TWOSIDED TILTING COMPLEXES FOR GREEN ORDERS AND BRAUER TREE ALGEBRAS

ALEXANDER ZIMMERMANN

Abstract. We give an explicit twosided tilting complex between two Green orders having the same structural data as they were defined by K. W. Roggenkamp in [13, 14]. This yields an explicit twosided tilting complex between two Brauer tree algebras over the same field associated to trees with the same number of edges and the same exceptional multiplicity. The present work also gives a certain generalization of a result of Gabriel and Riedtmann [3].

1. Introduction

Let Λ be a ring. The (bounded) derived module category D^{b}(Λ), or for short the derived category, of Λ is the category with objects being complexes of finitely generated projective modules which are bounded to the right and which have non zero homology only in finitely many degrees. Morphisms are complex morphisms up to homotopy. For details we refer to [5].

By the fundamental theorem [9] of Jeremy Rickard for any two rings Λ and Γ the derived module categories of Λ and of Γ are equivalent if and only if there is a complex T of finitely generated projective Λ modules which is bounded to the left and to the right and which has certain properties to be made more precise in Section 2. A complex satisfying these properties is called a onesided tilting complex, or for short, a tilting complex.

A more precise description can be given in case of R-projective algebras. Let R be an integral domain. An R–algebra is called an R–order if Λ is finitely generated projective as R–module and frac(R) ⊗_{R} Λ is semisimple, frac(R) being the field of fractions of R. Jeremy Rickard proved in [10] that if R is a Dedekind domain and if T is a tilting complex over the R–order Λ with endomorphism ring being the R–order Γ, then there is a bounded complex X of Λ − Γ–bimodules such that

X ⊗_{Γ}^{L} − : D^{b}(Γ) ⟶ D^{b}(Λ)

is an equivalence of triangulated categories. This complex of bimodules is called a twosided tilting complex between Λ and Γ. A more detailed description will be given in Section 2.

Given a twosided tilting complex between two Gorenstein R–orders Λ and Γ, then one can construct a Λ–Γ–bimodule M, projective if restricted to either side, and M ⊗_{Γ} − induces a stable equivalence between Λ and Γ.

This paper. The aim of the present paper is to construct a twosided tilting complex explicitly in purely ring theoretical terms for derived equivalences between two Green orders as they were introduced by Roggenkamp [13]. Roughly speaking, Green orders are an analogue of Brauer tree algebras replacing a field by a complete discrete valuation domain. Given a Brauer tree algebra A over a perfect field k then there is a complete discrete valuation domain R with residue field k and a Green order Λ such that k ⊗_{R} Λ ≃ A. Moreover, the ring theoretical structure of a Green order is determined by some combinatorial data and some ring theoretical structural data. Blocks of group rings RG of finite groups G with cyclic defect group are Green orders.

The author acknowledges financial support from the Deutsche Forschungsgemeinschaft.

1991 AMS subject classification: primary 20C05, secondary 16G30, 16S34, 18E30, 20C11

We shall construct explicitly a twosided tilting complex for any two Green orders having the same structure data. In [16] an iterative process is given to prove that two such Green orders are derived equivalent no matter what the combinatorial data is. The method was purely combinatorial. The construction and the proof was simplified by Steffen König and the author in [6].

The main difficulty one has to overcome is that there is no canonical way to produce a bimodule which induces a stable equivalence between the two Green orders. As mentioned above, a twosided tilting complex would provide us with such a bimodule.

A main source for the construction is a careful examination of the construction of Raphaël Rouquier [15] in case of ℤ̂_{3}S_{3}, the group ring of the symmetric group of degree 3 over the 3-adic integers. Here it becomes clear how a twosided two term tilting complex works. We shall construct the complex out of its homology. To first construct the homology and then the complex having this homology is used in [16] in a very special case and in this more general situation it seems to be new.

For two Brauer tree algebras over a field corresponding to Brauer trees with the same number of edges and the same multiplicity of the exceptional vertex, Gabriel and Riedtmann gave a stable equivalence in [3]. They did not give a bimodule inducing the stable equivalence; our construction gives one.

We mention that our method is entirely combinatorial.

What happened before. One may become interested in derived equivalences with a conjecture of Michel Broué [1] saying that the derived categories of a block of a group ring of a finite group with abelian defect group D and its Brauer correspondent in the group ring of the normalizer in the group of the defect group are equivalent as triangulated categories.

In [2, Remark after 4.7] Michel Broué asked for an explicit construction of a twosided tilting complex between a block of a finite group and its Brauer correspondent and stated that in December 1992 no explicit construction of a twosided tilting complex, even in case of a cyclic defect group, was known. Progress has been fast since then.

In [15] Raphaël Rouquier constructed under some conditions a twosided tilting complex for symmetric orders out of a stable equivalence given by a bimodule. The twosided tilting complex is as explicit as is the bimodule inducing the stable equivalence. For blocks with cyclic defect, the well known stable equivalence comming from Green correspondence is in fact induced by tensoring with a bimodule. The hypotheses for the construction in [15] are fulfilled. Other examples are given by Jeremy Rickard in [11] for the situation of algebraic groups.

