TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 5, Pages 2389–2405 S 0002-9947(00)02232-7 Article electronically published on February 14, 2000

ON THE STABLE MODULE CATEGORY OF A SELF-INJECTIVE ALGEBRA

KARIN ERDMANN AND OTTO KERNER

Abstract. Let Λ be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable Λmodules which belong to Auslander-Reiten components of the form ZA_{∞} or ZA_{∞}/⟨τ^{k}⟩. The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type.

We are interested in homological properties of modules for self-injective algebras. An important invariant of a finite-dimensional algebra is its stable Auslander-Reiten quiver; and during the past years, it has played a crucial role for classification problems of self-injective algebras of finite or tame representation type.

We call a stable component quasi-serial, if it is of the form ZA_{∞} or ZA_{∞}/⟨τ^{k}⟩. If A is any tame algebra, then most of its Auslander-Reiten components are homogeneous tubes, by [CB], hence are quasi-serial. Moreover, it seems that for self-injective algebras of wild type most components of the stable Auslander-Reiten quiver are quasi-serial. For a block of a group algebra which is of wild representation type, all components are of this form (see [E2]). Therefore, it is important to understand homological properties of quasi-serial components.

If Λ is a finite-dimensional self-injective algebra and X, Y are Λ-modules, then we denote the stable homomorphisms from X to Y by \underline{Hom}(X, Y). Following the terminology in [R1], if M is a module in a quasi-serial stable component C, then the quasi-length of M is the number of the row to which M belongs. The module M is quasi-simple if it has quasi-length one, that is, if it lies at the end of the component.

The first two chapters contain basic facts on dimensions of spaces of stable homomorphisms; this is intended to provide tools which may be of more general use.

In Chapter 3 we prove the following general results. Suppose f is an equivalence of the stable module category of Λ, consider stable homomorphisms \underline{Hom}(M, fM) for modules M in a quasi-serial component C. We prove that the dimension of \underline{Hom}(M, fM) is weakly increasing as a function of the quasi-length of M, except possibly when τΩ^{−1} ≅ f ≅ τ̸ on C (see 3.3). Moreover, we study the cases when f = τ^{s} and f = Ω^{s} in more detail.

We show that if C is a tube, then the dimensions of \underline{Hom}(M, τ^{s}M) for M ∈ C are unbounded (3.5). In general, if C is a component of the form ZA_{∞} which is not fixed by Ω and the dimensions of \underline{Hom}(M, τ^{s}M) are bounded, then \underline{Hom}(X, τ^{−m}X) = 0

\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

Received by the editors June 4, 1996 and, in revised form, October 2, 1997.

2000 Mathematics Subject Classification. Primary 18G25; Secondary 16G70, 20C20.

Key words and phrases. Stable category of finite-dimensional self-injective algebras, quasiserial Auslander-Reiten components, blocks of wild type, endo-trivial modules of p-group algebras.

for m large where X ∈ C is quasi-simple (see 3.6). Following [P], an indecomposable module M with \underline{End}(M) ≅ K is called a stable brick. As a consequence we obtain that if M is a stable brick in a quasi-serial component C as above, then all modules in C of quasi-length less or equal to the quasi-length of M are also stable bricks.

These results are valid for group algebras. Most important is the case when Λ = KG where G is a finite p-group and K is a field of characteristic p; and for these we obtain stronger results. Assume C is a quasi-serial component of such an algebra Λ. We show that then the dimension of \underline{Hom}(M, fM) is actually strictly increasing as a function of the quasi-length, if f is a stable equivalence fixing C, except possibly when Ω ≅ f ≅ Ω̸^{2} on C (4.2); this uses Carlson’s theory of rank varieties of modules. It follows that a stable brick in a quasi-serial component must be quasi-simple. Moreover, using properties of the Ext-algebra we show that ψ is at least unbounded in the general case of a wild block of a group algebra. We note that similar properties hold for restricted enveloping algebras.

In [D], Dade introduced endo-trivial modules; these occur in various contexts in the modular representation theory of finite groups. Endo-trivial modules for p-group algebras are stable bricks. Hence we have as a consequence that any endotrivial module of a p-group algebra of wild representation type is quasi-simple.

Suppose Λ is an arbitrary group algebra and C is a quasi-serial stable Auslander-Reiten component of Λ. We show that then the dimensions of \underline{Hom}(M, τ^{s}M) for M ∈ C are always unbounded (4.7).

A motivation for this work was the observation that there are analogies for representations of hereditary algebras and the stable category of group algebras. If A is a wild hereditary algebra, then all Auslander-Reiten components except for two are of the form ZA_{∞}; see [R1]. On the other hand, all stable Auslander-Reiten components for a block of a group algebra of wild type are quasi-serial; see [E2]. In [B], [K1] properties of \underline{Hom}(M, τ^{s}N) for regular modules of wild hereditary algebras were obtained which turned out to be of great importance; this suggested a study of \underline{Hom}(M, τ^{s}N) for self-injective algebras. Moreover, some of our results on stable bricks for self-injective algebras are analogues of results in [K2]. The methods used here are, however, quite different as those for hereditary algebras.

We assume that the field K is algebraically closed. All algebras have finite dimension over K and are self-injective. Modules are finite-dimensional left modules, and we write homomorphisms to the right. For general properties of the Auslander-Reiten quiver and of stable categories for self-injective algebras we refer to [ARS], [B].

Acknowledgement

This work was supported by an EPSRC Visiting Fellowship for the second author; this was held at Oxford and we are grateful to the Mathematical Institute at Oxford for their hospitality.

1. Self-injective algebras

1.1. In this section we will give an outline of general homological properties of the stable category of a self-injective algebra and relate these to the Auslander-Reiten quiver. Recall that the stable category \underline{mod}Λ has objects such as the Λ-modules, and if X, Y are Λ-modules, then the space of morphisms from X to Y in \underline{mod}Λ is defined to be

\underline{Hom}(X, Y) := Hom(X, Y)/P(X, Y)

where P(X, Y) is the subspace of Hom(X, Y) of maps factoring through projectives. Since Λ is self-injective, we have P(X, Y) = I(X, Y), the space of maps factoring through injectives and hence \underline{Hom}(X, Y) = \overline{Hom}(X, Y)(= Hom(X, Y)/I(X, Y)). For f ∈ Hom(X, Y) we denote by \underline{f} its image in \underline{Hom}(X, Y).

