TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 7, Pages 3411–3428 S 0002-9947(00)02530-7 Article electronically published on March 21, 2000

A GENERALIZED BRAUER CONSTRUCTION AND LINEAR SOURCE MODULES

ROBERT BOLTJE AND BURKHARD KÜLSHAMMER

Abstract. For a complete discrete valuation ring O with residue field F, a subgroup H of a finite group G and a homomorphism φ : H → O^{×}, we define a functor V ↦ \overline{\overline{V}} (H, φ) from the category of OG-modules to the category of F N_{G}(H, φ)-modules and investigate its behaviour with respect to linear source modules.

Introduction

Let O be a complete discrete valuation ring with residue field F = O/πO of characteristic p > 0, and let G be a finite group. For a p-subgroup P of G, R. Brauer defined a homomorphism of algebras Br_{P} : Z(F G) → Z(F C_{G}(P)) between the centers of the corresponding group algebras (see [CR87, §58]). This Brauer homomorphism plays an important role in representation theory, in particular for the formulation and proof of Brauer’s three main theorems on blocks.

The Brauer homomorphism was later generalized from group algebras to modules (see [Br]). If V is an OG-lattice and H is a subgroup of G, then V^{H} denotes the set of fixed points of H on V, tr_{I}^{H} : V^{I} → V^{H} denotes the trace map, for I ⩽ H, and

V (H) = V^{H}/( ∑_{I<H} tr_{I}^{H}(V^{I}) + πV^{H})

is called the Brauer construction. It is known that V (H) = 0 if H is not a p-group. When H = P is a p-group and V is the group algebra OG endowed with the conjugation action then V (P) can be identified with the group algebra F C_{G}(P), and the canonical map V^{P} → V (P) restricts to the classical Brauer homomorphism Br_{P} : Z(F G) → Z(F C_{G}(P)), composed with the reduction map Z(OG) → Z(F G).

In this paper we extend the applicability of the Brauer construction further, in two directions. First we modify the definition of V (H) in order to get nontrivial results also in cases where H is not a p-group. This is achieved by setting

\overline{\overline{V}} (H) := V^{H}/( ∑_{{I<H \atop p|[H:I]}} tr_{I}^{H}(V^{I}) + πV^{H}).

Then \overline{\overline{V}} (H) = V (H) whenever H is a p-group, but \overline{\overline{V}} (H) ≠ 0 also in many cases where H is not a p-group.

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

Received by the editors April 28, 1998.

2000 Mathematics Subject Classification. Primary 20C11, 20C20.

The first author’s research was supported by the DFG.

Our second modification takes into account homomorphisms φ : H → O^{×}. For such a homomorphism φ, we can consider a generalized fixed point space

V^{(H,φ)} := {v ∈ V | hv = φ(h)v for h ∈ H},

a generalized trace map

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

whenever (I, ψ) ⩽ (H, φ), i.e. I ⩽ H and ψ = φ|_{I}, and a generalized Brauer construction

\overline{\overline{V}} (H, φ) := V^{(H,φ)}/( ∑_{{(I,ψ)<(H,φ) \atop p|[H:I]}} tr_{(I,ψ)}^{(H,φ)}(V^{(I,ψ)}) + πV^{(H,φ)}).

Just as the “classical” Brauer construction V (P) is most useful with trivial source modules, our modified Brauer construction \overline{\overline{V}} (H, φ) is most useful with linear source modules. We show that, for a linear source module V, the dimension of \overline{\overline{V}} (H, φ) counts the multiplicity of the rank one module O_{φ} as a direct summand of Res_{H}^{G}(V) and that the dimensions of \overline{\overline{V}} (H, φ) determine V. We show also that, when V has vertex P and source O_{ψ} for a homomorphism ψ : P → O^{×}, then \overline{\overline{V}} (P, ψ) coincides with the reduction mod π of the Green correspondent W of V in N_{G}(P, ψ). As a consequence, we obtain that the functor V ↦ \overline{\overline{V}} (P, ψ) induces a one-to-one correspondence between the set of isomorphism classes of indecomposable linear source OG-modules with vertex P and source O_{ψ} and the set of isomorphism classes of indecomposable projective F [N_{G}(P, ψ)/P]-modules.

A nontrivial p-group P will, in general, have many homomorphisms P → O^{×} but only one homomorphism P → F^{×}. In order to avoid such phenomena, we study the reduction of linear source modules modulo high powers π^{n}. We show that, for sufficiently large n, reduction modulo π^{n} induces an injection (but not a surjection, in general) from the set of isomorphism classes of indecomposable linear source modules over O into the set of isomorphism classes of indecomposable linear source modules over O/π^{n}O. This injection preserves vertices, sources and Green correspondents. We also show that it suffices to choose n in such a way that π^{n}O = pO if p is odd, and π^{n}O = 4O if p = 2. In this way we obtain an explicit form of Maranda’s theorem (cf. [CR81, §30]).

Part of this article was written while the first author was visiting the University of Chicago to which he is grateful for the hospitality he enjoyed.

1. Notations and preliminaries

A GENERALIZED BRAUER CONSTRUCTION AND LINEAR SOURCE MODULES 3413

1.1. Throughout this article, G denotes a finite group and E its trivial subgroup. Moreover, R always denotes a commutative ring, R^{×} its group of units, J(R) its Jacobson radical, and RG the group algebra of G over R. We set Ĝ(R) := Hom(G, R^{×}), which is an abelian group under multiplication, and we denote by M_{G}(R) the set of all pairs (H, φ), where H ⩽ G and φ ∈ Ĥ(R). Note that M_{G}(R) is a partially ordered set, with (I, ψ) ⩽ (H, φ) if and only if I ⩽ H and ψ = φ|_{I}. Moreover, M_{G}(R) is a G-set by conjugation: For g ∈ G and (H, φ) ∈ M_{G}(R) let g(H, φ) := (^{g}H, ^{g}φ), with ^{g}H := gHg^{−1} and (^{g}φ)(x) := φ(g^{−1}xg) for x ∈ ^{g}H. By N_{G}(H, φ) we denote the stabilizer of (H, φ) in G. Note that the G-action on M_{G}(R) respects the partial order. For a prime p we denote by M_{G}(R)_{p} ⊆ M_{G}(R) the subset consisting of those pairs whose first component is a p-group. For φ ∈ Ĝ(R) we denote by R_{φ} the RG-module which has R itself as underlying R-module and carries the G-action given by g · α = φ(g)α for g ∈ G and α ∈ R. If φ = 1 is the trivial map, we just write R instead of R_{φ}. Note that φ ↦ R_{φ} induces a bijection between Ĝ(R) and the set of isomorphism classes of RG-modules which are free of rank one as R-modules.

If Λ is an R-order, we denote by _{Λ}mod the category of left Λ-modules which are finitely generated as R-modules, and by _{Λ}lat ⊆ _{Λ}mod the full subcategory of Λ-lattices. These are the finitely generated Λ-modules which are projective as Rmodules. If V is a free R-module, we denote its rank by rk_{R}(V). For two Λ-modules V and W we write V | W if V is isomorphic to a direct summand of W.

By ℕ, ℕ_{0}, and ℤ we denote the set of positive, non-negative, and all integers, and by ℚ the field of rational numbers. For a prime p, we write ℤ_{p} and ℚ_{p} for the ring of p-adic integers and the field of p-adic numbers, the completions of ℤ and ℚ with respect to the p-adic valuation.

1.2. Definition. For V ∈ _{RG}mod and any (H, φ) ∈ M_{G}(R) let

V^{(H,φ)} := {v ∈ V | hv = φ(h)v for all h ∈ H}

denote the R-submodule of (H, φ)-fixed points of V. Obviously, for g ∈ G and (I, ψ) ⩽ (H, φ) in M_{G}(R), one has

V ^{^{g}(H,φ)} = gV^{(H,φ)} ,  V^{(H,φ)} ⊆ V^{(I,ψ)} ,  V^{(E,1)} = V .

Thus V^{(H,φ)} can be considered as an RN_{G}(H, φ)-submodule of V.

For (I, ψ) ⩽ (H, φ) in M_{G}(R) we define the trace map

tr_{(I,ψ)}^{(H,φ)} : V^{(I,ψ)} ⟶ V^{(H,φ)} , v ↦ ∑_{h∈H/I} φ(h^{−1})hv ,

where h runs through a set of representatives for H/I. It is easy to check that the sum does not depend on the chosen representatives and that it is an (H, φ)-fixed point. The following properties of the trace map are immediate.

1.3. Lemma. Let V ∈ _{RG}mod, and let (J, μ) ⩽ (I, ψ) ⩽ (H, φ) and (L, λ) ⩽ (H, φ) be in M_{G}(R).

(a) For g ∈ G and v ∈ V^{(I,ψ)} one has

g tr_{(I,ψ)}^{(H,φ)}(v) = tr _{^{g}(I,ψ)}^{^{g}(H,φ)}(gv) .

