Contents

A. ZIMMERMANN: Structure of blocks with cyclic defect group and Green correspondence 1

1. Introduction 1 2. Green correspondence 3 2.1. Motivation 3 2.2. Adjoint functors 3 2.3. Some more background from modular representation theory 5 2.4. The Green correspondence for adjoint pairs of functors 8 3. Classical theory of blocks with cyclic defect groups and Green’s walk

around the Brauer tree 22 3.1. The theory of Dade on blocks with cyclic defect group 22 3.2. Green’s walk around the Brauer tree 23 3.3. Dade’s description for blocks with normal cyclic defect groups 26 3.4. Definition of the ’walk’ δ 27 3.5. Turning to characteristic 0 30 4. Blocks with cyclic defect groups are Green orders 33 4.1. A small example 33 4.2. Defining Green orders 34 4.3. The rational components; isotypic orders 36 4.4. Structure theorem for Green orders 39 References 43 STRUCTURE OF BLOCKS WITH CYCLIC DEFECT GROUP

AND GREEN CORRESPONDENCE

ALEXANDER ZIMMERMANN

1. Introduction

This paper is a based on a course I gave at the Ovidius university of Constanta on ”Structure of blocks with cyclic defect group and Green correspondence”. However, I added many details to the script of the course I have given there. We shall divide the material into three parts.

• In the first part we will present a new way due to Auslander and Kleiner [1] to get a form of Green correspondence. The classical Green correspondence follows easily from that and this more general point of view might have some impact to other fields of interest. The classical Green correspondence will be used in the following sections with good success.

• In the second part we shall present the classical and very beautifully written paper of Green [5]. Here, the Green correspondence in its classical form is used intensively. We just cite from the classical paper of Dade the fundamental properties of blocks with cyclic defect group. We also give results of Michler [13, 14] on the structure of blocks with cyclic defect groups.

• The third part deals with K. W. Roggenkamp’s paper [16] on Green orders in which he firstly defined Green orders, secondly used the results of Green to prove that a block with cyclic defect group is a Green order, and thirdly determines the structure of a Green order in great detail. As far as is known to the author, this is the most far reaching result on the structure of blocks with cyclic defect group. For the proof we follow [16].

The reader is assumed to know very basic facts on categories, not much more than the definition of a category, functors and natural transformations. A good reference is [15, 12]. Also the basic notions in noetherian ring theory are assumed such as the notion of a radical of a ring and a module, a socle and a top. As a good reference we give here [7]. Furthermore, some very basic algebraic number theory is assumed such as the basic definitions of a Dedekind domain and the ramification index for local fields. Certainly [6] covers more than what is enough as reference. Besides the deep theory of Dade, all proofs are included. In this sense the paper is self–contained.

The course, I gave in Constanta is contained, for the part dealing with the Green correspondence in the Sections 2.1, 2.2, 2.3, 2.4.1, 2.4.2 and the beginning parts of 2.4.4. The reader who is only interested in the classical Green correspondence is adviced to read only Sections 2.1, 2.3 and 2.4.1. Of course, the proof is given, as is done in [1], in the abstract terms. The nicest part of the abstract Green correspondence is collected in the Section 2.4.2 and 2.4.3. For the classical theory on blocks with cyclic defect groups, the material presented in Constanta is located in the Sections 3.1, 3.2 and 3.3. Section 3.4 and 3.5 are devoted to prove Green’s walk around the Brauer tree, in the way as Green did. For Roggenkamp’s description of blocks with cyclic defect groups the parts which were presented in Constanta are located in Section 4.1 and 4.2. The rest of the sections contain mainly proofs, which especially in Section 2 are uncomfortly technical, and which are not needed for the understanding of the other parts.

Date: November 1995.

The author acknowledges financial support from the Deutsche Forschungsgemeinschaft.

Acknowledgement. I want to thank the organizers of the “Workshop and Meeting on the Theory of Groups, Algebras and Orders” for having given me the opportunity to give this series of lectures. I also want to thank the Équipe des Groupes Finis de l’Univérsité de Paris 7 for their hospitality during the time this paper is written.

STRUCTURE OF BLOCKS WITH CYCLIC DEFECT GROUPAND GREEN CORRESPONDENCE 3

2. Green correspondence

2.1. Motivation. Let G be a finite group and let R be a complete discrete valuation ring of characteristic 0 with residue field k of characteristic p. A large part of modular representation theory of finite groups deals with trying to relate the representation theory of G to the representation theory of the ”p–local structure“ of G. The first and perhaps most elementary attempt to do so is the Green correspondence.

By RF − \overline{mod}^{∘} we denote for any finite group F the stable module category. The objects of_{RF} \overline{mod}^{∘} are the same as those of RG−mod^{∘}, namely finitely generated R– projective RF −modules, called in the sequel RF −lattices. To define the morphism set we have to put an equivalence relation onto the morphism sets of RF − mod^{∘}. Let M and N be two RF –lattices. Two morphisms f, g ∈ Hom_{RF} (M, N) are called equivalent, f ≅ g, if and only if f − g factors through a projective RF –module. Then,

Hom_{RF −\overline{mod}^{∘}}(M, N) =: \overline{Hom}_{RF} (M, N) := Hom_{RF} (M, N)/≅.

Theorem 2.1. (Green) Let D be a Sylow p−subgroup of G and let H := N_{G}(D) be the normalizer of D in G. Assume that for all g ∈ G \ H we get gDg^{−1} ∩ D = {1}. Then, induction

ind_{H}^{G} := RG ⊗_{RH} − : RH − \overline{mod}^{∘} ⟶ RG − \overline{mod}^{∘}

and restriction

res_{H}^{G} : RG − \overline{mod}^{∘} ⟶ RH − \overline{mod}^{∘}

are mutually inverse equivalences of categories. These equivalences of categories preserve the indecomposability of modules. More precisely, for every indecomposable non projective RG–module M there is an indecomposable non projective RH– module f(M) such that f(M)|res_{H}^{G}(M) and M/f(M) is a projective RH–module. For every indecomposable non projective RH–module N there is an indecomposable non projective RG–module g(N) such that g(N)|ind_{H}^{G}(N) and (ind_{H}^{G}N)/g(N) is a projective RG–module.

Remark 2.2. • The situation described by the hypotheses of Theorem 2.1 is commonly known and will be referred to as the TI–situation. Here TI stands for ’trivial intersection’.

• One should note that even the classical Green correspondence is much more general than expressed here. The theorem above is just a very special case where the correspondence appears in the most illustrative way.

This theorem is a very special case of a much more general statement which was proven by M. Auslander and M. Kleiner in [1]. They give a categorical and much more general approach to the theory of J. A. Green [4] which establishes an equivalence between certain quotient categories of finitely generated RG–modules and finitely generated RH–modules where H is a subgroup of G containing the normalizer in G of a certain p–subgroup D of G. The theorem above is the case for which the categorical equivalence of Green is most easily formulated.

2.2. Adjoint functors. The method of Auslander and Kleiner intensively use adjoint functors. We shall give a very brief account on what this is about.

We assume the reader to be familiar with the notion of a category, functors and of natural transformations between functors. As basic reference one might see [15] or [12].

Definition 2.3. Let A and B be two categories and let F : A ⟶ B and G : B ⟶ A be two functors. The category of sets is called E\∫.

If there is an equivalence

B(F −, −) ≃ A(−, G−)

of bifunctors

A × B ⟶ E\∫

then the functor F is said to be left adjoint to the functor G and the functor G is said to be right adjoint to F. The pair (F, G) is said to be an adjoint pair.

Let (F, G) be an adjoint pair. By the defining relation we get an isomorphism of bifunctors

η′ : B(F −, F −) ≃ A(−, GF −)

and hence we get a natural transformation

η : 1_{A} ⟶ GF

by just putting η(A) := η′(id_{F A}). The natural transformation η is called the unit of the adjointness. Of course, it depends not only on the two functors F and G but also on the choice of the isomorphisms in the defining relation.

We give an example. Let R be a commutative ring and let G be a finite group with subgroup H. We denote by ι : RH ⟶ RG the canonical embedding. Since an RG– module M is nothing else than a R–module M together with a ring homomorphism RG ⟶ End_{R}(M), one defines the restriction res_{H}^{G}(M) just as RH ⟶ RG ⟶ End_{R}(M). The corresponding mapping is denoted by ι_{∗}. One should observe that this amounts to saying that the RH–module structure of res_{H}^{G}(M) is just M as R–module and now only H operates on M in the same way as G does.

One defines functors

ind_{H}^{G} := RG ⊗_{RH} − : RH − mod ⟶ RG − mod

and

res_{H}^{G} := ι_{∗} : RG − mod ⟶ RH − mod

We claim that (ind_{H}^{G}, res_{H}^{G}) is an adjoint pair. This fact is commonly known as Frobenius reciprocity.

We have to give for all RG–modules M and for all RH–modules N natural isomorphisms

Hom_{RG}(ind_{H}^{G}(N), M) ≃ Hom_{RH}(N, res_{H}^{G}(M)).

We define

Hom_{RG}(ind_{H}^{G}(N), M) ≃ Hom_{RH}(N, res_{H}^{G}(M))
ϕ ⟶^{Φ} (n ⟶ ϕ(1 ⊗ n))  ∀ n ∈ N
(g ⊗ n ⟶ g · ψ(n)) ⟵^{Ψ} ψ  ∀ n ∈ N, g ∈ G

and observe that the second mapping is well defined since ψ is RH–linear. Now, one immediately checks that ΦΨ(ψ) = ψ and ΨΦ(ϕ) = ϕ for all ϕ ∈ Hom_{RG}(ind_{H}^{G}(N), M) and ψ ∈ Hom_{RH}(N, res_{H}^{G}(M)). For the functoriality of Φ and Ψ we observe that for a homomorphism α : N ⟶ N′ and for a homomorphism β : M ⟶ M′ we get

β([Φ(ϕ)](α(n))) = [β ∘ ϕ](1 ⊗ α(n))
= [Φ(β ∘ ϕ ∘ (id_{RG} ⊗ α))](n)

and

β[Ψ(ψ)](g ⊗ α(n)) = β(g · (ψ ∘ α)(n))
= (g · [β ∘ ψ ∘ α](n))
= [Ψ(β ∘ ψ ∘ α))](n) .

This proves the functoriality.

2.3. Some more background from modular representation theory. Though we do not need to know of modular representation theory for formulating and proving the Green correspondence for adjoint functors we shall give some background to see what the Green correspondence is about and to be able to give examples.

In this subsection we shall use the following notation.

• R is a commutative Noetherian ring.

• G is a finite group.

• For any subgroup S of G we set mod(G, S) := mod(RG, S) the full^{1}, additive subcategory of RG − mod whose objects are finitely generated RG−modules M for which there is an RS–module L such that M is a direct summand of RG ⊗_{RS} L.

With these notations we state the following results which also provide a brief introduction into some of the elementary techniques in modular representation theory.

1. We claim that choosing R a local complete discrete valuation ring of characteristic 0 with pR ≠ R or a field of characteristic p, for a prime number p, and choosing D a Sylow p−subgroup of G, mod(G, D) = RG − mod.

Proof. Given M ∈ mod(G, D). Then, the mapping

RG ⊗_{RD} M ⟶ M
g ⊗ m ⟶ gm

is split by

M ⟶ RG ⊗_{RD} M
m ⟶ \frac{1}{|G : D|} ∑_{Dh∈D\G} h^{−1} ⊗ hm

The last is clearly well defined and is a G–linear map since

g · ∑_{Dh∈D\G} h^{−1} ⊗ hm = ∑_{Dh∈D\G}(hg^{−1})^{−1} ⊗ (hg^{−1})gm

and to run over a coset {h} is the same as to run over the coset {hg^{−1}} = {h}g^{−1}.

Given a finitely generated RG module M we call a D which is of minimal order amongst all the subgroups D′ with M ∈ mod(G, D′) the vertex of M.

1A functor F : C′ ⟶ C is full, if it is surjective on the morphism sets.

2. (D. Higman) Let M be a finitely generated RG–module and let S be a subgroup of G. We claim that M is a direct summand of ind_{S}^{G}(L) for some finitely generated RS–module L if and only if M is a direct summand of ind_{S}^{G}res_{S}^{G}(M).

Proof. Clearly, if M|ind_{H}^{G}res_{S}^{G}(M), then there is the RS–module L = res_{H}^{G}M such that M|ind_{S}^{G}(L).

Conversely, let L be a finitely generated RS−module such that M|ind_{S}^{G}L. Then, by Mackey’s formula,

ind_{S}^{G}res_{S}^{G}M = ind_{S}^{G}res_{S}^{G}ind_{S}^{G}L
= ind_{S}^{G}( ⨁_{HgH∈H\G/H} ind_{H∩ ^{g}H}^{H}res_{H∩ ^{g}H}^{H} ^{g}L)
= ind_{H}^{G}L ⊕ others
= M ⊕ others

How unique the vertices are is the subject of the following item.

3. We assume now that R is a complete discrete valuation ring of characteristic 0 with pR ≠ R for a prime number p or a field of characteristic p. Given an indecomposable RG−module M we claim that vertices of M are conjugate to each other.

Proof. M|ind_{D}^{G}V and M|ind_{D′}^{G}W with V ∈ Ob(RD − mod) and W ∈ Ob(RD′ − mod), D and D′ being both vertices of M. But, using Mackey’s formula,

res_{D′}^{G}ind_{D}^{G}V = ⨁_{DgD′} ind_{^{g}D∩D′}^{D′}res_{^{g}D∩D′}^{^{g}D} ^{g}V

and

res_{D′}^{G}ind_{D′}^{G}W = ⨁_{D′hD′} ind_{^{h}D∩D′}^{D′}res_{^{h}D′∩D′}^{^{h}D} ^{h}W = W ⊕ others .

res_{D′}^{G}M is a direct summand of both modules. Direct summands X in the above equation have vertices smaller than D′ or are isomorphic to W. If X|res_{D′}^{G}M for X ≄ W then M|ind_{D′′}^{G} for some smaller subgroup D′′ of D′ and we reach a contradiction. Hence there must be some g_{0} with

W |ind_{^{g_{0}} D∩D′}^{D′}res_{^{g_{0}} D∩D′}^{^{g_{0}} D} ^{g_{0}}V

Hence, M|ind_{^{g_{0}} D∩D′}^{G}res_{^{g_{0}} D∩D′}^{^{g_{0}} D} ^{g_{0}}V and by the minimality of D, we get ^{g_{0}}D ∩ D′ = D′.

4. An indecomposable ring direct factor B of RG is called a block of RG. Of course, then B is an R(G × G)–module by putting (g, h) · m = gmh^{−1} where (g, h) ∈ G × G and m ∈ B.

We claim that there is always a vertex of B in {(g, g) ∈ G × G|g ∈ G} =: Δ(G).

Proof. We view R(G × G) as RG–right–module by letting G act as Δ(G).

R(G × G) ⊗_{RΔ(G)} R ⟶ RG
((g, h) ⊗ r) ⟶ grh^{−1} = rgh^{−1}

for r ∈ R, (g, h) ∈ G × G, is split by

R(G × G) ⊗_{RΔ(G)} R ⟵ RG
(∑_{g∈G} r_{g}(g, 1)) ⊗ 1 ⟵ ∑_{g∈G} r_{g}g

The splitting is a module homomorphism as one immediately verifies using that we tensor over RΔ(G).

The vertex of a block as R(G × G)–module is called a defect group of the block.

We assume now that R is a complete discrete valuation ring of characteristic 0 with pR ≠ R for a prime number p or a field of characteristic p, then the defect groups D are p-groups and the integer log_{p}(|D|) is called the defect of the block.

5. Now we assume again that R is a complete discrete valuation ring of characteristic 0 with pR ≠ R for a prime number p or a field of characteristic p. If B is a block of G with defect group D and if M is an indecomposable B–module, then there is a vertex of M contained in D.

