is a k-basis {z 0 , z 1 , . . . , z n } of M (C) such that the action of Λ on M (C) is given by the following representation ϕ C : Λ → Mat(n + 1, k). Let v(i) = e(w i+1 ) for 0 ≤ i ≤ n − 1 and v(n) = s(w n ). Then for each vertex u ∈ {0, 1} and for each arrow ξ ∈ {α, β, γ, η} in Q We will call ϕ C the canonical representation and {z 0 , z 1 , . . . , z n } the canonical k-basis for M (C) relative to the representative C. Note that M (C) ∼ = M (C −1 ). The string modules for the empty words are isomorphic to the simple Λ-modules, namely M (1 0 ) ∼ = S 0 and M (1 1 ) ∼ = S 1 .
Then bands are representatives w ∈ W r of the equivalence classes under ∼ r with the following property: w = w 1 · · · w n , n ≥ 1, with w i = w −1 i+1 and w n = w −1 1 , such that w is not a power of a smaller word, and, for all positive integers m, no subword of w m or its formal inverse belongs to J.
Let B = w 1 · · · w n be a band of length n. Then for each integer m > 0 and each λ ∈ k * there exists an indecomposable Λ-module M (B, λ, m) which is called the band module corresponding to the band B, λ and m. Note that for all i, j Each component of the stable Auslander-Reiten quiver of Λ consists either entirely of string modules or entirely of band modules. The band modules all lie in 1-tubes. The components consisting of string modules are one 1-tube, one 3-tube and infinitely many non-periodic components of type ZA ∞ ∞ .
The irreducible morphisms between string modules can be described using hooks and cohooks. For our algebra Λ, these are defined as follows. Let M be the set of maximal directed strings, i.e. M = {γβ, βα, αγ, η}. Definition 3.3. Let S be a string. We say that S starts on a peak (resp. starts in a deep) if S = S ′ C (resp. S = S ′ C −1 ) for some string C in M. Dually, we say that S ends on a peak (resp. ends in a deep) if S = D −1 S ′′ (resp. S = DS ′′ ) for some string D in M.
If S does not start on a peak (resp. does not start in a deep), there is a unique arrow ζ and a unique M ∈ M such that S h = SζM −1 (resp. S c = Sζ −1 M ) is a string. We say S h (resp. S c ) is obtained from S by adding a hook (resp. a cohook) on the right side.
Dually, if S does not end on a peak (resp. does not end in a deep), there is a unique arrow ξ and a unique N ∈ M such that h S = M ξ −1 S (resp. c S = N −1 ξS) is a string. We say h S (resp. c S) is obtained from S by adding a hook (resp. a cohook) on the left side.
In [15], all homomorphisms between string and band modules have been determined. The following remark describes the homomorphisms between string modules using the canonical bases defined in Definition 3.1. to the relative position of C in S and T , respectively. If e.g. S = s 1 s 2 · · · s m , T = t 1 t 2 · · · t n , and C = s i+1 s i+2 · · · s i+ℓ = t −1 j+ℓ t −1 j+ℓ−1 · · · t −1 j+1 , then σ C (x i+t ) = y j+ℓ−t for 0 ≤ t ≤ ℓ, and σ C (x u ) = 0 for all other u.
Note that there may be several choices of S ′ , S ′′ (resp. T ′ , T ′′ ) in (i) (resp. (ii)). In other words, there may be several k-linearly independent homomorphisms factoring through M (C). By [15], every Λ-module homomorphism σ : M (S) → M (T ) is a k-linear combination of homomorphisms which factor through string modules corresponding to strings C satisfying (i) and (ii).
The next result is an easy consequence of [15] and the symmetric shapes of the radical series of P 0 and P 1 as given in (3.2).