Thus the image V_{(I,ψ)}^{(H,φ)} of V^{(I,ψ)} under tr_{(I,ψ)}^{(H,φ)} is an R[N_{G}(H, φ) ∩ N_{G}(I, ψ)]- submodule of V^{(H,φ)}.

(b) One has tr_{(J,μ)}^{(H,φ)} = tr_{(I,ψ)}^{(H,φ)} ∘ tr_{(J,μ)}^{(I,ψ)}.

(c) If V = V_{1} ⊕ V_{2} is a decomposition into RH-submodules, then

V^{(H,φ)} = V_{1}^{(H,φ)} ⊕ V_{2}^{(H,φ)} and V_{(I,ψ)}^{(H,φ)} = (V_{1})_{(I,ψ)}^{(H,φ)} ⊕ (V_{2})_{(I,ψ)}^{(H,φ)}

as R-modules.

(d) For each v ∈ V^{(H,φ)} one has

tr_{(I,ψ)}^{(H,φ)}(v) = [H : I] · v .

(e) For v ∈ V^{(I,ψ)}, h ∈ H and x ∈ L ∩ ^{h}I one has xhv = φ(x)hv and

tr_{(I,ψ)}^{(H,φ)}(v) = ∑_{h∈L\H/I} φ(h^{−1})tr_{(L∩^{h}I,λ)}^{(L,λ)}(hv) .

1.4. We call an RG-module V monomial if there exists a monomial decomposition V = ⨁_{x∈X} V_{x} of V into R-submodules V_{x} which are free of rank one over R and permuted under the action of G. Of course, this is equivalent to saying that V is isomorphic to a finite direct sum of modules of the form Ind_{H}^{G}(R_{φ}) for various (H, φ) ∈ M_{G}(R). If V = ⨁_{x∈X} V_{x} is a monomial decomposition, then the bijection x ↦ V_{x} induces an action of G on X. By G_{x} we denote the stabilizer in G of an element x ∈ X, and by φ_{x}: G_{x} → R^{×} we denote the homomorphism by which G_{x} acts on V_{x}. Note that G_{^{g}x} = ^{g}G_{x} and φ_{^{g}x} = ^{g}φ_{x} for g ∈ G. If H ⩽ G, we set H_{x} := H ∩ G_{x} and often write φ_{x} instead of φ_{x}|_{H_{x}}. For any subset A ⊆ R, let

Ann_{R}(A) := {α ∈ R | αβ = 0 for all β ∈ A}

denote the annihilator ideal of A in R. For (H, φ), (I, ψ) ∈ M_{G}(R) we will have to consider the ideal

𝔞_{(H,φ)}^{(I,ψ)} := Ann_{R}({φ(h) − ψ(h) | h ∈ H ∩ I})

of R. The following result will be used repeatedly.

1.5. Lemma. Let V ∈ _{RG}mod be monomial with monomial decomposition V = ⨁_{x∈X} V_{x}, and let (H, φ) ∈ M_{G}(R). Then

V^{(H,φ)} = ⨁_{x∈H\X} tr_{(H_{x},φ|_{Hx})}^{(H,φ)}(𝔞_{(H_{x},φ_{x})}^{(H,φ)} · V_{x}) ,

where x runs through a set of representatives for the H-orbits of X. In particular, if R is an integral domain, one has

V^{(H,φ)} = ⨁_{{x∈H\X \atop φ|_{Hx} =φx}} tr_{(H_{x},φ_{x})}^{(H,φ)}(V_{x}) .

Proof. For x ∈ X, v ∈ V_{x}, α ∈ 𝔞_{(H_{x},φ_{x})}^{(H,φ)}, and h ∈ H_{x} we have

h(αv) = α(hv) = αφ_{x}(h)v = αφ(h)v = φ(h)(αv) .

Therefore, αv is an (H_{x}, φ|_{H_{x}})-fixed point. Moreover, it is easy to check that the R-module

tr_{(H_{x},φ|_{Hx})}^{(H,φ)}(𝔞_{(H_{x},φ_{x})}^{(H,φ)} · V_{x})

does not depend on the choice of x within its H-orbit. The sum on the right hand side in the first equation of the lemma is a direct sum, since the single R-modules have support in pairwise distinct H-orbits of X. So, it suffices to show that V^{(H,φ)} is contained in the right hand side. Let v ∈ V^{(H,φ)}, and write v = ∑_{x∈X} v_{x} with v_{x} ∈ V_{x} for x ∈ X. Then, for h ∈ H, we have

∑_{x∈X} φ(h)v_{x} = φ(h)v = hv = ∑_{x∈X} hv_{x}.

Comparing components, we conclude that φ(h)v_{hx} = hv_{x} for x ∈ X. In particular, φ(h)v_{x} = φ_{x}(h)v_{x} for h ∈ H_{x}, so v_{x} ∈ 𝔞_{(H_{x},φ_{x})}^{(H,φ)} · V_{x}. Thus

v = ∑_{x∈H\X} ∑_{h∈H/H_{x}} v_{hx} = ∑_{x∈H\X} ∑_{h∈H/H_{x}} φ(h^{−1})hv_{x} = ∑_{x∈H\X} tr_{(H_{x},φ|_{Hx})}^{(H,φ)}(v_{x}),

and the lemma is proved.

The following corollary is now immediate.

1.6. Corollary. Let (H, φ), (I, ψ) ∈ M_{G}(R). Then

(Ind_{I}^{G}(R_{ψ}))^{(H,φ)} = ⨁_{g∈H\G/I} tr_{(H∩^{g}I,φ)}^{(H,φ)}(g ⊗_{RI} 𝔞_{(H,φ)}^{^{g}(I,ψ)}) .

In particular, if R is an integral domain, one has

(Ind_{I}^{G}(R_{ψ}))^{(H,φ)} = ⨁_{{g∈H\G/I \atop φ|_{H∩^{g}I} =^{g}ψ|_{H∩^{g}I}}} R · tr_{(H∩^{g}I,φ)}^{(H,φ)}(g ⊗_{RI} 1) .

1.7. Remark. We note that Hom_{RH}(R_{φ}, V) ≅ V^{(H,φ)} for each V ∈ _{RH}mod under the map f ↦ f(1). Together with the adjointness of induction and restriction this yields an isomorphism of R-modules

Hom_{RG}(Ind_{H}^{G}(R_{φ}), Ind_{I}^{G}(R_{ψ})) → ⨁_{g∈H\G/I} tr_{(H∩^{g}I,φ)}^{(H,φ)}(g ⊗_{RI} 𝔞_{(H,φ)}^{^{g}(I,ψ)}) ,
f ↦ f(1 ⊗_{RH} 1) .

Thus Corollary 1.6 can also be used to compute Hom-sets and endomorphism rings of monomial modules.

2. The generalized Brauer construction

2.1. Throughout this section we fix a complete discrete valuation ring O with field of fractions K of characteristic 0 and with residue field F of prime characteristic p. Let v_{K}: K^{×} → ℤ denote the surjective valuation with valuation ring O and let π ∈ O be an element with v_{K}(π) = 1. Then O is a local principal ideal domain and each ideal of O is of the form π^{n}O with n ∈ ℕ_{0} ∪ {∞}, where π^{∞} := 0. We may assume that ℤ_{p} ⊆ O and ℚ_{p} ⊆ K as the closures of ℤ and ℚ. We denote by e = e_{K/ℚ_{p}} the ramification index of K/ℚ_{p}. Thus v_{K}(p) = e or, equivalently, pO = π^{e}O. We set k := O/pO and denote by ¯· (resp. ¯ ¯·) the reduction modulo pO (resp. πO or π̄k).

2.2. Definition. For V ∈ _{OG}mod and (H, φ) ∈ M_{G}(O) we set

\overline{V} (H, φ) := V^{(H,φ)}/( ∑_{{(I,ψ)<(H,φ) \atop p|[H:I]}} V_{(I,ψ)}^{(H,φ)} + pV^{(H,φ)})

and

\overline{\overline{V}} (H, φ) := V^{(H,φ)}/( ∑_{{(I,ψ)<(H,φ) \atop p|[H:I]}} V_{(I,ψ)}^{(H,φ)} + πV^{(H,φ)}) .

We call \overline{V} (H, φ) ∈ _{kN_{G}(H,φ)}mod and \overline{\overline{V}} (H, φ) ∈ _{F N_{G}(H,φ)}mod the generalized Brauer construction of V with respect to (H, φ). Moreover, we call the canonical maps

\overline{Br}_{(H,φ)} : V^{(H,φ)} → \overline{V} (H, φ)

and

\overline{\overline{Br}}_{(H,φ)} : V^{(H,φ)} → \overline{\overline{V}} (H, φ)

the generalized Brauer maps with respect to (H, φ).