Proof^{2}. Since B has defect group D, by 3. we see that

B|[R(G × G) ⊗_{RΔ(D)} B].

Hence,

M = B ⊗_{RG} M|R(G × G) ⊗_{RΔ(D)} B ⊗_{RG} M
= RG ⊗_{RD} M

With these preparations we shall illustrate the Green correspondence in the situation of Theorem 2.1.

Example. We fix a prime number p and set G := SL_{2}(p) the group of 2 by 2 matrices over the prime field of characteristic p with determinant 1.

We look at the modular representations of G over k being the prime field of characteristic p.^{3}

We set GL_{2}(p) the group of invertible 2 by 2 matrices over the prime field of characteristic p. Then we get an exact sequence

1 ⟶ SL_{2}(p) ⟶ GL_{2}(p) ⟶^{det} k^{∗} ⟶ 1.

Now, k^{∗} has order p − 1 and GL_{2}(p) has order (p^{2} − 1) · (p^{2} − p) since an invertible matrix is determined by its action on the 2-dimensional natural module and the first basis vector can be mapped to all of k^{2} besides the zero element, the second basis vector can be mapped to k^{2} besides the one dimensional space which is already spanned by the image of the first basis vector.

Hence, |SL_{2}(p)| = (p − 1) · p · (p + 1). The Sylow p–subgroup of G is hence cyclic of order p. In fact, it is easy to find

one explicitly:

D := \{( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) |y ∈ k\}

2This proof was pointed out to me by M.Linckelmann.

3This characteristic is commonly called the describing characteristic and in the theory of algebraic groups the describing characteristic always provides a hughe framework of techniques coming from algebraic geometry.

The normalizer H of D in G is

H := \{( \begin{matrix}x & y \\ 0 & x^{−1} \end{matrix} ) |x ∈ k^{∗} ; y ∈ k\}

We shall illustrate the Green correspondence on the (natural) module

M = \binom{k}{k}

on which G acts by matrix multiplication.

Clearly, M is indecomposable. By Theorem 2.1 we know that res_{H}^{G}(M) ≃ f(M) ⊕ P_{H} where P_{H} is a projective RH–module.

Claim 2.4. Let P be a p–group and let R be a complete discrete valuation ring of characteristic 0 with residue field of characteristic p. Then, RP is a local ring. Every projective RP –module is free.

Proof. We have to show that R/radR is the only simple RP –module. We proceed by induction on |P |.

The statement is true for the group with 1 element.

Let P be arbitrary. Let 1 ≠ c be a central element of P of order p, which exists by the cojugacy class number formula, just counting the size of the conjugacy classes and observing that their order equals the index of the stabilizer of an element which is a subgroup, and let V be a simple RP –module. Then, since c is central, V_{1} := (c − 1) · V is also an RP –module. If V_{1} = 0, then V is an indecomposable R(P/ < c >) module and V is isomorphic to R/radR by the induction hypothesis. Else, V_{1} = V by the simplicity of V. Hence,

V = (c − 1) · V = (c − 1)^{2} · V = ⋯ = (c − 1)^{p} · V = (c^{p} − 1) · V = 0.

We reached a contradiction.

The above claim shows that a projective kG–module has k–rank at least order of a Sylow p–subgroup. In fact, the restriction of a projective RG–module to a Sylow p–subgroup is again projective, hence free.

This argument (or by elementary computations) shows, that the 2–dimensional kH–module res_{H}^{G}(M) is indecomposable.

Conversely, let N be the natural two dimensional kH–module. Then, N = res_{H}^{G}(M). We look for its Green correspondent in kG. As is done above,

ind_{H}^{G}N = ind_{H}^{G}res_{H}^{G}M ⟶ M
g ⊗ m ⟶ gm

is split and M|ind_{H}^{G}N. Hence, the Green correspondent of N in kG is M = g(N).

But, since the index of H in G is p + 1, dim_{k}(ind_{H}^{G}N) = 2 · (p + 1) and dim_{k}(ind_{H}^{G}(N)/M) = 2p and ind_{H}^{G}(N)/M is a projective module of dimension 2p. Observe that this matches our observation in Claim 2.4.

2.4. The Green correspondence for adjoint pairs of functors. We shall follow the paper [1].

All categories we deal with are assumed to be additive ^{4}. We start with three (additive) categories D, H and G and functors

D ⟶^{S′} H ⟶^{S} G

as well as

D ⟵^{T′} H ⟵^{T} G

where (S, T) and (S′, T′) will form adjoint pairs.

In our later application to group theory these categories and functors will be specialized as follows.

Let k be a field of characteristic p ≥ 0, let G be a finite group, and assume for simplicity that kG is indecomposable as ring, let D be the Sylow p–subgroup of G, let H be a group with D ≤ H ≤ G. The results we get become non trivial only if we assume that in this example H ≥ N_{G}(D) := {g ∈ G|gD = Dg}. One should think of

D = kD − mod, H = kH − mod, G = kG − mod

and

S′ = kH ⊗_{kD} −; S = kG ⊗_{kH} −;
T = res_{H}^{G}(−); T′ := res_{D}^{H}(−)

where res_{H}^{G} and res_{D}^{H} are the restriction functors of kG−modules to kH–modules or of kH–modules to kD–modules respectively, and the adjointness is just Frobenius reciprocity as explained earlier in Subsection 2.2.

For technical reasons in later applications we fix isomorphisms, natural in both variables,

α(N, M) : G(SN, M) ⟶ H(N, T M); ∀ N ∈ Ob(H), M ∈ Ob(G)

and

γ(L, N) : H(S′L, N) ⟶ D(L, T′N); ∀ N ∈ Ob(H), L ∈ Ob(D) .

Throughout Section 2.4 we assume that

T S = 1_{H} ⊕ U

for an endofunctor U of H and that the induced natural transformation

η_{I} : 1_{H} ⟶^{η} T S = 1_{H} ⊕ U ⟶^{proj} 1_{H}

is an isomorphism.

Notation 1. • All subcategories in Section 2.4 are meant to be full and additive. If A and B are full subcategories of the category C, then we say that A divides B if for all M ∈ Ob(A) there is a X ∈ Ob(B) such that M|X, i.e. M is a direct summand of X. If Ob(A) has only one element M, then we also say that M divides B. We use the notation A|B.

4We remind the reader that a category is called additive if it has a zero object, there are finite products and coproducts, finite products over a set of objects and finite coproducts over this set are isomorphic by the natural map, and for every object A there is an endomorphism s_{A} of A such that, denoting by Δ_{A} the diagonal mapping and by ∇_{A} the codiagonal mapping, Δ_{A}(1_{A} ⊕s_{A})∇_{A} = 0. In additive categories the set of morphisms carries a structure of an abelian group by setting f + g = Δ_{A}(f ⊕ g)∇_{B} for f, g ∈ Mor(A, B)

• Let C′ be a subcategory of the category C. We denote by C/C′ the category whose objects are the same as those of C and the morphisms are equivalence classes of morphisms of C. Two morphisms are said to be equivalent if their difference factors through an object of C′.

• Let E and F be categories and let U : E ⟶ F be a functor. For any subcategory Y of F let U^{−1}(Y) be the full additive subcategory of E generated by objects M ∈ Ob(E) with U(M)|Y.

2.4.1. The theorem in group theoretical terms. We shall give the Green correspondence in the classical situation, before we turn to the more abstract setting.

Theorem 2.5. (Green) Let G be a finite group and let R be a complete discrete valuation ring of characteristic 0 with residue field of characteristic p > 0 or let R be a field of characteristic p. Let D be a p-subgroup of G and let H ≥ N_{G}(D). Set

X := {X ≤ D ∩ gDg^{−1}|g ∈ G \ H}
Y := {Y ≤ H ∩ gDg^{−1}|g ∈ G \ H}
Z := {Z ≤ D}

Set mod(G, F) the category of finitely generated RG−modules with vertex in F for F ∈ {X , Y, Z}.

Then,

ind_{H}^{G} : mod(H, Z)/mod(H, X) ⟶ mod(G, Z)/mod(G, X)

is an equivalence of categories and

res_{H}^{G} : mod(G, Z)/mod(G, X) ⟶ mod(H, Z)/mod(H, Y)

is an equivalence of categories.

For every indecomposable object M in mod(H, Z) \ mod(H, X) there is an indecomposable object g(M) in mod(G, Z) \ mod(G, X) which is a direct summand of ind_{H}^{G}(M).

For every indecomposable object N in mod(G, Z) \ mod(G, X) there is an indecomposable object f(N) in mod(H, Z) \ mod(H, Y) which is a direct summand of res_{H}^{G}(N).

We shall prove the theorem in the sequel.

2.4.2. The general situation. We are ready to state the most important theorem of this subsection. The remaining part deals with the particular situation of Krull– Schmidt categories. But even without this assumptions we are able to prove an equivalence of certain quotient categories. In the next subsection we shall explain how one can derive the usual Green correspondence from this rather abstract setting.

Theorem 2.6. [1](Green correspondence for adjoint functors) Let there be three additive categories D, H and G and functors

D ⟶^{S′} H ⟶^{S} G

as well as

D ⟵^{T′} H ⟵^{T} G

where (S, T) and (S′, T′) will form adjoint pairs. Assume that T S = 1_{H} ⊕ U and that for the unit η : 1_{H} ⟶ T S we get η_{I} := η · proj_{1} is an isomorphism. Let Y be a full, additive subcategory of H such that

S′T′Y|Y and S′T′Y|U^{−1}Y.

Then, 1. S, T induce functors

S : H/S′T′Y ⟶ G/SS′T′Y
T : G/SS′T′Y ⟶ H/Y

2. For Z := (US′)^{−1}Y, the restrictions of the functors S and T

S : (addS′Z)/S′T′Y ⟶ (addSS′Z)/SS′T′Y
T : (addSS′Z)/SS′T′Y ⟶ (addS′Z)/Y

are equivalences of categories and

T S : (addS′Z)/S′T′Y ⟶ (addS′Z)/Y

is isomorphic to the functor induced by the identity functor.

Before we prove Theorem 2.6 we shall see what this says for the group theoretical situation.

The question if the assumptions of the theorem are satisfied in the group theoretical situation will cover subsection 2.4.5 and we postpone this question until then.

Again let R be a complete discrete valuation ring of characteristic 0 with residue field k of characteristic p > 0 or let R be a field of characteristic p. Let G be a finite group and let D be a p− subgroup of G. Let H be a subgroup of G with D ≤ N_{G}(D) ≤ H ≤ G. We set

G := RG − mod^{0}, H := RH − mod^{0}, D := RD − mod^{0}

and

S = ind_{H}^{G}, S′ = ind_{D}^{H}, T = res_{H}^{G}, T′ = res_{D}^{H}.

Furthermore, we set

S := {V ≤ G| ∃ g ∈ G \ H : V ≤ g · D · g^{−1} ∩ H}

and let Y be the full additive subcategory of RH − mod^{0} whose objects are finite direct sums of indecomposable finitely generated RH–lattices which have vertex in S.

We compute S′T′Y. Let V ∈ S and L ∈ Ob(RV − mod^{0}). A generating object of S′T′Y is of the form

ind_{D}^{H}res_{D}^{H}ind_{V}^{H}L = ind_{D}^{H}( ⨁_{V hD∈V \H/D} ind_{^{h}V ∩D}^{D}res_{^{h}V ∩D}^{^{h}V} ^{h}L)
= ⨁_{V hD∈V \H/D} ind_{^{h}V ∩D}^{H}res_{^{h}V ∩D}^{^{h}V} ^{h}L

If we now set

X := {V ≤ D| ∃ g ∈ G \ H : V ≤ g · D · g^{−1} ∩ D}

we observe that the above generating modules are direct sums of modules which have vertices in X. We set U the full additive subcategory of RH − mod^{0} whose objects are finite direct sums of indecomposable finitely generated RH–lattices which have vertex in X. Clearly, H/(S′T′Y) = H/U since if a morphism factors through a direct summand it also factors through the whole direct sum.

In a similar vein, with somewhat more effort but still simply using Mackey’s formula one proves that Z is the full additive subcategory of RH − mod^{0} whose indecomposable objects are modules with vertex being a subgroup of D.

2.4.3. The proof. The proof of Theorem 2.6 will proceed in several steps. This will cover this subsubsection. The next subsubsection will deal with the special situation when we are given a Krull–Schmidt category.

The proof of Theorem 2.6 relies mainly on the following observation.

Proposition 2.7. Let Y′ be a subcategory of H and let X′ be a subcategory of G such that SY′|X′ and T X′|Y′. Then, S, T extend naturally to functors between H/Y′ and G/X′ and (S, T) form again an adjoint pair as functors between these quotient categories. The adjointness homomorphism α for the adjoint pair (S, T) of functors between H and G induces an adjointness homomorphism for the adjoint pair (S, T) of functors between H/Y′ and G/X′.

Proof. First we prove that S extends to the quotient categories. Let there be given two objects M and N in H and a morphism f ∈ H(M, N) which factors through an object Y of Y′. Then, there are f_{1} ∈ H(M, Y) and f_{2} ∈ H(Y, N) such that f = f_{1}f_{2}. Therefore, Sf = (Sf_{1})(Sf_{2}) and Sf factors through SY. But, SY′|X′ and therefore, there is an X ∈ Ob(X′) such that SY |X. Hence, Sf factors through an object in X′.

The argument that T extends to the quotient categories is absolutely analogous. We show that for any N ∈ Ob(H), M ∈ Ob(G) the mapping α(N, M) :

G/X′(SN, M) ⟶ H/Y′(N, T M) is an isomorphism. Let f : N ⟶ T M be a morphism factoring through Y ∈ Y. Then, there are f_{1} ∈ H(M, Y) and f_{2} ∈ H(Y, N) such that f = f_{1}f_{2}. Now, α^{−1}(f) = α^{−1}(f_{1})α^{−1}(f_{2}) with Sf_{1} = α^{−1}(f_{1}) ∈ G(SN, SY) and α^{−1}(f_{2}) ∈ G(SY, M). However, SY |X for an object X ∈ X′. Hence, α^{−1}(f) factors through an object of X′. Therefore, α^{−1} is defined over the quotient categories. Analogously, also α is defined over the quotient categories. It is clear that then α(N, M) is a natural isomorphism.

This proves the proposition.

Corollary 2.8. Let Y be a subcategory of H. If S′T′Y|Y, then

1. (S′, T′) is an adjoint pair as functors between H/Y and D/T′Y. The isomorphisms γ induce adjunctions also in the quotient categories.

2. (S′, T′) is an adjoint pair as functors between H/S′T′Y and D/T′Y. The isomorphisms γ induce adjunctions also in the quotient categories.

3. The functor 1_{H} induces a functor 1_{H} : H/S′T′Y ⟶ H/Y and gives rise to

an isomorphisms of bifunctors (H/S′T′Y)(S′−, −) ≃ (H/Y)(S′−, −) If T′T SS′T′Y|T′Y, then

4. (SS′, T′T) is an adjoint pair between the categories G/SS′T′Y and D/T′Y with adjunction induced by γα.

5. If moreover S′T′Y|Y, then the inverse of α induce isomorphisms functorial in both variables L ∈ Ob(D), M ∈ Ob(G),

H/S′T′Y(S′L, T M) ⟶ G/SS′T′Y(SS′L, M).

Proof:

Part 1. follows from Proposition 2.7 by just setting X′ := T′Y and Y′ := Y. Part 2. Set X′ := S′T′Y and Y′ := T′Y. Then, S′T′Y|Y ⟹ T′S′(T′Y)|(T′Y)

and Proposition 2.7 applies.

Part 3. We apply first 2. and then 1. to get the isomorphisms

H/S′T′Y(S′−, −) ˜⟶^{γ(−,−)} D/T′Y(−, T′−) ˜⟶^{γ^{−1}(−,−)} H/Y(S′−, −)

Part 4. This is just an application of Proposition 2.7 with X′ := SS′T′Y and Y′ := T′Y and as functors one takes just T′T.