We will very often write (X, Y) for Hom_{Λ}(X, Y), and we write (X, −) and (−, Y) for the corresponding functors.

1.2. Assume Λ is a self-injective algebra over K. If M is a module. then ΩM is the kernel of a minimal projective cover for M, and Ω^{−1}M is the cokernel of an injective hull. Then Ω induces an equivalence of the stable category of Λ, and so does the Auslander-Reiten translate τ. In particular, Ω induces a graph isomorphism of the stable Auslander-Reiten quiver of Λ.

1.3. We shall use the following functorial isomorphisms:

(1) If X, Y are indecomposable, then

D Ext^{1}(X, Y) ≅ \underline{Hom}(τ^{−1}Y, X) ≅ \overline{Hom}(Y, τX)

where D = Hom(−, K) denotes the usual duality. This is the Auslander-Reiten formula. We shall use the fact that Ext^{1}(X, Y) and \underline{Hom}(Y, τX) and \underline{Hom}(τ^{−1}Y, X) have the same dimension, for Λ self-injective; and for simplicity we only use isomorphisms over K and omit D. In particular, we have isomorphism over K

\underline{End}(X) ≅ Ext^{1}(X, τX) ≅ Ext^{1}(τ^{−}X, X).

(2) Further we will use that

\underline{Hom}(A, B) ≅ Ext^{1}(Ω^{−1}A, B) ≅ Ext^{1}(A, ΩB),

is natural in A and in B.

(3) Moreover, Ext^{n}(M, N) ≅ Ext^{n−s}(Ω^{s}M, N) ≅ Ext^{n−s}(M, Ω^{−s}N).

1.4 Lemma. Assume 0 → A →^{ϵ} B →^{π} C → 0 is an exact sequence of Λ-modules, and suppose W is any Λ-module. Then the induced sequence

\underline{Hom}(W, A) →^{ϵ̄} \underline{Hom}(W, B) →^{π̄} \underline{Hom}(W, C)

is exact. Moreover, (1) ϵ̄ is 1-1 if and only if (Ω^{−1}W, π) is onto.

(2) π̄ is onto if and only if (W, π) is onto.

Proof. Apply Hom(Ω^{−1}W, −) to the given exact sequence; since \underline{Hom}(W, −) is naturally isomorphic to Ext^{1}(Ω^{−1}W, −) we get the first statement from the long exact sequence of homology, and also part (1) follows.

For the second part, we have π̄ is onto if and only if Ext^{2}(Ω^{−1}W, ϵ) is 1-1. But Ext^{2}(Ω^{−1}W, −) is naturally isomorphic to Ext^{1}(W, −); and Ext^{1}(W, ϵ) is 1-1 if and only if (W, π) is onto.

1.4.1. For convenience, we state the dual of 1.4.

There is an exact sequence

\underline{Hom}(C, W) →^{π̄} \underline{Hom}(B, W) →^{ϵ̄} \underline{Hom}(A, W).

(1) The map π̄ is 1-1 if and only if (ϵ, ΩW) is onto.

(2) The map ϵ̄ is onto if and only if (ϵ, W) is onto.

1.4.2 Lemma. Let 0 → X →^{f} Y →^{g} Z → 0 be a short exact sequence.

(a) If α : Y → M is a homomorphism with \underline{fα} = 0, then there exists β : Z → M with \underline{α} = \underline{gβ}.

(b) If α : M → Y is a homomorphism with \underline{αg} = 0, then there exists β : M → X with \underline{α} = \underline{βf}.

Proof. (a) From Lemma 1.4 we get an exact sequence

\underline{Hom}(Z, M) →^{g̃} \underline{Hom}(Y, M) →^{f̃} \underline{Hom}(X, M).

By assumption \underline{α} is in the kernel of f̃, hence in the image of g̃, which is the claim. Part (b) is dual.

1.5. Suppose W is a Λ-module, and denote by d_{W} the function

d_{W} = dim\underline{Hom}(W, −).

It follows from (1.4) that for a short exact sequence 0 → A → B → C → 0 that d_{W} (B) ≤ d_{W} (A) + d_{W} (C) always holds.

1.5.1. Then d_{W} is additive on an Auslander-Reiten sequence

0 → τX →^{ϵ} E →^{π} X → 0

if and only if X is not a summand of W or Ω^{−1}W; this was proved in [ES]. It also follows from 1.4; since by the hypotheses on W and by the almost split property, both (Ω^{−1}W, π) and (W, π) are onto.

There is a dual of this result; which follows from 1.4.1. Namely, let d′_{W} := dim\underline{Hom}(−, W), then d′_{W} is additive on Auslander-Reiten sequences

0 → Y → F → τ^{−}Y → 0

for which Y is not a summand of W or ΩW.

We have for indecomposable modules W, X that d_{W} (X) = d_{τW} (τX) and d_{W} (X) = d_{ΩW} (ΩX).

1.5.2. Suppose W is arbitrary and

0 → τX →^{ϵ} E →^{π} X → 0

is an Auslander-Reiten sequence; this induces always an exact sequence

\underline{Hom}(W, τX) → \underline{Hom}(W, E) → \underline{rad}(W, X) → 0.

This follows from 1.5.1; the image of π̄ in 1.5.1 is \underline{rad}(W, X).

2. Stably quasi-serial components

2.1. We call a stable component quasi-serial if it is of the form ZA_{∞} or ZA_{∞}/⟨τ^{k}⟩, in analogy to the terminology in [R1]. Moreover, an Auslander- Reiten component is said to be stably quasi-serial if its stable part is quasi-serial.

Let C be a stable component which is quasi-serial. A module X in C is called quasi-simple if the middle term of the AR-sequence of X is indecomposable (or indecomposable modulo projectives). That is, X lies at the end of C.

Now suppose that X is quasi-simple. In C there is a unique infinite sectional path

⋯ → [r]X → [r − 1]X → ⋯ → [2]X → [1]X = X.

Then any indecomposable in the component which is not projective has the form τ^{i}[r]X, for some i ∈ Z. The quasi-length of a module M in C is the number of the row containing M, so [r]X has quasi-length r. We write ql(M) = r. The quasi-top of [r]X is X.

Dually, the module X(r) is defined via the unique infinite sectional path

X = X(1) → X(2) → ⋯ → X(r) → X(r + 1) → · · · .

Then X(r) has quasi-length r; and we call X the quasi-socle of X(r). Let X be the quasi-socle of X(r), then the quasi-top of X(r) is Y where Y ≅ τ^{−r+1}X and moreover, X(r) ≅ [r]Y.

