Endo-monomial modules over p-groups and their classification in the abelian case

Robert Hartmann Birkenfeldstr. 6 86157 Augsburg Germany hartmann@math.ucsc.edu

September 16, 2002

Abstract

The class of endo-monomial modules is introduced, generalizing endopermutation modules. They can be classified by determining the monomial Dade group, and for abelian p-groups, this group is proven to be isomorphic to the ordinary Dade group of endo-permutation modules.

1 Introduction

For a prime p and a p-group P, denote by RP the group algebra over R, where R may be either an algebraically closed field F of characteristic p or a complete discrete valuation ring O with residue field F and maximal ideal ℘. An RGmodule M is called an endo-permutation module if its endomorphism ring is a permutation RP -module under the conjugation action of P. These modules had been introduced by Dade in [7], and he also showed that they can be classified by determining the structure of an abelian group D_{R}(P), which is now called the Dade group of P. Since then, endo-permutation modules appeared in various contexts in representation theory of finite groups. They occur as source modules for nilpotent blocks, see Puig, [10], and in [11], Rickard uses them to construct endo-split permutation resolutions, which give rise to splendid equivalences, thus proving Broué’s conjecture for p-nilpotent groups with abelian Sylow p-subgroups. For abelian P, Dade classified endo-permutation modules completely, both over F P and OP. For general P, the classification over F is now close to completion, by work of Bouc-Thévenaz [5], Alperin [1], [2], and Carlson-Thévenaz [6].

However, plenty of questions remain open. In particular, it is not clear whether any given endo-permutation F P -module V can be lifted to an endopermutation OP -module M, i.e., such that F ⊗ M ≅ V.

We will suggest to consider slightly more general objects. Instead of endopermutation modules we define endo-monomial OP -modules to be finitely generated, O-free OP -modules whose endomorphism algebra is monomial as OP - module, i.e., it is a direct sum of induced modules Ind_{Q}^{P}X, where Q is a subgroup of P, and X is an OQ-module of O-rank one. Note that, over F, the notions of endo-monomial and endo-permutation OP -modules conincide. The main result of this article is that for abelian P every indecomposable endo-monomial OP - module is in fact an endo-permutation module. We hope to convince the reader that in some sense, endo-monomial OP -modules are the natural objects lifting endo-permutation F P -modules.

In more detail, we will proceed as follows.

In Section 2, we start defining endo-monomial modules, and prove some elementary properties. The results are in almost complete analogy to the results of Dade in [7] for endo-permutation modules. In particular, we define capped endo-monomial modules to be those who have an indecomposable direct summand with vertex P, and we call any such summand a cap of the module. However, the cap is now not unique up to isomorphism anymore, but up to tensoring with a rank one module. We define the monomial Dade group DO^{mon}(P) as the set of equivalence classes of capped endo-monomial OP -modules under the equivalence relation

M ∼ N ⟺ cap(M) ≅ X ⊗ cap(N)

for some OP -module X of rank one. Again, this becomes an abelian group under

[M] + [N] = [M ⊗ N],

where [M] denotes the class of M under the above relation. There is an obvious group homomorphism

η : DO^{mon}(P) ⟶ D_{F}(P),

mapping [M] to [F ⊗ M]. The next section is dedicated to the generalized Brauer construction. This is a generalization of the classical Brauer map, which is a useful tool for monomial and endo-monomial modules, as the classical Brauer map is for permutation and endo-permutation modules. Instead of the usual Q-fixed points M^{Q} of an OP -module M, one uses generalized (Q, ψ)-fixed points M^{(Q,ψ)}, which are those elements of M on which the element x of a subgroup Q acts just by multiplication with the value ψ(x) of the linear character ψ of Q. This idea was first introduced by Boltje and Külshammer in [4] where they used it to examine linear source modules. While their generalized Brauer construction (using generalized fixed points) uses only the module structure, we will extend it to P -algebras. It will be applied later to the P -algebra E = End_{O}(M), where M is an endo-monomial OP -module.

In section 4, we will use the generalized Brauer constructions to investigate the structure of E = End_{O}(M) for an indecomposable endo-monomial OP - module M. Of particular interest are the relations between generalized Brauer kernels and the radical of E^{P} and related algebras. In the final section, the structural results are used to prove the main result, giving a complete classification for endo-monomial modules, in the case that P is abelian. It turns out that every indecomposable endo-monomial OP -module is already an endo-permutation module. In fact, there is an isomorphism between the monomial Dade group (over O) and the ordinary Dade group (over F). This suggests that, when working over O, the compatibility condition for endomonomial modules is more natural than the one for endo-permutation modules. For the lifting problem mentioned earlier, this would mean that one should rather try to lift to endo-monomial modules. However, we cannot generalize the results of this article to the non-abelian case. In particular, it is not known whether there exist indecomposable endo-monomial OP -modules with vertex P, which are not endo-permutation. In a sequel to this article, we will further examine the non-abelian situation.

The contents of this paper are part of the author’s PhD thesis at the University of California in Santa Cruz under the supervision of Professor Boltje. The author wants to thank Robert Boltje for suggesting the topic and giving helpful advice.

Notation Throughout, p is a prime number, and P will always denote a p-group. We fix a p-modular system (K, O, F) , which is large enough for P, and we also assume that F = O/℘ is algebraically closed. Here ℘ denotes the maximal ideal of the local ring O. All OP -modules are assumed to be finitely generated and O-free. Whenever we use the symbol ⊗, we mean the tensor product over the ground ring, usually O or F, which should be clear from the context. By P̂ we denote the group of linear characters of P, i.e., P̂ = Hom(P, O^{×}). For φ ∈ P̂, we denote by O_{φ} the OP -module of rank one, on which an element x ∈ P acts by multiplication with φ(x). We define M_{P} to be the set of all pairs (Q, ψ), where Q is a subgroup of P and ψ ∈ Q̂. Then P acts by conjugation on M_{P}, and we write M_{P} /P for a set of representatives for the P -orbits. M_{P} has the structure of a poset via the ordering

(R, λ) ≤ (Q, ψ) :⇔ R ≤ Q and ψ|_{R} = λ.

2 Endo-monomial Modules

After defining endo-monomial modules, we prove some basic properties. All the results in this section as well as their proofs are easy generalizations of [7].

Recall that if M is an OP -module, and if E := End_{O}(M) denotes its endomorphism ring, then E is a P -algebra and, in particular, an OP -module itself with P -action given by

(^{x}f)(m) = x · f(x^{−1} · m) for f ∈ E, x ∈ P and m ∈ M.

2.1 Definition An OP -module M is called endo-monomial if its endomorphism algebra End_{O}(M), viewed as an OP -module, is monomial.

If M is an endo-monomial OP -module and E := End_{O}(M), then E has a decomposition

(2.1.a)  E = ⨁_{(Q,ψ)∈M_{P} /P} E[Q, ψ],

where E[Q, ψ] ≅ n_{(Q,ψ)} × Ind_{Q}^{P}O_{ψ} and n_{(Q,ψ)} ≥ 0.

Note that the decomposition (2.1.a) is not unique, though the numbers n_{(Q,ψ)} are. Unless otherwise mentioned, in the following we will fix any such decomposition.

The following proposition contains some basic properites of the class of endomonomial OP -modules. Their proofs are straight forward, and therefore omitted.