Part 5. We have

G/SS′T′Y(SS′−, −) ˜⟶^{γα(−,−)} D/T′Y(−, T′T −) ˜⟶^{γ^{−1}(−,−)} H/S′T′Y(S′−, T −)

where the last part is due to 2. and the first is due to 4.

We come to the actual proof of Theorem 2.6. We need a lemma.

Lemma 2.9. Under the assumptions of Theorem 2.6 we get the following.

(S′T′Y|Y and S′T′Y|U^{−1}(Y)) ⟺ T SS′T′Y|Y

Proof. T S = 1_{H} ⊕ U ⟹ T SS′T′ = S′T′ ⊕ US′T′ and inserting Y ∈ Y gives the result.

We can now prove Part 1 of Theorem 2.6. In fact, for S the statement is clear and for T it follows from Lemma 2.9.

Lemma 2.10. Under the assumptions of Theorem 2.6 we get the following. 1. For all L ∈ Ob(D), B ∈ Ob(U^{−1}Y),

S : (H/S′T′Y)(S′L, B)˜⟶(G/SS′T′Y)(SS′L, SB)

gives an isomorphism.

2. For all L ∈ Ob((US′)^{−1}(D)), A ∈ G,

T : (G/SS′T′Y)(SS′L, A)˜⟶(H/Y)(T SS′L, T A)

gives an isomorphism.

Remark 2.11. We notice at once that if S′T′(US′D)|(US′D), then Y := (US′D) satisfies each of the equivalent conditions in part 1.

Proof of Lemma 2.10.

The conditions to apply Corollary 2.8 1.–5. are satisfied. We have η : 1_{H} ⟶ T S and η_{B} ∈ H(B, T SB). The following diagram is commutative.

\frac{H}{S′T′Y} (S′L, B) ⟶^{η_{B}_{∗}}  \frac{H}{S′T′Y} (S′L, T SB)  ⟶^{α^{−1}}  \frac{G}{SS′T′Y} (SS′L, SB)
↓ 1_{H}  ≃↓ 1_{H}
\frac{H}{Y}(S′L, B) ⟶^{η_{B}_{∗}}  \frac{H}{Y}(S′L, T SB) = \frac{H}{Y}(S′L, B)

where the very left hand side vertical 1_{H} is an isomorphism by Corollary 2.8. 3. α^{−1} is an isomorphism by Corollary 2.8. 5. The equality in the lower right corner follows from the fact that T S = 1_{H} ⊕ U and UB|Y. But, η_{B}_{∗} · pro_{1} = η_{I} is an isomorphism. Therefore, going down, right, up we conclude that the upper η_{B}_{∗} is an isomorphism.

Furthermore, ∀ h ∈ H(S′L, B), we get

α^{−1}(η_{B} ∘ h) = α^{−1}(η_{B}) ∘ S(h) = 1_{SB} ∘ S(h) = S(h)

where the first equation is just the functoriality, the second is the definition of η by η_{N} = α(N, T SN)^{−1}(1_{SN}). This shows the statement for S.

The statement for the functor T is shown analogously.

We can now also prove Part 2 of Theorem 2.6. By Lemma 2.10 we see that the restriction of S to add S′Z/S′T′Y and of T to add SS′Z/SS′T′Y is full and faithful.

The restrictions of S and of T are dense to the image of a functor. This is precisely what we did.

Since T SS′Z = S′Z ⊕ US′Z and since by the definition of Z for all Z ∈ Ob(Z) we get US′(Z)|Y, we see that add T SS′Z/Y = add S′Z/Y.

Moreover, U makes all the occuring terms vanishing and henceforth by our assumption that η_{I} is an isomorphism, T S is just the natural projection.

This finishes the proof of the theorem.

2.4.4. The Krull–Schmidt situation.

Notation 2. If E and F are subcategories of a common Krull–Schmidt category ^{5} G, then F_{E} denotes the full additive subcategory of F generated by objects M ∈ Ob(F) such that no non zero direct summand of M divides E. One should think of F_{E} as the part of F which has nothing to do with E.

Lemma 2.12. Let E be a subcategory of a Krull–Schmidt category F. Then, the identity functor induces a functor F_{E} ⟶ F_{E}/E which is full^{6}, dense^{7} and reflects isomorphisms^{8}.

Proof. 1_{F} is clearly full and dense. Take an isomorphism X ⟶^{f} Y in F_{E}/E. Then, there is a Y ⟶^{g} X with gf = 1_{X} and fg = 1_{Y} in F_{E}/E. Take preimages f_{0} and g_{0} of f and g in F. Then, f_{0}g_{0} = 1_{X} + k_{X} where k_{X} is an endomorphism of X which factors through an object of E. No summand of X divides E and therefore, k_{X} ∈ radEnd(X). But, the Jacobson radical radEnd(X) has the property that 1+radEnd(X) is a subgroup of the unit group. Similarly, g_{0}f_{0} is invertible. Hence, f_{0} is an isomorphism.

Proposition 2.13. Let Y be a subcategory of H. Then T S : H ⟶ H/Y satisfies

T S(U^{−1}Y) ≤ (U^{−1}Y)/Y

and

T S : U^{−1}Y ⟶ (U^{−1}Y)/Y is isomorphic to 1_{H}.

If H is a Krull–Schmidt category, then T S : (U^{−1}Y)_{Y} ⟶ (U^{−1}Y)_{Y}/Y is full, dense and reflects isomorphisms.

5A Krull–Schmidt category is an additive category such that every object is a finite direct sum of indecomposable objects and endomorphism rings of indecomposable objects are local. It follows then that the decomposition into direct summands is unique up to isomorphisms.

6F : C′ ⟶ C is full, if it is surjective on the morphism sets.

7F : C′ ⟶ C is dense, if every object in C is of the form F C′ for a C ∈ Ob(C′).

8F : C′ ⟶ C is reflects isomorphisms, if

F (f) ∈ Mor_{C}(F C_{1}, F C_{2}) is an isomorphism iff f ∈ Mor(C_{1}, C_{2}) is an isomorphism.

Proof. We know that η_{I} is an isomorphism.

B ∈ U^{−1}Y ⇔ U(B)|Y
⇔ U(B) ≃ 0 ∈ Ob(H/Y)
⇒ η : 1_{H}|_{U^{−1}Y} → T S|_{U^{−1}Y} really is η_{I} which is an isomorphism.

The second statement follows immediately from Lemma 2.12.

Proposition 2.14. Let Y be a full additive subcategory of the Krull–Schmidt category H.

1.

S : U^{−1}(Y)_{Y} ⟶ S(U^{−1}(Y)_{Y})/T^{−1}Y

is dense, reflects isomorphisms, and N is indecomposable in U^{−1}(Y)_{Y} if and only if SN is indecomposable in G/T^{−1}Y.

2.

T : S(U^{−1}(Y)_{Y})/T^{−1}Y ⟶ U^{−1}(Y)_{Y}/Y

is full, dense, reflects isomorphisms and M is indecomposable in S(U^{−1}(Y)_{Y})/T^{−1}Y if and only if T M is indecomposable in H/Y.

Proof. \underline{S is dense} by definition. \underline{SN_{1} ≃ SN_{2}}⇒ T SN_{1} ≃ T SN_{2} ⇒ N_{1} ≃ N_{2} since T S reflects isomorphisms by Proposition 2.14. \underline{T is dense} since T S is dense. \underline{T is full} since given T X ⟶^{f} T Y, then there is a X′, Y′ mapping to X, Y by S such that given T SX′ ⟶^{f} T SY′. But, T S is full, again using Proposition 2.14, hence,

f = T Sf′ for X′ ⟶^{f′} Y′ and T (Sf′) = f and Sf is a preimage. \underline{T M_{1} ≃ T M_{2}}⇒ ∃_{N_{1},N_{2}}M_{i} = SN_{i} ⇒ T SN_{1} ≃ T SN_{2} ⇒ N_{1} ≃ N_{2} ⇒ M_{1} ≃ SN_{1} ≃ SN_{2} ≃ M_{2}. \underline{N is decomposable in U^{−1}(Y)_{Y}} ⇒ take N_{1}|N ⇒ SN_{1}|SN. But, SN_{1} = 0 ⇒ N_{1} = 0. Therefore, SN is decomposable. \underline{Let M be decomposable in S(U^{−1}(Y)_{Y})/T^{−1}Y.} Take 0 ≠ M_{1}|M ⇒ T M_{1}|T M. But, T M_{1} = 0 ∈ U^{−1}(Y)_{Y}/Y ⇒ T M_{1}|Y ⇒ M_{1} ∈ T^{−1}Y ⇒ M_{1} = 0 ∈ S(U^{−1}(Y)_{Y})/T^{−1}Y. Hence, T M is decomposable. \underline{SN decomposable}⇒ T SN decomposable in U^{−1}(Y)_{Y}/T^{−1}Y ⇒ N is decomposable since T S is full, dense and reflects isomorphisms. \underline{T M is decomposable} with M ∈ S(U^{−1}(Y)_{Y})/T^{−1}Y ⇒ ∃_{N}M ≃ SN ⇒ T SN ≃ T M is decomposable ⇒ N is decomposable since T S is full, dense, and reflects isomorphisms ⇒ M = SN is decomposable.

Putting all the pieces together, we have proved the proposition.

Corollary 2.15. Let H and G be Krull–Schmidt categories and let Y be a full additive subcategory of H.

1. For all indecomposable objects N ∈ U^{−1}(Y)_{Y} the object SN has exactly one indecomposable summand g(N) which is not contained in T^{−1}Y.

2. For all indecomposable objects M ∈ (add S(U^{−1}(Y)_{Y}))_{T^{−1}Y} the object T M has exactly one indecomposable summand f(M) that does not divide Y.

3. f(g(N)) ≃ N

4. g(f(M)) ≃ M

Remark 2.16. One should note that this establishes one part of the Green correspondence, namely, the bijective correspondence between parts of the two module categories of the group rings.

However, one should read this carefully. If we wanted to apply this to our group theoretic situation, one does not need for the statement that H ≥ N_{G}(D). It will become clear that if this is not the case, then there is no indecomposable object as required.

Proof of Corollary 2.15.

1. SN = M_{1} ⊕ ⋯ ⊕ M_{s} for indecomposables M_{i} in G with i = 1, ..., s and s ∈ ℕ. By Lemma 2.12, M_{j} is indecomposable or zero in G/T^{−1}Y. By Proposition 2.14 SN is indecomposable in G/T^{−1}Y. Hence, there is exactly one j_{0} with M_{j_{0}} not being contained in T^{−1}Y.

2. There is an indecomposable N ∈ U^{−1}(Y)_{Y} and some M ∈ G with SN = M ⊕ M′. Since M is not contained in T^{−1}Y we get SN ≃ M in G/T^{−1}Y. Let T SN = T M = N_{1} ⊕ ⋯ ⊕ N_{t} for indecomposable objects N_{i} in H and i = 1, ..., t and t ∈ ℕ. But we know that T SN is indecomposable in H/Y by Proposition 2.14. Hence there is exactly one i_{0} where N_{i_{0}} does not divide Y.

3. g(N) = SN in G/T^{−1}Y. f(g(N)) ≃ T SN ≃ N in H/Y since T S ≃ 1 as functor U^{−1}Y ⟶ (U^{−1}Y)/Y.

4.

fgf(M) ≃ f(M) by 3.  ⇒  T gf(M) ≃ T M in H/Y
⟹^{P rop.2.14.2} gf(M) ≃ M in G/T^{−1}Y
⟹^{Lemma2.12} gf(M) ≃ M in G.

Now we combine Theorem 2.6 and Corollary 2.15 to state the result.

Corollary 2.17. Assume we are in the situation of Theorem 2.6 and let in addition be G and H Krull–Schmidt categories. Then^{9} ,

1. ∀ N ∈ ind(add S′Z)_{S′T′Y}, the object SN has precisely one indecomposable summand g(N) not dividing SS′T′Y.

2. ∀ M ∈ ind(add SS′Z)_{SS′T′Y}, the object T M has precisely one indecomposable summand f(M) not dividing Y.

3. f(g(N)) ≃ N

4. g(f(M)) ≃ M

2.4.5. The situation for group rings. Again we shall follow [1] closely.

We shall apply Theorem 2.6 to the case mentioned at the beginning of Subsection 2.4. We fix the following setting.

1. Let R be a commutative Noetherian ring.

2. Let G be a finite group and let D ≤ H < G. ^{10}

3. ∀_{F ≤G} ind_{F}^{G} := RG ⊗_{RF} − : RF − mod ⟶ RG − mod ; res_{F}^{G} : RG − mod ⟶ RF − mod is the restriction functor.

9indC means the class of indecomposable objects in the category C.

10To avoid technical difficulties we assume that H ≠ G. Otherwise we shall have to deal with unpleasant exceptions arising from empty set discussions in our formulas.

4.

mod(G, F) = add(ind_{F}^{G}(RF − mod))
= ’direct summands of RG–modules induced from F ’.

The objects are called relatively F –projective modules.

5. If F is a set of subgroups of G, then ind_{F}^{G} is the least full additive subcategory of RG − mod containing all modules of the form ind_{F}^{G}(V) with V ∈ Ob(RF − mod) and F ∈ F. mod(G, F) is the least full additive subcategory of RG − mod containing mod(G, F) with F ∈ F.

6. If g ∈ G and F ≤ G. Then, ^{g}F = gF g^{−1} and for M ∈ RF − mod one forms the R ^{g}F –module ^{g}M by f · m =: gfg^{−1}m for all m ∈ M, f ∈ F.

7. We take a disjoint union

G = ⋃_{i=1}^{s} Hg_{i}H for g_{i} ∈ G; g_{1} = 1

Then, as RH–RH–bimodule,

RG = ⨁_{i=1}^{s} RHg_{i}H.

8. p : RG ⟶ RH · 1 · RH is RH − RH–linear and an epimorphism.

9. i : RH ⟶ RH · 1 · RH is RH − RH–linear and a monomorphism.

10. Set G := RG−mod, H := RH − mod, D := RD−mod. S := ind_{H}^{G}, S′ := ind_{D}^{H}, T := res_{H}^{G}, T′ := res_{D}^{H}.

11. For all N ∈ Ob(RH − mod), M ∈ Ob(RG − mod) set

α(N, M) : Hom_{RG}(ind_{H}^{G}(N), M) ˜⟶ Hom_{RH}(N, res_{H}^{G}M)

by Frobenius’ reciprocity, as explained in Section 2.2. 12.

η : 1_{RH−mod} ⟶ res_{H}^{G} ∘ ind_{H}^{G}

by means of

η_{N} = α(N, ind_{H}^{G}N)(1_{ind_{H}^{G}(N)}) = (n ⟶ 1 ⊗ n)).

with N ∈ RH − mod, n ∈ N.

13. U := ⨁_{i=2}^{s} RHg_{i}RH ⊗_{RH} −.

14. Since pi = 1_{RH}, we get T S = 1_{RH} ⊕ U.

15. η_{I} is the identity, hence an isomorphism. To apply the theorem, one has to try to produce a subcategory Y of RH − mod

such that

S′T′Y|Y and S′T′Y|U^{−1}Y.

Notation 3. We fix for any set S of subgroups of H

X := {D ∩ Y |Y ∈ S} and Y := ind_{S}^{H}.

Remark 2.18. Since ind and res result to isomorphic modules when passing to conjugate subgroups, one may assume that S is closed under conjugation. Since res and ind are transitive, one may furthermore assume that S is closed under subgroups.

Proposition 2.19. 1. S′T′Y is a subcategory of ind_{X}^{H}. Furthermore, ind_{X}^{H}|S′T′Y.

2. RH − mod/S′T′Y = RH − mod/ind_{X}^{H}

3. S′T′Y is a subcategory of Y.

Proof.