We use the same terminology for the non-projective indecomposable modules in a stably quasi-serial component.

If C is is quasi-serial, then so is ΩC; moreover, the quasi-length is invariant under Ω. In the special case of symmetric algebras, if C ≅ ZA_{∞}, then ΩC ≠ C; otherwise, one would have Ω ≅ τ^{m} on C and τ ≅ Ω^{2} ≅ τ^{2m}.

2.2. Suppose C is a quasi-serial stable component and M = [r]X belongs to C. Modifying [R2, (3.3)] slightly, the full subquiver W(M) of C whose vertices are given by τ^{l}([s]X) with 1 ≤ s ≤ r and 0 ≤ l ≤ r − s is called the wing spanned by M.

Suppose W(X(r + 1)) is a wing and W is an indecomposable module with W, Ω^{−}W ∉ W(τ^{−}X(r)). As in 1.5 d_{W} denotes the function dim \underline{Hom}(W, −). Then we have

d_{W} (X(r + 1)) = ∑_{i=0}^{r} d_{W} (τ^{−i}X).

Consider the Auslander-Reiten sequence

0 → X(r) → X(r + 1) ⊕ τ^{−}X(r − 1) ⊕ P → τ^{−}X(r) → 0

where P is projective or zero. Then d_{W} (X(r + 1)) = d_{W} (X(r)) + d_{W} (τ^{−}X(r)) − d_{W} (τ^{−}X(r − 1)) by 1.5.1. The claim follows by induction on r.

2.2.1. We call a full subquiver of a quasi-serial component C a ladder if its vertices are given by all [s]X for 1 ≤ s ≤ r together with τ[t]X for 1 ≤ t < r (or dually by the vertices X(s) for 1 ≤ s ≤ r together with τ^{−}X(t) for 1 ≤ t < r.

Lemma. Given any ladder, there is at most one projective attached to its interior:

Proof. Assume for some m > 1 there is an Auslander-Reiten sequence in C, say

0 → τX(m) →^{ϵ} τX(m + 1) ⊕ P ⊕ X(m − 1) →^{π} X(m) → 0

where P is indecomposable projective and where ϵ = (ϵ_{i}) and π = (π_{i})^{t}. We will show that P is the unique indecomposable projective in the ladder starting at τX(m+1) and X(m), respectively τX(m) and τX(m+1). Then P ≠ 0 if and only if π_{1} and π_{3} are mono; moreover, if this happens, then ϵ_{1}, ϵ_{3} are epi. If ϵ_{3} is epi, then it follows by induction on i that all the irreducible maps τX(m−i) → X(m−i−1) are surjective for 0 ≤ i ≤ m − 2. Similarly, π_{3} : X(m − 1) → X(m) mono gives irreducible monos τ^{−i}X(m−i−1) → τ^{−i}X(m−i) for 0 ≤ i ≤ m−2, too. Therefore, there is no non-zero projective in the ladder starting with τX(m) → X(m − 1) as well as in the ladder starting at τX(m) and τX(m + 1).

2.3. If A is a representation-infinite hereditary algebra and C a regular component in the Auslander-Reiten quiver of A, then C is either a tube or of type ZA_{∞}. All irreducible maps X(i) → X(i + 1) are injective and one may consider them as inclusions. If A is tame hereditary, then C is a tube of period k with quasisimple modules X_{1}, …, X_{k}. The category add C is a serial abelian category, that is, any indecomposable object in add C has a unique composition series in add C and the simple objects in this category are the modules X_{1}, …, X_{k}; see [R2]. This does not hold if A is wild hereditary. But one gets at least that for any quasisimple module X in C and any natural numbers i, j there is a short exact sequence 0 → X(i) → X(i + j) → τ^{−i}X(j) → 0; see [R1]. Hence C was called quasi-serial in this case. The following result explains why we call a component for a self-injective algebra stably quasi-serial, if its stable part is of type ZA_{∞} or is a tube.

Proposition. Let C be a component which is stably quasi-serial, let X be quasisimple in C and i, j > 0. Then there exists a short exact sequence

0 → X(i) →^{ϵ} X(i + j) ⊕ P →^{π} τ^{−i}X(j) → 0

where P is projective, possibly decomposable or zero, and π = (π_{1}, π_{2})^{t}, ϵ = (ϵ_{1}, ϵ_{2}) with ϵ_{1}, π_{1} chains of irreducible maps corresponding to the sectional paths; and the components of π_{2}, ϵ_{2} are compositions of irreducible maps.

Proof. The proof is done by induction on i and j. The first case is formulated as a separate lemma.

2.3.1 Lemma. Let C be a component which is stably quasi-serial, and let X(r) = [r]Y be indecomposable in C.

(a) There exists a short exact sequence

0 → τX →^{ϵ} τX(r + 1) ⊕ P →^{π} X(r) → 0

where π = \binom{π_{1}}{π_{2}} and ϵ = (ϵ_{1}, ϵ_{2}) with π_{1} irreducible, and where P is zero or the indecomposable projective in the ladder starting at τX(r + 1), X(r) and π_{2}, ϵ_{i} are compositions of irreducible maps. Hence ϵ_{1}π_{1} is in P(τX, X(r)).

(b) There exists a short exact sequence

0 → X(r) →^{ϵ} X(r + 1) ⊕ P →^{π} τ^{−1}Y → 0

where P is zero or the indecomposable projective in the ladder starting at X(r), X(r + 1) and where ϵ = (ϵ_{1}, ϵ_{2}) and π = \binom{π_{1}}{π_{2}} with ϵ_{1} irreducible and ϵ_{2}, π_{i} compositions of irreducible maps. Hence ϵ_{1}π_{1} is in P(X(r), τ^{−}Y).

Proof. (a) For i = 1, 2, ... we fix irreducible maps corresponding to Auslander-Reiten sequences which satisfy the mesh relations, as follows. Let

f_{i} : X(i) → X(i + 1), f′_{i} : τX(i) → τX(i + 1), g_{i} : τX(i + 1) → X(i).

If there is a projective attached to the ladder, say between X(m) and τX(m), then we have also such irreducible maps α : τX(m) → P and β : P → X(m). (This can be done, by using [ARS] V.5.3, V.2.3.)