2.2 Proposition (i) Let M, N be endo-monomial OP -modules. Then also M^{∗}, M ⊗ N amd Hom_{O}(M, N) are endo-monomial OP -modules. (ii) Let M be an endo-monomial OP -module, and let Q ≤ P. Then Res_{Q}^{P}M is an endo-monomial OQ-module. (iii) Direct summands of endo-monomial modules are endo-monomial. (iv) Monomial and endo-permutation modules are endo-monomial.

As it is for endo-permutation modules, direct sums of endo-monomial modules need not be endo-monomial.

2.3 Lemma Let M and N be endomonomial OP -modules. The following are equivalent:

(i) M ⊕ N is endo-monomial. (ii) Hom_{O}(M, N) is monomial. (iii) Hom_{O}(N, M) is monomial. In this case we say that M and N are compatible as endo-monomial modules.

Proof The second and third condition are equivalent, since

Hom_{O}(N, M) ≅ N^{∗} ⊗ M ≅ (M^{∗} ⊗ N)^{∗} ≅ Hom_{O}(M, N)^{∗}

and a module is monomial if and only if its dual is. Now, if M and N are endo-monomial then End_{O}(M) and End_{O}(N) are monomial. Hence

End_{O}(M ⊕ N) ≅ End_{O}(M) ⊕ End_{O}(N) ⊕ Hom_{O}(M, N) ⊕ Hom_{O}(N, M)

is monomial if and only if both Hom_{O}(M, N) and Hom_{O}(N, M) are. □

Note that this compatibility condition is weaker than the one for endopermutation modules. For example, if O_{φ} and O_{λ} are two non-isomorphic rank-one OP -modules (φ, λ ∈ P̂, φ ≠ λ), then they are not compatible as endopermutation modules, but compatible as endo-monomial modules, since

Hom_{O}(O_{φ}, O_{λ}) ≅ (O_{φ})^{∗} ⊗ O_{λ} ≅ O_{φ^{−1}λ}

is monomial, but not a permutation module. The following lemma now follows using an easy induction argument:

2.4 Lemma Let K_{1}, …, K_{r} be endo-monomial OP -modules, where r ≥ 2. Then K_{1}⊕⋯⊕K_{r} is endo-monomial if and only if each pair K_{i}, K_{j} (1 ≤ i, j ≤ r) is compatible. □

We also give a condition for induced modules to be endo-monomial:

2.5 Lemma If L is an endo-monomial OQ-module, Q ≤ P, then Ind_{Q}^{P}L is an endo-monomial OP -module if and only if the following holds:

(2.5.a)  The endo-monomial O(Q ∩^{x}Q)-modules Res_{Q∩^{x}Q}^{Q}(L)
and Res_{Q∩^{x}Q}^{^{x}Q}(^{x}L) are compatible for all x ∈ P.

Proof One has OP -isomorphisms

End_{O}(Ind_{Q}^{P}L) ≅ Ind_{Q}^{P}(L^{∗}) ⊗ Ind_{Q}^{P}(L) ≅

≅ Ind_{Q}^{P} (L^{∗} ⊗ Res_{Q}^{P}(Ind_{Q}^{P}L)) ≅

≅ Ind_{Q}^{P} ( ⨁_{x∈Q\P/Q} L^{∗} ⊗ Ind_{Q∩^{x}Q}^{Q}(Res_{Q∩^{x}Q}^{^{x}Q}(^{x}L))) ≅

≅ Ind_{Q}^{P} ( ⨁_{x∈Q\P/Q} Ind_{Q∩^{x}Q}^{Q} (L^{∗}|_{Q∩^{x}Q} ⊗ (^{x}L)|_{Q∩^{x}Q})) ≅

≅ ⨁_{x∈Q\P/Q} Ind_{Q∩^{x}Q}^{P}(L^{∗}|_{Q∩^{x}Q} ⊗ (^{x}L)|_{Q∩^{x}Q}) ≅

≅ ⨁_{x∈Q\P/Q} Ind_{Q∩^{x}Q}^{P} (Hom_{O}(L^{∗}|_{Q∩^{x}Q}, (^{x}L)|_{Q∩^{x}Q})) .

Now Ind_{Q}^{P}L is endo-monomial if and only if Hom_{O}(L^{∗}|_{Q∩^{x}Q}, (^{x}L)|_{Q∩^{x}Q}) is monomial for all x ∈ P, which is equivalent to condition (2.5.a) by Lemma 2.3 and Lemma 2.4. □

3 Generalized Fixpoints and Brauer Constructions

In this section, we introduce a generalized Brauer construction. It can be described in the more general context of G-algebras, where G denotes an arbitrary finite group.

3.1 In [4] Boltje and Külshammer introduced the notion of generalized fixed points as follows: If V is any OG-module, one can define

V^{(H,ψ)} = {v ∈ V | h · v = ψ(h)v for all h ∈ H}

for H ≤ G and ψ ∈ Ĥ. As an OH-module, V^{(H,ψ)} is isomorphic to a direct sum of copies of O_{ψ}. However, since V^{(H,ψ)} is stable under N_{G}(H, ψ), it is a ON_{G}(H, ψ)-module.

Also, one has a generalized trace map

tr_{(I,λ)}^{(H,ψ)} : V^{(I,λ)} → V^{(H,ψ)}, v ↦ ∑_{h∈H/I} ψ(h^{−1})h · v

for (I, λ) ≤ (H, ψ) in M_{G}. We write V_{(I,λ)}^{(H,ψ)} for tr_{(I,λ)}^{(H,ψ)}(V^{(I,λ)}) and define the generalized Brauer construction by

V (H, ψ) = V^{H}/BK(H, ψ; V),

where

BK(H, ψ; V) = ∑_{(I,λ)<(H,ψ)} V_{(I,λ)}^{(H,ψ)} + ℘V^{(H,ψ)}

is the generalized Brauer kernel of V with respect to (H, ψ). Then the generalized Brauer map with respect to (H, ψ) is the canonical surjection

Br_{(H,ψ)} : V^{(H,ψ)} → V (H, ψ),

this is an F N_{G}(H, ψ)-module homomorphism. If H is not a p-group these Brauer constructions are always zero. Note that this is not true for the original definition in [4]; however, in our applications we are only dealing with p-groups and then both definitions coincide.

3.2 Now, if A is a G-algebra, A^{(H,ψ)} is still an F N_{G}(H, ψ)-module, but not multiplicatively closed, since for ψ_{1}, ψ_{2} ∈ Ĥ we have

A^{(H,ψ_{1})} · A^{(H,ψ_{2})} ⊆ A^{(H,ψ_{1}ψ_{2})}.

However, if we set

A^{(H,Ĥ)} := ⨁_{ψ∈Ĥ} A^{(H,ψ)}

then A^{(H,Ĥ)} becomes an F N_{G}(H)-algebra, with the multiplication

(a_{ψ})_{ψ∈Ĥ} · (b_{ψ})_{ψ∈Ĥ} = (∑_{θ∈Ĥ} a_{θ}b_{θ^{−1}ψ})_{ψ∈Ĥ}.

To see that the sum is direct, assume that 0 = ∑_{ψ∈Ĥ} a_{ψ}, with a_{p}si ∈ A^{(H,ψ)}. Then, for any x ∈ H^{ab} = H/H′, one has

0 = ^{x}0 = ^{x}( ∑_{ψ∈Ĥ} a_{ψ}) = ∑_{ψ∈Ĥ} ψ(x)a_{ψ}.