2.3. Remark. (a) It is clear from Definition 2.2 that the generalized Brauer construction induces functors

¯·(H, φ): _{OG}mod → _{kN_{G}(H,φ)}mod

and

¯¯·(H, φ): _{OG}mod → _{F N_{G}(H,φ)}mod

which commute with direct sums. Note that, by definition, the F N_{G}(H, φ)-module \overline{\overline{V}} (H, φ) arises from \overline{V} (H, φ) by reduction modulo π̄k.

(b) Let φ = 1 be the trivial homomorphism. If H is a p-subgroup of G, then \overline{\overline{V}} (H, φ) in Definition 2.2 coincides with the usual Brauer construction, cf. [Br] or [Th, §11]. If H is not a p-subgroup of G, then the usual Brauer construction with respect to H is identically zero whereas \overline{\overline{V}} (H, φ) is not, in general.

(c) Note that by Lemma 1.3 (d), one has

pV^{(H,φ)} ⊆ ∑_{{(I,ψ)<(H,φ) \atop p|[H:I]}} V_{(I,ψ)}^{(H,φ)} , (2.3.a)

if p divides |H|. Note also that \overline{V} (H, φ) = V^{(H,φ)}/pV^{(H,φ)} in the case that H is a p′-group. We will see that \overline{V} (H, φ) is free as a k-module if V is monomial. There is no immediate need to go over to \overline{\overline{V}} (H, φ) instead of using \overline{V} (H, φ). In fact, the inclusion in (2.3.a) indicates that the construction \overline{V} (H, φ) is more natural than \overline{\overline{V}} (H, φ).

2.4. Proposition. Let V be a monomial OG-module with monomial decomposition V = ⨁_{x∈X} V_{x}, and let (H, φ) ∈ M_{G}(O). Then

∑_{{(I,ψ)<(H,φ) \atop p|[H:I]}} V_{(I,ψ)}^{(H,φ)} + pV^{(H,φ)} = ⨁_{x∈H\X ; {φ|_{Hx} =φx \atop p|[H:Hx]}} tr_{(H_{x},φ_{x})}^{(H,φ)}(V_{x}) ⊕ ⨁_{{x∈H\X \atop φ|_{Hx} =φx} ; p |̸ [H:Hx]} p · tr_{(H_{x},φ_{x})}^{(H,φ)}(V_{x}).

Thus \overline{V} (H, φ) is a monomial kN_{G}(H, φ)-module with monomial decomposition

\overline{V} (H, φ) = ⨁_{{x∈H\X \atop φ|_{Hx} =φx} ; p |̸ [H:Hx]} \overline{V} (H, φ)\overline{x},

where \overline{x} denotes the H-orbit of x and \overline{V} (H, φ)\overline{x} := \overline{Br}_{(H,φ)}(tr_{(H_{x},φ_{x})}^{(H,φ)}(V_{x})), for x ∈ X. The subgroup N_{G}(H, φ) of G permutes the k-submodules \overline{V} (H, φ)\overline{x} of \overline{V} (H, φ) in the same way as the H-orbits \overline{x}. Thus N_{G}(H, φ)\overline{x} = N_{G}(H, φ)_{x}H, and N_{G}(H, φ)\overline{x} acts on \overline{V} (H, φ)\overline{x} via the homomorphism \overline{φ_{x}φ} : N_{G}(H, φ)\overline{x} → k^{×} defined by (φ_{x}φ)(ab) := φ_{x}(a)φ(b) for a ∈ N_{G}(H, φ)_{x}, b ∈ H.

Proof. We recall that

V^{(H,φ)} = ⨁_{\begin{matrix}x∈H\X \\ φ|_{Hx} =φ_{x} \end{matrix}} tr_{(H_{x},φ_{x})}^{(H,φ)}(V_{x}),

where each summand is free of rank one over O. Therefore the right hand side of the first equation to be proved is indeed a direct sum contained in the left hand side, and pV^{(H,φ)} is contained in the right hand side. Let (I, ψ) < (H, φ) be such that p|[H : I], and let v ∈ V_{(I,ψ)}^{(H,φ)}. Then v is a sum of elements of the form

w_{x} = tr_{(I,ψ)}^{(H,φ)}(tr_{(I_{x},φ_{x})}^{(I,ψ)}(v_{x})) = tr_{(I_{x},φ_{x})}^{(H,φ)}(v_{x}) = [H_{x} : I_{x}]tr_{(H_{x},φ_{x})}^{(H,φ)}(v_{x}),

where v_{x} ∈ V_{x} and ψ|_{I_{x}} = φ_{x}. If p|[H_{x} : I_{x}] then w_{x} ∈ pV^{(H,φ)}, which is contained in the right hand side. Thus suppose that p |̸[H_{x} : I_{x}]. Then p|[H : H_{x}], for otherwise we would have p |̸[H : I_{x}], contradicting our assumption p|[H : I]. Therefore w_{x} is also contained in the right hand side in this case.

From this we get the decomposition of \overline{V} (H, φ) into its k-submodules \overline{V} (H, φ)\overline{x} which are free of rank one over k. It is immediate that these submodules are permuted according to the canonical action of N_{G}(H, φ) on H\X. A Frattini argument shows that N_{G}(H, φ)\overline{x} = N_{G}(H, φ)_{x}H. If φ|_{H_{x}} = φ_{x}, then, for v ∈ V_{x}, a ∈ N_{G}(H, φ)_{x} and b ∈ H, we have

abtr_{(H_{x},φ_{x})}^{(H,φ)}(v) = aφ(b)tr_{(H_{x},φ_{x})}^{(H,φ)}(v) = φ(b)tr_{(^{a}H_{x},^{a}φ_{x})}^{(H,φ)}(av)
= φ_{x}(a)φ(b)tr_{(H_{ax},^{a}φ|H_{ax})}^{(H,φ)}(v) = φ_{x}(a)φ(b)tr_{(H_{x},φ|_{Hx})}^{(H,φ)}(v)
= φ_{x}(a)φ(b)tr_{(H_{x},φ_{x})}^{(H,φ)}(v).

The result follows.

For any V ∈ _{OG}lat and any (H, φ) ∈ M_{G}(O) let m_{(H,φ)}(V) denote the multiplicity with which the OH-module O_{φ} occurs as a direct summand in Res_{H}^{G}(V). Our next aim is to show that m_{(H,φ)}(V) is related to the k-rank of \overline{V} (H, φ) whenever V is monomial.

The following terminology will also be useful. For V ∈ _{OG}lat, we will call an indecomposable direct summand of the OG-End_{OG}(V)-bimodule V a block of V. The motivation for this terminology comes from the fact that, when V is the group algebra OG, considered as a left OG-module under multiplication, then End_{OG}(V) can be identified with OG, acting by right multiplication, so the blocks of V are just the blocks of OG in the usual sense.

It is clear that every V ∈ _{OG}lat has a decomposition

V = V_{1} ⊕ ⋯ ⊕ V_{r}

into blocks V_{1}, …, V_{r}. This decomposition gives rise to a decomposition

1_{E} = id_{V} = e_{1} + ⋯ + e_{r}

of the identity element 1_{E} = id_{V} of E := End_{OG}(V) into pairwise orthogonal idempotents e_{1}, …, e_{r} in E, where each e_{i} is the projection onto the direct summand V_{i}. One checks that e_{1}, …, e_{r} are contained in the center of E. It follows that e_{1}, …, e_{r} are just the block idempotents of E. Thus we obtain a corresponding decomposition

E = End_{OG}(V) = E_{1} ⊕ ⋯ ⊕ E_{r}

of E into blocks E_{1} = e_{1}E, …, E_{r} = e_{r}E. In this way the blocks of the OG-lattice V correspond to the blocks of the O-order E. Since V_{i} = E_{i} · V for i = 1, …, r, the blocks V_{1}, …, V_{r} are uniquely determined by V up to their order (and not just up to isomorphism). It is easy to see that if W is a direct summand of V as OG-module, then also W = e_{1}W ⊕ ⋯ ⊕ e_{r}W is a decomposition of (not necessarily indecomposable) OG-End_{OG}(W)-bimodules.

2.5. Lemma. Let φ ∈ Ĝ(O) and (I, ψ) ∈ M_{G}(O). Then

m_{(G,φ)}(Ind_{I}^{G}(O_{ψ})) = \begin{cases}1, & if ψ = φ|_{I} and p |̸ [G : I], \\ 0, & otherwise. \end{cases}

Moreover, if m_{(G,φ)}(Ind_{I}^{G}(O_{ψ})) = 1, then Ind_{I}^{G}(O_{ψ})^{(G,φ)} is a block of Ind_{I}^{G}(O_{ψ}).