Take ϵ_{1} = f′_{1}f′_{2}...f′_{r} and π_{1} = g_{r}. If there is no projective attached to the ladder, take ϵ_{2} = 0 = π_{2}. Otherwise, take ϵ_{2} = ±f′_{1}f′_{2}...f′_{m−1}α and π_{2} = βf_{m}f_{m+1}...f_{r−1}; the mesh relations imply that ϵπ = 0 for the appropriate choice of the sign.

The map ϵ is injective by definition. We show that π is surjective. This is obvious if P = 0 or m = r, so we assume P ≠ 0 and we use induction on r − m. Assume m < r and π is not surjective. Take x ∈ X(r) \ Im(π). Choose x_{1} ∈ τX(r + 1), x_{2} ∈ X(r − 1) such that x_{1}g_{r} + x_{2}f_{r−1} = x. By induction, the map \binom{g_{r−1}}{π′_{2}} : τX(r) ⊕ P → X(r − 1) is surjective; hence x_{2} = yg_{r−1} + zπ′_{2}. But then we get x = x_{1}g_{r} + yg_{r−1}f_{r−1} + zπ_{2} = (x_{1} + yf′_{r})g_{r} + zπ_{2}, a contradiction.

From the construction it follows that dim(τX(r + 1) ⊕ P) = dim(τX ⊕ X(r)), hence the sequence is exact. Part (b) is proved similarly.

2.3.2. We continue with the proof of Proposition 2.3. We fix the irreducible maps in the relevant part of the component C such that all mesh-relations are satisfied (this can be done locally; see [ARS] V5.3, V2.3). We denote by f_{s} : X(i + s − 1) → X(i + s) for s ≥ 1 the irreducible maps, and we take ϵ_{1} = f_{1} ⋯ f_{j}. Similarly, for π_{1} we take the composition of the irreducible maps corresponding to the sectional path X(i + j) → τ^{−i}X(j).

Consider the set of indecomposable projective modules {P_{1}, …, P_{r}} such that there are paths α_{t} from X(i) to P_{t} and β_{t} from P_{t} to τ^{−i}X(j) in C and the composed path has length i + j. We may additionally assume that there are no further projective modules on the composed path α_{t}β_{t}. Let e_{t} : X(i) → P_{t} and p_{t} : P_{t} → τ^{−i}X(j) be the chain of irreducible maps corresponding to the chosen paths α_{t} and β_{t}, define P = ⨁ P_{t} and ϵ_{2} = (±e_{1}, …, ±e_{r}) and π_{2} = (p_{1}, …, p_{r})^{t}.

First, observe the following. If γ is a path from X(i) to τ^{−i}X(j) such that no projective occurs in the part of C between γ and the sectional paths X(i) → τ^{−i+1}X and τ^{−i}X → τ^{−i}X(j), then any composition of irreducible maps corresponding to γ is zero. Using this and the mesh relations one sees that there is a choice of signs such that ϵπ = 0. (See Figure 1.)

Since dim(X(i + j) ⊕ P) = dim(X(i) ⊕ τ^{−i}X(j)), it suffices to show that ϵ is injective and π is surjective.

Consider the ladder starting at X(i) and X(i + 1). If no projective module is in the interior of the ladder, clearly f_{1} is injective and additionally we may assume that all the maps e_{i} are of the form f_{1}e′_{i}. Since i + j = (i + 1) + (j − 1) we may assume by induction that ϵ′ = (f_{2} ⋯ f_{i}, e′_{1}, … e′_{r}) : X(i + 1) → X(i + j) ⊕ P is injective. But ϵ = f_{1}ϵ′ in this case.

If there is a projective occurring in the ladder, then by 2.2.1 there is only one, P_{1} say; and by 2.3.1(b) the map (f_{1}, e_{1}) : X(i) → X(i + 1) ⊕ P_{1} is injective. For j > 1 we may assume e_{j} = f_{1}e′_{j}. Again, we get by induction that the map (f_{2} ⋯ f_{j}, e′_{2}, …, e′_{r}) : X(i + 1) → X(i + j) ⊕ (⨁_{j=2}^{r} P_{j}) is injective. Hence ϵ is injective as well.

Dually one shows the surjectivity of π, using 2.3.1(a) and induction.

2.4 Lemma. Assume C is a quasi-serial regular component and X, Y in C are quasi-simple with X(r) ≅ [r]Y.

Figure 1

(a) Suppose π : [r + 1]Y → [r]Y is an irreducible epi. If W is indecomposable and not isomorphic to X(s) for 1 ≤ s ≤ r, then π induces an epimorphism

Hom(W, [r + 1]Y) → Hom(W, [r]Y) → 0.

(b) Suppose ϵ : X(r) → X(r +1) is an irreducible mono. If W is indecomposable and not isomorphic to [s]Y for 1 ≤ s ≤ r, then ϵ induces an epimorphism

Hom(X(r + 1), W) → Hom(X(r), W) → 0.

This is proved in [R1]; the proof given there does not use that the algebra is hereditary. In fact, it holds for an arbitrary finite-dimensional algebra.

The hypotheses in 2.4 are also necessary. With the notation as in 2.4(a), consider M = X(s) where 1 ≤ s ≤ r, then the map (M, π) is not surjective. If r = s, then id_{M} does not factor through π. Moreover, for 1 ≤ s < r there is a monomorphism ϵ : M → X(s) which is a composition of irreducible maps. Then ϵ does not factor through π. (Suppose ϵ = ψπ for ψ : M → [r + 1]Y. By applying 2.4(b) repeatedly we see that ϵ induces an epimorphism

Hom(X(s), [r + 1]Y) → Hom(X(r), [r + 1]Y) → 0.

Hence ψ = ϵη for η : X(s) → [r+1]Y. Now ϵ = ϵηπ; since 1−ηπ is an isomorphism, it follows that ϵ = 0, a contradiction.)

Similarly, if M = [s]Y for some s with 1 ≤ s ≤ r, then (ϵ, M) is not surjective.

2.4.1. Lemma 2.4 has a stable version.

Lemma. Assume C is a stably quasi-serial component and X, Y are quasi-simple in C with X(r) ≅ [r]Y. Let W be indecomposable.

(a) Suppose π : [r + 1]Y → [r]Y is irreducible. If W is not isomorphic to X(s) for 1 ≤ s ≤ r, then π induces an epimorphism

\underline{Hom}(W, [r + 1]Y) → \underline{Hom}(W, [r]Y) → 0.