Part 1. Let N ∈ Y. Then, N ≃ ind_{V}^{H}W with W ∈ Ob(RV − mod); V ∈ S. We apply Mackey’s formula to obtain

T′N = res_{D}^{H}ind_{V}^{H}W = ⨁_{V gD∈V \G/D;g_{1}=1} ind_{D∩ ^{g}Y}^{D} res_{D∩ ^{g}Y}^{^{g}Y} ^{g}W ∈ Ob(ind_{X}^{D})

Since ind_{D}^{H}ind_{X}^{D} = ind_{X}^{H}, we get the first statement.

Let W ∈ Ob(D ∩ Y − mod), Y ∈ Y, then again using Mackey’s formula,

res_{D}^{H}ind_{D∩Y}^{H} W = ind_{D∩Y}^{D} W ⊕ ⨁_{D≠DgD∈D\H/D} ind_{^{g}(D∩Y)∩H}^{H}res_{^{g}(D∩Y)∩H}^{^{g}(D∩Y)} ^{g}W

Therefore,

ind_{D∩Y}^{D} W |res_{D}^{H}ind_{D∩Y}^{H} W = res_{D}^{H}ind_{D∩Y}^{D} W = T′(ind_{D∩Y}^{D} W)

and with the transitivity of ind, one gets that

ind_{D∩Y}^{H} W |S′T′(ind_{D∩Y}^{H} W).

Part 2. follows from 1. since by the first inclusion, every morphism factoring through an object of S′T′Y factors also through an object of ind_{X}^{H}. On the other hand side, by the second statement, a morphism factoring through an object of ind_{X}^{H} factors also through an object of S′T′Y, this having the object from before as direct summand.

Part 3. follows from the transitivity of the induction and part 1.

Just to be able to write the result in a more concise form we introduce a new

Notation 4. Let F be a set of subgroups of H. We set

F′ := {H ∩ ^{g}F |g ∈ G \ H and F ∈ F}

Lemma 2.20. ∀ F ∈ F with F ≤ H is

U(ind_{F}^{H}) ⊆ ind_{F′}^{H}|U(ind_{F}^{H})

Proof. The ’source’ of the lemma is entirely Mackey’s formula. This fact makes the proof somewhat unpleasantly technical.

Of course, it is sufficient to prove the statements for F = {F }, a set with cardinality 1. We start with proving this statement and hence assuming that we are given a V ∈ RF − mod and set N := ind_{F}^{H}V.

By definition,

U(N) = ⨁_{i=2}^{s} RHg_{i}H ⊗_{RH} N.

It is hence enough to prove that

RHgH ⊗_{RH} N ∈ ind_{F′}^{H} for any g ∈ G \ H.

We first discuss what is meant by RHgH. We see that gRH is isomorphic to the R( ^{g}H) − RH−bimodule _{g}RH which is RH as R−module and on which from the left ^{g}h ∈ ^{g}H acts by multiplication by ^{g}h on RH. Now, precisely those objects hgh′ ∈ HgH belong to {1}gH for which h ∈ H ∩ ^{g}H. Therefore, RHgH = RH ⊗_{R(H∩ ^{g}H)} _{g}RH as bimodule. We compute

RHgH ⊗_{RH} N = RH ⊗_{R(H∩ ^{g}H)} _{g}RH ⊗_{RH} N
= RH ⊗_{R(H∩ ^{g}H)} ^{g}N
= ind_{H∩ ^{g}H}^{H}res_{H∩ ^{g}H}^{^{g}H} ^{g}N
= ind_{H∩ ^{g}H}^{H}res_{H∩ ^{g}H}^{^{g}H}ind_{^{g}F}^{^{g}H} ^{g}V
= ind_{H∩ ^{g}H}^{H} ⨁_{(H∩ ^{g}H)t ^{g}F} ind_{[^{tg}F ∩H∩ ^{g}H]}^{[^{g}H∩H]}res_{[^{tg}F ∩H∩ ^{g}H]}^{[^{tg}F]} ^{tg}V

Furthermore, for n := g^{−1}tg,

H ∩ ^{g}H ∩ ^{tg}F = H ∩ ^{gn}F

since x ∈ H ∩ ^{gn}F ⇒ x ∈ H and x ∈ ^{gn}F and ^{gn}F = {gn · f · n^{−1}g^{−1}} ⊆ ^{g}H, taking into account that t ∈ ^{g}H ⇒ n ∈ H. Hence,

RHgH ⊗_{RH} N = ⨁_{n} ind_{H∩ ^{gn}F}^{H} (res_{[H∩ ^{gn}F]}^{^{gn}F} ^{gn}V)
∈ ind_{F′}^{H}

We hence proved the first statement.

We have to prove the second statement. Let g ∈ G and n ∈ H, let V ∈ Ob(RF − mod) and N := ind_{F}^{H}V. Then, by the above calculation,

ind_{[^{tg}F ∩H∩ ^{g}H]}^{[^{g}H∩H]}res_{[^{tg}F ∩H∩ ^{g}H]}^{[^{tg}F]} ^{tg}V | RHgH ⊗_{RH} N.

We just have to show that a W = ind_{H∩ ^{g}F}^{H} Q for Q ∈ R(H ∩ ^{g}F) − mod is a direct summand of a module of the above form.

Set ind_{^{g^{−1}} H∩ F}^{F} ^{g^{−1}}Q =: V ∈ Ob(RF − mod). Then again, applying Mackey’s formula,

ind_{H∩ ^{g}F}^{H} res_{H∩ ^{g}F}^{^{g}F} (ind_{H∩ ^{g}F}^{^{g}F} Q) = ind_{H∩ ^{g}F}^{H} (Q ⊕ modules from lower subgroups)
= W ⊕ modules from lower subgroups

We hence have shown that W is a direct summand of a module of the above type which in turn divides U(ind_{F}^{H}). We have shown the lemma.

We remind the reader that we are given the set of subgroups S and we have set Y := ind_{S}^{H} and X := {V ∩ D|V ∈ S}.

Corollary 2.21. If X′ ⊆ S, then S′T′Y|Y and S′T′Y|U^{−1}Y.

Proof.

• We have just to show that U(ind_{X}^{H})|ind_{S}^{H}, since by Proposition 2.19, S′T′Y is a subcategory of Y and hence obviously S′T′Y|Y automatically. Also, S′T′Y|ind_{X}^{H}. Hence, U(S′T′Y)|U(ind_{X}^{H}) and if we could show that U(ind_{X}^{H})|ind_{S}^{H} = Y, then we also had the second condition.

• But, U(ind_{X}^{H}) ⊆ ind_{X′}^{H} by Lemma 2.20.

• ind_{X′}^{H}|U(ind_{X}^{H}) by Lemma 2.20.

• ind_{X′}^{H} ⊆ ind_{S}^{H} since X′ ⊆ S.

• Hence, ind_{X′}^{H}|ind_{S}^{H} and even U(ind_{X}^{H})|ind_{S}^{H}.

Remark 2.22. We immediately check two situations where we may verify the condition in Corollary 2.21.

1. If S = {V | there is a g ∈ G \ H : V ≤ H ∩ ^{g}D}, then X = {X| there is a g ∈ G \ H : X ≤ D ∩ ^{g}D} and X′ ⊆ S.

2. If E is a normal subgroup of H and D ∩ E is a normal subgroup of G, set

S = {V |V ≤ E}. Then, X = {X|X ≤ D ∩ E} and X′ ≤ S. The first situation leads to the classical Green correspondence contrary to the second which is a new application and leads to a theorem due to Auslander–Kleiner [1].

Summarizing the results we just apply Theorem 2.6 to the above situation with the knowledge we have obtained for it up to now.

Theorem 2.23. Let Z be the largest set of subgroups of D such that Z′ ⊆ S. If X′ ⊆ Y, then

1.

ind_{H}^{G} : \frac{mod(H, Z)}{mod(H, X)} ˜⟶\frac{mod(G, Z)}{mod(G, X)}

is an equivalence of categories. 2.

res_{H}^{G} : \frac{mod(G, Z)}{mod(G, X)} ˜⟶\frac{mod(H, Z)}{mod(H, S)}

is an equivalence of categories.

3. res_{H}^{G} ∘ ind_{H}^{G} is induced by the identity functor on RH − mod.

Corollary 2.17 now translates to

Corollary 2.24. Assume we are in the situation of Theorem 2.23 and furthermore assume that RG − mod and RH − mod are Krull–Schmidt categories. Given M ∈ mod(G, Z) \ mod(G, X) indecomposable and N ∈ mod(H, Z) \ mod(H, X) indecomposable. Then,

1. ind_{H}^{G}(N) has a unique indecomposable direct summand g(N) in mod(G, Z) \ mod(G, X).

2. res_{H}^{G}(M) has a unique indecomposable direct summand f(M) in mod(H, Z)\ mod(H, S).

3. fg(N) ≃ N.

4. gf(M) ≃ M.

An application is the definition of a Brauer correspondent. Assume that we are in the situation of Corollary 2.24.

1. The syzygy-operator Ω_{G} on the stable category of RG − mod and the syzygyoperator Ω_{H} on the stable category of RH − mod commutes with g and commutes with f. More precisely:

Ω_{G}g ≃ gΩ_{H} and Ω_{H}f ≃ fΩ_{G}.

This follows since ind_{H}^{G} and res_{H}^{G} are exact and send projective modules to projective modules, hence a projective resolution to a projective resolution. Then, applying Schanuel’s Lemma, we realize that syzygies are well defined up to projective direct summands. This gives the result.

2. In the situation of the first part of Corollary 2.21 we look at the various sets of subgroups of G more closely.

S = {V | ∃ g ∈ G \ H : V ≤ H ∩ ^{g}D}
X = {X| ∃ g ∈ G \ H : X ≤ D ∩ ^{g}D}
Z = {Z ≤ D| ∃ g, g′ ∈ G \ H : ^{g′}Z ∩ H ≤ H ∩ ^{g}D}

Since D ≤ H, we always get D ∈ Z choosing g = g′. If N_{G}(D) \ H ≠ ∅, then there is a g ∈ N_{G}(D) \ H. Taking this g, we conclude that D ∈ X. But then Z = X = S and Theorem 2.23 establishes a bijection between the empty sets and in Corollary 2.24 there is no indecomposable module satisfying the assumptions. If H ≥ N_{G}(D), then trivially this never happens and the theorem is non trivial.

3. Let B be a block of RG with defect group D. Let H ≥ N_{G}(D). There is a Green correspondence for G = R(G × G) and H = R(H × H) and D = R(D × D) since N_{G×G}(D × D) = N_{G}(D) × N_{G}(D). Furthermore, as usual the functors S, S′ are the induction functors. T, T′ are the corresponding restriction functors. Now,

res_{H×H}^{G×G} : RG ⟶ [ ⨁_{HgH∈H\G/H} RHgH] = [RH ⊕ ⨁_{HgH∈H\G/H; g∉H} RHgH]

and so a block B of RG with defect group D has a Green correspondent b which is a direct summand of the right hand side. It is now easy to see that the Green correspondent of B is a direct summand of RH. The Green correspondent f(B) of B is a block of RH and is called the Brauer correspondent of B.

3. Classical theory of blocks with cyclic defect groups and Green’s walk around the Brauer tree

In this section we shall present the results of Green [5], Dade [2] and Michler [13, 14].

Throughout this section we use the following notations.

1. R is a complete discrete valuation ring of characteristic 0 with residue field k of characteristic p. The field of fractions of R is K.

2. G is a finite group.

3. B is a block of RG with defect d.

4. D is a defect group of B with order q = p^{d}.

5. D_{1} is the subgroup of D of order p.

6. H = N_{G}(D_{1}) ≥ N_{G}(D).

7. B′ is the Brauer correspondent of B in RH.

8. C_{G}(D_{1}) =: C.

3.1. The theory of Dade on blocks with cyclic defect group.

Definition 3.1. (Michler [13, 14]; Feit [3])

• The number e of isomorphism classes of simple B′–modules is called the inertial index of G.

• There is a finite Galois extension K̂ of K such that for the ring of integers R̂ in K̂ over R all the primitive |G|^{th} roots of unity are contained in R̂/rad R̂. Let B̂ be one indecomopsable factor of R̂⊗_{R} B′. (The others are Galois conjugate to this.) The number ê is defined to be the number of isomorphism classes of simple B̂–modules.

Michler shows [13, 14] that e divides p − 1.

Set I := {0, 1, …, (e − 1)}.

The main theorem of Dade describes the structure of the composition series of projective k ⊗_{R} B–modules in terms of combinatorial data, a Brauer tree. Janusz and independently Kupisch [8], [10, 11] prove that not only the composition series of the indecomposable projective modules are determined but also those of all indecomposable modules. We do not need this description for Roggenkamp’s description of blocks of cyclic defect group and so we refrain from presenting this theory as well.

The theory of Dade on the structure of blocks with cyclic defect groups is one of the most beautiful in the theory of blocks. It is a complete answer to the questions on the module structure of blocks with cyclic defect groups in terms of a very ingenious combinatorial description. One of the key tools is the Green correspondence.

Theorem 3.2. (Dade [2]) We assume that k contains all |G|–th roots of unity. There is a set Λ of simple KG–modules, called the exceptional k ⊗_{R} B–modules with the following properties.

1. The graph which consists of the following data is a tree:

• The vertices of the graph are the isomorphism classes of the non exceptional simple K ⊗_{R} B−modules and an additional vertex; representing the set of exceptional modules, called the exceptional vertex.

• There is an edge between two vertices v, w if and only if there is an indecomposable projective B–module P such that K ⊗_{R} P has the modules which correspond to the vertices v and w as direct summands.

The graph is called a Brauer tree and in case frac R is a splitting field for B, the cardinality of Λ is called the multiplicity of the exceptional vertex and equals μ = (|D| − 1)/e.

2. Let P be a projective indecomposable k ⊗_{R} B–module. Then, rad P/soc P is a direct sum of two uniserial modules S_{P} and T_{P}.

3. There is an embedding of the Brauer tree in the plane^{11} such that one can get the composition series of S_{P} and T_{P} by the following algorithm. S_{P} (and T_{P}) correspond to a vertex v (and w), say. By symmetry we describe the algorithm just for S_{P}. Since the tree is embedded into the plane, one has an ordering of the n_{P} projective RG−modules Q such that KQ has composition factor KS_{P} by a counterclockwise numeration of the edges adjacent to v. Now, rad^{i}(S_{P})/rad^{i+1}(S_{P}) ≃ Q_{i}/rad(Q_{i}) for i = 1, 2, …, n(P) · e(P) − 1 where Q_{i} is the projective indecomposable module which is i positions after P in the counterclockwise ordering. If v is the exceptional vertex, then e(P) = e and if v is not the exceptional vertex then e(P) = 1.

We describe the algorithm by a simple example. We are given the following Brauer tree

•
|1
• −_{2} • −_{3} • 2 −_{4} •
|5
•

We impose to the second right vertex a multiplicity 2, as indicated in by a box. We shall give the composition series of the projective indecomposable modules for this example.

P_{1} = ( \begin{matrix}1 \\ 2 \\ 5 \\ 3 \\ 1 \end{matrix} ); P_{2} = ( \begin{matrix}2 \\ 5 \\ 3 \\ 1 \\ 2 \end{matrix} ); P_{5} = ( \begin{matrix}5 \\ 3 \\ 1 \\ 2 \\ 5 \end{matrix} ); P_{4} = ( \begin{matrix}4 \\ 3 \\ 4 \\ 3 \\ 4 \end{matrix} ); P_{3} = ( \begin{matrix}& 3 \\ 1 & & 4 \\ 2 & & 3 \\ 5 & & 4 \\ & 3 \end{matrix} ) .

3.2. Green’s walk around the Brauer tree. After Dade’s paper, Green proved the following theorem. This theorem is of fundamental importance not only for the proof of Roggenkamp’s theory of Green orders.

Let W_{i}; i = 0, …, (e − 1) be the projective indecomposable B–modules.

Theorem 3.3. (Green [5]) We assume that k contains all |G|–th roots of unity. Let G be a finite group, let B be an RG − block, let D be a cyclic defect group of B and let Γ be the Brauer tree of B.