In [13, 14] Klaus W. Roggenkamp introduced a certain class of orders, Green orders, and clarified their structure completely. Moreover, he proved that blocks of cyclic defect groups of group rings of finite groups over any finite extension of the p–adic integers are Morita equivalent to Green orders of a special shape. Furthermore, reducing modulo the radical of the coefficient domain, every Brauer tree algebra over a perfect field of finite characteristic is an image of a suitably chosen Green order. In [13, 14] the only missing part in the complete description of the blocks of cyclic defect by Green orders is the structure of the ’exceptional vertex’. For the field of fractions of the coefficient domain being a splitting field for the block, Markus Linckelmann’s result [7] published in his thesis answers the remaining question of the structure of the centre of the block. For more general coefficient domains, Plesken [8] gives information for the structure of the exceptional vertex.

In Section 2 we state the Theorems of J.Rickard. The onesided tilting complexes are described in Section 3, making a summary of the relevant parts of [16] and of [13]. The complex is then constructed in Section 4.

Acknowledgement: I want to thank Jeremy Rickard for his kind permission to include his result stated in this paper as Lemma 1. Je remercie l’Équipe des Groupes Finis de l’Université Paris 7 pour leur aide et pour l’hospitalité qu’ils m’ont accordée tout au long de ce travail. Tout particulièrement, j’aimerais remercier Raphaël Rouquier pour de nombreuses discussions et remarques.

2. Rickard’s theorems

We state, for the reader’s convenience, Rickard’s two main theorems. The first of which deals with onesided tilting complexes, the second deals with twosided tilting complexes.

Objects of the category D^{b}(Λ) are right bounded complexes of finitely generated projective Λ-modules and with homology concentrated in only finitely many degrees. Morphisms are complex morphisms modulo homotopy.

The category K^{b}(P_{Λ}) is the full subcategory of D^{b}(Λ) generated by bounded complexes.

Theorem 1. (Rickard [9]) Let Λ and Γ be two rings. Then, the following conditions are equivalent:

(1) D^{b}(Λ) ≃ D^{b}(Γ) as triangulated categories

(2) There is a T ∈ K^{b}(P_{Λ}) such that the ring End_{K^{b}(P_{Λ})}(T) is isomorphic to Γ as rings and Hom_{K^{b}(P_{Λ})}(T, T [i]) = 0 for all i ≠ 0 and the rank one free module is contained in the triangulated category generated by direct summands of finite direct sums of T.

Remark 1. A complex T as in Theorem 1 is called a tilting complex from Λ to Γ. If such a T exists then Λ and Γ are called derived equivalent.

The second theorem deals with a different type of complexes making the equivalence of the derived categories more explicit.

Theorem 2. (Rickard [10]) Let R be a commutative ring and let A and B be two R-projective R–algebras. If A and B are derived equivalent by a functor F, then the functor F induces also a derived equivalence between D^{b}(A⊗_{R} A^{op}) and D^{b}(A⊗_{R} B^{op}) and the image of A as A⊗_{R} A^{op} module under this functor is a complex X in D^{b}(A ⊗_{R} B^{op}). Furthermore,

X ⊗_{B}^{L} − : D^{b}(B) ⟶ D^{b}(A)

is an equivalence of triangulated categories.

Remark 2. The complex X as in Theorem 2 is called a twosided tilting complex. Note that X is defined in the derived category and not in the homotopy category.

We finish with a yet unpublished lemma of Jeremy Rickard.

Lemma 1. (J. Rickard [12]) Let R be a complete discrete valuation ring and let Λ and Γ be two R–orders in semisimple artinian frac(R)–algebras. Let X be a complex of Λ–Γ bimodules bounded from the left and bounded from the right. Let X be isomorphic in D^{b}(Λ) to a tilting complex with endomorphism ring Γ and let X be isomorphic in D^{b}(Γ^{op}) to a tilting complex with endomorphism ring Λ, then X is a twosided tilting complex.

Proof. (J. Rickard)

Let more generally A and B be additive categories and let L_{i} : A ⟶ B with i = 1, 2 be functors with right adjoints R_{1} and R_{2}. Now, a natural transformation

L_{1} ⟶ L_{2}

gives rise to a natural transformation

R_{2} ⟶ R_{1}L_{1}R_{2} ⟶ R_{1}L_{2}R_{2} ⟶ R_{1}

which is induced by the unit 1 ⟶ R_{1}L_{1}, the above natural transformation and the counit L_{2}R_{2} ⟶ 1. This correspondence is called the conjugate map. This is characterized by the commutativity of the diagram

Hom_{B}(L_{2}−, −) ≃ Hom_{A}(−, R_{2}−)
↓  ↓
Hom_{B}(L_{1}−, −) ≃ Hom_{A}(−, R_{1}−)

Given a bounded complex of functors

L^{∗} : … ⟶ L_{0} ⟶ L_{1} ⟶ L_{2} ⟶ …

by the commutativity of the diagram defining the adjoint map above, this gives rise to a complex of functors

R^{∗} : … ⟶ R_{2} ⟶ R_{1} ⟶ R_{0} ⟶ …

by taking the conjugate maps. Now, L^{∗} and R^{∗} induce functors L^{∗} : C^{b}(A) ⟶ C^{b}(B) and R^{∗} : C^{b}(B) ⟶ C^{b}(A) by taking the total complex of the resulting double complex. Of course, homotopies map to homotopies this way such that our functors L^{∗} and R^{∗} carries over to functor

L^{∗} : K^{b}(A) ⟶ K^{b}(B)

and

R^{∗} : K^{b}(B) ⟶ K^{b}(A)

We observe that (L^{∗}, R^{∗}) is an adjoint pair. In fact, for any X and Y in A we get an isomorphism of the triple complexes

Hom_{B}(L^{∗}(X), Y) ≃ Hom_{A}(X, R^{∗}(Y))

natural in X and in Y. If one takes now total complexes and observes that the zero homology of this complex then gives just the homomorphisms in the homotopy categories, we get by the naturality of the construction in both variables the adjointness property.