(b) Suppose ϵ : X(r) → X(r + 1) is irreducible. If W is not isomorphic to [s]Y for 1 ≤ s ≤ r, then ϵ induces an epimorphism

\underline{Hom}(X(r + 1), W) → \underline{Hom}(X(r), W) → 0.

Proof. (a) There is a module P which is projective or zero and an irreducible map

P ⊕ [r + 1]Y →^{π_{1}} [r]Y.

This induces

\underline{Hom}(W, P ⊕ [r + 1]Y) →^{π̄_{1}} \underline{Hom}(W, [r]Y).

We proceed by induction on r. If r = 1, then P ⊕ [r + 1]Y is the whole middle term of the Auslander-Reiten sequence and then (W, π_{1}) is surjective and hence π̄_{1} is surjective, by 1.4.

Now assume r > 1. Let f : W → [r]Y, then f is not a split epimorphism, so we have f = g_{1}π_{1} + g_{2}π_{2} where the Auslander-Reiten sequence is given by

0 → τ[r]Y →^{(ϵ_{1},ϵ_{2})} (P ⊕ [r + 1]Y) ⊕ τ[r − 1]Y \underrightarrow{\binom{π_{1}}{π_{2}}} [r]Y → 0.

Note that ϵ_{1}π_{1} + ϵ_{2}π_{2} = 0.

Consider the irreducible map ϵ_{2}. By the inductive hypothesis, ϵ_{2} induces a surjective map

\underline{Hom}(W, τ[r]Y) → \underline{Hom}(W, τ[r − 1]Y) → 0.

This gives \underline{g_{2}} = \underline{ηϵ_{2}} for η : W → τ[r]Y, that is,

g_{2} = ηϵ_{2} + ρ, ρ ∈ P(W, τ[r]Y)

and therefore f = g_{1}π_{1} + (ηϵ_{2} + ρ)π_{2} = (g_{1} − ηϵ_{1})π_{1} + ρπ_{2} and \underline{f} = \underline{hπ_{1}} for some h, as required.

Part (b) is similar.

2.5. We combine 1.4 , 2.3 and 2.4, 2.4.1 and obtain the following:

(a) Assume 0 → τX →^{ϵ} τX(r + 1) ⊕ P →^{π} X(r) → 0 is the short exact sequence as in 2.3.1(a) and let X(r) = [r]Y.

Suppose W is indecomposable, then we have an exact sequence

\underline{Hom}(W, τX) →^{ϵ̄} \underline{Hom}(W, [r + 1]Y) →^{π̄} \underline{Hom}(W, [r]Y).

(1) If Ω^{−1}W ≅ X̸(s) for 1 ≤ s ≤ r, then ϵ̄ is 1-1.

Namely, by 2.4.1 we know \underline{Hom}(Ω^{−}W, π) is onto, and then by 1.4(2) also (Ω^{−1}W, π) is onto. Now 1.4(1) implies that ϵ̄ must be 1-1.

(2) If W ≅ X̸(s) for 1 ≤ s ≤ r, the π̄ is onto.

Hence, if W, Ω^{−1}W ≅ X̸(s) for 1 ≤ s ≤ r, then

d_{W} (τX) + d_{W} ([r]Y) = d_{W} ([r + 1]Y).

(b) Assume 0 → X(r) →^{ϵ} X(r + 1) ⊕ P →^{π} τ^{−1}Y → 0 is exact, as in 2.3.1(b)

Suppose W is indecomposable, then we have an exact sequence

\underline{Hom}(τ^{−1}Y, W) →^{π̄} \underline{Hom}(X(r + 1), W) →^{ϵ̄} \underline{Hom}(X(r), W).

(1) If ΩW ≅ [s]Y for 1 ≤ s ≤ r, then π̄ is 1-1.

(2) If W ≅ [s]Y for 1 ≤ s ≤ r, then ϵ̄ is onto.

In particular, if W, ΩW ≅ [s]Y for 1 ≤ s ≤ r, then

d′_{W} (τ^{−}Y) + d′_{W} (X(r)) = d′_{W} (X(r + 1)).

2.6 Lemma. Let C be a stably quasi-serial component. If [r]X ∈ C has quasilength r and quasi-top X, then a chain of irreducible maps π : [r]X → [s]X (ϵ : τ^{r−s}[s]X → [r]X, respectively) for r > s ≥ 1 does not factor through a projective module.

Proof. We may assume that s = 1. First, it will be shown that \underline{π} = 0 implies \underline{Hom}([r]X, X) = 0. For 0 ≠ \underline{f} ∈ \underline{Hom}([r]X, X) we get \underline{f} = \underline{gπ} by 2.4.1, with g ∈ End([r]X), hence \underline{π} ≠ 0.

Suppose next \underline{Hom}([r]X, X) = 0. By 2.2 we have

0 < dim \underline{End}([r]X) = d_{[r]X}([r]X) = ∑_{i=0}^{r−1} d_{[r]X}(τ^{i}X).

Hence we get d_{[r]X}(τ^{i}X) ≠ 0 for some i and from the assumption i > 0. Let 0 ≠ \underline{f} ∈ \underline{Hom}([r]X, τ^{i}X). Again, by 2.4.1 we deduce the existence of a map g ∈ Hom([r]X, τ^{i}[r]X) with \underline{f} = \underline{gπ′}, where π′ : τ^{i}[r]X → τ^{i}X is a chain of irreducible maps. But \underline{π′} = 0 gives the contradiction. Dually one shows \underline{ϵ} ≠ 0.

2.6.1 Corollary. In 2.4.1 the converse also is true: With the notation as in 2.4.1 we have:

(a) If W ≅ X(s) for some s with 1 ≤ s ≤ r, then the map induced by π

\underline{Hom}(W, [r + 1]Y) → \underline{Hom}(W, [r]Y)

is not surjective.

(b) If W ≅ [s]Y for some s with 1 ≤ s ≤ r, then the map induced by ϵ

\underline{Hom}(X(r + 1), W) → \underline{Hom}(X(r), W)

is not surjective.

Proof. (a) Let f = id if r = s and f = ϵ_{1} : W → [r]Y the sectional chain of irreducible maps, otherwise. We claim that \underline{f} does not lie in the image of the map induced by π. Otherwise, we would have

f = ψπ + ρ, for ψ ∈ Hom(W, [r + 1]Y)ρ ∈ P(W, [r]Y).

In case f = id we have ρ is in the radical of End([r]Y), hence is nilpotent and π splits, a contradiction.