1. There is a family (A_{n})_{n∈ℤ} of RG–lattices and a permutation δ of I = {0, …, (e − 1)} such that there exist short exact sequences of RG–modules

E_{2i} : 0 ⟶ A_{2i+1} ⟶ W_{δ(i)} ⟶ A_{2i} ⟶ 0

11This is just another way of saying that one imposes to each vertex v of the tree a cyclic ordering of the edges v −^{e} w which are incident to the vertex v.

E_{2i+1} : 0 ⟶ A_{2i+2} ⟶ W_{i} ⟶ A_{2i+1} ⟶ 0

with W_{i} ≃ W_{i+e} and A_{i} ≃ A_{i+2e} for all i ∈ ℤ.

2. The A_{0}, A_{1}, …, A_{2e−1} are mutually non isomorphic.

3. KA_{n} is a vertex of Γ.

We should remind the reader that it is possible to reconstruct the Brauer tree from the permutation δ. One forms a path

• ⟶^{1} • ⟶^{δ(1)} • ⟶^{2} ⋯ ⟶^{δ(e−1)} • ⟶^{e} • ⟶^{δ(e)} •

which closes to an oriented circle. Then, one glues • ⟶^{i} • with • ⟵^{δ(j)} • if i = δ(j) and the result is the Brauer tree. Obviously it is a graph but by Dade’s theorem, this is in fact a tree and looking at the isomorphism classes of the vertices even the Brauer tree.

One may define conversely a permutation δ of the set of edges for every embedded tree out of which it is possible to reconstruct the tree in the above way. This permutation depends not only on the tree but also on a starting point:

1. One starts at a certain edge e and declares this edge to be 1.

2. Take a vertex v which is adjacent to the edge taken. The edge following e in the circular ordering at v is defined to be δ(1).

3. The other extremity (not the vertex v) of δ(1) is w.

4. The edge following δ(1) in the circular ordering at w is 2.

5. To find δ(2) one proceeds as in 2.

6. One stops after having determined δ(e). In our example

•
|
• − • − • − •
|
•

starting with the upper vertical edge one gets the assignment

•
1|^{δ(5)}
• ^{δ(1)}−_{2} • ^{5}−_{δ(3)} • ^{δ(4)}−_{4} •
δ(2)|^{3}
•

and the permutation (1 2 3 5).

If one starts with the right most edge, one gets the assignment

•
δ(2)|^{2}
• ^{3}−_{δ(3)} • ^{δ(1)}−_{5} • ^{1}−_{δ(5)} •
4|^{δ(4)}
•

and the permutation (1 5).

Michler generalized Theorem 3.3 to the case where there is a ring R as in the introduction, without assuming that the residue field is large enough.

Theorem 3.4. (Michler [13, 14]) Let R be as in the introduction to this section. Let G be a finite group, let B be an RG − block and let D be a cyclic defect group of B.

1. There are precisely e pairwise non isomorphic indecomposable k⊗_{R}B–modules M_{i} with source k, the trivial kD–module.

2. Let, for all i = 1, … e, \overline{P}_{i} be the k ⊗_{R} B–projective cover of M_{i} and let

0 ⟶ \overline{Ω}M_{i} ⟶ \overline{P}_{i} ⟶ M_{i} ⟶ 0

be exact. Then, (1 − α)kD is the source of \overline{Ω}M_{i}.

3. Let, for all i = 1, … e, \overline{Q}_{i} be the k ⊗_{R} B–projective cover of \overline{Ω}M_{i} and let

0 ⟶ \overline{Ω}^{2}M_{i} ⟶ \overline{Q}_{i} ⟶ \overline{Ω}M_{i} ⟶ 0

be exact. Then, one can find a numbering for the M_{i} such that \overline{Ω}^{2}M_{i} ≃ M_{i+1}.

4. There are e pairwise non isomorphic indecomposable B–lattices W_{i} with source R, the trivial RD–lattice and k ⊗_{R} W_{i} = M_{i}. For each i there is, up to isomorphism, only one B–lattice with these properties.

5. Let, for all i = 1, … e, P_{i} be the projective cover of W_{i} and let

0 ⟶ ΩW_{i} ⟶ P_{i} ⟶ W_{i} ⟶ 0

be exact. Then, (1 − α)RD is the source of ΩW_{i}.

6. Let, for all i = 1, … e, Q_{i} be the projective cover of ΩW_{i} and let

0 ⟶ Ω^{2}W_{i} ⟶ Q_{i} ⟶ ΩW_{i} ⟶ 0

be exact. Then, one can find a numbering for the W_{i} such that Ω^{2}W_{i} = W_{i+1} where the indices are taken modulo e, and k ⊗_{R} ΩW_{i} = \overline{Ω}M_{i}.

7. In the set {Ω^{k}W_{1}|k ∈ ℕ} a maximal subset of pairwise non isomorphic modules has cardinality 2e.

Following Feit [3, Chapter VII Remark after Theorem 2.11] we define the property

(∗) The number of characters of the group H which are afforded by irreducible frac(R) ⊗_{R} B–modules is equal to (q − 1)/ê.

As is proved in Feit [3, Chapter VII, Corollary 6.8], the definition for a Brauer tree as in Theorem 3.2 works also for a more general R satisfying condition (∗). Feit gives also an example that it is in general not enough to adjoin all q^{th} roots of unity.

As is proved in Feit [3, Chapter VII Theorem 10.6] one can prove a theorem which is analogous to Theorem 3.3 also for more general R satisfying the assumption (∗) from above.

Green shows Theorem 3.3 by first showing Theorem 3.5 below. He applies Green correspondence with G = kG − mod, H = kH − mod and D = kD − mod. Clearly, D_{1} is the only minimal subgroup of D and one chooses H such that each g ∈ G \ H satisfies ^{g}D_{1} ∩ D_{1} = {1}, hence, X = {1}. The block B′ of kH is the Brauer correspondent of B. The simple B′–modules are called S_{1}, …, S_{e}.

Theorem 3.5. 1. B contains e simple kG-modules V_{i}; i ∈ I such that every simple kG–module in B is isomorphic to exactly one V_{i}. Let \overline{W}_{i} be the projective cover of V_{i} as kG–module for all i ∈ I.

2. There is a numbering of the V_{i} such that

Hom_{kH}(fV_{j}, S_{i}) ≃ Hom_{kG}(V_{j}, gS_{i}) = \{ \begin{matrix}k & if i = j \\ 0 & if i ≠ j \end{matrix}

and there is a permutation δ of I such that

Hom_{kH}(S_{i}, fV_{j}) ≃ Hom_{kG}(gS_{i}, V_{j}) = \{ \begin{matrix}k & if δ(i) = j \\ 0 & if δ(i) ≠ j \end{matrix}

3. For all i ∈ I there are non split exact sequences

F_{2i} : 0 ⟶ Ω_{G}(gS_{i}) ⟶ \overline{W}_{δ(i)} ⟶ gS_{i} ⟶ 0

and

F_{2i+1} : 0 ⟶ gS_{i+1} ⟶ \overline{W}_{i} ⟶ Ω_{G}(gS_{i}) ⟶ 0

We should take some few lines to interpret Theorem 3.5. The theorem says in other words that the permutation δ can be read off from the Green correspondents of the simple B′–modules. In fact, the Green correspondent g(S_{i}) has the property

top(g(S_{i})) = V_{δ(i)} and soc(g(S_{i})) = V_{i} for all i = 1, … e

of course after a renumeration. By the discussion of the permutation δ one gets the tree back from the permutation. Therefore, the Brauer tree as abstract tree is determined by the Green correspondence.

We shall prove Theorem 3.3 in detail in the following subsections.

3.3. Dade’s description for blocks with normal cyclic defect groups. As illustration on the degree of completeness of the description of the module structure as well as preparation for the proof of Theorem 3.3 we give Dade’s results for the special case of a normal cyclic defect group D of of the block B in this subsection.

Then, the Brauer tree is a star and the exceptional vertex is in the centre. This is the subject of the following Lemma.

We introduce some notation before. As above, the Brauer correspondent of B in kH is called B′.

Lemma 3.6. (Dade) We assume that k contains all |G|–th roots of unity. B′ contains e simple modules S_{0}, …, S_{e−1} with projective covers T_{0}, …, T_{e−1}.

1. There is a multiplicative isomorphism : D ⟶ Centre(kC) such that taking a generator α of D and defining a := \overline{α} − 1 the only composition series, which is also the radical series of each T_{i} with i = 1, … e is

T_{i} > T_{i} · a > T_{i} · a^{2} > ⋯ > T_{i}a^{q} = 0 .

2. Every indecomposable kH−module is isomorphic to one of the T_{i,ν} = T_{i}/(T_{i}a^{ν}) ; i = 0, 1, …, (e − 1); ν = 0, 1, …, (q − 1).

3. There is a kC_{G}(D_{1})–block b such that kH ⊗_{kC_{G}(D_{1})} b = B′ and all such blocks are conjugate in H. Moreover, the stabilizer of b in H is of the form C_{G}(D_{1}) ×E for a subgroup E of N_{G}(D). The group E operates on D_{1} by conjugation and is E-linear.

We take α_{1} = α^{p^{d−1}}.

Then, D_{1} =< α_{1} >. Since H = N_{G}(D_{1}) for all h ∈ H there is a number n(h) defined uniquely modulo p such that,

h^{−1} · α_{1}h = α_{1}^{n(h)}.

ψ : H ⟶ (k \ {0}, ·)
h ⟶ n(h)

is a homomorphism and gives rise to one dimensional module. Since C = C_{G}(D_{1}), we have C ∈ ker(ψ) and hence ψ^{|H:C|} = 1.

We do similar computations with E and D. We define ∀_{z∈E}z^{−1}αz = α^{n(z)}. We use the same symbols for D_{1} as well as for D since we used a compatible choice for the generators of D and D_{1}.

We compute

\overline{α}^{z} =^{by 3.} \overline{α^{z}} =^{by Def.} \overline{α^{n(z)}} =^{by 3.} \overline{α}^{n(z)}.

Lemma 3.7. (Green)

1. S_{i,ν} := T_{i,ν}/T_{i,(ν+1)} ≃ ψ^{ν} ⊗_{k} S_{i}

2. For all n ∈ ℤ set S_{n} := S_{0,n} and then {S_{0}, S_{1}, …, S_{e−1}} is a complete set of representators of isomorphism classes of simple kH−modules.

3. The composition factors of T_{i} are S_{i}, S_{i+1}, …, S_{i+q−1} ≃ S_{i}.

Proof. Part 1. ∀_{t∈T_{i};z∈E} : t·a^{ν}·z = t·z·(\overline{α}^{n(z)}−1)^{ν} = t·z·a^{ν}·(1+\overline{α}+\overline{α}^{2}+⋯+\overline{α}^{n(z)−1})^{ν} ≡

t · z · a^{ν} · n(z)^{ν} since (1 + \overline{α} + \overline{α}^{2} + ⋯ + \overline{α}^{n(z)−1}) ≡ n(z) modulo T_{1} · a.

Part 2. Since ψ^{|H:C|} = 1 ⇒ S_{m} ≃ S_{n} if m ≡ n mod |H : C|. But, |H : C| | (p−1), then follows by part 1 that all S_{n} are composition factors of T_{0}. Hence, S_{n} all belong to B′.

Let S be a simple B′−module. Then, there is a sequence i_{0}, i_{1}, …, i_{r} ∈ I such that S_{i_{0}} ≃ S_{0} and S_{i_{r}} ≃ S and S_{i,j} is a composition factor of T_{i_{j−1}}.^{12} We know that S_{i_{j}} ≃ ψ^{ν_{j}} ⊗ S_{i_{j−1}} for all j. Hence, there is an integral number x such that S ≃ ψ^{x} ⊗ S_{0}. We know, that there are precisely e simple modules and therefore we found all of them.

Part 3. follows from Part 1, Part 2. and Lemma 3.6.

Corollary 3.8. (Green) Let i ∈ I ; ν ∈ {1, …, q}.

1. T_{i,ν} is projective if and only if ν = q.

2. There are non split exact sequences

0 ⟶ T_{i+1,q−ν} ⟶ T_{i,q} ⟶ T_{i,ν} ⟶ 0

3. ∀_{1≤ν≤q−1}Ω_{H}(T_{i,ν}) ≃ T_{i+ν,q−ν}.

4. Ω_{H}(Ω_{H}S_{i}) ≃ S_{i+1}

The proof is clear.

3.4. Definition of the ’walk’ δ. In this subsection we follow closely Green [5]. We shall prove in this subsection Theorem 3.5.

In the following we first examine the situation over k and pass then, in the next subsection, over to R.

For the proof of Theorem 3.5 we proceed in several lemmata. Let {V_{j}} be a complete set of simple kG–modules. For proving Part 1 of the

theorem we have to show that there is a bijection between I and J.

12This is an alternative method of describing blocks. In fact, we need only the necessity. If there was not such a sequence, then we can divide the projective indecomposables into two disjoint sets P_{1} and P_{2} such that for all P_{1} ∈ P_{1} and all P_{2} ∈ P_{2} Hom(P_{1}, P_{2}) = Hom(P_{2}, P_{1}) = 0 and hence B′ = End(⊕_{P_{1}∈P_{1}}P_{1} ⊕ ⊕_{P_{2}∈P_{2}}P_{2}) = End(⊕_{P_{1}∈P_{1}}P_{1}) ⊕ End(⊕_{P_{2}∈P_{2}}P_{2}) decomposes.

Claim 3.9. fV_{j} is indecomposable and non projective and belongs to B′. gS_{i} is indecomposable and non projective and belongs to B.

Proof. The only thing one has to show is that fV_{j} and gS_{i} belong to the blocks as claimed. Since fg(N) = N and gf(M) = M for all N and M, we just have to show one of the statements.

We have the Green correspondence with G = kG − mod and H = kH − mod and D = kD − mod on the level of the modules. The Brauer correspondence is a Green correspondence with G = k(G × G) − mod and H = k(H × H) − mod and D = k(D × D) − mod. Since G × G ⟶ G × 1 ≃ G is an epimorphism, we can view each kG–module as k(G × G)–module. The analogous holds for H and D. The Green correspondent for a kG–module V is the same as the Green correspondent of V as k(G × G)−module. This proves the statement since belonging to a block means for a module that the corresponding idempotent of the block acts as identity on the module. Then using the functoriality gives the statement.

We now turn to prove Part 2 of the theorem.

For this purpose we prove that

Hom_{kH}(S_{i}, fV_{j}) = (Hom_{kH}/mod(H, 1))(S_{i}, fV_{j})

and similarly

Hom_{kG}(gS_{i}, V_{j}) = (Hom_{kG}/mod(G, 1))(gS_{i}, V_{j}) .

More generally, let X be a non projective indecomposable module, then

Hom_{kH}(S_{i}, X) = (Hom_{kH}/mod(H, 1))(S_{i}, X)

Let ϕ : S_{i} ⟶ X be a map which is zero on the right side of the equation. Then, ϕ factors through a projective module. However, group rings are selfinjective algebras^{13}. Since S_{i} is simple, the projective module over which the mapping factors has as direct summand the injective hull P of S_{i} and the mapping actually factors over P. But, soc(P) = S_{i} and therefore, if the mapping is not zero, it is injective. However, then the injective module P is a submodule and hence even a direct summand of X. This gives a contradiction.

We proved the

AA ⟶ Hom_{R}(A_{A}, R)
a ⟶ b ⟶ λ(ab))

13An R–algebra A is called selfinjective if each projective A–module is injective. Group rings are selfinjective since there is a linear map λ : A ⟶ R such that kerλ contains no non zero left nor right ideal and ∀_{a,b∈A}λ(ab) = λ(ba). Such algebras are called symmetric, which is a slightly stronger condition. A group algebra RG is symmetric since we put λ(∑_{g∈G} r_{g}g) := r_{1}. Since λ((∑_{g∈G} r_{g}g)g^{−1}) = r_{g} there is no ideal in kerλ. Taking we realize that this mapping is injective since an element in the kernel would induce an ideal in the kernel of λ generated by this element. Going to the residue field of R we see that this mapping is also surjective, hence an isomorphism. Injective modules are hence also projective and vice versa.