We now specialize to L_{i} = − ⊗_{Γ} X_{i} for our bimodules X_{i} which are finitely generated projective on both sides. Then, R_{i} = Hom_{Γ}(X_{i}, −). By the fact that X_{i} is finitely generated projective as Γ–module,

Hom_{Γ}(X_{i}, −) ≃ − ⊗_{Γ} Hom_{Γ}(X_{i}, Γ).

We get a unit

1_{?(Λ)} ⟶ − ⊗_{Λ} X ⊗_{Γ} Hom_{Γ}(X, Γ)

and a counit

− ⊗_{Γ} Hom_{Γ}(X, Γ) ⊗_{Λ} X ⟶ 1_{?(Γ)}

for ? being C, the complex category or equally well K, the homotopy category. Therefore we get natural maps of complexes of right modules

Λ ⟶ X ⊗_{Γ} Hom_{Γ}(X, Γ)

and

Hom_{Γ}(X, Γ) ⊗_{Λ} X ⟶ Γ

which are actually maps of complexes of bimodules by letting the rings act via endomorphisms of the free rank one right modules, using the functoriality of the construction.

Take D_{Γ}(X) = Hom_{Γ}(X, Γ) as complex of Γ − Λ–bimodules. Now, − ⊗_{Λ} X is left adjoint to − ⊗_{Λ} D_{Γ}X as functors C^{b}(Λ) ⟶ C^{b}(Γ). We get natural maps of complexes of bimodules

Λ ⟶^{α} X ⊗_{Γ} D_{Γ}X and D_{Γ}(X) ⊗_{Λ} X ⟶^{β} Γ

We now show, that α is an isomorphism in the derived category of bounded complexes of projective modules.

Set k = R/rad R.

We may assume, using Lemma 2, that X consists of bimodules whose restrictions to the left and to the right are projective.

Obviously, \overline{M} := k ⊗_{R} (X ⊗_{Γ} D_{Γ}(X)) is isomorphic to a module in D^{b}(Λ ⊗ Λ^{op}).

We denote the kernel of the mapping induced by α on the degree zero homology by \overline{K}. We observe that

k ⊗_{R} X ⊗_{Γ} D_{Γ}(X) ⟶^{α⊗_{Λ}1_{M}}  k ⊗_{R} X ⊗_{Γ} D_{Γ}(X) ⊗_{Λ} X ⊗_{Γ} D_{Γ}(X)
∥  ↓ 1_{X} ⊗_{Γ} β ⊗_{Γ} 1_{D_{Γ}(X)}
k ⊗_{R} X ⊗_{Γ} D_{Γ}(X)  =  k ⊗_{R} X ⊗_{Γ} D_{Γ}(X)

is a commutative diagram.

Since now α ⊗ 1_{X⊗D_{Γ}(X)} is a split monomorphism, \overline{K} ⊗_{Λ} X ⊗_{Γ} D_{Γ}(X) = 0.

Since the complex D_{Γ}(X) is a complex of projective Γ–modules,_{Γ}|D_{Γ}(X) is a tilting complex. In fact, Hom_{Γ}(−, Γ) is faithfully flat on projective Γ–modules and it is now easy to derive the defining conditions of a tilting complex from the property of X being a tilting complex as complex of Γ–modules.

Since _{Γ}|D_{Γ}(X) is a tilting complex, \overline{K} ⊗_{Λ} X = 0. Since X|_{Λ} is a tilting complex, \overline{K} = 0. Hence, we know that α induces an injective mapping on the degree 0–homology. As R–

modules we have

H_{0}(X ⊗_{Γ} D_{Γ}(X)) ≃ End_{K^{b}(P_{Γ})}(X|_{Γ}).

Since every monomorphism between finite dimensional vector spaces of equal dimension is an isomorphism, α induces an isomorphism

R/rad R ⊗_{R} Λ ≃ R/rad R ⊗_{R} H_{0}(X ⊗ D_{Γ}(X))

as Λ − Λ–bimodules. Since we assumed Λ and Γ to be finitely generated over the noetherian complete discrete valuation ring R, we know that α induces an isomorphism between Λ and the degree zero homology of X ⊗ D_{Γ}(X)..

We have to finish with proving that also β is an isomorphism.

The composition

X ⟶^{α⊗id_{X}} X ⊗_{Γ} D_{Γ}(X) ⊗_{Λ} X ⟶^{id_{X}⊗β} X

equals the identity and α is an isomorphism, so id_{X} ⊗β is an isomorphism. Forming the triangle in the homotopy category of complexes of finitely generated projective modules

D_{Γ}(X) ⊗_{Γ} X ⟶^{β} Γ ⟶ C ⇝ …,

we get X ⊗_{Γ} C is acyclic, and since X as complex of right modules is isomorphic to a tilting complex, C is acyclic.

This completes the proof of Lemma 1.

In order to prove that a complex of Λ–Γ–bimodules is a twosided tilting complex, for Λ and Γ being as in the lemma, we only have to compute a complex X of bimodules which restricts on either sides to a tilting complex with correct endomorphism rings.

3. Recapitulation of the onesided situation

3.1. Green orders. K.W. Roggenkamp defined Green orders to explain the structure of blocks of group rings over a complete discrete valuation ring of characteristic 0 with cyclic defect group [13].

We use a suggestion of L. Puig to define Green orders in a different, equivalent, way.