As x runs through H^{ab}, we obtain q = [H : H′] equations, and the (q × q)- coefficient matrix Ψ = (ψ(x))_{ψ∈Ĥ,x∈H^{ab}} with entries in O ⊂ K is the character table of the abelian group H^{ab}, hence invertible in K. Thus a_{ψ} = 0 for all ψ ∈ Ĥ.

Since BK(H, ψ) ⊆ A^{(H,ψ)} for every ψ ∈ Ĥ, certainly

BK(H, Ĥ; A) := ⨁_{ψ∈Ĥ} BK(H, ψ; A)

is contained in A^{(H,Ĥ)}. Even more, we have

3.3 Lemma BK(H, Ĥ; A) is a two-sided ideal in A^{(H,Ĥ)}.

Proof We show that the generalized Brauer kernel is a left ideal. A very similar calculation shows it is also a right ideal. It is sufficient to show that if f ∈ BK(H, ψ; A) for some ψ ∈ Ĥ and g ∈ A^{(H,Ĥ)}, then gf ∈ BK(H, Ĥ; A). If f ∈ ℘A^{(H,ψ)} then certainly also gf ∈ ℘A^{(H,Ĥ)}, so we can assume that f ∈ ∑_{(K,η)<(H,ψ)} A_{(K,η)}^{(H,ψ)}. It is even enough to consider f ∈ A_{(K,η)}^{(H,ψ)} for some fixed

(K, η) < (H, ψ), so let f = tr_{(K,η)}^{(H,ψ)}(h) with h ∈ A^{(K,η)}. Finally, it suffices to

consider g = g_{λ} ∈ A^{(H,λ)} for some fixed λ ∈ Ĥ. Then an easy calculation shows that

g_{λ}f = tr_{(K,(ψλ)|_{K})}^{(H,ψλ)}(g_{λ}h) ∈ BK(H, ψλ; A) ⊆ BK(H, Ĥ; A).

□

3.4 Now we can define

A(H, Ĥ) := A^{(H,Ĥ)}/BK(H, Ĥ; A),

which is an algebra over F. Moreover, since both A^{(H,Ĥ)} and BK(H, Ĥ; A) are stable under N_{G}(H), and since the commutator subgroup H′ is contained in ker(ψ) for every ψ ∈ Ĥ and thus acts trivially on every A^{(H,ψ)}, the factor group N_{G}(H)/H′ acts on A(H, Ĥ). We call the canonical surjection

(3.4.a)  Br_{(H,Ĥ)}^{A} : A^{(H,Ĥ)} → A^{(H,Ĥ)}/BK(H, Ĥ; A)

the generalized Brauer map with respect to H. We write just Br_{(H,Ĥ)} whenever it is clear which algebra we are talking about.

4 Capped modules and the monomial Dade groups

4.1 Let M be any endo-monomial OP -module, set E := End_{O}(M) and fix a decomposition (2.1.a).

For Q ≤ P we have the usual trace map

tr_{Q}^{P} : E^{Q} → E^{P} , g ↦ ∑_{x∈P/Q}^{x}g

and the ideal

BK(P; E) := ∑_{Q<P} tr_{Q}^{P}E^{Q} + ℘E^{P}

in E^{P}. Of course the decomposition (2.1.a) implies that

(4.1.a)  E^{P} = E[P, 1] ⊕ ⨁_{Q∈S_{P}^{<}/P}(E[Q, 1])^{P} ,

since the direct summands of the form Ind_{Q}^{P}O_{ψ} with nontrivial ψ ∈ Q̂ do not have any P -fixed points. Here S_{P}^{<} denotes the set of all proper subgroups of P, and as before, S_{P}^{<}/P denotes a set of representatives for the P -conjugacy classes of this set.

In terms of the decomposition (2.1.a), the Brauer kernel BK(P; E) has the following description:

4.2 Lemma For any φ ∈ P̂ one has

(4.2.a) BK(P, φ; E) = ℘E[P, φ] ⊕ ⨁_{(Q,ψ)∈M_{P} /P, (Q,ψ)<(P,φ)}(E[Q, ψ])^{(P,φ)},

and

(4.2.b)  E^{(P,φ)} = E[P, φ] + BK(P, φ; E).

In particular, for φ = 1, the first statement says

(4.2.c)  BK(P; E) = ℘E[P, 1] ⊕ ⨁_{Q∈S_{P}^{<}/P}(E[Q, 1])^{P} .

Proof Clearly, for λ ∈ P̂, we have that (O_{λ})^{(P,φ)} = 0 unless λ = φ and (O_{φ})^{(P,φ)} = O_{φ}. So the second statement will be immediate from the first, which we will prove. By definition, ℘E^{(P,φ)} ⊆ BK(P, φ; E). Also we have

E^{(P,φ)} = E[P, φ] ⊕ ⨁_{(Q,ψ)∈M_{P} /P, Q<P} n_{(Q,ψ)} × (Ind_{Q}^{P}O_{ψ})^{(P,φ)}.

By Corollary 2.16 in [4], (Ind_{Q}^{P}O_{ψ})^{(P,φ)} = 0 unless φ|_{Q} = ψ, and in the latter case it equals (Ind_{Q}^{P}O_{ψ})^{(P,φ)} = O · tr_{(Q,ψ)}^{(P,φ)}(1 ⊗_{OQ} 1), which clearly lies in BK(P, φ; E), and the result follows. □

In the following we will restrict our attention to endo-monomial OP -modules which have an indecomposable direct summand with vertex P. A justification for that will be given in Lemma 4.9 at the end of this section.

The following lemma will allow us to distinguish these modules easily.

4.3 Lemma Let M be an indecomposable endo-monomial OP -module, and fix a decomposition (2.1.a) of its endomorphism algebra E. Then: M has vertex P if and only if E[P, 1] ≠ 0, i.e., if the trivial OP -module O is a direct summand of E.

Proof By Higman’s criterion, M has vertex P if and only if the identity map 1_{M} is not in E_{Q}^{P} for any Q < P. By Rosenberg’s Lemma, this is equivalent to saying that 1_{M} is not in ∑_{Q<P} E_{Q}^{P}. Certainly 1_{M} is not in ℘E^{P}, so that M has vertex P if and only if 1 ∉ BK(P; E). This in turn is equivalent to E^{P} = 1·E^{P} ≠ BK(P; E). Hence, M has vertex P if and only if BK(P; E) ≠ E^{P} which, by (4.1.a) and (4.2.c), is equivalent to ℘E[P, 1] ≠ E[P, 1]. But ℘E[P, 1] = E[P, 1] is only possible if E[P, 1] = 0. This proves the lemma. □

4.4 Proposition Two indecomposable endo-monomial OP -modules M and N with common vertex P are compatible if and only if there exists φ ∈ P̂ such that N ≅ O_{φ} ⊗ M.

Proof If N ≅ O_{φ} ⊗ M for some φ ∈ P̂ then

Hom_{O}(M, N) ≅ M^{⋆} ⊗ N ≅ M^{⋆} ⊗ M ⊗ O_{φ} ≅ End_{O}(M) ⊗ O_{φ},

which is monomial, since End_{O}(M) is.