Now suppose f = ϵ_{1}. By applying 2.4.1(b) repeatedly we have that the map

ϵ̄_{1} : \underline{Hom}(X(r), [r + 1]Y) → \underline{Hom}(X(s), [r + 1]Y)

is surjective. So ψ = ϵ_{1}η + ρ_{1} with ρ_{1} ∈ P(W, [r + 1]Y). It follows that

ϵ_{1} = (ϵ_{1}η + ρ_{1})π + ρ

and ϵ_{1}(1−ηπ) factors through a projective. Since 1−ηπ is a unit, ϵ_{1} factors through a projective module, a contradiction to 2.6. Part (b) is similar.

3. On dimensions of spaces of stable homomorphism

3.1. In this section, Λ is an arbitrary finite-dimensional represenation infinite connected self-injective algebra, and we assume that f is some equivalence of the stable category \underline{mod}Λ; this includes f = τ^{s} or f = Ω^{s}. We study the function

ψ_{f}(M) = dim\underline{Hom}_{Λ}(M, fM)

for M a module. Since f commutes with τ, Ω (see [ARS] X.1) it is constant on τ-orbits and on Ω-orbits.

3.2 Lemma. Given an exact sequence in some quasi-serial component

0 → τ[r − 1]Z →^{ϵ} [r]Z ⊕ P →^{π} Z → 0

with Z quasi-simple, as in 2.3.1. Suppose W is indecomposable such that neither W nor ΩW is a module in the same component of quasi-length < r, then d′_{W} is additive on such an exact sequence.

In particular, for W = f[r]Z we get

d_{τ[r−1]Z}(f[r]Z) ≤ ψ_{f}([r]Z).

Proof. The first part follows from 1.4.1. For the second part W = f[r]Z has quasilength r and by the first part d′_{W} (τ[r − 1]Z) ≤ d′_{W} ([r]Z). Now, d′_{W} (τ[r − 1]Z) = d_{τ[r−1]Z}(W) = d_{τ[r−1]Z}(f[r]Z) and d′_{W} (([r]Z) = ψ_{f}(([r]Z).

3.3 Theorem. Let C be a stable component which is quasi-serial and assume that it is not the case that τΩ^{−1} ≅ f ≅ τ̸ on C. Suppose that M, Q ∈ C and ql(Q) < ql(M), then

ψ_{f}(Q) ≤ ψ_{f}(M).

Proof. Let r = ql(M); it suffices to consider the case when there is an irreducible map Q → M and ql(Q) = r − 1. Let Y be the quasi-top of fM, then fM = [r]Y and fQ = τ[r − 1]Y. We consider the values of the function d_{Q} on the ladder starting with fQ → fM and ending at Y. Let

a_{i} = d_{Q}([i]Y), 1 ≤ i ≤ r, and b_{i} = d_{Q}(τ[i]Y), 1 ≤ i ≤ r − 1.

Then b_{r−1} = d_{Q}(fQ) = ψ_{f}(Q) and moreover, we have a_{r} = d_{Q}(fM) and hence by 3.2 we know that a_{r} ≤ ψ_{f}(M). So it suffices to show that b_{r−1} ≤ a_{r}.

By considering quasi-lengths one sees that Q, Ω^{−1}Q cannot occur in the lower part of the ladder and hence d_{Q} is additive on this part. Therefore,

b_{1} + a_{1} = a_{2}, b_{i} + a_{i} = a_{i+1} + b_{i−1}, 2 ≤ i ≤ r − 2.

This implies

b_{i} = a_{i+1} − a_{1}, 1 ≤ i ≤ r − 2.

Case 1. Assume f ≅ τ̸ on C, then by the hypothesis also f ≅ τ̸Ω^{−1} on C. Hence by 1.5.1, d_{Q} is additive on the whole ladder and

b_{r−1} + a_{r−1} = a_{r} + b_{r−2}.

It follows that b_{r−1} = a_{r} − a_{1} ≤ a_{r} ≤ ψ_{f}(M), as required.

Case 2. Assume f ≅ τ on C. Then the remaining Auslander-Reiten sequence of the ladder is of the form

0 → τQ →^{ϵ} τM ⊕ M′ →^{π} Q → 0

(where M′ = τ[r − 2]Y ⊕ P for P projective or zero; for r = 2 the first summand of M′ is zero). By 1.5.2 this induces an exact sequence

η : \underline{Hom}(Q, fQ) →^{ϵ̄} \underline{Hom}(Q, fM ⊕ M′) → \underline{rad}(Q, Q) → 0.

Assume first that Q ≠ Ω(Q). Then (Ω^{−1}Q, π) is onto and by 1.4 we have that ϵ̄ is 1-1 and

a_{r} + b_{r−2} = b_{r−1} + (a_{r−1} − 1)

hence a_{r} − b_{r−1} = a_{1} − 1. Now, a_{1} = dim\underline{Hom}(Q, Y) here and this is non-zero by 2.6. Therefore, a_{r} − b_{r−1} ≥ 0, as required.

Now assume Q ≅ Ω(Q). Then in the above exact sequence η, the kernel of ϵ̄ has dimension 1. So we get a_{r} − b_{r−1} = a_{1} − 2. As before, we know that a_{1} ≥ 1 and hence b_{r−1} = ψ_{f}(Q) ≤ a_{r} + 1. If we can show that a_{r} < ψ_{f}(M), then we are done. From 3.2 we know that

a_{r} + dim \underline{Hom}(Z, fM) = ψ_{f}(M)

where Z is the quasi-top of M. So we must show that

\underline{Hom}(Z, fM) ≠ 0.

Since f ≅ τ and Ω ≅ id on C we have K-isomorphisms

\underline{Hom}(Z, fM) ≅ Ext^{1}(M, Z) ≅ \underline{Hom}(M, Z) ≠ 0

again by 2.6.

3.3.1. Consider a quasi-serial component C such that τΩ^{−1} ≅ f ≅ τ̸ on C.

(1) Suppose f fixes C, then this is quite exceptional. Namely, f must fix each τ-orbit in C and hence there is some integer s such that

τΩ^{−1}M ≅ fM ≅ τ^{s}M (M ∈ C).

Since τ ≅ Ω^{2}ν where ν is a Nakayama twist, it follows that Ω^{2s−1} ≅ ν^{1−s} on C. This preserves dimensions of modules. So if this happens, then all modules in C have bounded projective resolutions. If in addition Λ is symmetric, then ν ≅ id. It is easy to see that τ^{s−1} ≅ Ω^{−1} ≅ i̸d on C if and only if C is a tube fixed by Ω of rank t > 1 where t divides 2s − 1.