Let R be an integral domain with field of fractions K and let Λ be an indecomposable R–order in the semisimple K–algebra A = K ⊗_{R} Λ.

Let I be a complete set of primitive idempotents of Λ (i.e. End_{Λ}(⊕_{i∈I}Λ · i) is Morita equivalent to Λ and the modules Λ·i for all i ∈ I are projective indecomposable left–Λ–modules for all i ∈ I).

Definition 1. The indecomposable R–order Λ in the semisimple algebra A is called a Green order if

• there is a set E of central idempotents of A with ∑_{e∈E} e = 1 such that

T := {(i, e) ∈ I × E|i · e ≠ 0}

is a tree (i.e. defines a connected relation on I × E, |E| = |I| + 1 and for all i ∈ I we get |T ∩ ({i} × E)| = 2).

• noting by π : T ⟶ I and θ : T ⟶ E the natural projections, there are a transitive permutation ω of T and for all t ∈ T a Λ–module homomorphism

g_{t} : Λ · π(t) ⟶ Λ · π(ω(t)) with g_{t}(Λ · π(t)) = ker(g_{ω(t)}) ≃^{σ_{t}} Λ · π(t) · θ(t) .

where for all λ ∈ Λπ(t) we have σ_{t}g_{t}(λ) = λ · θ(t).

Remark 3. We should provide a link to the definition in [13].

(1) It is immediate to see that a finite, connected, unoriented graph T with one edge less than it has vertices does not have cycles (use an induction). Hence, the three conditions on the cardinalities of I and E in relation with T defines in fact a tree.

(2) The transitive permutation ω of T is the ’walk around the Brauer tree’ which was invented in Green’s paper [4]. Green gives reference to this permutation in the introduction, however, without mentioning the permutation explicitly.

(3) The walk around the Brauer tree manifests in a projective resolution

0 ← Λ · π(t) · θ(t) ← Λπ(t) ← Λπ(ω(t)) ← Λπ(ω^{2}(t)) ← ⋯ ← Λπ(t) ← …

(4) A tree may be realized by a graph in the plane, meaning a complex of 1-simplices in IR^{2}. One associates to the plane an orientation, and the permutation ω is the ’walk around the tree’ as described in [4].

We abbreviate Λ · t = Λπ(t)θ(t) and t · Λ = π(t)θ(t)Λ. The main theorem of Roggenkamp in [13] may be reformulated as follows.

Theorem 3. (Roggenkamp [13]) We assume that Λ is basic.

There is an R–torsion R–algebra \overline{Ω} and a family (f_{t} : tΛt ⟶ \overline{Ω}) of R–algebra homomorphisms with kernel being a principal ideal a_{t}tΛt.

T is totally ordered by ω and a first element. There is a (equivalent) set of primitive idempotents I such that one can choose the first element of T such that the Pierce decomposition

Λ = ( \begin{matrix}i_{1}Λi_{1} & i_{1}Λi_{2} & … & i_{1}Λi_{k} \\ i_{2}Λi_{1} & i_{2}Λi_{2} & … & i_{2}Λi_{k} \\ ⋮ & ⋮ & & ⋮ \\ i_{k}Λi_{1} & i_{k}Λi_{2} & … & i_{k}Λi_{k} \end{matrix} )

has the following properties (1) For all t < t′ and θ(t) = θ(t′) with t, t′ ∈ T we have tΛt = π(t)Λπ(t′) = t′Λt′.

(2) f_{t} depends only on θ(t) and we denote f_{θ(t)} := f_{t} and a_{θ(t)} := a_{t}.

(3) For all t > t′ and θ(t) = θ(t′) with t, t′ ∈ T we have a_{θ(t)} · π(t′)Λπ(t) = π(t)Λπ(t′).

(4) If t, t′ ∈ T with π(t) = π(t′) then π(t)Λπ(t) is a pullback

π(t)Λπ(t) ⟶ tΛt
↓  ↓ f_{t}
t′Λt′  ⟶^{f_{t′}}  \overline{Ω}

Moreover, if for an R–order Λ′ in K ⊗_{R} Λ with I being also a complete set of primitive idempotents of Λ′ the tree T and the transitive permutation ω of T from above have the property that there is an element t_{1} ∈ T defining together with ω a linear ordering on T such that the Pierce decomposition has the properties (1), (2), (3) and (4) then Λ′ is a Greenorder.

Remark 4. (1) One should observe that since the e in E are central and pairwise orthogonal, most of the entries in the above matrix are zero. The rest of the entries falls naturally into matrix rings. Furthermore, by the first and the third points of the theorem, the matrix rings in question are upper triangular over Ω_{θ(t)} := tΛt which does only depend on θ(t) und the lower diagonal entries is in a principal ideal a_{t}tΛt, again depending only on θ(t). These matrix rings are linked among each other by the pullbacks with the main diagonal entries mentioned in point 4 of the theorem. This is the interpretation given by Roggenkamp in [13]. We may hence label the vertices of the tree by the pairs (Ω_{θ(t)}, a_{θ(t)}). Mostly we only write down the labels Ω_{θ(t)}, understanding that the ideal a_{θ(t)}Ω_{θ(t)} is fixed once for all, if this does not cause confusion.

(2) With I = {i_{1}, i_{2}, …, i_{k}} being the above set of primitive idempotents, one obtains that g_{t} is just multiplication by t and σ_{t} is the identity.