Proof. Since

Hom_{OG}(Ind_{I}^{G}(O_{ψ}), O_{φ}) ≅ Hom_{OI}(O_{ψ}, O_{φ|_{I}}),

the multiplicity m_{(G,φ)}(Ind_{I}^{G}(O_{ψ})) can only assume the values 0 and 1, and it is zero if ψ ≠ φ|_{I}. So we assume ψ = φ|_{I} from now on. If O_{φ} | Ind_{I}^{G}(O_{ψ}), then I contains a Sylow p-subgroup of G, since O_{φ} has the Sylow p-subgroups of G as vertices. Therefore, p |̸ [G : I]. Conversely, assume that p |̸ [G : I]. Then the maps

ϑ : Ind_{I}^{G}(O_{ψ}) → O_{φ}, ∑_{g∈G/I} g ⊗_{OI} α_{g} ↦ ∑_{g∈G/I} φ(g)α_{g},

and

η : O_{φ} → Ind_{I}^{G}(O_{ψ}), α ↦ [G : I]^{−1} ∑_{g∈G/I} φ(g^{−1})g ⊗_{OI} α,

are independent of the choice of the transversal for G/I and OG-linear such that ϑ ∘ η = id_{O_{φ}}. Thus ϵ := η ∘ ϑ is an idempotent in E := End_{OG}(Ind_{I}^{G}(O_{ψ})) with image

ϵ(Ind_{I}^{G}(O_{ψ})) = η(O_{φ}) ≅ O_{φ}.

Hence, we get ϵ(Ind_{I}^{G}(O_{ψ})) = Ind_{I}^{G}(O_{ψ})^{(G,φ)}, which is an OG-E-subbimodule of Ind_{I}^{G}(O_{ψ}). It is easy to check that ker(ϑ) = ∑_{g∈G}[(g − φ(g)1)Ind_{I}^{G}(O_{ψ})]. Thus,

Ind_{I}^{G}(O_{ψ}) = η(O_{φ}) ⊕ ker(ϑ)

is a decomposition of OG-E-bimodules, and the result follows.

The following result is now immediate from Proposition 2.4 and Lemma 2.5.

2.6. Proposition. Let V be a monomial OG-module and (H, φ) ∈ M_{G}(O). Then

m_{(H,φ)}(V) = rk_{k}\overline{V} (H, φ) = dim_{F} \overline{\overline{V}} (H, φ).

2.7. Example. If V ∈ _{OG}lat is not monomial, then \overline{V} (H, φ) need not be free as a k-module, and the dimension of \overline{\overline{V}} (H, φ) need not be the multiplicity of O_{φ} in Res_{H}^{G}(V). In fact, let G = ⟨x⟩ be the group of order 3, p = 3, let O contain a primitive third root of unity ζ, let H = G, and let φ ∈ Ĝ(O) be defined by φ(x) = ζ. Furthermore, let V = {α + βx + γx^{2} ∈ OG | α + β + γ = 0} be the augmentation ideal of OG. Then V is indecomposable, since its reduction modulo π is the radical of F G. Moreover, one has V^{(H,φ)} = O · (1 + ζ^{2}x + ζx) and V_{(E,1)}^{(H,φ)} = (1−ζ)O ·(1+ζ^{2}x+ζx), so \overline{V} (H, φ) ≅ O/(1−ζ)O and also \overline{\overline{V}} (H, φ) ≅ F, whereas the multiplicity of O_{φ} in V is zero. Note that, since (1 − ζ)^{2}O = 3O, the k-module \overline{V} (H, φ) is not free.

In the next section, we will show that Proposition 2.6 remains valid for OGlattices which are direct summands of monomial modules.

3. Linear source modules

3.1. We continue with the notation introduced in 2.1. In this section we are going to study linear source modules, i.e. direct summands of monomial modules. In view of the preceding section, we will work with the coefficient rings O, k and F. In order to handle these coefficient rings simultaneously, we fix n ∈ ℕ ∪ {∞} and set R := O/π^{n}O. Then many of the usual theorems for OG-lattices hold also for RG-lattices: The Krull-Schmidt Theorem and the theory of vertices and sources are valid, as well as the Green correspondence and the Higman criterion (see [Be, Sections 3.10, 3.12, and Proposition 3.6.4]). Moreover, in special situations also Green’s indecomposability theorem holds: Let I ⊴ H be subgroups of G such that H/I is a p-group and let ψ ∈ Î(R). Then Ind_{I}^{H}(R_{ψ}) is indecomposable. In fact, if Ind_{I}^{H}(R_{ψ}) were decomposable, then so would be its reduction modulo J(R), which is isomorphic to Ind_{I}^{H}(Fψ̄). But the latter module certainly is indecomposable by Green’s original theorem [Gr, Theorem 8]. An RG-lattice is called a linear source RG-module (resp. trivial source RG-module), if it is a direct sum of indecomposable RG-lattices which have sources of R-rank one (resp. trivial sources). We write RGtriv ⊆ _{RG}lin ⊆ _{RG}lat for the full subcategories of linear source RG-modules and trivial source RG-modules, respectively. A pair (P, ψ) ∈ M_{G}(R)_{p} is called a defect pair of an indecomposable linear source RG-module V if P is a vertex and R_{ψ} a source of V. Note that the defect pairs of V are unique up to G-conjugacy. The following lemma is proved in precisely the same way as in [Bo98, Proposition 1] (resp. [Br, (0.2), (0.4)]).

3.2. Lemma. Let n ∈ ℕ ∪ {∞} and let R = O/π^{n}O.

(a) For V ∈ _{RG}lat and a Sylow p-subgroup P of G the following statements are equivalent:

(i) The lattice V is a linear (resp. trivial) source RG-module. (ii) The lattice Res_{P}^{G}(V) is a monomial (resp. permutation) RP -module. (iii) The lattice V is isomorphic to a direct summand of a monomial (resp. per-

mutation) RG-module.

(b) For H ⩽ G, for linear (resp. trivial) source RG-modules V, V′, and for a linear (resp. trivial) source RH-module W one has:

(i) The RG-lattices V ⊕ V′, V ⊗ V′, and V^{∗} = Hom_{R}(V, R) are again linear (resp. trivial) source RG-modules.

(ii) The RH-lattice Res_{H}^{G}(V) is a linear (resp. trivial) source RH-module. (iii) The RG-lattice Ind_{H}^{G}(W) is a linear (resp. trivial) source RG-module. (iv) Every direct summand of V is a linear (resp. trivial) source RG-module.

Our first main result in this section is a Clifford type theorem for linear source modules. Before we state it, we would like to recall some elementary facts in Clifford theory for which we do not know a suitable reference.

Let V ∈ _{OG}lat be indecomposable, and let H⊴G. We consider the decomposition

Res_{H}^{G}(V) = V_{1} ⊕ ⋯ ⊕ V_{r}

of Res_{H}^{G}(V) into blocks V_{1}, …, V_{r}. Then, for g ∈ G, we have

V = gV = gV_{1} ⊕ ⋯ ⊕ gV_{r},

where each gV_{i} is also a block of Res_{H}^{G}(V), as is easily checked. This shows that the action of G on V permutes the blocks of Res_{H}^{G}(V). Since V is indecomposable, it must permute them transitively. Thus, if we denote the stabilizer of V_{1} in G by I, then V_{1} is an OI-lattice such that

Res_{H}^{G}(V) = ⨁_{g∈G/I} gV_{1} and V ≅ Ind_{I}^{G}(V_{1}).

3.3. Theorem. Let V ∈ _{OG}lin be indecomposable, and let (H, φ) ∈ M_{G}(O) be such that H ⊴ G and O_{φ} | Res_{H}^{G}(V). Then V has a defect pair (P, ψ) with the following properties:

(i) P is a Sylow p-subgroup of HP.

(ii) There exists a homomorphism λ : HP → O^{×} such that λ|_{H} = φ and λ|_{P} = ψ; in particular, φ is P -stable and HP ⩽ N_{G}(H, φ).

(iii) V ≅ Ind_{N_{G}(H,φ)}^{G}(V^{(H,φ)}) and V^{(H,φ)} is an indecomposable linear source ON_{G}(H, φ)-module with defect pair (P, ψ).

Proof. Let (P, ψ) be an arbitrary defect pair of V. Since V | Ind_{P}^{G}(O_{ψ}), there exists an indecomposable direct summand W of Ind_{P}^{HP} (O_{ψ}) such that V | Ind_{HP}^{G} (W). It is immediate that (P, ψ) is a defect pair of W as well. Moreover, we have

O_{φ} | Res_{H}^{G}(V) | Res_{H}^{G}(Ind_{HP}^{G} (W))

≅ ⨁_{g∈H\G/HP} Ind_{H∩^{g}(HP)}^{H}(Res_{H∩^{g}(HP)}^{^{g}(HP)}(^{g}W)) ≅ ⨁_{g∈G/HP} ^{g}Res_{H}^{HP} (W).