Conversely, suppose M and N are compatible, then Hom_{O}(M, N) is monomial, and we have

Hom_{O}(M, N) ≅ M^{⋆} ⊗ N ≅ ⨁_{(Q,ψ)∈M_{P} /P} m_{(Q,ψ)} × Ind_{Q}^{P}O_{ψ}

for certain integers m_{(Q,ψ)} ≥ 0. Hence

M ⊗ M^{⋆} ⊗ N ≅ ⨁_{(Q,ψ)∈M_{P} /P} m_{(Q,ψ)} × (M ⊗ Ind_{Q}^{P}O_{ψ}) ≅

≅ ⨁_{(Q,ψ)∈M_{P} /P} m_{(Q,ψ)} × Ind_{Q}^{P}(M|_{Q} ⊗ O_{ψ})

and any indecomposable direct summand of M ⊗ M^{⋆} ⊗ N with vertex P must be isomorphic to M ⊗ O_{φ} for some φ ∈ P̂. Now, by Lemma 4.3, End_{O}(M) ≅ M^{⋆} ⊗ M has a direct summand isomorphic to O. Therefore N ≅ O ⊗ N is an indecomposable direct summand of M ⊗ M^{⋆} ⊗ N, so that N ≅ M ⊗ O_{φ} □

4.5 Remark If M is an indecomposable endo-permutation OP -module, and if φ ∈ P̂ is such that M and N := M ⊗ O_{φ} are non-isomorphic, then they are not compatible as endo-permutation modules, but they are compatible as endomonomial modules. However, after reduction modulo ℘, the endo-permutation F P -modules \overline{M} and \overline{N} become isomorphic, thus compatible. This is a first indication that the compatibility condition for endo-monomial OP -modules defined above lifts the compatiblity of endo-permutation F P -modules in a more natural way than the compatibility of endo-permutation OP -modules as defined in [7].

4.6 Now let L be an arbitrary endo-monomial OP -module. Proposition 4.4 says that any two direct summands M and N of L with common vertex P satisfy M ≅ N ⊗ O_{φ} for some φ ∈ P̂. If L has at least one indecomposable direct summand with vertex P then we call L capped. Also, we call any such summand M the cap of L and write M = cap(L). Thus the cap of L is defined only up to isomorphism and tensoring with some O_{φ}.

We define an equivalence relation

(4.6.a)  M ∼ M′ :⇔ M ≅ M′ ⊗ O_{φ} for some φ ∈ P̂

on the class of indecomposable endo-monomial OP -modules with vertex P.

Then there is a well defined map from the class of capped endo-monomial OP -modules to the set of equivalence classes under this relation, mapping L to [M], if M is an indecomposable direct summand of L with vertex P. This induces an equivalence relation on the class of capped endo-monomial OP - modules

(4.6.b)  L ∼ L′ :⇔ cap(L) ≅ cap(L′) ⊗ O_{φ} for some φ ∈ P̂.

By DO^{mon}(P) we denote the set of equivalence classes of capped endo-monomial OP -modules under this relation. Clearly, any class [L] contains precisely one equivalence class [M] of indecomposable endo-monomial OP -modules under the first relation, namely the class of M = cap(L). In other words, one can always choose an indecomposable OP -module as a representative for a class.

4.7 Lemma Let K, L be capped endo-monomial OP -modules. Then also the endo-monomial modules K ⊗L, K^{⋆}, Hom_{O}(K, L), Res_{Q}^{P}(K) for any Q ≤ P and K ⊕ L (if compatible) are capped.

Proof By Lemma 4.3, an endo-monomial OP -module is capped if and only if its endomorphism algebra has a trivial direct summand.

Set E_{K} = End_{O}(K) and E_{L} = End_{O}(L), then we have

E_{K} ≅ O ⊕ X and E_{L} ≅ O ⊕ Y

for some monomial OP -modules X and Y. Thus

End_{O}(K ⊗ L) ≅ E_{K} ⊗ E_{L} ≅ (O ⊕ X) ⊗ (O ⊕ Y) ≅ O ⊕ X ⊕ Y ⊕ (X ⊗ Y).

Thus K ⊗ L is capped. Also,

End_{O}(K^{∗}) ≅ (E_{K})^{∗} ≅ O ⊕ X^{∗},

so that K^{∗} is capped, and therefore also Hom_{O}(K, L) ≅ K^{∗} ⊗ L. Certainly End_{O}(K|_{Q}) ≅ (E_{K})|_{Q} ≅ O ⊕ X|_{Q}, so that K|_{Q} is capped. Finally, if K and L are compatible, then

End_{O}(K ⊕ L) ≅ E_{K} ⊕ E_{L} ⊕ Hom_{O}(K, L) ⊕ Hom_{O}(L, K) ≅
≅ O ⊕ X ⊕ O ⊕ Y ⊕ Hom_{O}(K, L) ⊕ Hom_{O}(L, K),

thus K ⊕ L is capped. □

Completely analogous to the Dade group, the set DO^{mon}(P) has the structure of an abelian group, given by

[L_{1}] + [L_{2}] := [L_{1} ⊗ L_{2}].

The identity element is the class of the trivial module O, and the inverse of [L] is [L^{⋆}], since [L ⊗ L^{⋆}] = [End_{O}(L)] = [O] by Lemma 4.3. We call DO^{mon}(P) the monomial Dade group of P.

Equivalently, one can define DO^{mon}(P) as the set of equivalence classes of indecomposable endo-monomial OP -modules with vertex P under the relation (4.6.a). Then the group operation is given by

[M_{1}] + [M_{2}] := [cap(M_{1} ⊗ M_{2})].

Here it is necessary to choose a cap, since the module M_{1} ⊗ M_{2} will in general not be indecomposable. It is clear from the construction that this yields an isomorphic group.

4.8 Now let L be an arbitrary endo-monomial OP -module, and fix a decomposition (2.1.a) of E = End_{O}(L). Then the natural homomorphism

L → F ⊗ L, l ↦ 1 ⊗ l

is surjective with kernel ℘L, so F ⊗ L ≅ L/℘L =: \overline{L}. We also have natural isomorphisms

\overline{E} := E/℘E ≅ F ⊗ E ≅ End_{F} (\overline{L})

of F P -modules, and the decomposition (2.1.a) implies that

\overline{E} = ⨁_{(Q,ψ)∈M_{P} /P} \overline{E[Q, ψ]},

where

\overline{E[Q, ψ]} ≅ n_{(Q,ψ)} × (F ⊗ Ind_{Q}^{P}O_{ψ}) ≅ n_{(Q,ψ)} × Ind_{Q}^{P}F.

Hence \overline{L} is an endo-permutation F P -module. In other words, reduction modulo ℘ induces a group homomorphism

DO^{mon}(P) → D_{F}(P),

where D_{F} (P) denotes the Dade group of F P. It is easy to see that this map is well defined.

We conclude this section with the promised justification for focusing our interest on modules with full vertex P. This result is completely analogous to Theorem 6.6 in [7].

4.9 Lemma If L is an indecomposable endo-monomial OP -module with vertex Q then Res_{Q}^{P}(L) is a capped endo-monomial OQ-module, and any cap S of Res_{Q}^{P}(L) is a source for L, i.e., L ≅ Ind_{Q}^{P}(S).