(3) By the last of the 4 properties one can see that the the projective indecomposable modules Λi for i ∈ I are pullbacks, where t · i ≠ 0 ≠ t′ · i for two different t, t′ in T,

Λi  →^{g_{t}}  Λt
↓ g_{t′}  ↓ f̃_{t}
Λt′  →^{f̃_{t′}} (\overline{Ω}_{j})_{j=1}^{k}

where

f̃_{t}|_{t′′Λt} = \{ \begin{matrix}f_{t} & if & t′′ = t \\ 0 & if & t′′ ≠ t \end{matrix} and f̃_{t′}|_{t′′Λt′} = \{ \begin{matrix}f_{t′} & if & t′′ = t′ \\ 0 & if & t′′ ≠ t′ \end{matrix}

Moreover,

\overline{Ω}_{j} = \{ \begin{matrix}\overline{Ω} & if & i = i_{j} \\ 0 & if & i ≠ i_{j} \end{matrix}

As a consequence, we get that

ker(f̃_{t}) ≃ ker(σ_{t′}g_{t′}) ≃ ker(g_{t′})

3.2. Tilting Green orders. We recall the following facts from [16] and from the refinement [6]. We are given a Green order Λ with Brauer tree

[figure: ⋮; ⋮; …; 𝔘; …; ⋮; 𝔏; …; Ω_{1}; Ω_{3}; Ω_{2}]

and data (Ω_{i}, f_{i}), where we denote

𝔏 := { vertices v| the shortest path in the tree from v to Ω_{2} passes Ω_{1}}

𝔘 := { vertices v| the shortest path in the tree from v to Ω_{1} passes Ω_{2}} \ {Ω_{3}}

The orientation in the plane is meant to be counterclockwise.

The indecomposable projective corresponding to the edge linking Ω_{1} and Ω_{2} is denoted by P = Λi_{1,2}, for an idempotent i_{1,2} of Λ and the indecomposable projective corresponding to the edge linking Ω_{2} and Ω_{3} is denoted by Q = Λi_{2,3} for an idempotent i_{2,3} of Λ. We identify edges with indecomposable projectives and denote the indecomposable projective corresponding to the edge e by P_{e}. Set

L := ⨁ ; \binom{edges e involving}{only vertices in 𝔏} P_{e} and U := ⨁ ; \binom{edges f involving}{only vertices in 𝔘} P_{f}

The complex

T := (0 ⟶ L ⊕ U ⊕ P ⊕ P ⟶^{(0,0,0,g_{i_{1,2}e_{2}} )} Q ⟶ 0)

is a tilting complex. Its endomorphism ring is a Green order Γ associated to the following tree with obvious notations.

[figure: ⋮; ⋮; …; 𝔏; …; ⋮; 𝔘; …; Ω_{2}; Ω_{3}; Ω_{1}]

The argument applied to right Γ–modules implies that, denoting by P′^{∗} the indecomposable projective corresponding to the edge linking Ω_{1} and Ω_{2} and by Q′^{∗} the projective indecomposable corresponding to the edge linking Ω_{1} and Ω_{3}, defining L′^{∗} and U′^{∗} analogously as above, the complex

T′ := (0 ⟶ L′^{∗} ⊕ U′^{∗} ⊕ P′^{∗} ⊕ P′^{∗} ⟶^{(0,0,0,g_{j_{1,2}e_{1}} )} Q′^{∗} ⟶ 0)

is a tilting complex with endomorphism ring Λ.

We denote by i_{U} the idempotent corresponding to U, by i_{L} the idempotent corresponding to L, by i_{1,2} the idempotent corresponding to P and by i_{2,3} the idempotent corresponding to Q.

Analogously, we denote by j_{U} the idempotent corresponding to U′^{∗}, by j_{L} the idempotent corresponding to L′^{∗}, by j_{1,2} the idempotent corresponding to P′^{∗} and by j_{1,3} the idempotent corresponding to Q′^{∗}.

By abuse of language we denote both for Λ and for Γ the central idempotents corresponding to Ω_{1} by e_{1}, those corresponding to Ω_{2} by e_{2}, those corresponding to U by e_{U} and those corresponding to L by e_{L}.

We choose the ’beginning element’ of T_{Λ} and of T_{Γ} such that

T_{Λ} = {i_{U}e_{2} < i_{1,2}e_{1} < i_{L}e_{L} < i_{L}e_{1} < i_{1,2}e_{2} < i_{2,3}e_{3} < i_{2,3}e_{2} < i_{U}e_{U}}

and

T_{Γ} = {j_{1,3}e_{1} < j_{1,3}e_{3} < j_{U}e_{2} < j_{U}e_{U} < j_{1,2}e_{2} < j_{1,2}e_{1} < j_{L}e_{L} < j_{L}e_{1}}