Projective modules for symmetric artinian algebras over a field have the property that the socle and the head are isomorphic.

Lemma 3.10.

Hom_{RH}(S_{i}, fV_{j})  =^{above}  (Hom_{RH}/mod(H, 1))(S_{i}, fV_{j})
=^{Greencorr.} (Hom_{RG}/mod(G, 1))(gS_{i}, V_{j})
=^{above}  Hom_{RG}(gS_{i}, V_{j})

and analogously

Hom_{RH}(fV_{j}, S_{i}) = Hom_{RG}(V_{j}, gS_{i}).

Lemma 3.11. There is a bijection h : J ⟶ I such that

∀_{i∈I,j∈J}h(j) = i ⟺ Hom_{RH}(fV_{j}, S_{i}) ≠ 0

Proof. fV_{j} ≃ T_{h(j),ν(j)} for h(j) ∈ I, ν(j) ∈ {1, …, q − 1} by its indecomposability. But then, it is even uniserial and

Hom_{kH}(fV_{j}, S_{i}) = \{ \begin{matrix}k & if h(j) = i \\ 0 & if h(j) ≠ i \end{matrix}

by Schur’s Lemma.^{14}

Given i ∈ I and S|soc(gS_{i}). Since S is a B–module, there is a j ∈ J such that V_{j} ≃ S. Hence, Hom_{kG}(V_{j}, gS_{i}) ≠ 0 and therefore, Hom_{kH}(fV_{j}, S_{i}) ≠ 0 by Lemma 3.10. This proves that h(i) = j and h is surjective.

Given j, j′ ∈ J with h(j) = h(j′) = i. Then, we may assume without loss of generality, interchanging j and j′ if necessary that f(V_{j}) = T_{i,ν} and f(V_{j′}) = T_{i,ν′} for some 1 ≤ ν′ ≤ ν ≤ q − 1. Hence, there is an epimorphism

T_{i,ν} ⟶ ⟶ T_{i,ν′}

If this mapping would factor through a projective module it would factor through T_{i} which is the projective cover of T_{i,ν′}. Hence,^{15} top(T_{i,ν}) is mapped to a subquotient of rad(T_{i,ν}) unless ν = 0 what we excluded. Therefore, the mapping was not surjective and this gives a contradiction. We conclude

0 ≠ (Hom_{kH}/mod(H, 1))(fV_{j}, fV_{j′}) =^{Green corr.} (Hom_{kG}/mod(G, 1))(V_{j}, V_{j′})
=^{Schur}  Hom_{kG}(V_{j}, V_{j′})
⇒  j = j′.

Hence, h is also injective which finishes the proof of Lemma 3.11.

From now on, we take I = J and h = id_{I} and have fV_{j} = T_{j,ν(j)} for all j ∈ I and certain ν(j) ∈ {1, …, q − 1}.

Now we use the same proof as in the lemma in the situation Hom_{kH}(S_{i}, fV_{j}) instead of Hom_{kH}(fV_{j}, S_{i}) to obtain a bijection δ : I ⟶ I, which is a permutation, such that Hom_{kH}(S_{i}, fV_{j}) = k if and only if δ(i) = j and 0 else.

This completes the proof of Part 2. of the theorem.

We are going to show Part 3.

Hom_{A}(S, T) = \{ \begin{matrix}a skewfield & if S ≃ T \\ 0 & else \end{matrix} .

14Schur’s Lemma says that given a ring A and simple A–modules S and T, then The proof is easy since a kernel and an image under an A–isomorphism are ideals which are either 0 or the whole module by the simplicity of S and T.

15We use the terminus ’top’ synonymous to ’head’.

By Part 2. we get soc(gS_{i}) ≃ V_{i} and top(gS_{i}) ≃ V_{δ(i)}. Therefore, there is a short exact sequence

0 ⟶ ΩgS_{i} ⟶ \overline{W}_{δ(i)} ⟶ gS_{i} ⟶ 0.

Since \overline{W}_{i+1} is also injective and since soc(gS_{i}) ≃ V_{i} there is a short exact sequence

0 ⟶ gS_{i+1} ⟶ \overline{W}_{i} ⟶ V ⟶ 0

with some kG–module V. But,

gS_{i+1} ≃ gΩ^{2}S_{i} ≃ Ω^{2}gS_{i}

and we see that there is a non split exact sequence

0 ⟶ gS_{i+1} ⟶ \overline{W} ⟶ ΩgS_{i} ⟶ 0

with a projective kG–module \overline{W}. Applying Schanuel’s Lemma gives V ≃ ΩgS_{i}. We have also proven Part 3. of the theorem.

Remark 3.12. The same proof, and statement, works for a stable equivalence between two selfinjective k–algebras A and B such that A is serial.

3.5. Turning to characteristic 0. We now prove the main result of this section. For the reader’s convenience we state it here again.

Theorem 3.13. (Green) Let G be a finite group, let B be an RG − block, let D be a cyclic defect group of B and let Γ be the Brauer tree of B.

1. There is a family (A_{n})_{n∈ℤ} of RG–lattices and a permutation δ of I = {0, …, (e − 1)} such that there exist short exact sequences of RG–modules

E_{2i} : 0 ⟶ A_{2i+1} ⟶ W_{δ(i)} ⟶ A_{2i} ⟶ 0
E_{2i+1} : 0 ⟶ A_{2i+2} ⟶ W_{i} ⟶ A_{2i+1} ⟶ 0

with W_{i} ≃ W_{i+e} are projective indecomposable B–modules and A_{i} ≃ A_{i+2e} for all i ∈ ℤ.

2. The A_{0}, A_{1}, …, A_{2e−1} are mutually non isomorphic.

3. KA_{n} is a vertex of Γ.

Proof. We can lift the projective indecomposable modules \overline{W}_{i} to projective indecomposable RG–modules W_{i} such that k ⊗_{R} W_{i} ≃ \overline{W}_{i} for all i ∈ I. We may extend this definition to i ∈ ℤ by requiring that W_{i} ≃ W_{i+e}.

In the situation of Theorem 3.5 we define,

B_{2i} := gS_{i} and B_{2i+1} := ΩgS_{i}.

Lemma 3.14. Let m ∈ ℤ and M be an RG−lattice such that k⊗_{R} M ≃ B_{m}. Then there are RG−lattices A_{n} with A_{m} ≃ M and short exact sequences E_{n} such that k ⊗_{R} E_{n} ≃ F_{n} for all n ∈ ℤ.

E_{n} : 0 ⟶ A_{n+1} ⟶ W_{n} ⟶ A_{n} ⟶ 0

Proof. Start with A_{m} = M.

0 ⟶ A_{m+1} ⟶  W_{m}  ⟶  A_{m}  ⟶ 0
↓ ϕ  ↓ k ⊗_{R} −  ↓ k ⊗_{R} −
0 ⟶ B_{m+1} ⟶  \overline{W}_{m}  ⟶  B_{m}  ⟶ 0

with A_{m+1} is just defined to be the kernel of W_{m} ⟶ A_{m}. ϕ is defined by the universal property of the kernel. Hence, there is an A_{m+1} which lifts B_{m+1} and inductively one gets all A_{n} for n ≥ m.

For n ≤ m one uses the following strategy. We apply Hom_{R}(−, R) to the first row and Hom_{k}(−, k) to the second. We get

0 ⟶  A_{m−1}^{∗}  ⟶ Hom_{R}(W_{m−1}, R) ⟶ Hom_{R}(A_{m}, R) ⟶ 0
↓ ϕ  ↓ k ⊗_{R} −  ↓ k ⊗_{R} −
0 ⟶ Hom_{k}(B_{m−1}, k) ⟶ Hom_{k}(\overline{W}_{m−1}, k) ⟶ Hom_{k}(B_{m}, k) ⟶ 0

with A_{m−1}^{∗} is just defined to be the kernel of Hom_{R}(W_{m−1}, R) ⟶ Hom_{R}(A_{m}, R) and ϕ is defined by the universal property of the kernel. Dualizing again,

0 ⟶ A_{m−1}^{∗} ⟶ Hom_{R}(W_{m−1}, R) ⟶ Hom_{R}(A_{m}, R) ⟶ 0

we get that

0 ⟶ A_{m} ⟶ W_{m−1} ⟶ Hom_{R}(A_{m−1}, R) ⟶ 0

is exact and inductively we get the statement.

Lemma 3.15. If KM is a vertex of Γ, then all KA_{n} are vertices and KA_{n} ≃ KA_{n+2e} for all n ∈ ℤ and KA_{n} ≃ KA_{n+2e}.

Proof. We get that

KW_{n} = KA_{n} ⊕ KA_{n+1} ∀ n ∈ ℤ

Hence, all KA_{n} are vertices. In the Grothendieck group K^{0}(KG) we take

∑_{i=n}^{n+2e}(−1)^{i}[KW_{i}] = 0 = (−1)^{n} · ([KA_{n}] − [KA_{n+2e}]).

Lemma 3.16. Let A_{n} and M be as above, then A_{n} ≃ A_{n+2e}.

Proof. Given n ∈ ℤ. We have a unique decomposition KW_{n} = Y_{n}(1) ⊕ Y_{n}(2) where Y_{n}(1) and Y_{n}(2) are both vertices of Γ. Define

X_{n}(i) := W_{n} ∩ Y_{n}(i) ; i = {1, 2}.

These are R-pure submodules of W_{n} since

W_{n}/(W_{n} ∩ Y_{n}(i)) ≃ (W (n) + Y_{n}(i))/Y_{n}(i) ≤ KW (n)/Y_{n}(i)

the latter of which is R–torsion free. Furthermore, they are the only two R–pure submodules X̃ of W_{n} with KX̃ is a vertex in Γ. In fact, let X̃ be a counterexample with KX̃ = Y_{n}(i). Then surely X̃ ≤ X_{n}(i). The following diagram is then commutative with exact rows:

0 ⟶ X̃ ⟶ W_{n} ⟶ Q_{n} ⟶ 0
↓ ∥ ↓
0 ⟶ X_{n}(i) ⟶ W_{n} ⟶ R_{n} ⟶ 0

where by the serpent lemma the right most vertical map is surjective. Also, we assumed Q_{n} to be R–free and again by serpent lemma also the cokernel of X̃ ⟶ X_{n}(i) is torsion free. But then, KX̃ ≠ X_{n}(i) and we reached a contradiction.

This proves the lemma.

Now the problem is just reduced to trying to find an M to start with. For the principal block we just take M = R, the trivial module.

The general case is more involved and uses a construction of Dade.

It remains to prove Part 2.

We have:

k ⊗_{R} A_{2n+1} ≃ ΩgS_{n}

k ⊗_{R} A_{2n} ≃ gS_{n}

If q = |D| ≥ 2 then S_{i} = T_{i,1} ≄ T_{i,q−1} = ΩS_{i} and clearly S_{i} ≄ S_{j} for i ≠ j mod e. If q = 2 then A_{0} ≃ A_{1} and since e|(q − 1) we get e = 1. But, KW_{0} ≃ Y_{0}(1) ⊕ Y_{0}(2) which are non isomorphic and also KW_{0} ≃ KA_{0} ⊕ KA_{1} where both are vertices of Γ. This gives a contradiction.

Hence, we proved now the whole theorem.

4. Blocks with cyclic defect groups are Green orders

We maintain the notation of Section 3.

In this section we shall define ’Green orders’ due to Roggenkamp and prove that a block of the group ring RG with cyclic defect group D is a Green order. Moreover, we shall describe the structure of Green orders in great detail.

We follow the exposition in [16] and mention that all the material is contained in [16].

4.1. A small example. Let p be a rational prime number. We shall discuss the integral group ring of the dihedral group D_{p} of order 2p. We remark that D_{p} fits well in Green’s framework at the prime p, where we mean that ℤ_{p}D_{p} is a group ring which satisfies all the assumptions of Theorem 3.3.

Let D_{p} =< a, b|a^{p} = b^{2} = baba = 1 > be a presentation of D_{p}. Then, < a > is a cyclic normal subgroup of index 2 in D_{p}. Hence, we get a surjective ring homomorphism

ℤD_{p} ⟶ ℤC_{2}.

This is induced by multiplication by the central idempotent

e = \frac{1}{p} ∑_{i=1}^{p} a^{i} ∈ QD_{p}.

Hence, one gets a pullback diagram

ℤD_{p}  ⟶^{e·}  ℤD_{p}e
↓ ·(1 − e)  ↓
ℤD_{p}(1 − e)  ⟶ ℤD_{p}e/(ℤD_{p}e ∩ ℤD_{p})

which becomes

ℤD_{p} ⟶^{e·} ℤC_{2}
↓ ·(1 − e)  ↓
Λ  ⟶ IF_{p}C_{2}

where IF_{p} is the prime field of characteristic p and the right hand vertical mapping is just reduction modulo p. In fact, pe ∈ ℤD_{p} and ℤD_{p}e ∩ ℤD_{p} = pℤD_{p}e.

We have to determine Λ. Multiplication of ℤD_{p} by (1 − e) amounts to saying that a acts on ℤ[ζ_{p}] as multiplication by ζ_{p}, where ζ_{p} is a primitive p^{th} root of unity. In fact, 1 + a + a^{2} + ⋯ + a^{p−1} acts as 0 and this is the only relation among the elements of ℤ < a >. The b however, it inverts the a and acts therefore as Galois automorphism ζ_{p} ⟶ ζ_{p}^{−1}. The element a acts as ζ_{p} and this means that over the fixed ring ℤ[ζ_{p} + ζ_{p}^{−1}] the element a satisfies the minimal polynomial

X^{2} − (ζ_{p} + ζ_{p}^{−1})X + 1

and with basis {1, ζ_{p}} the representation can be described by the accompanying matrices

a ⟶ ( \begin{matrix}0 & −1 \\ 1 & ζ_{p} + ζ_{p}^{−1} \end{matrix} ) and b ⟶ ( \begin{matrix}1 & ζ_{p} + ζ_{p}^{−1} \\ 0 & −1 \end{matrix} ) .

Conjugating by ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ) from the left one gets the representations

a ⟶ ( \begin{matrix}ζ_{p} + ζ_{p}^{−1} − 1 & 1 \\ ζ_{p} + ζ_{p}^{−1} − 2 & 1 \end{matrix} ) and b ⟶ ( \begin{matrix}−1 & 0 \\ ζ_{p} + ζ_{p}^{−1} − 2 & 1 \end{matrix} )

where π_{p} := ζ_{p} + ζ_{p}^{−1} − 2 generates the unique prime ideal above p in ℤ[ζ_{p} + ζ_{p}^{−1}] (see [6]).

After constructing the standard idempotents in the matrix ring, one gets

Λ = ( \begin{matrix}ℤ[π_{p}] & ℤ[π_{p}] \\ π_{p}ℤ[π_{p}] & ℤ[π_{p}] \end{matrix} )

and the mapping to IF_{p}C_{2} equals reduction modulo

J = ( \begin{matrix}π_{p}ℤ[π_{p}] & ℤ[π_{p}] \\ π_{p}ℤ[π_{p}] & π_{p}ℤ[π_{p}] \end{matrix} )

We write down

ℤD_{p} = \{(u, ( \begin{matrix}x & y \\ z & w \end{matrix} ) , v) ∈ ℤ × ( \begin{matrix}ℤ[π_{p}] & ℤ[π_{p}] \\ π_{p}ℤ[π_{p}] & ℤ[π_{p}] \end{matrix} ) × ℤ | x − u ∈ π_{p}ℤ[π_{p}] and
u − w ∈ π_{p}ℤ[π_{p}] and
u − v ∈ 2ℤ}

Localizing at the prime p this gives the Brauer tree

•− •^{\frac{p−1}{2}} −•

with exceptional vertex with multiplicity (p − 1)/2 in the centre. We shall show that this structure has a feature which is common for all blocks of finite groups with cyclic defect groups.