(2) Similarly, if f = Ω^{s}, then it follows that ν ≅ Ω^{s−1} on C and modules in this component have bounded projective resolutions. In case Λ is symmetric we have C must be a tube of rank t and t divides s − 1.

3.3.2. In general, it is possible that ql(Q) < ql(M) but ψ_{f}(Q) = ψ_{f}(M). An example can be found in [EKS, 6.6] with f = id: for any natural number n ≥ 2 there exists a symmetric algebra A with n simple modules which has a component C of the form ZA_{∞} containing stable bricks of quasi-length n − 1. Hence we have ψ_{f}(Q) = 1 for all Q ∈ C with ql(Q) ≥ n − 1.

3.4. In the situation of 3.3, we have a vanishing condition: Suppose ql(Q) + 1 = ql(M) and there is an irreducible map Q → M. Assume we are in Case 1 of the proof. If ψ_{f}(Q) = ψ_{f}(M), then d_{Q}(Y) = 0 where Y is the quasi-top of fM.

3.5. One may ask whether the function ψ_{f} is unbounded on a quasi-serial component. For tubes we get.

Lemma. Let T be a tube, and let f be a stable equivalence of \underline{mod}Λ. Then the set {ψ_{f}(M) : M ∈ T } is either unbounded, or ψ_{f}(M) = 0 for all M ∈ T . In the latter case we have fT ≠ T .

Proof. Let T be a tube and suppose k is the period of T .

Assume first fT = T . Then f = τ^{s} on the objects of T , for some s; see [ARS] X.1. Let r ≥ mk and M be an indecomposable module in T of quasi-length r. If fM = τ^{s}M = X(r) with quasi-socle X, then we have

ψ_{f}(M) = d_{M}(τ^{s}M) = d_{M}(X(r)).

Clearly M, Ω^{−}M do not belong to W(τ^{−}X(r − 1)), hence by 2.2 we have

d_{M}(X(r)) = ∑_{i=0}^{r−1} d_{M}(τ^{−i}X).

Let δ = ∑_{i=0}^{k−1} d_{M}(τ^{−i}X), then δ is non-zero (since the quasi-top of M occurs amongst the modules τ^{−i}X). It follows that d_{M}(X(r)) ≥ mδ ≥ m.

Suppose fT ≠ T and there is some indecomposable M = Y (r) in T with ψ_{f}(M) ≠ 0. Let fM = X(r). Again, we get

ψ_{f}(M) = ∑_{i=0}^{r−1} d_{M}(τ^{−i}X)

therefore δ = ∑_{i=0}^{k−1} d_{M}(τ^{−i}X) is non-zero. From (2.4.1(b)) we get d_{Y (r)}(τ^{−i}X)J ≤ d_{Y (r+k)}(τ^{−i}X). Hence we have ψ_{f}(Y (r + k)) ≥ ψ_{f}(M) + δ and the claim follows.

3.5.1. In the special case when f = id we get

Lemma. Let T be a tube of period k. If M is indecomposable in T with quasi-length r = mk + s with 1 ≤ s < k, then dim \underline{End}(M) ≥ m + 1.

Proof. Let X be the quasi-top of M. By 2.2 we have dim \underline{End}(M) = ∑_{i=0}^{r−1} d_{M}(τ^{i}X) ≥ (m + 1)d_{M}(X). But d_{M}(X) > 0 by 2.6.

3.6 Lemma. Let C be a stable component which is isomorphic to ZA_{∞} which is not fixed by Ω, and let f = τ^{s}. If {ψ_{f}(M) : M ∈ C} is bounded, then for m ≫ 0

\underline{Hom}(X, τ^{−m}X) = 0 = \underline{Hom}(τ^{m}X, X)

where X ∈ C is quasi-simple.

Proof. Since C is not fixed by Ω, the hypothesis for 3.3 is satisfied. By 3.3 there is some r_{0} such that for all Z ∈ C with ql(Z) ≥ r_{0} we have

ψ_{f}(Z) = d.

Fix a quasi-simple module X in C and set Y := τ^{s}X. We will first show that

for all r ≥ r_{0}, d′_{Y} ([r]τX) = 0.

Let r ≥ r_{0}. Clearly, τ^{s−1} ≠ id on C; we apply 3.4 with Q = τ[r]X (and M = [r + 1]X), and we get d_{Q}(Y) = 0; but d_{Q}(Y) = d′_{Y} (Q).

Let r_{1} = max{r_{0}, s + 1} and let (→ [r_{1}]τX) be the cone of all predecessors of [r_{1}]τX in C. The function d′_{Y} is additive on this cone by 1.5.1.

For a = 1, 2, 3, ..., r_{1} we set k_{a} := d′_{Y} ([r_{1} − a + 1]τ^{a}X). We claim that

d′_{Y} ([r]τ^{a}X) = k_{a}, constant, for r ≥ r_{1} − a + 1.

For a = 1 we have k_{1} = 0 and the statement was just proved; the general case follows by induction on a, by additivity. But then, again by the additivity of d′_{Y} on the cone (→ [r_{1}]τX) we get

d′_{Y} ([t]τ^{b}X) = 0 for all t ≥ 1 and b > r_{1}.

Hence \underline{Hom}(X, τ^{−m}X) = 0 for all large m, and the statement follows.

3.6.1. We do not know an example of a stable component of type ZA_{∞} on which ψ_{f} is bounded and we expect that this may not occur.

The converse of 3.6 is not true. For example, let H be a connected wild hereditary algebra, D a regular component in Γ_{H} and Y quasi-simple in D. Then we have Hom(Y, τ^{−m}Y) = 0 for m ≫ 0, by [K1]. It follows from [Ba] that for any integer s the sequence (dimHom(Y (m), τ^{s}Y (m)))_{m} grows exponentially. If A is the trivial extension of H by D(H) (or more generally a self-injective algebra of wild tilted type as studied in [EKS]), then it follows that the same holds for the stable homomorphisms of A for any stable component of ZA_{∞} of A (see [EKS], section 4).

3.7. The Ext-algebra Ext^{∗}(X, X) of a module X is of central interest. As an application of 3.3 we get for the dimensions of the spaces Ext^{i}(X, X).

Corollary. Let C be a stable component which is quasi-serial. Suppose M, Q are in C with ql(Q) < ql(M).