Then, the result in [16] affirms that

i_{U}Λi_{U} = j_{U}Λj_{U} with f_{i_{U} ,e_{2}}^{Λ} = f_{i_{U} ,e_{2}}^{Γ} and f_{i_{U} ,e_{U}}^{Λ} = f_{i_{U} ,e_{U}}^{Γ} ,
i_{1,2}Λi_{U} = j_{1,2}Γj_{U},
i_{L}Λi_{L} = j_{L}Γj_{L} with f_{i_{L},e_{1}}^{Λ} = f_{i_{L},e_{1}}^{Γ} and f_{i_{L},e_{L}}^{Λ} = f_{i_{L},e_{L}}^{Γ},
i_{1,2}Λi_{L} = j_{1,2}Γj_{L},
i_{1,2}Λi_{1,2} = j_{1,2}Γj_{1,2} with f_{i_{1,2}e_{1}}^{Λ} = f_{j_{1,2}e_{1}}^{Γ} and f_{i_{1,2}e_{2}}^{Λ} = f_{j_{1,2}e_{2}}^{Γ},
i_{1,2}Λi_{L} = j_{1,2}Γj_{L},
i_{U}Λi_{1,2} = j_{U}Γj_{1,2},
i_{L}Λi_{1,2} = j_{L}Γj_{1,2},
i_{L} ker(g_{(i_{1,2},e_{2})}) = j_{L}Γj_{1,3},
i_{1,2} ker(g_{(i_{1,2},e_{2})}) = j_{1,2}Γj_{1,3},
i_{2,3}Λi_{U} = ker(g_{(j_{1,2},e_{1})}) j_{U},
i_{2,3}Λi_{1,2} = ker(g_{(j_{1,2},e_{1})}) j_{1,2} .

In section 4 we shall construct a complex X of Λ–Γ–bimodules which is isomorphic in the derived category of Λ–left modules to T and which is isomorphic in the derived category of Γ–right modules to T′. The combinatorics. In [6] it is proven that iterating the procedure above, or otherwise said, taking as ’new’ Λ the ’old’ order Γ and choosing a numeration for Ω_{1}, Ω_{2} and Ω_{3}, it is possible to end up after a certain number of steps as Γ a Green order associated to a tree being a star with the same data we started with, and any association of the vertices of the star to the data (Ω_{i}, f_{i}).

In other words, it is possible to ’deform’ the tree to a star and to interchange any two vertices. So, tensoring the complexes we construct, this gives a twosided tilting complex between any

two Green orders with the same data, no matter the tree looks like as long as the data and the number of vertices are fixed.

3.3. An observation. In our onesided construction we have to deal with two terms tilting complexes. Let

X = (0 ⟶ X_{1} ⟶ X_{0} ⟶ 0)

be a twosided two term tilting complex and we may and will assume that X has homology concentrated in the degrees 0 and 1.

Using the arguments in the appendix we may furthermore assume that X_{0} is a projective bimodule.

We denote by Ω_{1}(H_{0}(X)) the first syzygy of H_{0}(X) as bimodule. We get short exact sequences

0 ⟶ Ω_{1}(H_{0}(X)) ⟶ X_{0} ⟶ H_{0}(X) ⟶ 0

and

0 ⟶ H_{1}(X) ⟶ X_{1} ⟶ Ω_{1}(H_{0}(X)) ⟶ 0.

Moreover, K = frac(R) being the field of fractions of R,

K ⊗_{R} X_{1} ≃ K ⊗_{R} H_{1}(X) ⊕ K ⊗_{R} Ω′_{1}(H_{0}(X)).

We are given two onesided tilting complexes T and S over Λ and Γ^{op} resp. and we want to find a twosided tilting complex X in D^{b}(Λ ⊗_{R} Γ^{op}) with

X ≃ T in D^{b}(Λ) and X ≃ S in D^{b}(Γ^{op})

It is therefore necessary to find a bimodule H_{1}(X) such that

Λ|H_{1}(X) ≃ H_{1}(T) and H_{1}(X)|_{Γ} ≃ H_{1}(S)

Furthermore, the restriction of the composition

X_{1} ⟶ Ω′_{1}(H_{0}(X)) ⟶ X_{0}

to the left and to the right have to coincide with the onesided complex, in the derived category. This small observation will allow us to construct X.

4. Constructing the complex

We shall give a twosided tilting complex in the situation above.

Again we have to compute the homology in degree 1 and find a bimodule which restricts to these homologies on either side.

We remind the reader that we denote

g_{Λ} := g_{i_{1,2}e_{2}} : Λi_{1,2} ⟶ Λi_{2,3}

and

g_{Γ} := g_{j_{1,2}e_{1}} : j_{1,2}Γ ⟶ j_{1,3}Γ

There is only one canonical way to produce a bimodule H_{1}(X): The bimodule is isomorphic to

i_{U}Λi_{U} i_{U}Λi_{1,2} i_{U}ker(g_{Λ}) i_{U}Λi_{L}
i_{1,2}Λi_{U} i_{1,2}Λi_{1,2} i_{1,2}ker(g_{Λ}) i_{1,2}Λi_{L}
i_{L}Λi_{U} i_{L}Λi_{1,2} i_{L}ker(g_{Λ}) i_{L}Λi_{L}
i_{2,3}Λi_{U} i_{2,3}Λi_{1,2} i_{2,3}ker(g_{Λ}) i_{2,3}Λi_{L}

which is equal to

j_{U}Γj_{U}  j_{U}Γj_{1,2}  j_{U}Γj_{1,3}  j_{U}Γj_{L}
j_{1,2}Γj_{U}  j_{1,2}Γj_{1,2}  j_{1,2}Γj_{1,3}  j_{1,2}Γj_{L}
j_{L}Γj_{U}  j_{L}Γj_{1,2}  j_{L}Γj_{1,3}  j_{L}Γj_{L}
ker(g_{Γ})j_{U} ker(g_{Γ})j_{1,2} ker(g_{Γ})j_{1,3} ker(g_{Γ})j_{L}

where we used the relations mentioned in Subsection 3.2 and that we know, by the fact that the idempotents e_{1}, e_{2}, e_{U} and e_{L} are central and pairwise orthogonal (both for Λ and for Γ), that

i_{U}ker(g_{Λ}) = j_{U}Γj_{1,3} = 0,
i_{U}Λi_{L} = j_{U}Γj_{L} = 0,
i_{L}Λi_{U} = j_{L}Γj_{U} = 0,
i_{2,3}ker(g_{Λ}) = ker(g_{Γ})j_{1,3} = 0.