After replacing (P, ψ) and W by suitable conjugates we may assume that O_{φ} | Res_{H}^{HP} (W). Then we have

O_{φ} | Res_{H}^{HP} (W) | Res_{H}^{HP} (Ind_{P}^{HP} (O_{ψ})) ≅ Ind_{H∩P}^{H} (O_{ψ|_{H∩P}}).

By Lemma 2.5, this implies that H ∩ P is a Sylow p-subgroup of H, that φ|_{H∩P} = ψ|_{H∩P} and that Ind_{H∩P}^{H} (O_{ψ|_{H∩P}})^{(H,φ)} is a block of Ind_{H∩P}^{H} (O_{ψ|_{H∩P}}). Therefore W^{(H,φ)} is a block of Res_{H}^{HP} (W), of O-rank 1. Let I denote the stabilizer of W^{(H,φ)} in HP. By the remarks preceding the theorem we have W ≅ Ind_{I}^{HP} (W^{(H,φ)}); in particular, W is I-projective. On the other hand, W has defect pair (P, ψ), where P is a Sylow p-subgroup of HP since H ∩ P is a Sylow p-subgroup of H. Hence we have p |̸[HP : I] | [HP : H] = [P : P ∩ H], which implies that I = HP. Thus Res_{H}^{HP} (W) = W^{(H,φ)} ≅ O_{φ}; in particular, we have rk_{O}W = 1. So there exists a homomorphism λ : HP → O^{×} such that W ≅ O_{λ}. Since Res_{H}^{HP} (W) ≅ O_{φ}, we get λ|_{H} = φ. Since O_{λ} ≅ W | Ind_{P}^{HP} (O_{ψ}), Lemma 2.5 implies that λ|_{P} = ψ. Finally, V | Ind_{HP}^{G} (W) ≅ Ind_{HP}^{G} (O_{λ}) implies that

Res_{H}^{G}(V) | Res_{H}^{G}(Ind_{HP}^{G} (O_{λ})) ≅ ⨁_{g∈G/HP} ^{g}Res_{H}^{HP} (O_{λ}) = ⨁_{g∈G/HP} ^{g}O_{φ}.

Thus Res_{H}^{G}(V) = ⨁_{g∈G/N_{G}(H,φ)} gV^{(H,φ)} and V ≅ Ind_{N_{G}(H,φ)}^{G}(V^{(H,φ)}); in particular, the ON_{G}(H, φ)-module V^{(H,φ)} is indecomposable. Now V | Ind_{HP}^{G} (O_{λ}), so by Corollary 1.6 we obtain

V^{(H,φ)} | Ind_{HP}^{G} (O_{λ})^{(H,φ)} = ⨁_{\begin{matrix}g∈H\G/HP \\ φ|_{H∩^{g} (HP)}=^{g}λ|_{H∩^{g} (HP)} \end{matrix}} O · tr_{(H∩^{g}(HP),φ)}^{(H,φ)}(g ⊗_{OHP} 1)

= ⨁_{\begin{matrix}g∈G/HP \\ φ=^{g}λ|_{H} \end{matrix}} O(g ⊗_{OHP} 1) = ⨁_{g∈N_{G}(H,φ)/HP} O(g ⊗_{OHP} 1)

≅ Ind_{HP}^{N_{G}(H,φ)}(O_{λ}) | Ind_{P}^{N_{G}(H,φ)}(O_{ψ}).

Thus the ON_{G}(H, φ)-module V^{(H,φ)} is P -projective. Let (Q, μ) be a defect pair of V^{(H,φ)} such that Q ⩽ P. Then V^{(H,φ)} | Ind_{Q}^{N_{G}(H,φ)}(O_{μ}) and

V ≅ Ind_{N_{G}(H,φ)}^{G}(V^{(H,φ)}) | Ind_{Q}^{G}(O_{μ}),

so V is Q-projective. Since V has defect pair (P, ψ), we conclude that |P | ⩽ |Q| ⩽ |P |. Thus P = Q, and (P, ψ) is a defect pair of V^{(H,φ)}.

We get the following consequence.

3.4. Corollary. Let (H, φ) ∈ M_{G}(O) with H ⊴ G, and let I := N_{G}(H, φ). Then the maps V ↦ V^{(H,φ)} and W ↦ Ind_{I}^{G}(W) induce mutually inverse bijections between the set of isomorphism classes of indecomposable linear source OG-modules V such that O_{φ} | Res_{H}^{G}(V) and the set of isomorphism classes of indecomposable linear source OI-modules W such that O_{φ} | Res_{H}^{I}(W). Moreover, the map W ↦ Ind_{I}^{G}(W) preserves defect pairs.

Proof. Let V be an indecomposable linear source OG-module such that O_{φ} | Res_{H}^{G}(V). Then, by Theorem 3.3, V^{(H,φ)} is an indecomposable linear source OImodule, and V ≅ Ind_{I}^{G}(V^{(H,φ)}). Moreover, it is immediate that O_{φ} | Res_{H}^{I}(V^{(H,φ)}).

Conversely, let W be an indecomposable linear source OI-module such that O_{φ} | Res_{H}^{I}(W). Then Theorem 3.3 implies that W = W^{(H,φ)}. Since W | Res_{I}^{G}(Ind_{I}^{G}(W)) there exists an indecomposable direct summand V of Ind_{I}^{G}(W) such that W | Res_{I}^{G}(V). Thus O_{φ} | Res_{H}^{I}(W) | Res_{H}^{G}(V). By Theorem 3.3, we have V ≅ Ind_{I}^{G}(V^{(H,φ)}), and

rk_{O}V = [G : I]rk_{O}V^{(H,φ)} ⩾ [G : I]rk_{O}W^{(H,φ)}
= [G : I]rk_{O}W = rk_{O}Ind_{I}^{G}(W) ⩾ rk_{O}V.

It follows that Ind_{I}^{G}(W) = V is an indecomposable linear source OG-module and W^{(H,φ)} ≅ V^{(H,φ)}; hence W = W^{(H,φ)} ≅ V^{(H,φ)} = Ind_{I}^{G}(W)^{(H,φ)}. Finally, the statement about defect pairs is an easy consequence of Theorem 3.3.

3.5. Theorem. Let V ∈ _{OG}lin and (H, φ) ∈ M_{G}(O). Then \overline{V} (H, φ) ∈ _{kN_{G}(H,φ)}lin, \overline{\overline{V}} (H, φ) ∈ _{F N_{G}(H,φ)}triv, and

dim_{F} \overline{\overline{V}} (H, φ) = rk_{k}\overline{V} (H, φ) = m_{(H,φ)}(V) .

In particular, one has

\overline{\overline{V}} (H, φ) ≠ 0 ⟺ \overline{V} (H, φ) ≠ 0 ⟺ O_{φ} | Res_{H}^{G}(V) .

Proof. By Proposition 2.4, the functor \overline{·}(H, φ) takes monomial OG-modules to monomial ON_{G}(H, φ)-modules, and it commutes with direct sums. Thus \overline{V} (H, φ) ∈ kN_{G}(H,φ)lin and \overline{\overline{V}} (H, φ) ∈ _{F N_{G}(H,φ)}lin; in particular, we have dim_{F} \overline{\overline{V}} (H, φ) = rk_{k}\overline{V} (H, φ). In order to prove that rk_{k}\overline{V} (H, φ) = m_{(H,φ)}(V) we may certainly assume that G = H and that U = V ⊕ W with U, W ∈ _{OG}lin and U monomial. Since O_{φ}(H, φ) ≅ O_{φ}/pO_{φ} has k-rank 1, the additivity of the Brauer construction implies m_{(H,φ)}(V) ⩽ rk_{k}\overline{V} (H, φ). Thus, from Proposition 2.6 we obtain

m_{(H,φ)}(U) = m_{(H,φ)}(V) + m_{(H,φ)}(W) ⩽ rk_{k}\overline{V} (H, φ) + rk_{k}\overline{W} (H, φ)
= rk_{k}\overline{U}(H, φ) = m_{(H,φ)}(U) .

This shows that m_{(H,φ)}(V) + m_{(H,φ)}(W) = rk_{k}\overline{V} (H, φ) + rk_{k}\overline{W} (H, φ), and then also m_{(H,φ)}(V) = rk_{k}\overline{V} (H, φ).

3.6. Remark. By [Bo98, Corollary 2.7], the multiplicities m_{(H,φ)}(V), (H, φ) ∈ M_{G}(O), determine V ∈ _{OG}lin up to isomorphism. Therefore, the above theorem implies that also the Brauer constructions \overline{V} (H, φ) (resp. \overline{\overline{V}} (H, φ)), (H, φ) ∈ M_{G}(O), and even their ranks (resp. dimensions) determine V. The proof of Corollary 2.7 in [Bo98] makes use of Proposition 2.6 in the same article. We noticed a gap in the proof of this proposition which will be filled in Section 5.