Lemma 3.5. Let ν be the bijection on the arrows in Q which fixes α and η and interchanges β and γ. Let S = w 1 w 2 · · · w n be a string of length n ≥ 1, and define ν(S) = ν(w 1 ) −1 ν(w 2 ) −1 · · · ν(w n ) −1 . Then ν(S) is a string, dim k End Λ (M (S)) = dim k End Λ (M (ν(S))), and dim k Ext Let C 0 be the component of the stable Auslander-Reiten quiver of Λ containing S 0 , and let U be a Λ-module belonging to C 0 ∪ Ω(C 0 ). Using the description of the components of the stable Auslander-Reiten quiver of Λ as in §3, we see that C 0 is of type ZA ∞ ∞ . Moreover, using hooks and cohooks (see Definition 3.3) we obtain the following. There is an i ∈ Z such that Ω i (U ) is isomorphic to one of the following string modules: Lemma 4.1. For n ≥ 1, let V n be an indecomposable kS 4 -module which, under the Morita equivalence, corresponds to the Λ-module A n or B n from (4.1). Then End kS4 (V n ) ∼ = k ∼ = Ext 1 kS4 (V n , V n ). Proof. We prove this for V n corresponding to the Λ-module B n . The case when V n corresponds to the Λ-module A n follows then from Lemma 3.5.
We first show how the results in [2] imply that V n is endo-trivial, in the sense that the kS 4 -module End k (V n ) is isomorphic to a direct sum of the trivial simple kS 4 -module T 0 = k and a projective kS 4 -module. Note that if M and N are endo-trivial kS 4 -modules, then so are M ⊗ k N and Ω i (M ) for all i ∈ Z. Consider the almost split sequence ending in T 0 where, by our assumptions, V 1 corresponds to the Λ-module B 1 , and W 1 corresponds to the Λmodule Ω 2 (A 1 ). Since Ω −1 (V 1 ) ⊕ Ω −1 (W 1 ) is isomorphic to the heart rad(P T0 )/soc(P T0 ) of the projective indecomposable kS 4 -module P T0 with top T 0 , it follows from [ is the almost split sequence ending in M modulo projective direct summands. Applying this inductively to M = V n and using that V n+1 is a direct summand of the middle term of the almost split sequence ending in V n , we obtain that V n is endo-trivial for all n ≥ 1.
Since V n is endo-trivial, it follows that End kS4 (V n ) ∼ = k and that Proof. We prove this for V n corresponding to the Λ-module A n , the case of V n corresponding to B n being similar. We use the description in [14,Lemma V.2.5] of the Morita equivalence between kS 4 and Λ = kQ/I with Q and I as in (3.1). From this it follows that we can take the arrow α to be (1 + h)e 0 where h ∈ S 4 is a transposition and e 0 is the idempotent corresponding to the projective indecomposable kS 4 -module P T0 with top T 0 . Let C be the cyclic subgroup of S 4 of order two generated by h. Let T 00 be the uniserial kS 4 -module with radical series T 0 , T 0 . Because of our choice of h, it follows that restricting the action of S 4 on T 00 to C results in a 2-dimensional kC-module with non-trivial action. Considering a representation of the simple kS 4 -module T 1 , it follows that restricting the action of S 4 on T 1 to C also results in a 2-dimensional kC-module with non-trivial action. Hence

Hence it follows by induction that
Res S4 C V n ∼ = k ⊕ (kC) 2n where k stands for the trivial simple kC-module. Therefore, Res S4 C V n is a kC-module with stable endomorphism ring k which has universal deformation ring R(C, Res S4 C V n ) ∼ = W [Z/2]. Let (U Vn,C , φ n,C ) be a universal lift of Res S4 C V n over W [Z/2], and let (U Vn , φ n ) be a universal lift of V n over R(S 4 , V n ). Then there exists a unique W -algebra homomorphism f n : W [Z/2] → R(S 4 , V n ) in C such that the lift (Res S4 C U Vn , Res S4 C φ n ) is isomorphic to the lift (R(S 4 , V n )⊗ W [Z/2],fn U Vn,C , φ n,C ). We next show that f n is surjective for all n ≥ 1. For this, we consider all morphisms ρ : R(S 4 , V n ) → k[ǫ]/(ǫ 2 ). There is a non-split short exact sequence of kS 4 -modules (4.5) 0 Hence E n = (P T0 ) n ⊕ T 00 defines a non-trivial lift of V n over k[ǫ]/(ǫ 2 ). Since Res S4 C E n ∼ = (kC) 4n ⊕ kC, Res S4 C E n defines a non-trivial lift of Res S4 C V n over k[ǫ]/(ǫ 2 ). Because Ext 1 kS4 (V n , V n ) ∼ = k by Lemma 4.1, this implies that as ρ runs through the morphisms R(S 4 , V n ) → k[ǫ]/(ǫ 2 ), the composition ρ • f n runs through the morphisms W [Z/2] → k[ǫ]/(ǫ 2 ). Hence f n is surjective.