Proof Since L has vertex Q, there exists an indecomposable direct summand S of Res_{Q}^{P}(L), which is a source for L, so that S has vertex Q and L is isomorphic to a direct summand of Ind_{Q}^{P}(S). By Proposition 2.2, S is endo-monomial, and by Green’s indecomposability theorem, since P is a p-group, Ind_{Q}^{P}(S) is indecomposable, hence L ≅ Ind_{Q}^{P}(S). □

5 The Brauer kernels

Throughout, let L be an indecomposable endo-monomial OP -module with vertex P, and fix a decomposition (2.1.a) of E = End_{O}(L). In this section we will study the (generalized) Brauer kernels BK(P, φ; E) for φ ∈ P̂. We start with the case φ = 1:

5.1 Lemma BK(P; E) ⊆ rad(E^{P})

Proof E^{P} = End_{OP} (L) is a local ring since L is indecomposable, hence rad(E^{P}) is the unique maximal ideal in E^{P}. But also BK(P; E) is an ideal in E^{P}, and the description (4.2.c) of BK(P; E) and Lemma 4.3 imply that BK(P; E) ≠ E^{P}, therefore BK(P; E) ⊆ rad(E^{P}). □

5.2 Since E ≅ M_{n}(O) where n = rk_{O}(L), the trace form

(5.2.a)  < −, − >: E × E → O, < a, b >:= tr(ab) for a, b ∈ E,

defines a symmetric non-degenerate P-invariant bilinear form on E. Thus the map

Γ : E → E^{⋆} = Hom_{O}(E, O), a ↦ (b ↦< a, b >)

is an injective homorphism of OP -modules. To see that it is even an isomorphism, denote by e_{ij} the matrix in Mat_{n}(O) ≅ E with all entries zero except the (i, j)-th which is one. Then the elements f_{ij} (1 ≤ i, j ≤ n) defined by

f_{ij}(e_{kl}) = δ_{ik}δ_{jl}

form a basis for E^{⋆}. Now Γ(e_{ji}) = f_{ij}, which shows that Γ is surjective.

5.3 Lemma rad(E^{P}) = BK(P; E)

Proof The inclusion BK(P; E) ⊆ rad(E^{P}) has been shown in Lemma 5.1. Conversely, suppose there exists j ∈ rad(E^{P}) ∖ BK(P; E). We show that we can choose j such that Oj is a trivial direct summand of E. Write E = T ⊕ N with T ≅ r × O and N having no trivial direct summand as OP -modules. Then E^{P} = T ⊕ N^{P} and

E(P) = (T^{P} ⊕ N^{P})/BK(P; E) = (T ⊕ N^{P})/BK(P; E).

But N^{P} is contained in BK(P; E) by (4.2.c), hence

E(P) ≅ T/(BK(P; E) ∩ T) = T/BK(P; T)

Since j ∉ BK(P; E), the image of j in E(P) is non-zero, and if we write j = j_{N} + j_{T} with j_{N} ∈ N and j_{T} ∈ T then Br_{P} (j_{T}) ≠ 0 and Br_{P} (j_{N}) = 0, so that j_{T} ∉ BK(P; E), but j_{T} = j − j_{N} ∈ rad(E^{P}), since j ∈ rad(E^{P}) and j_{N} ∈ BK(P; E) ⊆ rad(E^{P}) by Lemma 5.1. Hence we can replace j by j_{T} and assume that j ∈ T. Now T ≅ O ⊕ ⋯ ⊕ O and BK(P; T) ≅ ℘ ⊕ ⋯ ⊕ ℘, so that

Br_{P}^{T} : T = T^{P} → T (P) ≅ F ⊕ ⋯ ⊕ F

and Br_{P}^{T} (j) ≠ 0 in T (P), hence j ∉ ℘T. Choose an O-basis x_{1}, ⋯, x_{r} of T and write j = α_{1}x_{1} +⋯+α_{r}x_{r}. Then at least one coefficient is not in ℘, say α_{1} ∉ ℘. Now we can replace x_{1} by j in the basis. Since x_{1} = α_{1}^{−1}(j − α_{2}x_{2} − ⋯ − α_{r}x_{r}), this is still a basis of T and

E^{P} = T ⊕ N = Oj ⊕ Ox_{2} ⊕ ⋯ ⊕ Ox_{r} ⊕ N

Now we can write E = Oj ⊕ R for some OP -submodule R of E. By 5.2, there exists i ∈ E such that < i, j >= 1 and < i, r >= 0 for r ∈ R. Then i ∈ E^{P} since Γ is an OP -isomorphism and

Γ(i) ∈ (Hom_{O}(E, O))^{P} = Hom_{OP} (E, O)

by the choice of i. Since j ∈ rad(E^{P}) and rad(E^{P}) is an ideal, also ij ∈ rad(E^{P}). Now denote by \overline{ij} the image of ij under the canonical epimorphism E → \overline{E}, which maps E^{P} onto F ⊗ E^{P}, thus it maps rad(E^{P}) to rad(F ⊗ E^{P}). Since F ⊗E^{P} is a finite dimensional F -algebra, every element in rad(F ⊗E^{P}) is nilpotent, hence \overline{ij} is nilpotent and tr(\overline{ij}) = 0. But tr(\overline{ij}) = 1 by the choice of i, which is a contradiction. □

5.4 Corollary If L is an indecomposable endo-monomial OP -module with vertex P then E = End_{O}(L) has precisely one trivial direct summand.

Proof Since L is indecomposable, E^{P} = End_{OP} (L) is a local ring and

End_{OP} (L)/rad(End_{OP} (L))

is a division algebra over F which must be isomorphic to F, since we assumed that F is algebraically closed (see for example [12], 1.7 and 4.6). But by Lemma 5.3 this implies that

(5.4.a)  E^{P} /BK(P; E) ≅ F.

Finally, by (4.1.a) and (4.2.c) this is equivalent to

E[P, 1]/℘E[P, 1] ≅ F

which implies that E[P, 1] ≅ O. □

5.5 Lemma One can choose the decomposition (2.1.a) in such a way that E[P, 1] = O1_{M} where 1_{M} is the identity on M.

Proof We can write E = Of ⊕ R_{0} for some f ∈ E^{P} and some OP -submodule R_{0} of E. Then write 1_{M} = αf +r_{0} with α ∈ O and r_{0} ∈ R_{0}. Now E^{P} = Of ⊕R_{0}^{P} and r_{0} ∈ R_{0}^{P} since 1_{M}, f ∈ E^{P}. For g ∈ E^{P} denote by \overline{g} the image under the Brauer map

E^{P} → E^{P} /BK(P; E),

and for γ ∈ O let \overline{γ} ∈ F be the image under the map O → O/℘ = F. Then we have

\overline{1_{M}} = \overline{α}\overline{f} + \overline{r_{0}} ≠ 0.

If \overline{α}\overline{f} = 0 then \overline{r_{0}} ≠ 0. But by (4.1.a) and (4.2.c) we obtain that R_{0}^{P} ⊆ BK(P; E) and therefore \overline{r_{0}} = 0. Hence \overline{α}\overline{f} ≠ 0. So we have α ∈ O^{×} and we can write f = α^{−1}1_{M} − α^{−1}r_{0}, which shows that O1_{M} + R_{0} = E. To see that it is a direct sum, let x ∈ O1_{M} ∩ R_{0}, then x = γ1_{M} = r_{1} ∈ R_{0} for some γ ∈ O, hence γ(αf + r_{0}) = r_{1} or γαf = r_{1} − γr_{0} ∈ R_{0} ∩ Of = 0. But α ≠ 0, hence γ = 0 and x = 0. □ 5.6 Lemma Assume that f ∈ E^{(P,φ)} generates an OP -direct summand of E isomorphic to O_{φ} for some φ ∈ P̂, φ ≠ 1. Then there exists an element g ∈ E^{(P,φ^{−1})} such that < f, g >= 1, and g generates an OP -direct summand of E isomorphic to O_{φ^{−1}}.