3.7. Proposition. Let V ∈ _{OG}lin be indecomposable with defect pair (P, ψ). Moreover, let (H, φ) ∈ M_{G}(O) and let Q be a Sylow p-subgroup of H. Then, \overline{V} (H, φ) ≠ 0 implies that (Q, φ|_{Q}) ⩽ ^{g}(P, ψ) for some g ∈ G. If H is a p-subgroup of G, also the converse is true.

Proof. By Theorem 3.5, \overline{V} (H, φ) ≠ 0 is equivalent to O_{φ} | Res_{H}^{G}(V). Since V | Ind_{P}^{G}(O_{ψ}), Mackey’s decomposition formula shows that O_{φ} | Res_{H}^{G}(V) implies

O_{φ} | Ind_{H∩^{g}P}^{H} (O_{(^{g}ψ)|_{H∩^{g} P}})

for some g ∈ G. By Lemma 2.5 this implies that H ∩^{g}P is a Sylow p-subgroup of H and that φ|_{H∩^{g}P} = ^{g}ψ|_{H∩^{g}P}. Thus (H ∩^{g}P, φ|_{H∩^{g}P}) ⩽ ^{g}(P, ψ), and there exists h ∈ H such that ^{h}(H ∩^{g}P) = Q. Then ^{h}(H ∩^{g}P, φ|_{H∩^{g}P}) = (Q,^{h}φ|_{Q}) = (Q, φ|_{Q}). Finally, if H = Q and (Q, φ|_{Q}) ⩽ ^{g}(P, ψ), then

O_{φ} | Res_{Q}^{^{g}P} (O_{^{g}ψ}) | Res_{Q}^{^{g}P} (Res_{^{g}P}^{G} (V)) ,

and V (H, φ) ≠ 0 by Theorem 3.5.

3.8. We recall the Green correspondence as explained, for example, in [Th, Proposition (20.8)]. Thus let (P, ψ) ∈ M_{G}(O)_{p}. If V ∈ _{OG}lin is indecomposable with defect pair (P, ψ), then Res_{N_{G}(P,ψ)}^{G}(V) has a unique (up to isomorphism) indecomposable direct summand W ∈ _{ON_{G}(P,ψ)}lin with defect pair (P, ψ), W occurs with multiplicity one, and all other indecomposable direct summands W′ of Res_{N_{G}(P,ψ)}^{G}(V) have defect pairs (Q, μ) such that (Q, μ) ⩽ ^{g}(P, ψ) for some g ∈ G. It may happen that Q = P. Conversely, if W ∈ _{ON_{G}(P,ψ)}lin is indecomposable with defect pair (P, ψ), then Ind_{N_{G}(P,ψ)}^{G}(W) has a unique (up to isomorphism) indecomposable direct summand V with defect pair (P, ψ), V occurs with multiplicity one, and all other indecomposable direct summands V′ of Ind_{N_{G}(P,ψ)}^{G}(W) have defect pairs strictly smaller than (P, ψ). In this way one obtains mutually inverse bijections between IInd^{(P,ψ)}(OG), the set of isomorphism classes of indecomposable linear source OGmodules with defect pair (P, ψ), and the corresponding set IInd^{(P,ψ)}(ON_{G}(P, ψ)). These bijections are called the Green correspondence with respect to (P, ψ).

3.9. Theorem. Let V ∈ _{OG}lin be indecomposable with defect pair (P, ψ), and let W ∈ _{ON_{G}(P,ψ)}lin be the Green correspondent of V with respect to (P, ψ). Then

\overline{V} (P, ψ) ≅ W/pW and \overline{\overline{V}} (P, ψ) ≅ W/πW.

Proof. We write Res_{N_{G}(P,ψ)}^{G}(V) = W ⊕ W′ with W′ ∈ _{ON_{G}(P,ψ)}lin, and fix an indecomposable direct summand U of W′. Let (P′, ψ′) be a defect pair of U. Then (P′, ψ′) ⩽ ^{g}(P, ψ) for some g ∈ G, and (P′, ψ′) ≠ (P, ψ) by the remarks above. Thus, by Proposition 3.7, we have \overline{U}(P, ψ) = 0. Hence \overline{V} (P, ψ) = \overline{W} (P, ψ). On the other hand, Proposition 3.7 implies that \overline{W} (P, ψ) ≠ 0. Thus O_{ψ} | Res_{P}^{N_{G}(P,ψ)}(W) by Theorem 3.5, and W = W^{(P,ψ)} by Theorem 3.3. Hence \overline{W} (P, ψ) = W/pW, and the result follows.

3.10. Remark. Let (P, ψ) ∈ M_{G}(O)_{p}. Then by Theorem 3.3 the indecomposable direct summands of Ind_{P}^{N_{G}(P,ψ)}(O_{ψ}) all have defect pair (P, ψ). Thus the set IInd^{(P,ψ)}(ON_{G}(P, ψ)) is in bijection with the set of isomorphism classes of indecomposable projective modules over E := End_{ON_{G}(P,ψ)}(Ind_{P}^{N_{G}(P,ψ)}(O_{ψ})). But E is a twisted group algebra of N_{G}(P, ψ)/P over O, and every element in E induces an F N_{G}(P, ψ)-endomorphism of

Ind_{P}^{N_{G}(P,ψ)}(O_{ψ})/π · Ind_{P}^{N_{G}(P,ψ)}(O_{ψ}) ≅ Ind_{P}^{N_{G}(P,ψ)}(O_{ψ}/πO_{ψ}) ≅ Ind_{P}^{N_{G}(P,ψ)}(F).

Moreover,

End_{F N_{G}(P,ψ)}(Ind_{P}^{N_{G}(P,ψ)}(F)) ≅ F [N_{G}(P, ψ)/P],

so we obtain E/πE ≅ F [N_{G}(P, ψ)/P]. Thus reduction modulo π yields a bijection between IInd^{(P,ψ)}(ON_{G}(P, ψ)) and IInd^{pr}(F [N_{G}(P, ψ)/P]), the set of isomorphism classes of indecomposable projective F [N_{G}(P, ψ)/P]-modules. Combining this bijection with Theorem 3.9, we see that the functor V ↦ \overline{\overline{V}} (P, ψ) induces a bijection

IInd^{(P,ψ)}(OG) → IInd^{pr}(F [N_{G}(P, ψ)/P]).

4. Reduction of linear source modules

4.1. We assume the notation established in 3.1. Let R = O/π^{n}O with n ∈ ℕ ∪ {∞}. We denote by T_{R}(G) ⊆ L_{R}(G) the subrings of the Green ring of G over R, generated by the linear source and trivial source RG-modules. More precisely, we may consider L_{R}(G) (resp. T_{R}(G)) as the free abelian group on the isomorphism classes of indecomposable linear source (resp. trivial source) RG-modules. Note that T_{R} and L_{R} are Green functors on G (see for example [Bo96] for a definition).

Lemma 3.2 implies that, for R′ = O/π^{m}O with m ⩽ n, reduction modulo π^{m}, π^{n}, and π induces functors

[figure: _{OH}lin; _{RH}lin; _{R′H}lin; _{F H}lin; ⋃ |; ⋃ |; ⋃ |; _{_{OH}triv}; _{_{RH}triv}; _{_{R′H}triv}; _{_{F H}triv}]

for all H ⩽ G which commute with direct sums, tensor products, restriction, induction and conjugation. Thus, we obtain morphisms of Green functors

[figure: L_{O}; L_{R}; L_{R′}; L_{F}; ⋃ |; ⋃ |; ⋃ |; T_{O}; T_{R}; T_{R′}; T_{F}] (4.1.a)

on G. Note that for n = 1 we have T_{F} = L_{F}, and recall that the reduction map T_{O} → T_{F} is an isomorphism (see for example [Br, (3.5)]) which preserves indecomposablility and vertices. One can also see easily that each map in the bottom row of Diagram (4.1.a) is an isomorphism preserving indecomposability and vertices. In fact, it suffices to show that this is true for T_{O} → T_{R} and any R = O/π^{n}O with n ∈ ℕ. By Remark 1.7, for each p-subgroup P of G, the natural ring homomorphism End_{OG}(Ind_{P}^{G}(O)) → End_{RG}(Ind_{P}^{G}(R)) is surjective with kernel consisting of those endomorphisms f satisfying f(Ind_{P}^{G}(O)) ⊆ π^{n}Ind_{P}^{G}(O). Therefore, the kernel is contained in the radical of End_{OG}(Ind_{P}^{G}(O)) and the lifting theorem of idempotents applies to show that reduction modulo π^{n} preserves indecomposability of relatively P -projective trivial source modules and induces a bijection between the set of isomorphism classes of indecomposable direct summands of Ind_{P}^{G}(O) and of Ind_{P}^{G}(R). Since vertices can only become smaller under reduction, the preservation of vertices under reduction modulo π^{n}O follows from its preservation under reduction modulo πO.