(a) We have dim \underline{End}(Q) ≤ dim \underline{End}(M) unless possibly τΩ^{−1} ≅ id ≅ τ̸ on C.

(b) We have dim Ext^{s}(Q, Q) ≤ dim Ext^{s}(M, M) for s ≥ 1, unless possibly τΩ^{−1} ≅ Ω^{s} ≅ τ̸ on C.

Proof. For part (a), take f = id in 3.3. In (b) take f = Ω^{s}, recall that Ext^{s}(M, M) ≅ \underline{Hom}(M, Ω^{s}M).

3.8. A module whose endomorphism ring is K usually is called a brick. We call an indecomposable module M a stable brick if \underline{End}(M) = K. For example, any τ-translate of a simple non-projective module is a stable brick. Although a stable brick is not normally a brick, we prefer this terminology. We get from 3.7

Corollary. Let C be a quasi-serial component such that it is not the case that τ^{−} ≅ Ω^{−1} ≠ id on C. Assume M = X(r) is a stable brick in C, of quasi-length r. Then for 1 ≤ s ≤ r the modules X(s) are stable bricks.

4. Applications to group algebras

4.1. Assume that Λ = KG where G is a finite group. Then Λ and, more generally, any block B of Λ is a symmetric algebra, and our results may be applied. We may assume Λ is not semi-simple, that is, the characteristic of K divides the order or G. When G is a p-group we have, in fact, strict inequality in 3.3.

4.2 Theorem. Let Λ = KG where G is a p-group and charK = p. Assume f is a stable equivalence of Λ and C is a quasi-serial component of Γ_{s}(Λ) which is fixed by f. Suppose it is not the case that Ω ≅ f ≅ Ω̸^{2} on C. Suppose M, Q ∈ C with ql(Q) < ql(M), then

dim \underline{Hom}(Q, fQ) < dim \underline{Hom}(M, fM).

Proof. Recall that τ ≅ Ω^{2} for Λ. It follows from 3.3 that the dimensions in question are weakly increasing as functions of the quasi-length. Assume (for a contradiction) that equality holds for Q, M such that there is an irreducible mono Q → M in C. By 3.4 we have that d_{Q}(Y) = 0 where Y is the quasi-top of fM. Hence, using a well-known identity

0 = \underline{Hom}_{Λ}(Q, Y) ≅ \underline{Hom}_{Λ}(K, Q^{∗} ⊗ Y)

where K is the trivial module. Now, Λ is a local algebra, and it follows that Q^{∗} ⊗ Y is projective. This is not possible for non-projective modules in the same Auslander-Reiten-component.

Namely, let V (M) denote the variety of a Λ-module M (see [B] or [C]). The relevant properties are as follows. One has V (M) = {0} if and only if M is projective. Moreover, V (M) = V (M^{∗}) and V (X ⊗ Y) = V (X) ∩ V (Y); and the variety of modules is constant on stable Auslander-Reiten components.

In the case considered we deduce {0} = V (Q^{∗} ⊗ Y) = V (Q^{∗}) ∩ V (Y) = V (Y) and Y is projective, a contradiction.

4.3 Remarks. (1) Recall that the hypothesis of 4.2 excludes precisely the case when C is a tube fixed by Ω such that f ≅ τ^{s} on C and where C has rank t > 1 where t divides 2s − 1 (see 3.3.1). Modules in such a tube have the property that 2dimM is divisible by the order of G. This follows from the projective resolution. Namely, there is an exact sequence

0 → M → P_{m} → ⋯ → P_{0} → M → 0

with m even and where P_{i} is free, hence dim(P_{i}) is divisible by |G|. By the exactness, 2dimM = ∑(−1)^{i}(dim(P_{i})).

(2) If f = id, then the conditions in 4.2 are vacuously satisfied.

4.4 Corollary. Let Λ = KG where G is a p-group and charK = p. Assume that Λ is of wild representation type. If M is indecomposable and \underline{End}(M) ≅ K, then M is quasi-simple.

Proof. Apply 4.2 with f = id; all components of Γ_{s}(Λ) are quasi-serial (by [E2]).

4.5. Let G be a p-group and charK = p. It follows from 3.3 that M is a stable brick if and only if M^{∗} ⊗ M ≅ X_{M} ⊕ P where P is projective and X_{M} has a simple socle and is non-projective.

Following Dade [D], a KG-module M is defined to be endo-trivial if M ⊗ M^{∗} ≅ K ⊕ P where P is projective. Hence an endo-trivial module is a stable brick, and we obtain

Corollary. Let Λ = KG where G is a p-group and charK = p. If Λ is of wild type, then every endo-trivial module M has an Auslander-Reiten sequence whose middle term has a unique non-projective summand.

If p = 2 and Λ = KD where D is a dihedral 2-group, then there are endotrivial modules where the middle term of the Auslander-Reiten sequence has two non-projective indecomposable summands.

4.6. If G is a p-group, we were not able to find an example of a KG-module which is a stable brick but which is not endo-trivial. After asking various people for examples it turned out that Jon Carlson already knew that stable bricks must always be endo-trivial; this will appear in [CR].

4.7 Theorem. Assume Λ = KG where G is a finite group and C is quasi-serial, and let f = τ^{s}. Then the set of dimensions {ψ_{f}(M)|M ∈ C} is unbounded.

Proof. The statement holds by 3.5 for tubes. So we may assume that C = ZA_{∞}. Then C is not fixed by Ω (see 2.1). Assume (for a contradiction) that the dimensions are bounded; then it follows from 3.6 that for m ≫ 0 and X ∈ C quasi-simple we have

\underline{Hom}(X, τ^{−m}X) = 0.

Hence Ext^{2m}(X, X) = 0 for m ≫ 0. Let Ext^{∗}(X, X) be the Ext-algebra, then it follows that every element in this algebra of positive even degree is nilpotent, and then also every element of positive degree. By [B2, 5.2.3] this happens only when X is projective; and this is a contradiction.

Added in proof. The result mentioned in 4.6 has appeared in Jon F. Carlson, A characterization of endotrivial modules over p-groups, Manuscripta Math. 97 (1998), 303–307. MR 99h:20006

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Mathematical Institute, University of Oxford, 24-29 St. Giles, Oxford OX1 3LB, United Kingdom

   E-mail address: erdmann@maths.ox.ac.uk

Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany

   E-mail address: kerner@cs.uni-duesseldorf.de