We now use Lemma 2.4 to prove that V n has always two non-isomorphic lifts over W . We first show that the kS 4 -module M corresponding to the Λ-string module M (γβ) has two non-isomorphic lifts over W . For this we consider the representation ϕ : S 4 → Mat 4 (W ) which is the natural permutation representation of S 4 , i.e. ϕ is given by Reducing these matrices modulo 2, results in a representation ϕ : S 4 → Mat 4 (k) which is the representation of a 4-dimensional indecomposable kS 4 -module corresponding to the Λ-string module M (γβ). We can order the ordinary irreducible characters of S 4 in (3.3) such that χ 1 is the trivial character, χ 2 is the sign character and the F -character of ϕ is equal to χ 1 + χ 3 . Twisting ϕ with the sign character χ 2 , we obtain a new representation ϕ ′ : S 4 → Mat 4 (W ) with F -character χ 2 + χ 4 which modulo 2 defines a representation of S 4 over k that is also equivalent to ϕ. This means that we get two non-isomorphic lifts (X, φ X ) and (X ′ , φ X ′ ) of M over W such that the F -character of X is χ 1 + χ 3 and the F -character of X ′ is χ 2 + χ 4 .
Let V 0 = T 0 . Then it follows from the decomposition matrix (3.3) that there are two nonisomorphic W S 4 -modules U W 0,1 and U W 0,2 which define lifts of V 0 over W and whose F -characters are given by χ 1 and χ 2 , respectively.
Let n ≥ 1. Then Hom kS4 (M, V n−1 ) ∼ = k n , Ext 1 kS4 (M, V n−1 ) ∼ = k, and we have a short exact sequence of kS 4 -modules Assume by induction that there are two non-isomorphic W S 4 -modules U W n−1,1 , and U W n−1,2 which define lifts of V n−1 over W such that the F -character of U W n−1,1 is Similarly, if n − 1 = 2s then dim F Hom F S4 (X F , U F n−1,2 ) = n − 1, and if n − 1 = 2s + 1 then dim F Hom F S4 (X ′ F , U F n−1,2 ) = n − 1. Hence by Lemma 2.4, there are two non-isomorphic W S 4modules U W n,1 and U W n,2 which define lifts of V n over W and whose F -characters are as in (4.8) and (4.7) when n − 1 is replaced by n.
Summarizing, we see that for each n ≥ 1, we have a surjective morphism f n : W [Z/2] → R(S 4 , V n ) in C and there are two distinct morphisms R(S 4 , V n ) → W in C. Hence Spec(R(S 4 , V n )) contains both points of the generic fiber of Spec(W [Z/2]). Since the Zariski closure of these points is all of Spec(W [Z/2]), this implies that R(S 4 , V n ) must be isomorphic to W [Z/2]. This completes the proof of Lemma 4.2.
Since by [19, §1.4 In this section, we prove part (ii) of Theorem 1.1. As before, let Λ = kQ/I where k, Q and I are as in (3.1).
Let C 1 be the component of the stable Auslander-Reiten quiver of Λ containing S 1 . Using the description of the components of the stable Auslander-Reiten quiver of Λ as in §3, we see that C 1 is of type ZA ∞ ∞ . Moreover near S 1 , C 1 looks as in Figure 5.1. where S 00 is the uniserial Λ-module with radical series S 0 , S 0 and S 0011 is the string module M (α −1 γη −1 ). Hence C 1 = Ω(C 1 ).
Lemma 5.1. The only Λ-modules belonging to C 1 which have stable endomorphism ring k are of the form Ω i (S 1 ) for some i ∈ Z. S 00 ( ( P P P P P P Proof. Let U be a Λ-module belonging to C 1 which is not in the Ω-orbit of S 1 . Using hooks and cohooks (see Definition 3.3) we obtain the following because of Figure 5.1. There exist integers i and n ≥ 1 such that Ω i (U ) is isomorphic to one of the following string modules: Let {x ℓ,n r } n ℓ r=0 be the canonical k-basis for C ℓ,n , ℓ ∈ {1, 2, 3}. Then the Λ-module endomorphism of C ℓ,n which sends x ℓ,n 0 to x ℓ,n 1 and all other x ℓ,n r to zero does not factor through a projective Λ-module.