4.2. Defining Green orders. We shall define a class of orders with a structure like in the above example. These orders are introduced by Roggenkamp [16] and who called them Green orders.

Throughout this subsection let R be a local Dedekind domain.

• Let Γ be a tree^{16} embedded in the plane^{17}.

• Choose a local R–torsion R–algebra k finitely generated as R–module.

• Associate to each vertex v of Γ a pair (Ω_{v}, f_{v}) where Ω_{v} is a local R–order in a semisimple algebra A_{v}, and where f_{v} is a surjective ring homomorphism f_{v} : Ω_{v} ⟶ k with kernel being a principal ideal a_{v}Ω_{v}.

• If v −^{e} w is an edge, then put ν_{e}(v) := w and ν_{e}(w) := v the mapping giving the other extremity of the edge e. Of course, ν_{e} is an involution.

• If e is an edge incident to a vertex v of the graph Γ, then set α_{v}(e) the edge which follows e in the cyclic ordering at the vertex v. We set α_{v}^{i}(e) = α_{v}^{i−1}(α_{v}(e)) and α_{v}^{1} = α_{v}.

• n_{v} := #{α_{v}^{i}(e)|i ∈ ℕ; e is an edge incident to v} for any vertex v of the graph

Γ.

A leaf of Γ is a vertex v with n_{v} = 1.

16A tree is understood to be a finite, connected, undirected graph without cycles.

17This is equivalent to saying that one imposes to each vertex a cyclic ordering of the edges incident to the vertex.

• Attach to each vertex v of Γ the order

Λ_{v} := ( \begin{matrix}Ω_{v} & … & … & … & Ω_{v} \\ (a_{v}) & Ω_{v} & & & ⋮ \\ ⋮ & (a_{v}) & ... & & ⋮ \\ ⋮ & & ... & ... & ⋮ \\ (a_{v}) & … & … & (a_{v}) & Ω_{v} \end{matrix} )_{n_{v}×n_{v}}

• We denote the pullback

(∗) \begin{matrix}Ω_{v} − Ω_{w} & ⟶ & Ω_{v} \\ ↓ & & ↓ f_{w} \\ Ω_{w} & ⟶^{f_{v}} & k \end{matrix} .

• We form the iterated pullback of the orders Λ_{v} for each vertex v under the following iterative procedure. Set Λ_{g} := ⨁_{v∈Γ_{vertex}} Λ_{v}

1. Fix a leaf^{18} v. The edge leaving v is e and w := ν_{e}(v). Form the pullback (∗) between^{19} (Λ_{w})_{(1,1)} and Ω_{v}. Set Λ_{g} the subring of the old Λ_{g} given by this pullback.

2. For each i = 2, …, n_{w} form the subring of Λ_{g} by the pullback (∗) between (Λ_{w})_{(i,i)} and (Λ_{ν_{α{i−1 \atop w}(e)}(w)})_{(1,1)}. Put Λ_{g} the new subring formed by these pullbacks. Call the vertices ν_{α_{w}^{i−1}(e)}(w) reached. Call w saturated.

3. If there is no vertex which is not yet saturated, then we define the generic Green order to the tree Γ with data (Ω_{v}, f_{v}) to be Λ_{g}. Stop the algorithm!

4. Else there is a vertex v which is reached and not saturated. Since v is reached, Λ_{g} contains a pullback between (Λ_{v})_{(1,1)} and a Ω_{w}. Set e to be the edge v − w. Proceed with 2.

Definition 4.1. The resulting order Λ_{g} which occurs after the algorithm executed point 3 in the algorithm is called generic Green order to the tree Γ with data (Ω_{v}, f_{v})_{v∈Γ_{vertex}}.

The reader might like to construct the generic Green order to the tree in Section 3.1.

We remark that the isomorphism type of the generic Green order depends only on the embedded graph and the data. This can be proved since Λ_{v} contains an automorphism conjugation by

( \begin{matrix}0 & 1 & 0 & … & 0 \\ ⋮ & 0 & 1 & ... & ⋮ \\ ⋮ & & ... & ... & 0 \\ 0 & & & ... & 1 \\ a_{v} & 0 & … & … & 0 \end{matrix} )_{n×n}

This induces a cyclic permutation of the diagonal entries, the last becoming the first.

In the following section we shall elaborate on the orders Λ_{v}.

18A leaf of a graph is a vertex v ith n_{v} = 1.

19The notation M_{(i,i)} means the (i, i)−entry of the matrix M.

Definition 4.2. (Roggenkamp [16]) Let R be a Dedekind domain with field of fractions K. An R–order Λ in a separable K–algebra A is called a Green order if there is a finite connected tree with vertices {v_{i}}_{i=0}^{n} and edges {e_{k}}_{k=1}^{n}.

1. The vertices {v_{i}}_{i=0}^{n} correspond to (not necessarily primitive) central idempotents {η_{i}}_{i=1}^{n} of A with 1 = ∑_{i=0}^{n} η_{i}.

2. The edges {e_{k}}_{k=1}^{n} correspond to a full set of indecomposable projective Λ– lattices {P_{k}}_{k=1}^{n}.

3. The tree and a starting vertex determines^{20} a permutation δ of {1, …, e} and there is a set of Λ–lattices {A_{i}}_{i=1}^{n} such that

(a) KA_{i} ≃ Aη_{i} for all i = 0, 1, …, n

(b) for all i = 0, …, n there are short exact sequences

E_{2i} : 0 ⟶ A_{2i+1} ⟶ P_{δ(i)} ⟶^{η_{δ(i)}·} A_{2i} ⟶ 0

E_{2i+1} : 0 ⟶ A_{2i+2} ⟶ P_{i} ⟶^{η_{i}·} A_{2i+1} ⟶ 0 .

The term generic Green order is used since in Theorem 4.3 (see also the proof of Lemma 3.16) it will be proven that all Green orders are Morita equivalent to generic Green orders.

Example. Let G be a finite group and let R be a complete discrete valuation ring of characteristic 0 with residue field of characteristic p containing all the |G|^{th} roots of unity. Let B be a block of RG with cyclic defect group D. By Theorem 3.3 B is a Green order.

Theorem 4.3. (Roggenkamp) [16] Let Λ be a Green order with tree Γ. Then Λ is Morita equivalent to a generic Green order with tree Γ.

We shall give Roggenkamp’s proof of Theorem 4.3 in the sequel. For this purpose we shall introduce in the next section another type of orders, which Roggenkamp calls in [17] isotypic orders. These are the orders Λ_{v} in the definition for a generic Green order.

4.3. The rational components; isotypic orders. Throughout this subsection let R be a Dedekind domain with field of fractions K. Let Λ be an R–order in a separable K–algebra A.

Definition 4.4. (Roggenkamp) [16] The order Λ is called isotypic order provided there is a twosided Λ–ideal J such that

1. K · J = A

2. J is projective as left Λ–module.

3. Λ/J is a direct product of local R–algebras.

4. Λ is nilpotent modulo the Higman ideal H(Λ).^{21} Then, J is called associated to Λ or defining ideal of the isotypic order.

One first property is almost immediate:

Λ is isotypic if and only if R̂_{℘} ⊗ Λ is isotypic for all prime ideals ℘ of R. Here we denote by R̂_{℘} the completion of R at ℘.

20in the sense described in the discussion in Section 3.2.

21The Higman ideal of an R–order Λ is the R–annihilator of Ext_{Λ⊗_{R}Λ^{op}}^{1}(Λ, −). For orders Λ in a separable algebra we have K · H(Λ) = 0 . [18, V. 3.5]

Proof. If Λ is isotypic, then R̂_{℘} ⊗ Λ is isotypic. In fact, 1. and 3. are clear. 2. and 4. are consequences of the ’change of rings’ theorem.

If R̂_{℘} ⊗Λ is isotypic for all prime ideals ℘ of R, then we use the following general property for orders Λ in a separable algebra A over a Dedekind domain R.

If M and L are full R–lattices in the K–vector space V, then M_{℘} = L_{℘} almost everywhere. Furthermore, for each ℘ let there be given for each ℘ a full R_{℘}–lattice X(℘) such that X(℘) = M_{℘}. Then N := ⋂ X(℘) has the property that N_{℘} = X(℘) for all ℘ ∈ Spec(R).

We apply this to J. We have ideals J(℘) for all ℘ by the definition of isotypic orders. Set J(℘) = Λ̂_{℘} whenever Λ_{℘} is a maximal order. We form J := ⋂_{℘∈Spec(R)} J(℘). Now, J is projective since J(℘) = J_{℘} is projective for all ℘. (This is observed most easily by seeing that Hom_{Λ}(J, −) is exact. This in turn is seen by the ’change of rings theorem’. KJ = A since this holds locally and Λ_{℘}/J_{℘} is R_{℘}–torsion since KΛ_{℘} = KJ_{℘}. Hence,

Λ/J ≃ ∏_{℘} R_{℘} ⊗ Λ/J ≃ ∏_{℘}(R_{℘} ⊗ Λ)/(R_{℘} ⊗ J) ≃ ∏_{℘} Λ_{℘}/J_{℘}

the latter of which was assumed to be a direct product of local algebras. This finishes the proof of the observation.

We may assume by this observation, in clarifying the structure of isotypic orders, that R is a complete discrete valuation domain with residue field IF and radical πR.

Assuming this, the Higman ideal of Λ is a power of πR. By a general property of Jacobson radicals for artinian algebras gives us that J is nilpotent modulo πR, if and only if J ≤ rad Λ. However, J is nilpotent modulo πR if and only if J is nilpotent modulo a certain power of πR. (Just multiply the nilpotency degree by the power which was fixed at the beginning.) So Condition 4. translates to J ≤ rad Λ.

Theorem 4.5. (Roggenkamp [16]) Let R be a complete discrete valuation domain. Assume that Λ is a basic^{22} isotypic R–order with associated ideal J. Let furthermore Λ be indecomposable as ring.

Then, there is a local R–order Ω and a regular non unit a ∈ Ω such that aΩ = Ωa =: (a) and Ω/(aΩ) is a local algebra, and a natural number n such that

Λ ≃ Λ_{0} = Λ_{0}(Ω, a, n) := ( \begin{matrix}Ω & … & … & … & Ω \\ (a) & Ω & & & ⋮ \\ ⋮ & (a) & ... & & ⋮ \\ ⋮ & & ... & ... & ⋮ \\ (a) & … & … & (a) & Ω \end{matrix} )_{n×n}

Conversely, every such order is isotypic.

22A Noetherian ring is basic if under each decomposition into a direct sum of indecomposable left projective modules Λ = ⊕_{i∈I}P_{i}, no two projective summands P_{i} and P_{j} with i ≠ j; i, j ∈ I are isomorphic.

Remark 4.6. 1. For Λ_{0} one can take the associated ideal

J = ( \begin{matrix}0 & 1 & 0 & … & 0 \\ ⋮ & 0 & 1 & ... & ⋮ \\ ⋮ & & ... & ... & 0 \\ 0 & & & ... & 1 \\ a & 0 & … & … & 0 \end{matrix} )_{n×n} · ( \begin{matrix}Ω & … & … & … & Ω \\ (a) & Ω & & & ⋮ \\ ⋮ & (a) & ... & & ⋮ \\ ⋮ & & ... & ... & ⋮ \\ (a) & … & … & (a) & Ω \end{matrix} )_{n×n}

which is principal. We call the generating element above ω.

2. The product of local algebras as in 3. in the definition of ’isotypic order’ ranges over a set of pairwise isomorphic local algebras Ω/(a).

Proof. Λ_{0} is isotypic. In fact,

• a is regular, hence Part 1. of the Definition of an isotypic order.

• J is principal, generated by a regular element, hence free and Part 2. follows.

• Λ/J = ∏_{i=1}^{n} Ω/(a) is a direct product of local orders, hence we get also Part

3. a is not a unit. Since Ω is local, J is contained in the radical of Λ. The radical of Λ is the ideal with radΩ in the main diagonal and in the lower triangular matrix, Ω in the rest of the entries.

We have to prove the converse. We may order a complete set of projective indecomposable modules P_{1}, P_{2}, …, P_{n} such that rank_{R}P_{i} ≤ rank_{R}P_{i+1} for all i = 1, 2, …, n − 1.

Since for any i the module P_{i} is projective, there is a Q_{i} and an integer m(i) such that P_{i} ⊕ Q_{i} ≃ Λ^{m(i)}. But then, J ⊗_{Λ} P_{i} ⊕ J ⊗_{Λ} Q_{i} ≃ J^{m} which is projective since J is projective. Hence, J ⊗_{Λ} P_{i} is projective. Then, for any i the set

P_{i} := { isomorphism classes of the modules J^{j} ⊗_{Λ} P_{i}|j ∈ ℕ}

consists of projective modules. The cardinality of this set is k(i).

Claim 4.7. J^{k(1)} ⊗_{Λ} P_{1} ≃ P_{1}. Furthermore, k(1) = n and hence P_{1} is a complete set of isomorphism classes of projective modules.

Proof. Since KJ = A, we get rank_{R}P_{1} = rank_{R}J^{j} ⊗_{Λ} P_{1} and we may order the projective indecomposable modules such that J^{j} ⊗_{Λ} P_{1} = P_{1+j}. Let Q = {Q_{1}, Q_{2}, …, Q_{s}} be a set of representatives of isomorphism classes of indecomposable projective modules such that the isomorphism class of no element of Q is contained in P_{1}.

Since Λ is indecomposable, there is a Q_{i} with Hom_{Λ}(Q_{i}, P_{1+j}) ≠ 0. If not, the endomorphism ring of Λ would be the direct product of the endomorphism rings of the direct sum of modules in Q and that of ⊕_{i=0}^{k(1)}P_{1+i}.

We now reduce to artinian algebras. Take 0 ≠ ϕ ∈ Hom_{Λ}(Q_{i}, P_{1+j}). Then, there is a ν ∈ ℕ such that the composition

Q_{i} ⟶^{ϕ} P_{1+j} ⟶ (P_{1+j})/(π^{ν}P_{1+j})

is non zero. Since J ≤ rad Λ, there is a μ such that J^{μ} ≤ π^{ν}Λ. But, for all k, J^{k}/J^{k+1} is a projective Λ/J module. However, this is a direct product of local algebras. Hence,

J^{k} · P_{1+j}/J^{k+1} · P_{1+j}

is a local Λ/J–module and has all composition factors isomorphic to P_{1+j}/rad P_{1+j}. Hence, P_{1+j}/π^{ν}P_{1+j} has composition factors all isomorphic to P_{k}/rad P_{k} for k = 1, …, k(1). But, the top of Q_{i} is isomorphic to one of them, hence the isomorphism class of Q_{i} is in P_{1}, a contradiction and Q is empty.

It remains to prove that J^{k(1)} ⊗_{Λ} P_{1} ≃ P_{1}.

If J^{k(1)} ⊗_{Λ} P_{1} ≃ P_{k} for some k, then one forms the set P_{k} instead of P_{1} and {P_{1}, …, P_{k−1}} would be in Q. The analogous arguments as above give a contradiction and we are done.

By Morita theory, ⊕_{i=1}^{n}P_{i} gives a Morita bimodule inducing a Morita equivalence between Λ and End_{Λ}(⊕_{i=1}^{n}P_{i}). Since we have the Krull–Schmidt theorem for Λ and since Λ is basic, Λ ≃ End_{Λ}(⊕_{i=1}^{n}P_{i}).

Claim 4.8. End_{Λ}(⊕_{i=1}^{n}P_{i}) ≃ Λ_{0} for Ω := End_{Λ}(P_{1}) and (a) := Hom_{Λ}(P_{1}, J^{n} ⊗_{Λ} P_{1}).