On this bimodule Λ acts by multiplication by

i_{U}Λi_{U} i_{U}Λi_{1,2} i_{U}Λi_{L} i_{U}Λi_{2,3}
i_{1,2}Λi_{U} i_{1,2}Λi_{1,2} i_{1,2}Λi_{L} i_{1,2}Λi_{2,3}
i_{L}Λi_{U} i_{L}Λi_{1,2} i_{L}Λi_{L} i_{L}Λi_{2,3}
i_{2,3}Λi_{U} i_{2,3}Λi_{1,2} i_{2,3}Λi_{L} i_{2,3}Λi_{2,3}

from the left and Γ acts by matrix multiplication by

j_{U}Γj_{U} j_{U}Γj_{1,2} j_{U}Γj_{1,3} j_{U}Γj_{L}
j_{1,2}Γj_{U} j_{1,2}Γj_{1,2} j_{1,2}Γj_{1,3} i_{1,2}Γi_{L}
j_{1,3}Γj_{U} j_{1,3}Γj_{1,2} j_{1,3}Γj_{1,3} j_{1,3}Γj_{L}
j_{L}Γj_{U} j_{L}Γj_{1,2} j_{L}Γj_{1,3} j_{L}Γj_{L}

The homology in degree 0 is easy to compute, it is Λe_{3} = e_{3}Γ as Λ ⊗ Γ^{op}–module. The projective cover of this module as bimodule is

X_{0} := Λi_{2,3} ⊗_{R} j_{1,3}Γ.

For constructing the bimodule extension of the homology with the first syzygy of

Ω_{3} = e_{3}i_{2,3}Λi_{2,3} = e_{3}j_{1,3}Λj_{1,3}

we have to explicit this module. (Again, not to overload the notation, we write ”⊗” for ”⊗_{R}”.)

Λi_{2,3} ⊗_{R} j_{1,3}Γ =

= ( \begin{matrix}i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{U} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{U} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{L}Λi_{2,3} ⊗ j_{1,3}Γj_{U} & i_{L}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{L}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{L}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{U} & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \end{matrix} )

= ( \begin{matrix}i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \end{matrix} )

since j_{1,3}Γj_{U} = 0 and i_{L}Λi_{2,3} = 0.

This module maps by a mapping π̂ onto Ω_{3} by mapping i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} onto its e_{3}– component. We denote this last mapping by π. We hence have an exact sequence

0 ⟶ ( \begin{matrix}i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & ker(π) & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \end{matrix} ) ⟶^{ι}

⟶^{ι} ( \begin{matrix}i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \end{matrix} ) ⟶^{π̂} Ω_{3} ⟶ 0

Now, we have to find the common extension of the kernel of this sequence and the homology. We modify the homology-bimodule to do so. The special structure of the Green order Λ

(namely the first three properties in Roggenkamp’s theorem) and this choice of the first element as done above implies that i_{U}Λi_{2,3} = i_{U}Λi_{1,2} and i_{2,3}Λi_{1,2} = e_{2}i_{1,2}Λi_{1,2}.

We shall use

i_{U}Λi_{U}  i_{U}Λi_{2,3}  e_{1}i_{U}Λi_{1,2} i_{U}Λi_{L}
i_{1,2}Λi_{U} e_{1}i_{1,2}Λi_{1,2} ⊕ i_{1,2}Λi_{2,3} e_{1}i_{1,2}Λi_{1,2} i_{1,2}Λi_{L}
i_{L}Λi_{U}  i_{L}Λi_{1,2}  e_{1}i_{L}Λi_{1,2} i_{L}Λi_{L}
i_{2,3}Λi_{U}  e_{2}i_{2,3}Λi_{2,3}  e_{1}i_{2,3}Λi_{1,2} i_{2,3}Λi_{L}

as Λ–left–module which is equal to

j_{U}Γj_{U}  j_{U}Γj_{1,2}  j_{U}Γj_{1,3}  j_{U}Γj_{L}
j_{1,2}Γj_{U} e_{1}j_{1,2}Γj_{1,2} ⊕ e_{2}j_{1,2}Γj_{1,2} j_{1,2}Γj_{1,3} j_{1,2}Γj_{L}
j_{L}Γj_{U}  j_{L}Γj_{1,2}  j_{L}Γj_{1,3}  j_{L}Γj_{L}
e_{2}j_{1,2}Γj_{U}  e_{2}j_{1,2}Γj_{1,2}  e_{2}j_{1,2}Γj_{1,3} e_{2}j_{1,2}Γj_{L}

as Γ–right–module, using the ordering and the special choice of a first element in T_{Γ}. We need one more observation concerning the construction of the differential.

The theorem of Roggenkamp gives us a canonical Λ-module structure on ( 0 \overline{Ω} 0 \overline{Ω} ) by the

following fact. Since in the chosen ordering of T_{Λ} we have i_{2,3}e_{2} > i_{1,2}e_{2} > i_{U}e_{2}, there are inclusions

g_{i_{U} e_{2}}(Λi_{U}) < e_{2}Λi_{1,2} and g_{i_{1,2}e_{2}}(Λi_{1,2}) < e_{2}Λi_{2,3} .