Proof Write E = Of ⊕ R for some OP -submodule R of E. The isomorphism Γ : E → E^{⋆} from 5.2 gives us an element g ∈ E such that Γ(g) maps f to 1 and R to 0. Then for x ∈ P we have

1 = Γ(g)(f) =< g, f >=< ^{x}g,^{x}f >=< ^{x}g, φ(x)f >=
= φ(x) < ^{x}g, f >= φ(x)Γ(^{x}g)(f)

and for r ∈ R

Γ(^{x}g)(r) =< ^{x}g, r >=< ^{x^{−1}x}g,^{x^{−1}}r >=< g,^{x^{−1}}r >= 0,

since also ^{x^{−1}}r ∈ R. So we have Γ(^{x}g)(f) = φ^{−1}(x) and Γ(^{x}g)(r) = 0 for all r ∈ R. Hence under the isomorphism Γ the two elements ^{x}g and φ^{−1}(x)g are both mapped to h ∈ End_{O}(E, O) defined by

h(f) = φ^{−1}(x) and h(r) = 0 for r ∈ R.

Therefore ^{x}g = φ^{−1}(x)g, which means that g ∈ E^{(P,φ^{−1})}. It remains to show that Og is a direct summand of E. Choose an O-basis {f, r_{1}, …, r_{n−1}} of E such that

E = Of ⊕ R as OP -module
and R = Or_{1} ⊕ ⋯ ⊕ Or_{n−1} as O-module,

and denote by {f̂, r̂_{1}, …, r̂_{n−1}} the basis of E^{⋆} defined by

f̂(f) = 1, f̂(r_{i}) = 0
and r̂_{j}(f) = 0, r̂_{j}(r_{i}) = δ_{ij}

for i, j = 1, …, n − 1, and let R̂ = Or̂_{1} ⊕ ⋯ ⊕ Or̂_{n−1}. Note that Γ(g) = f̂. Then for x ∈ P and i = 1, …, n − 1

(^{x}r̂_{i})(f) = r̂_{i}(^{x^{−1}}f) = r̂_{i}(φ(x^{−1})f) = φ(x^{−1})r̂_{i}(f) = 0,

so R̂ is an OP -submodule of E^{⋆} and so is Of̂, hence E^{⋆} = Of̂ ⊕ R̂ as OP - module. But Of̂ ≅ O_{φ^{−1}}, and

E^{⋆} ≅ OΓ(g) ⊕ R̂,

which implies that under the isomorphism Γ^{−1} we obtain

E ≅ Og ⊕ Γ^{−1}(R̂) ≅ O_{φ^{−1}} ⊕ Γ^{−1}(R̂).

□ 5.7 Proposition In the decomposition (2.1.a) the numbers n_{(Q,ψ)} for (Q, ψ) ∈ M_{P} /P satisfy

n_{(P,1)} = 1 and n_{(Q,ψ)} = n_{(Q,ψ^{−1})}.

Proof In Corollary 5.4 we showed already that n_{(P,1)} = 1. In 5.2, we constructed an OP -isomorphism Γ : E → E^{∗}. But

E ≅ ⨁_{(Q,ψ)∈M_{P} /P} n_{(Q,ψ)} × Ind_{Q}^{P}O_{ψ},

hence

E^{∗} ≅ ⨁_{(Q,ψ)∈M_{P} /P} n_{(Q,ψ)} × Ind_{Q}^{P}O_{ψ^{−1}},

and the result follows by comparing the two decompositions. □

6 Crossed products

We need to recall some facts about group-graded rings.

6.1 Let G be any finite group. A G-graded ring is a ring A together with a decomposition

A = ⨁_{g∈G} A_{g}

into additive subgroups A_{g} such that A_{g}A_{h} ⊆ A_{gh} for all g, h ∈ G. The 1component A_{1} is always a unitary subring of A. A graded ideal of A is an ideal I with the property that

I = ⨁_{g∈G} I ∩ A_{g}.

Then A/I becomes a G-graded ring itself, with the decomposition

A/I = ⨁_{g∈G} A_{g}/(I ∩ A_{g}).

A G-graded ring A is called fully graded if A_{g}A_{h} = A_{gh} for all g, h ∈ G. A crossed product is a G-graded ring A such that A_{g} ∩A^{×} ≠ ∅ for all g ∈ G. Every crossed product is fully graded. For an arbitrary G-graded ring A we set

G_{A} = {g ∈ G | A_{g} ∩ A^{×} ≠ ∅}.

The following lemma is proved in [9], Lemma 7-9:

6.2 Lemma (Fan, Külshammer) Let A be a G-graded ring with the decomposition A = ⨁_{g∈G} A_{g}.

(i) If I = ⨁_{g∈G} I_{g} is a graded ideal of A with the additional property that I_{1} ⊆ rad(A_{1}), then I ⊆ rad(A).

(ii) G_{A} is a subgroup of G.

(iii) If A_{1} is a division ring, then the G_{A}-graded ring ⨁_{g∈G_{A}} A_{g} is a crossed product, and ⨁_{g∈G∖G_{A}} A_{g} is a nilpotent graded ideal of A.

Next we quote a result about crossed products over p-groups from [3]. To avoid the introduction of new notations which will not be needed in the following, we will state only a less general version of the original result.

6.3 Proposition (Betsch, Schmid, [3]) Let A = ⨁_{g∈P} A_{g} be a crossed product such that A_{1} = F is an algebraically closed field of characteristic p, and P is a finite p-group. Then

A/rad(A) ≅ F.

We just remark that our algebraically closed field F of characteristic p certainly is perfect (i.e. F^{p} = F) and primary as an F -algebra, i.e. F/rad(F) is simple artinian (we even have rad(F) = 0), and finally that P acts trivially on F = A_{1}. With that in mind, the above stated proposition can be easily derived from the original version in [3].

6.4 We continue to examine the indecomposable endo-monomial OP -module L with vertex P and its endomorphism algebra E = End_{O}(L) with a fixed decomposition (2.1.a). Recall that

BK(P, P̂; E) = ⨁_{φ∈P̂} BK(P, φ; E),

and

E(P, P̂) = E^{(P,P̂)}/BK(P, P̂; E) ≅ ⨁_{φ∈P̂} E^{(P,φ)}/BK(P, φ; E).

Obviously, BK(P, P̂; E) is a graded ideal in the P̂-graded ring E^{(P,P̂)}, so that E(P, P̂) is a P̂-graded ring itself, and the 1-component E(P, 1) is isomorphic to the field F by (5.4.a). Therefore, by Lemma 6.2 (iii), we can write

(6.4.a)  E(P, P̂) = N ⊕ C

where N is a nilpotent graded ideal in E(P, P̂) and

(6.4.b)  C = ⨁_{φ∈X} E(P, φ)

for some subgroup X of P̂, in the notation of Lemma 6.2

X = P̂_{E(P,P̂)} = {φ ∈ P̂ | E(P, φ) contains a unit }.

Also note that by Lemma 5.3 we have

BK(P, 1; E) = rad(E^{(P,1)}),

so we can apply Lemma 6.2 (i) and obtain that

(6.4.c)  BK(P, P̂; E) ⊆ rad(E^{(P,P̂)}).

6.5 Lemma E(P, P̂) is a crossed product.

Proof By the above it suffices to show that the nilpotent part N in (6.4.a) is zero. So assume that there exists 0 ≠ \overline{f} ∈ E(P, φ), where φ ∉ X. In particular, φ ≠ 1. Let f be a preimage of \overline{f} under the canonical epimorphism Br_{(P,φ)}, then f ∈ E^{(P,φ)}. Now fix a decomposition (2.1.a) of E; by (4.2.b) we have

E^{(P,φ)} = E[P, φ] + BK(P, φ; E).