The stable Auslander-Reiten component of kS 4 which is a 3-tube
In this section, we prove part (iii) of Theorem 1.1. As before, let Λ = kQ/I where k, Q and I are as in (3.1).
Let C be a component of the stable Auslander-Reiten quiver of Λ containing a uniserial module X of length 3 with non-isomorphic top and socle. Then X is either X 1 with descending radical series S 0 , S 0 , S 1 or X 2 with descending radical series S 1 , S 0 , S 0 . In particular, X 2 = Ω(X 1 ) = Ω −2 (X 1 ), which implies that C is the unique 3-tube of the stable Auslander-Reiten quiver of Λ with X 1 and X 2 belonging to its boundary, and C = Ω(C). The third Λ-module at the boundary of C is Ω 2 (X 1 ) which is the uniserial Λ-module with radical series S 1 , S 1 .
Proof. Let U be a Λ-module belonging to C but not to its boundary. Using hooks and cohooks (see Definition 3.3) it follows that there exist integers n ≥ 1 and i ∈ {0, 1, 2} such that Ω i (U ) is isomorphic to one of the following string modules: One now goes through all the cases for S in (7.1) and determines suitable endomorphisms that do not factor through a projective Λ-module. We provide a few examples. Let {z r } ℓ(S) r=0 be the canonical k-basis of the string module M (S) relative to the chosen representative S.
If S = α −1 γ · · · then the endomorphism sending z 0 to z 1 and all other z r to zero does not factor through a projective Λ-module, and the same is true for M (S h···h ) and M (S c···c ).
Next, we consider all components of the stable Auslander-Reiten quiver which are one-tubes. Except for one 1-tube consisting of string modules, all the other 1-tubes consist of band modules. It follows from [15] that if B is a band, λ ∈ k − {0} and n ≥ 2 is an integer, then End Λ (M (B, λ, n)) has k-dimension at least 2. Hence we can concentrate on the band modules M (B, λ, 1). We need the following two definitions.
where T 1 , T ′ 1 are strings and ξ 1 , ξ 2 are arrows in Q; and ii. B ∼ r ST 2 with T 2 = ζ 1 T ′ 2 ζ −1 2 , where T 2 , T ′ 2 are strings and ζ 1 , ζ 2 are arrows in Q. Then by [15] there exists a non-zero endomorphism of M B,λ which factors through M (S). We will call such an endomorphism to be of string type S. Note that there may be several choices of T 1 (resp. T 2 ) in (i) (resp. (ii)). In other words, there may be several k-linearly independent endomorphisms of string type S. By [15], every endomorphism of M B,λ is a k-linear combination of the identity homomorphism and of endomorphisms of string type S for suitable choices of strings S satisfying (i) and (ii).
where s ≥ 1 is odd and C 0 , C 1 , . . . , C s are top-socle pieces of B.
In the proof of the following proposition, we will often use the equality sign instead of the more precise ∼ r . Proof. Consider first the 1-tube consisting of string modules. If X is a string module in this 1-tube, then X = M (β −1 γ −1 (αβ −1 γ −1 ) n ) for some n ≥ 0. Let {z r } 3n+2 r=0 be the canonical k-basis of the string module X. Then the endomorphism sending z 0 to z 3n+2 and all other z r to zero does not factor through a projective Λ-module.
All other 1-tubes consist of band modules, and it follows from [15] that the only possible band modules with stable endomorphism ring k are those lying at the boundary. Hence one only needs to consider the band modules M B,λ = M (B, λ, 1) where B is a band for Λ and λ ∈ k − {0}. Suppose there is a band B such that M B,λ has stable endomorphism ring equal to k.
Since every band contains either β −1 γ −1 or η −1 as top-socle piece, one successively eliminates these two cases by determining endomorphisms of suitable string types which do not factor through a projective Λ-module. We provide a few examples in each case.
Altogether, it follows that B contains neither β −1 γ −1 nor η −1 as top-socle piece, which means that B does not exist and hence Proposition 7.4 follows.