4.2. We will study the reduction morphism L_{O} → L_{R} for R = O/π^{n}O with n ∈ ℕ large enough. Note that each element in R^{×} which is congruent to 1 modulo J(R) has a finite p-power order. Hence, in general, the reduction map cannot be surjective. However, injectivity of the reduction map is guaranteed for large n by Maranda’s theorem (cf. [CR81, § 30]). We proceed to investigate the question of for which n ∈ ℕ the reduction map L_{O} → L_{R} is injective in more detail. We start by investigating the reduction of rank one modules.

Recall that the ramification index e_{ℚ_{p}(ζ)/ℚ_{p}} of ℚ_{p}(ζ) over ℚ_{p}, for a primitive d-th root of unity ζ, with d = p^{r}m where p |̸ m, is given by p^{r−1}(p − 1) if r ⩾ 1, and by 1 if r = 0, and that v_{ℚ_{p}(ζ)}(1 − ζ^{m}) = 1. Also recall that reduction modulo πO induces an isomorphism between the p′-part of the group μ_{K} of roots of unity of K and the corresponding group μ_{F}. So if μ_{K} is a p′-group, then reduction modulo π^{n}O is injective for arbitrary n ∈ ℕ. The remaining case is treated in the following lemma.

4.3. Lemma. Assume that K contains a primitive p-th root of unity and let n ∈ ℕ. Then the reduction map μ_{K} → (O/π^{n}O)^{×} is injective if and only if n > e/(p − 1), where e denotes the ramification index of K/ℚ_{p}.

Proof. Let 1 ≠ ζ ∈ μ_{K} and write ζ = ζ_{p^{r}} · ζ_{m} with roots of unity ζ_{p^{r}} and ζ_{m} of respective orders p^{r} and m, where p |̸ m. Since F^{×} has no elements of p-power order except 1, we have ζ_{p^{r}} ≡ 1 mod πO. Thus, if ζ_{m} ≠ 1, then v_{K}(1 − ζ) = 0. If ζ_{m} = 1, then with K′ := ℚ_{p}(ζ) we have v_{K}(1 − ζ) = e_{K/K′}v_{K′}(1 − ζ) = e_{K/K′} = e/p^{r−1}(p − 1). Hence, μ_{K} → (O/π^{n}O)^{×} is injective if and only if v_{K}(1 − ζ) < n for each ζ ∈ μ_{K} ∖ {1}, which in turn is equivalent to n > e/p^{r−1}(p − 1) for all r ∈ ℕ such that K has a primitive p^{r}-th root of unity. Since e/p^{r−1}(p − 1) assumes its minimum for r = 1, the proof is complete.

4.4. Lemma. Let n ∈ ℕ be such that n > e/(p − 1), and let R := O/π^{n}O. Let ¯· (resp. ¯ ¯·) denote the functor of reducing modulo π^{n}O (resp. πO). Then for each (H, φ) ∈ M_{G}(O), the rings End_{OG}(Ind_{H}^{G}(O_{φ})) and End_{RG}(Ind_{H}^{G}(Rφ̄)) have ideals contained in their respective radicals such that the corresponding factor rings are isomorphic to F ⊗_{O} End_{OG}(Ind_{H}^{G}(O_{φ})).

Proof. By Remark 1.7 we have a commutative diagram

where the upper horizontal map is the obvious natural ring homomorphism. Note that the right diagonal map sends to zero those summands which are parametrized

by g ∈ H\G/H with φ|_{H∩^{g}H} ≠ ^{g}φ|_{H∩^{g}H}, since, by Lemma 4.3, we have 𝔞_{(H,φ̄)}^{^{g}(H,φ̄)} = R if and only if φ|_{H∩^{g}H} = ^{g}φ|_{H∩^{g}H}.

Obviously both diagonal maps are surjective, the kernels consisting of those endomorphisms whose images are contained in πInd_{H}^{G}(O_{φ}), resp. π̄Ind_{H}^{G}(Rφ̄). Therefore, the kernels are ideals contained in the respective radicals. Obviously the left diagonal map is a ring homomorphism, since it is induced by the canonical map End_{OG}(Ind_{H}^{G}(O_{φ})) → F ⊗_{O} End_{OG}(Ind_{H}^{G}(O_{φ})). Since the triangle commutes, since End_{RG}(Ind_{H}^{G}(Rφ̄)) is the sum of the image of the horizontal map and the kernel of the right diagonal map, and since the left diagonal map is a ring homomorphism, also the right diagonal map is a ring homomorphism, and the proof is complete.

4.5. Theorem. Let n ∈ ℕ be such that n > e/(p − 1), let R := O/π^{n}O, let ¯· denote reduction modulo π^{n}O, and let V ∈ _{OG}lin be indecomposable with defect pair (P, ψ) ∈ M_{G}(O)_{p} and with Green correspondent W ∈ _{ON_{G}(P,ψ)}lin with respect to (P, ψ).

(a) The reduction \overline{V} ∈ _{RG}lin of V is indecomposable.

(b) The pair (P, ψ̄) is a defect pair of \overline{V}, and N_{G}(P, ψ) = N_{G}(P, ψ̄).

(c) The module \overline{W} ∈ _{RN_{G}(P,ψ)}lin is the Green correspondent of \overline{V} with respect to (P, ψ̄).

(d) Reduction modulo π^{n}O induces a bijection IInd^{(P,ψ)}(OG) → IInd^{(P,ψ̄)}(RG).

(e) Two modules X, X′ ∈ _{OG}lin are isomorphic if and only if their reductions X̄, X̄′ ∈ _{RG}lin are isomorphic.

Proof. (a) This follows immediately from Lemma 4.4.

(b), (c) By Lemma 4.3 we have N_{G}(P, ψ) = N_{G}(P, ψ̄), and by part (a) the module \overline{W} is indecomposable. Moreover, it is easy to see that \overline{W} has defect pair (P, ψ̄), since \overline{W} | Ind_{P}^{N_{G}(P,ψ)}(Oψ̄). Hence, its Green correspondent is the unique indecomposable direct summand of Ind_{N_{G}(P,ψ)}^{G}(\overline{W}) with defect pair (P, ψ̄). Since

Ind_{N_{G}(P,ψ)}^{G}(\overline{W}) ≅ \overline{Ind_{N_{G}(P,ψ)}^{G}(W)} ≅ \overline{V ⊕ V′} ≅ \overline{V} ⊕ \overline{V′}

with V′ ∈ _{OG}lin having only indecomposable direct summands with vertex of order smaller than |P |, the module \overline{V} has to be the Green correspondent of \overline{W} with respect to (P, ψ̄) and has the same defect pair (P, ψ̄) as \overline{W}.

(d) This follows from part (b) and Lemma 4.4.

(e) This follows from part (c) and part (d).

4.6. Corollary. Let V ∈ _{OG}lin. The reduction map V ↦ V/pV for p ⩾ 3 and V ↦ V/4V for p = 2 induces an injective ring homomorphism L_{O}(G) → L_{O/pO}(G) for odd p, resp. L_{O}(G) → L_{O/4O}(G) for p = 2, which preserves indecomposability and vertices.

4.7. Remark. If V ∈ _{OG}lin is indecomposable, then in general V/πV need not be indecomposable: Let G = S_{3}, the symmetric group of degree 3, let p = 3, and let P ◁ G denote the Sylow p-subgroup. Moreover, let ψ ∈ P̂(O) be non-trivial. Then V := Ind_{P}^{G}(O_{ψ}) is indecomposable, since its character is irreducible. But V/πV ≅ Ind_{P}^{G}(F) splits into two one-dimensional F G-modules.