So we can write f = f_{0} + f_{B} with f_{0} ∈ E[P, φ] and f_{B} ∈ BK(P, φ; E). Since f ∉ BK(P, φ; E), we conclude that f_{0} ∉ ℘E[P, φ]. Therefore f_{0} generates a rank-one direct summand of E isomorphic to O_{φ}. Now we can replace f by f_{0}, and assume that f generates a rank-one direct summand of E isomorphic to O_{φ}. By Lemma 5.6 there exists g ∈ E^{(P,φ^{−1})} such that < f, g >= 1. Set \overline{g} = Br_{(P,φ^{−1})}(g). Since \overline{f} ∈ N, and N is an ideal in E(P, P̂), we obtain \overline{f}\overline{g} ∈ N. On the other hand, \overline{f}\overline{g} ∈ E(P, 1), so that \overline{f}\overline{g} ∈ E(P, 1) ∩ N = 0, and we conclude that fg ∈ BK(P, 1; E). Hence we can write

fg = h_{P} + ∑_{Q<P} tr_{Q}^{P}(h_{Q})

for certain h_{Q} ∈ E^{(Q,1)} and h_{P} ∈ ℘E^{(P,1)}. Now, recall that the ordinary trace map is invariant under the action of P on E, so we have

1 = tr(fg) = tr(h_{P} + ∑_{Q<P} tr_{Q}^{P}(h_{Q})) = tr(h_{P} + ∑_{Q<P} ∑_{x∈P/Q}^{x}h_{Q}) =

= tr(h_{P}) + ∑_{Q<P} ∑_{x∈P/Q} tr(^{x}h_{Q}) = tr(h_{P}) + ∑_{Q<P} ∑_{x∈P/Q} tr(h_{Q}) =

= tr(h_{P}) + ∑_{Q<P}[P : Q] · tr(h_{Q}) ∈ ℘ + pO = ℘,

which is a contradiction. □

The following lemma will be the most important tool to prove the main result.

6.6 Lemma

E^{(P,P̂)}/rad(E^{(P,P̂)}) ≅ F.

Proof By Lemma 6.5, E(P, P̂) is a crossed product, and by Lemma 6.3 this implies that

E(P, P̂)/rad(E(P, P̂)) ≅ F.

But the Brauer kernel BK(P, P̂; E) of the Brauer map is contained in the radical of E^{(P,P̂)} by (6.4.c), so that

E^{(P,P̂)}/rad(E^{(P,P̂)}) ≅
≅ (E^{(P,P̂)}/BK(P, P̂; E)) /rad (E^{(P,P̂)}/BK(P, P̂; E)) ≅
≅ E(P, P̂)/rad(E(P, P̂)) ≅ F.

□

7 Abelian P

Throughout this section we assume that P is an abelian p-group. By Dade’s classification of endo-permutation modules in [7], we have an isomorphism

P̂ × D_{F} (P) ≅ D_{O}(P),

and the structure of the Dade group can be completely described. For endomonomial modules we have

7.1 Proposition Suppose that P is an abelian p-group. Then every indecomposable endo-monomial OP -module with vertex P is an endo-permutation OP -module. Moreover, there is a group isomorphism

DO^{mon}(P) ˜→ D_{F}(P)

given by

[L] ↦ [\overline{L}]

where \overline{L} is the reduction mod ℘ of L.

Proof We resume the notation of the previous chapter; L denotes an indecomposable endo-monomial OP -module with vertex P, and E = End_{O} ( L ).

(i) We first show that

\overline{E^{(P,P̂)}} ≅ \overline{E}^{P} ,

which means that the reduction mod ℘ of the generalized fixed points of E gives precisely the fixed points in the reduced algebra \overline{E}. First note that, if f ∈ E^{(P,φ)} then ^{x}f = φ(x)f for all x ∈ P, hence ^{x}\overline{f} = \overline{φ(x)} · \overline{f} = \overline{f}, so that \overline{f} ∈ \overline{E}^{P}. Now fix a decomposition (2.1.a) of E, then \overline{E} has a corresponding decomposition

\overline{E} ≅ ⨁_{(Q,ψ)∈M_{P} /P} F ⊗ E[Q, ψ]

and since E[Q, ψ] ≅ n_{(Q,ψ)} × Ind_{Q}^{P}O_{ψ}, we obtain

F ⊗ E[Q, ψ] ≅ n_{(Q,ψ)} × Ind_{Q}^{P}F.

Now pick a direct summand of E isomorphic to Ind_{Q}^{P}O_{ψ}, then its reduction modulo ℘ is isomorphic to Ind_{Q}^{P}F. One can write

Ind_{Q}^{P}F ≅ ⨁_{x∈P/Q} x ⊗_{OQ} F,

then (Ind_{Q}^{P}F)^{P} is isomorphic to the one-dimensional trivial submodule generated by the element

w = ∑_{x∈P/Q} x ⊗ 1.

Recall from the proof of Lemma 4.2 that for Q ≤ P

(7.1.a)  (Ind_{Q}^{P}O_{φ})^{(P,φ)} = 0  for φ|_{Q} ≠ ψ

and it has O-rank one if φ|_{Q} = ψ. By that, we see that w is the reduction modulo ℘ of the element

h = ∑_{x∈P/Q} φ(x^{−1})x ⊗ 1 ∈ (Ind_{Q}^{P}O_{ψ})^{(P,φ)}

if φ ∈ P̂ with φ|_{Q} = ψ. Since P is abelian, every character ψ of a subgroup Q has such an extension φ ∈ P̂. Hence the summand (Ind_{Q}^{P}F)^{P} of \overline{E}^{P} is the reduction mod ℘ of the corresponding summand (Ind_{Q}^{P}O_{ψ})^{(P,φ)} of E^{(P,P̂)}. Moreover, if φ′ is another extension of ψ to P̂, then also the element

h′ = ∑_{x∈P/Q} φ′(x^{−1})x ⊗ 1,

which generates (Ind_{Q}^{P}O_{ψ})^{(P,φ′)}, reduces to w, thus \overline{h} = \overline{h′} in \overline{E^{(P,P̂)}}. (ii) A consequence of (i) is that for every indecomposable endo-monomial OP - module L, also its reduction \overline{L} is indecomposable. For that it suffices to show that

End_{F P} (\overline{L})/rad(End_{F P} (\overline{L})) ≅ F.

But by (i) and Lemma 6.6 we have that this quotient equals

\overline{E}^{P} /rad(\overline{E}^{P}) ≅ \overline{E^{(P,P̂)}}/rad(\overline{E^{(P,P̂)}}) ≅ F.