Proof. Since KP_{1} is simple by construction, the isomorphism P_{1} ⟶ J^{n}P_{1} is multiplication by a regular element a ∈ Ω. Hence,

Hom_{Λ}(P_{1}, J^{n} ⊗_{Λ} P_{1}) = Ω · a.

Any endomorphism ϕ ∈ Ω can be extended to an endomorphism of J^{i} ⊗_{Λ} P_{1} by id_{J^{i}} ⊗ ϕ. Hence,

Ω ≤ End_{Λ}(P_{2}) ≤ ⋯ ≤ End_{Λ}(P_{n−1}) ≤ End_{Λ}(P_{n}) ≤ End_{Λ}(J^{n} ⊗_{Λ} P_{1}) = a^{−1} · Ω · a.

But then, all of the endomorphism rings are equal, Ω being noetherian.

Since when tensored by K over R, all the P_{i} are isomorphic and simple, any non zero mapping ϕ : P_{i} ⟶ P_{i+j} for i = 1, …, n and j = 1 − i, …, n − i is injective. Looking at the tops of the modules P_{i+j}/J^{k}P_{i+j} by the arguments given in the proof of the preceding claim, Hom_{Λ}(P_{i}, P_{j}) = Ω if j ≤ i and Hom_{Λ}(P_{i}, P_{j}) = Ω · a if j > i.

This proves the claim and also the theorem.

4.4. Structure theorem for Green orders. Proof of Theorem 4.3. We assume that Λ is basic. The proof will be done in several steps.

Lemma 4.9. Let Λ be a basic Green order with connected tree Γ and let v be a leaf. Let e be the edge joining v with some vertex w. The projective indecomposable module associated with e is P_{0}. Then, End_{Λ}(Λ/P_{0}) is a Green order with tree Γ′ where Γ′_{vertex} = Γ_{vertex} \ {v} and Γ′_{edge} = Γ_{edge} \ {e}.

Proof. Without loss of generality we may set A_{0} = Λη_{0}. Set P := Λ/P_{0}. Apply F := Hom_{Λ}(P, −) to E_{n} for all n = 0, … 2n−1. Since P is projective, this functor is exact. However, since there is only one projective indecomposable module, namely P_{0}, F (A_{0}) = 0. But then F (E_{2i}) for i = 1, …, n and F (E_{2i+1}) for i = 0, …, n − 1 are the required exact sequences.

This finishes the proof of the lemma.

Lemma 4.10. Let Λ be a basic R–order in the separable algebra A. Let e be a central idempotent of A and let P |Λ with P · e = 0. Put End_{Λ}(Λ/P) =: Λ_{0} as a subring of Λ (not with the same unit!). Then,

Λ · e = Λ_{0} · e and Λ_{0} · e ∩ Λ = Λ_{0} · e ∩ Λ_{0}.

Proof. To prove the first statement one observes that

Λe = eΛe
= eEnd_{Λ}((Λ/P) ⊕ P)e
= e ( \begin{matrix}End_{Λ}(Λ/P) & Hom_{Λ}((Λ/P), P) \\ Hom_{Λ}(P, Λ/P) & End_{Λ}(P) \end{matrix} ) e
= ( \begin{matrix}eEnd_{Λ}(Λ/P)e & 0 \\ 0 & 0 \end{matrix} )
= Λ_{0}e

The second statement is proved as follows:

By the first statement,

Λ_{0} ∩ (Λ_{0}e) = Λ_{0} ∩ (Λe)

and clearly

Λ ∩ (Λe) ⊇ Λ_{0} ∩ (Λe)

Hence we have to show that

Λ ∩ (Λe) ⊆ Λ_{0} ∩ (Λe)

This is done by the following computation:

Λ ∩ (Λe) = ( \begin{matrix}End_{Λ}(Λ/P) & Hom_{Λ}((Λ/P), P) \\ Hom_{Λ}(P, Λ/P) & End_{Λ}(P) \end{matrix} )

∩e ( \begin{matrix}End_{Λ}(Λ/P) & Hom_{Λ}((Λ/P), P) \\ Hom_{Λ}(P, Λ/P) & End_{Λ}(P) \end{matrix} ) e

= ( \begin{matrix}End_{Λ}(Λ/P) & Hom_{Λ}((Λ/P), P) \\ Hom_{Λ}(P, Λ/P) & End_{Λ}(P) \end{matrix} )

∩ ( \begin{matrix}eEnd_{Λ}(Λ/P)e & 0 \\ 0 & 0 \end{matrix} )

= ( \begin{matrix}End_{Λ}(Λ/P) & 0 \\ 0 & 0 \end{matrix} ) ∩ ( \begin{matrix}eEnd_{Λ}(Λ/P)e & 0 \\ 0 & 0 \end{matrix} )
= Λ_{0} ∩ (Λ_{0}e)

Claim 4.11. Let Λ be a basic Green order with tree Γ. For the idempotent η corresponding to a vertex v the ring Λ/(Λ ∩ Λη) is a direct product of local algebras.

Proof. We use induction on the number of vertices.

If the Green order has only 2 vertices, then the statement is clear since there is just one projective indecomposable module and the Green order is local.

Let then v be a leaf of the tree Γ. Let e be the edge of the tree that links v with the rest of the tree. Let P_{0} be the projective indecomposable which corresponds to e. Set P = Λ/P_{0} and Λ_{0} = End_{Λ}(P). Then, by Lemma 4.10 the tree Γ′ defined by

Γ_{vertex} \ {v} =: Γ′_{vertex}; Γ_{edge} \ {e} =: Γ′_{edge}

defines a Green order structure on Λ_{0}.

Let v −^{e} w and let v′ with w ≠ v′ ≠ v be a vertex of Γ. Then, for v′ the statement is true by induction.

We need a proof for w and its central idempotent η_{w} only. Let η_{v} be the central idempotent corresponding to v. ϵ := 1 − η_{v} − η_{w}. We get the following pullback diagram.

Λ ⟶  Λ(ϵ + η_{v})
↓  ↓
Λη_{w} ⟶ Λ(ϵ + η_{v})/(Λ ∩ Λ(ϵ + η_{v}))

We see that

Λ(ϵ + η_{v}) = Λϵ ⊕ Λη_{v}

and

Λ(ϵ + η_{v})/(Λ ∩ Λ(ϵ + η_{v})) = [Λϵ/Λ ∩ Λϵ] ⊕ [Λη_{v}/Λ ∩ Λη_{v}].

But, by Lemma 4.10

Λϵ/(Λ ∩ Λ(ϵ)) = Λ_{0}ϵ/(Λ_{0} ∩ Λ_{0}(ϵ))

and since ϵ is also a central idempotent of KΛ_{0}, which gives rise to a pullback diagram itself with quotient A := Λ_{0}ϵ/(Λ_{0} ∩ Λ_{0}(ϵ)), the ring A is a direct product of local R–algebras by induction.

Now we use the fact that v is a leaf. If Λη_{v} was not local, it had two non isomorphic simple modules. Since Λη_{v} is an image of Λ, Λ itself has these two non isomorphic modules. However, there is only one simple Λ–module, namely the top of P_{0}, on which Hom_{Λ}(P_{0}, −) is non zero. The two simple Λ–modules constructed above however have this property, and hence they cannot exist. We conclude, Λη_{v} is local.

Since Λη_{v} is local, Λη_{v}/(Λη_{v} ∩ Λ) is local.

Lemma 4.12. Let Λ be a basic Green order with tree Γ. Let v be a vertex of Γ with corresponding central idempotent η_{v}. Then, Λη_{v} ∩ Λ is a free Λη_{v}–module.

Proof. Again we use induction on the number of vertices of Γ.

Assume that there are only two vertices. Hence, there are short exact sequences

0 ⟶ A_{1} ⟶ Λ ⟶^{η_{v}} A_{0} ⟶ 0

0 ⟶ A_{0} ⟶ Λ ⟶^{1−η_{v}} A_{1} ⟶ 0.

Thus, Λ(1 − η_{v}) ∩ Λ ≃ A_{0} ≃ Λη_{v}.

Assume we have more than two vertices.

If v is a leaf with idempotent η_{v}, let v −^{e} w be the edge of the tree linking v with the rest of Γ. The idempotent associated with w is denoted by η_{w}. Hence, there is an indecomposable projective Λ–module P_{0}, and there are short exact sequences

0 ⟶ A_{0} ⟶ P_{0} ⟶^{η_{w}} A_{−1} ⟶ 0

0 ⟶ A_{1} ⟶ P_{0} ⟶^{η_{v}} A_{0} ⟶ 0.

Let P_{0} = Λe_{0} for an idempotent e_{0} of Λ. Since v is a leaf, η_{v}P_{0} = η_{v}Λ; the proof is the same as that of proving that Λη_{v} is local. Now, since η_{w}P_{0} = (1 − η_{v})P_{0},

Λη_{v} ∩ Λ = η_{v}P_{0} ∩ Λ = η_{v}P_{0} ∩ P_{0} ≃ A_{0} = η_{v}P_{0}

If v_{2} is a vertex different from v and w, then let η_{v_{2}} be the central idempotent associated to v_{2}. Then, since

Λη_{v_{2}} ∩ Λ = Λ_{0}η_{v_{2}} ∩ Λ_{0} and Λη_{v_{2}} = Λ_{0}η_{v_{2}},

as usual for Λ_{0} = End_{Λ}(Λ/P_{0}), by Lemma 4.10, the statement is true by the induction hypothesis.

We have to prove the statement for w. Let {e_{1}, e_{2}, …, e_{n}} be the set of edges adjacent to w. Let {P_{1}, P_{2}, …, P_{n}} be the corresponding projective indecomposable modules. Set Δ := End_{Λ}(⨁_{i=1}^{n} P_{i}). Then, again by Lemma 4.10

Δη_{w} = Λη_{w} and Δη_{w} ∩ Δ = Λη_{w} ∩ Λ

Hence we may prove the statement for Γ being a star and w in the centre and number the edges in their cyclic ordering. Let {η_{1}, η_{2}, …, η_{n}} be the idempotents corresponding to the vertices being the end of {e_{1}, e_{2}, …, e_{n}} unequal to w. The idempotent corresponding to w is η. By the definition of a Green order we get short exact sequences

0 ⟶ A_{1} ⟶ P_{1} ⟶^{η_{1}·} A_{0} ⟶ 0

0 ⟶ A_{2} ⟶ P_{2} ⟶^{η·} A_{1} ⟶ 0

0 ⟶ A_{3} ⟶ P_{2} ⟶^{η_{2}·} A_{2} ⟶ 0

0 ⟶ A_{4} ⟶ P_{3} ⟶^{η·} A_{3} ⟶ 0

0 ⟶ A_{5} ⟶ P_{3} ⟶^{η_{3}·} A_{4} ⟶ 0

. . . . . .

0 ⟶ A_{2n−1} ⟶ P_{n} ⟶^{η_{n}·} A_{2n−2} ⟶ 0

0 ⟶ A_{0} ⟶ P_{1} ⟶^{η·} A_{2n−1} ⟶ 0

For computing Λη ∩ Λ we have to sum up every second kernel and get

Λη ∩ Λ ≃ ⨁_{i=1}^{n} A_{2i} = ⨁_{i=1}^{n} ηP_{i} = η ⨁_{i=1}^{n} P_{i}

is a progenerator. If we furthermore assume that the Green order is basic, Λη ∩ Λ is a free Λη–module.

This proves the lemma.

To prove the theorem we just have to paste the different pieces together.

We see that for each vertex v the order Λ_{v} := Λη_{v} is an isotypic order with defining ideal J_{v} := Λη_{v} ∩ Λ.

In fact,

• K ⊗_{R} (Λη_{v} ∩ Λ) = K ⊗_{R} Λη_{v},

• J_{v} is a free Λη_{v}–ideal by Lemma 4.12,

• and Λη_{v}/(Λη_{v} ∩ Λ) is a direct product of local R–algebras by Lemma 4.9.

• The ideal J_{v} for all vertices v is contained in the radical of Λ_{v}: Let S be a simple Λ_{v}–module. One has to show that J_{v} acts as 0 on S. Since Λ ⟶ ⟶ Λ_{v}, each simple Λ_{v}–module is also a simple Λ–module. But, we get n simple Λ–modules just by the local algebras which we get in the quotients Λ_{v}/J_{v}, where n is the number of edges of Γ. All these are annihilated by all the J_{v}. S is one of them.

We get the structure of Λ_{v} by Theorem 4.5. The Ω_{v} in the main diagonals of the matrices correspond to the projective indecomposable modules, as one sees from the proof of Theorem 4.5, and these are for Green orders just the η_{v}P for η_{v} being the idempotent corresponding to v and for P being a projective indecomposable module corresponding to an edge incident to v. The pullbacks linking the different Λ_{v} are as constructed in the example since this is the way the exact sequences corresponding to the tree are built.

Remark 4.13. In [19, 9] it is proven that two Green orders having the same data (Ω_{v}, f_{v}) but not necessarily the same underlying tree have equivalent bounded derived module categories. An explicit twosided tilting complex which provides this derived equivalence is given in [20].

References

[1] M. Auslander and M. Kleiner, Adjoint functors and an extension of the Green correspondence for group representations, Adv. in Math. 104 (1994), 297–314.

[2] E. C. Dade, Blocks with cyclic defect groups, Annals of Mathematics, 84 (1966) 20–48.

[3] W. Feit, The representation Theory of Finite Groups, North Holland, Amsterdam, New York, Oxford, 1982.

[4] J. A. Green, On the indecomposable representations of a finite group, Math. Zeitschrift 70 (1959), 430–445.

[5] J. A. Green, Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1972), 197–213.

[6] H. Hasse, Zahlentheorie, Akademie Verlag, Berlin 1963.

[7] I. N. Herstein, Nonnommutative rings, Carus Monographs Volume 15, 1968.

[8] G. J. Janusz, Indecomposable modules for finite groups, Annals of Mathematics 89 (1969) 209–241.

[9] S. König and A. Zimmermann, Tilting selfinjective algebras and Gorenstein orders, preprint (1995).

[10] H. Kupisch, Projektive Moduln endlicher Gruppen mit zyklischer p–Sylowgruppe, J. of Alg. 10 (1968) 1–7.

[11] H. Kupisch, Unzerlegbare Moduln endlicher Gruppen mit zyklischer p–Sylowgruppe, Math. Zeitschrift 108 (1969) 77–104.

[12] S. Mac Lane, Categories for the working mathematician, Springer Verlag, Berlin, Heidelberg, New York 1971.

[13] G. Michler, Green correspondence between blocks with cyclic defect groups I, Journal of Algebra 39 (1976) 26–51.

[14] G. Michler, Green correspondence between blocks with cyclic defect groups II, pages 210–235. in: Representations of Algebras, eds: V. Dlab and P. Gabriel; Springer Lecture Notes in Mathematics 488, Berlin, Heidelberg, New–York 1975.

[15] Bodo Pareigis, Kategorien und Funktoren, Teubner Verlag, Stuttgart 1969.

[16] K. W. Roggenkamp, Blocks of cyclic defect and Green orders, Communications in Algebra 20 (1992), 1715–1734.

[17] K. W. Roggenkamp, Blocks with cyclic defect (Green orders), in: V.Dlab and L.Scott (ed.), Finite dimensional algebras and related topics, Kluwer (1994), 265–276.

[18] K. W. Roggenkamp and V. Huber–Dyson, Lattices over Orders Volume 1, Springer Lecture Notes in Math. 115, Heidelberg 1970.

[19] A. Zimmermann, Derived equivalences of orders, to be published in Bautista, Martinez, de la Peña (eds.) Proceedings of the ICRA VII Mexico.

[20] A. Zimmermann, A twosided tilting complex for Green orders and Brauer tree algebras, preprint (1995).

Address of the author: Mathematisches Institut B, Universität Stuttgart, D–70550 Stuttgart, Germany Current address: Université de Paris 7; UFR de Mathḿatiques; 2, place Jussieu; F–75251 Paris Cedex 05; France