4.8. Remark. The collection of the functors ¯ ¯·(P, ψ) for (P, ψ) ∈ M_{G}(O)_{p} induces a map

α: L_{O}(G) →(  ∏_{(P,ψ)∈M_{G}(O)_{p}} T_{F} (N_{G}(P, ψ)/P))^{G}

≅ ∏_{(P,ψ)∈M_{G}(O)_{p}/G} T_{F} (N_{G}(P, ψ)/P),

where G acts on the product by conjugation. Note that after ordering M_{G}(O)_{p}/G according to the orders of the involved p-subgroups P, and ordering the indecomposable linear source modules in a similar way according to their defect pairs, Proposition 3.7 implies that the matrix associated to the map α is a block triangular matrix. The block with coordinates (P, ψ) and (Q, φ) from a set of representatives for the Gconjugacy classes of M_{G}(O)_{p} is indexed by IInd^{(P,ψ)}(OG) × IInd(_{F N_{G}(Q,φ)/Q}triv), where the latter factor denotes the set of isomorphism classes of indecomposable trivial source F N_{G}(Q, φ)/Q-modules. Associating to each trivial source module its Brauer character induces a map

∏_{(P,ψ)∈M_{G}(O)_{p}/G} T_{F} (N_{G}(P, ψ)/P) → ∏_{(P,ψ)∈M_{G}(O)_{p}/G} R_{F} (N_{G}(P, ψ)/P),

which composes with α to a map

β: L_{O}(G) → ∏_{(P,ψ)∈M_{G}(O)_{p}/G} R_{F} (N_{G}(P, ψ)/P)

mapping the class of a linear source OG-module V to the family of Brauer characters of \overline{\overline{V}} (P, ψ), (P, ψ) ∈ M_{G}(O)_{p}/G. Here, by R_{F} (G) we denote the Grothendieck group of the category of finitely generated F G-modules. The matrix representing the map β is again block triangular with the Cartan matrix of N_{G}(P, ψ)/P as diagonal block at position (P, ψ). Thus β is injective and invertible after tensoring with ℚ.

5. A correction

We conclude this article by filling a gap in the proof of Proposition 2.6 in [Bo98]. The proof there only reduced the statement of Proposition 2.6 to the proof of the following proposition. The following notation is needed. Let L′_{O}(G) ⊆ L_{O}(G) denote the span of the isomorphism classes of those indecomposable V ∈ _{OG}lin which have the Sylow p-subgroups of G as vertices, and let q_{G}: L_{O}(G) → L′_{O}(G) denote the obvious projection map. A finite group G is called p-hypoelementary if it has a normal p-subgroup with cyclic factor group.

5.1. Proposition. Let G be a p-hypoelementary group and assume that O contains a root of unity of order |G|. Let x ∈ L_{O}(G) be an element such that q_{H}(res_{H}^{G}(x)) = 0 for all H ⩽ G. Then x = 0.

Proof. Let V, W ∈ _{OG}lin be given such that, for all subgroups H ⩽ G, each indecomposable OH-module with the Sylow p-subgroup of H as vertex occurs with the same multiplicity as direct summand in Res_{H}^{G}(V) and in Res_{H}^{G}(W). It suffices to show that in this situation we have V ≅ W. We will prove this by induction on the order of G and then by induction on rk_{O}V + rk_{O}W.

We choose among all indecomposable direct summands of V and W one, say X, with maximal vertex Q, and assume by symmetry that X | V. Let (Q, ψ) be a defect pair of X, let N := N_{G}(Q), and let M := N_{G}(Q, ψ).

Assume that N < G. Then by induction Res_{N}^{G}(V) ≅ Res_{N}^{G}(W). Let X′ ∈ _{ON}lin denote a Green correspondent of X. So also X′ | Res_{N}^{G}(W). Let Y | W be indecomposable such that X′ | Res_{N}^{G}(Y). Then the maximality of Q and the Green correspondence imply that Y ≅ X. Hence, V ≅ X ⊕ V′ and W ≅ Y ⊕ W′ for some V′, W′ ∈ _{OG}lin. Then the hypothesis at the beginning of the proof is also satisfied for V′ and W′, and by induction on the ranks we have V′ ≅ W′, hence V ≅ W.

Therefore we assume from now on that N = G. Assume further that M < G. Then Res_{M}^{G}(V) ≅ Res_{M}^{G}(W) by induction. By Theorem 3.3 we know that X^{(Q,ψ)} | Res_{M}^{G}(X) and X ≅ Ind_{M}^{G}(X^{(Q,ψ)}). Let Y | W be an indecomposable direct summand such that X^{(Q,ψ)} | Res_{M}^{G}(Y). Then, again by Theorem 3.3, we have X^{(Q,ψ)} ≅ Y^{(Q,ψ)}, and then X ≅ Y. Therefore we have V ≅ W by induction on the ranks as before.

From now on we assume that M = G. Let I := ker ψ, let P ⊴ G be the Sylow p-subgroup of G, and let C ⩽ G be a complement of P in G, hence a cyclic p′- subgroup. Since (Q, ψ) is stable under G, it follows that the subgroups Q and I are normal in G, and G centralizes the factor group Q/I. We set H = QC. Then H/I decomposes as the direct product of CI/I ≅ C and Q/I. Let ψ̃ ∈ Ĥ(O) be such that ψ̃|_{Q} = ψ and ψ̃|_{C} = 1. Moreover, let φ_{1}, …, φ_{r} ∈ Ĥ(O) denote those homomorphisms which are trivial on Q. Then r = |C| and φ_{i}ψ̃, i = 1, …, r, are the different extensions of ψ to H. Therefore we have

Ind_{Q}^{G}(O_{ψ}) ≅ Ind_{H}^{G}(Ind_{Q}^{H}(O_{ψ})) ≅ ⨁_{i=1}^{r} Ind_{H}^{G}(O_{φ_{i}ψ̃}) .

The summands X_{i} := Ind_{H}^{G}(O_{φ_{i}ψ̃}), i = 1, …, r, are indecomposable, since Res_{P}^{G}(X_{i}) ≅ Ind_{Q}^{P}(O_{ψ}) is indecomposable. Let m_{i}, resp. n_{i}, i = 1, …, r, denote the multiplicities of X_{i} as direct summands in V, resp. W. Note that any indecomposable module Z ∈ _{OG}lin such that O_{ψ} | Res_{Q}^{G}(Z) is isomorphic to X_{i} for some i = 1, …, r, by the maximality of Q and by Theorem 3.3 (a). Thus X ≅ X_{j} for some j ∈ {1, …, r}, and m_{j} > 0 for this j. By our hypothesis we therefore know that

∑_{i=1}^{r} m_{i}res_{H}^{G}([X_{i}]) = ∑_{i=1}^{r} n_{i}res_{H}^{G}([X_{i}]) ∈ L′_{O}(H) .

Here, [X_{i}] denotes the element in L_{O}(G) associated to X_{i}. Now it suffices to show that the elements res_{H}^{G}([X_{i}]), i = 1, …, r, are linearly independent in L′_{O}(H). Let X′_{i} := Ind_{H}^{G}(O_{φ_{i}}) for i = 1, …, r. Then Res_{H}^{G}(X′_{i}) ≅ Oψ̃^{−1} ⊗ Res_{H}^{G}(X_{i}), and it suffices to show that the elements res_{H}^{G}([X′_{i}]), i = 1, …, r, are linearly independent in L′_{O}(H). We have Ind_{Q}^{G}(O) ≅ X′_{1} ⊕ ⋯ ⊕ X′_{r} and X′_{1}, …, X′_{r} are indecomposable by the same argument as for X_{1}, …, X_{r}. Since Q acts trivially on X′_{1}, …, X′_{r}, we may assume that Q = 1. Then X′_{1}, …, X′_{r} are all the projective indecomposable modules of G. They are pairwise distinct, since F G has precisely r simple modules. It is well-known that the characters χ′_{i} of X′_{i}, i = 1, …, r, are linearly independent and vanish on p-singular classes. Therefore, the restrictions of χ′_{1}, …, χ′_{r} to the p′-Hall subgroup H of G are still linearly independent, since all p′-Hall subgroups of G are conjugate. This completes the proof.

References

[Be] D. Benson: Representations and cohomology, vol. I. Cambridge University Press, Cambridge 1991. MR 92m:20005

[Bo96] R. Boltje: A general theory of canonical induction formulae. J. Algebra 206 (1998), 293– 343. MR 99k:20013

[Bo98] R. Boltje: Linear source modules and trivial source modules. Proc. Sympos. Pure Math. 63 (1998), 7–30. MR 99d:20016

[Br] M. Broué: On Scott modules and p-permutation modules: An approach through the Brauer morphism. Proc. Amer. Math. Soc. 93 (1985), 401–408. MR 86d:20010

[CR81] C.W. Curtis, I. Reiner: Methods of representation theory, Vol. 1. J. Wiley & Sons, New York 1981. MR 82i:20001

[CR87] C.W. Curtis, I. Reiner: Methods of representation theory, Vol. 2. J. Wiley & Sons, New York 1987. MR 88f:20002

[Gr] J.A. Green: On the indecomposable representations of a finite group. Math. Zeitschr. 70 (1959), 430–445. MR 24:A1304

[Th] J. Thévenaz: G-algebras and modular representation theory. Oxford University Press 1995. MR 96j:20017

Department of Mathematics, University of California, Santa Cruz, California 95064

   E-mail address: boltje@math.ucsc.edu

Mathematisches Institut, Universität Jena, 07 740 Jena, Germany

   E-mail address: kuelshammer@uni-jena.de