(iii) Next we show the claimed isomorphism of the Dade groups DO^{mon}(P) and D_{F} (P). The reduction map is a group homomorphism and certainly surjective since, by Dade’s theorem, every indecomposable endo-permutation F P -module is the reduction of an indecomposable endo-permutation OP -module which is also endo-monomial. To show injectivity, let [L] ∈ DO^{mon}(P) be in the kernel of the reduction map. Then [L] can be represented by an indecomposable endomonomial OP -module L with vertex P, and its reduction \overline{L} is still indecomposable. But [\overline{L}] = [F] implies that \overline{L} ≅ F, hence L ≅ O_{φ} for some φ ∈ P̂, and hence [L] = [O] in DO^{mon}(P), which shows that the reduction map is injective. (iv) Finally we conclude that every indecomposable endo-monomial OP -module L with vertex P is endo-permutation. Since [\overline{L}] has a unique preimage, and since L lifts to an endo-permutation OP -module L_{0} by Dade’s theorem, we have [L] = [L_{0}] in DO^{mon}(P), hence L ≅ O_{φ} ⊗ L_{0} which is an endo-permutation module. □

7.2 Theorem If P is abelian then every indecomposable endo-monomial OP - module is an endo-permutation module.

Proof Let L be an indecomposable endo-monomial OP -module, and let Q be a vertex of L. For Q = P the result is shown in Proposition 7.1, so assume Q < P. By Lemma 4.9 we have L ≅ Ind_{Q}^{P}(S), where S is a cap of Res_{Q}^{P}(L). Now S is an indecomposable endo-monomial OQ-module with vertex Q for the abelian group Q, hence S is an endo-permutation module. Then also L ≅ Ind_{Q}^{P}(S) is endo-permutation, since the condition from [7] (2.16) requires in this case only that the OQ-modules S and ^{x}S are compatible as endo-permutation modules, for all x ∈ P. But this is trivial, since S ≅ ^{x}S for all x ∈ P. □

However, there are (decomposable) endo-monomial OP -modules that are not endo-permutation, due to the weaker compatibility condition for endo-monomial modules. For example, if P is any non-trivial abelian p-group and φ, φ′ ∈ P̂ with φ ≠ φ′, then O_{φ} ⊕ O_{φ′} is monomial and hence endo-monomial, but not an endo-permutation module. Next we describe the compatibility condition for non-capped endo-monomial modules.

7.3 Lemma Let L and L′ be two indecomposable endo-monomial OP -modules with a common vertex Q. Then L and L′ are compatible if and only if

L′ ≅ O_{φ} ⊗ L for some φ ∈ P̂.

Proof By Lemma 4.9, we can write L ≅ Ind_{Q}^{P}S and L′ ≅ Ind_{Q}^{P}S′, where S and S′ are sources of L respectively L′, so S and S′ are indecomposable endomonomial OQ-modules with vertex Q. If L and L′ are compatible, then so are S and S′. Hence, by Proposition 4.4, we obtain that S′ ≅ O_{ψ} ⊗ S for some ψ ∈ Q̂. Since P is abelian there exists φ ∈ P̂ such that φ|_{Q} = ψ, and hence

L′ ≅ Ind_{Q}^{P}S′ ≅ Ind_{Q}^{P}(O_{ψ} ⊗ S) ≅
≅ Ind_{Q}^{P}(Res_{Q}^{P}(O_{φ}) ⊗ S) ≅ O_{φ} ⊗ Ind_{Q}^{P}S ≅ O_{φ} ⊗ L.

Conversely, it is clear that L and O_{φ} ⊗ L are compatible for every φ ∈ P̂. □

Finally we give a compatibility condition for modules with distinct vertices:

7.4 Proposition Let L and M be indecomposable endo-monomial OP -modules, and write L = Ind_{Q}^{P}(S) and M = Ind_{R}^{P}(T) for sources S and T and vertices Q and R of L respectively M as in Lemma 4.9. Then L and M are compatible if and only if the capped endo-monomial O(Q ∩ R)-modules Res_{Q∩R}^{P}(S) and Res_{Q∩R}^{P}(T) are compatible.

Proof We have:

L^{⋆} ⊗ M ≅ Ind_{Q}^{P}(S^{⋆}) ⊗ Ind_{R}^{P}(T) ≅ Ind_{R}^{P}(Res_{R}^{P}(Ind_{Q}^{P}(S^{⋆})) ⊗ T) ≅

≅ Ind_{R}^{P}( ⨁_{x∈R\P/Q} Ind_{Q∩R}^{R}(S^{⋆}|_{Q∩R}) ⊗ T) ≅

≅ |R\P/Q| × Ind_{Q∩R}^{P}(S^{⋆}|_{Q∩R} ⊗ T |_{Q∩R}).

Now L and M are compatible if and only if L^{⋆} ⊗ M is monomial. By the above, this happens if and only if

S^{⋆}|_{Q∩R} ⊗ T |_{Q∩R}

is monomial, which in turn is equivalent to saying that Res_{Q∩R}^{Q}(S) and Res_{Q∩R}^{R}(T) are compatible. □

Now we are ready to summarize the results in this section to give a complete description of capped endo-monomial modules:

7.5 Proposition Let P be an abelian p-group, and let C denote an indecomposable endo-permutation OP -module with vertex P. For every subgroup Q of P, let C_{Q} denote an indecomposable direct summand of Res_{Q}^{P}C with vertex Q. Then we have:

(i) If L is a capped endo-monomial OP -module with C as a cap then

L ≅ ⨁_{(Q,ψ)∈M_{P}} n_{(Q,ψ)} × Ind_{Q}^{P}(O_{ψ} ⊗ C_{Q}),

for some integers n_{(Q,ψ)} ≥ 0, (Q, ψ) ∈ M_{P}. (ii) Conversely, given integers n_{(Q,ψ)} ≥ 0 for (Q, ψ) ∈ M_{P} with n_{(P,1)} ≥ 1, then

⨁_{(Q,ψ)∈M_{P}} n_{(Q,ψ)} × Ind_{Q}^{P}(O_{ψ} ⊗ C_{Q})

is a capped endo-monomial OP -module with C as a cap.

7.6 Remark The question whether there are indecomposable endo-monomial OP -modules with vertex P, which are not endo-permutation, remains open for general P. However, there are such indecomposable modules with smaller vertices. For example, let P be quaternion or dihedral of order eight, let Q be a cyclic subgroup of order four, and choose a generator ψ of Q̂. Then Ind_{Q}^{P}O_{ψ} is monomial, and hence endo-monomial, but not endo-permutation, since after restricting to Q it is a direct sum of two non-isomorphic rank-one OQ-modules.

In a sequel to this article, we will further investigate the non-abelian situation.

References

[1] J. Alperin: Lifting endo-trivial modules. J. Group Theory 4 (2001) no.1, 1-2

[2] J. Alperin A construction of endo-permutation modules J. Group Theory 4 (2001) no.1, 3-10

[3] A. Betsch, P. Schmid: Crossed products of p-groups and Green’s indecomposability theorem. Journal of Algebra 168(1994), 868-876

[4] R. Boltje, B. Külshammer: A generalized Brauer construction and linear source modules. Trans. Amer. Math. Soc. 352 (2000), no.7, 3411-3428

[5] S.Bouc, J.Thévenaz: The group of endo-permutation modules Invent.Math. 139 (2000), no.2, 275-349

[6] J. Carlson, J. Thévenaz: Torsion endo-trivial modules. Algebr. Represent. Theory 3 (2000), no.4, 303-335

[7] E.C. Dade Endo-permutation modules over p-groups I Annals of Math. 107(1978), 459-494

[8] E.C. Dade Endo-permutation modules over p-groups II Annals of Math. 108(1978), 317-346

[9] Y. Fan, B. Külshammer: Group graded rings and finite block theory. Pacific J. Math. 196 (2000), no.1, 177-186

[10] L. Puig: Nilpotent blocks and their source algebras. Invent. Math. 93 (1988), no.1, 77-116

[11] J. Rickard: Splendid equivalences: derived categories and permutation modules. Proc. London Math. Soc. (3) 72 (1996), no.2, 331-358

[12] J.Thévenaz G-algebras and modular representation theory Oxford University Press, Oxford 1995