Hence we get an exact sequence

0 ⟶ e_{2}Λi_{U} ⟶^{g_{i_{U} e_{2}} g_{i_{1,2}e_{2}}} e_{2}Λi_{2,3} \underrightarrow{( \begin{matrix}0 \\ f_{e_{2}} \\ 0 \\ f_{e_{2}} \end{matrix} )} ( 0 \overline{Ω} 0 \overline{Ω} ) ⟶ 0 .

Likewise one finds a Λ–module structure by transport of structure of Λi_{1,2}e_{1} via f̃_{i_{1,2}e_{1}} on ( 0 \overline{Ω} 0 0 ).

This gives

( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & \overline{Ω} & \overline{Ω} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & \overline{Ω} & 0 & 0 \end{matrix} )

the structure of a Λ-left–module. The structure from the right as Γ-right-module is given by the fact that

f_{e_{2}}^{Λ} = f_{e_{2}}^{Γ} and f_{e_{1}}^{Λ} = f_{e_{1}}^{Γ}

The second line in the matrix gets its right-Γ-module structure by transport of structure of e_{1}j_{1,2}Γ via (0, −f_{e_{1}}^{Γ}, −f_{e_{1}}^{Γ}, 0) and the fourth line gets its Γ-right-module structure by transport of structure of e_{2}j_{1,2}Γ via f̃_{j_{1,2}e_{2}}.

As a whole, by the analogous arguments as in the case of the Λ-structure, we get a right Γ-structure using the same mapping

( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & (−f_{e_{1}}, f_{e_{2}}) & −f_{e_{1}} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & f_{e_{2}} & 0 & 0 \end{matrix} )

for the left and for the right.

With these preparations we can define our X_{1} and the differential as pullback diagram

( \begin{matrix}j_{U} Γj_{U} & j_{U} Γj_{1,2} & j_{U} Γj_{1,3} & j_{U} Γj_{L} \\ j_{1,2}Γj_{U} & e_{1}j_{1,2}Γj_{1,2} ⊕ e_{2}j_{1,2}Γj_{1,2} & j_{1,2}Γj_{1,3} & j_{1,2}Γj_{L} \\ j_{L}Γj_{U} & j_{L}Γj_{1,2} & j_{L}Γj_{1,3} & j_{L}Γj_{L} \\ e_{2}j_{1,2}Γj_{U} & e_{2}j_{1,2}Γj_{1,2} & e_{2}j_{1,2}Γj_{1,3} & e_{2}j_{1,2}Γj_{L} \end{matrix} )  ⟵ X_{1}

∥  ∥

( \begin{matrix}i_{U} Λi_{U} & i_{U} Λi_{2,3} & e_{1}i_{U} Λi_{1,2} & i_{U} Λi_{L} \\ i_{1,2}Λi_{U} & e_{1}i_{1,2}Λi_{1,2} ⊕ i_{1,2}Λi_{2,3} & e_{1}i_{1,2}Λi_{1,2} & i_{1,2}Λi_{L} \\ i_{L}Λi_{U} & i_{L}Λi_{1,2} & e_{1}i_{L}Λi_{1,2} & i_{L}Λi_{L} \\ i_{2,3}Λi_{U} & e_{2}i_{2,3}Λi_{2,3} & e_{1}i_{2,3}Λi_{1,2} & i_{2,3}Λi_{L} \end{matrix} )  ⟵ X_{1}

↓ χ  ↓ φ

( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & \overline{Ω} & \overline{Ω} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & \overline{Ω} & 0 & 0 \end{matrix} ) ⟵^{ψ}  ( \begin{matrix}0 & i_{U} Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{U} Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{U} Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ 0 & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} & i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \\ 0 & 0 & 0 & 0 \\ 0 & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{1,2} & ker(π) & i_{2,3}Λi_{2,3} ⊗ j_{1,3}Γj_{L} \end{matrix} )

where

ψ = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & f_{e_{2}}^{Λ} · f_{e_{1}}^{Γ} & f_{e_{2}}^{Λ} · f_{e_{1}}^{Γ} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & f_{e_{2}}^{Λ} · f_{e_{1}}^{Γ} & 0 & 0 \end{matrix} )

and

χ = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & (−f_{e_{1}}, f_{e_{2}}) & −f_{e_{1}} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & f_{e_{2}} & 0 & 0 \end{matrix} )

We define the complex

X := (0 ⟶ X_{1} ⟶^{ι∘ϕ} Λi_{2,3} ⊗_{R} j_{1,3}Γ ⟶ 0)

Proposition 1. The complex X in D^{b}(Λ) is isomorphic to a tilting complex with endomorphism ring Γ and X in D^{b}(Γ^{op}) is isomorphic to a tilting complex with endomorphism ring Λ.

Theorem 4. Let Λ and Γ be Green orders associated to the Brauer trees

[figure: ⋮; …; …; Ω_{2}; Ω_{3}; Ω_{1}; ⋮; …; …; 𝔘; 𝔏; for Λ; and; ⋮; …; …; Ω_{2}; Ω_{3}; Ω_{1}; ⋮; …; …; 𝔘; 𝔏; for Γ]

with the same data (Ω_{i}, f_{i})_{i∈I} for the same suitable index set I of vertices.

Let e_{3}^{Λ} be the central idempotent of K ⊗_{R} Λ corresponding to the leaf Ω_{3} of Λ and let e_{3}^{Γ} be the central idempotent of K ⊗_{R} Γ corresponding to the leaf Ω_{3} of Γ.

• Then, the complex X as defined above is a twosided tilting complex.

• The term X_{0} of X in degree 0 is the projective cover of Ω_{3} as Λ–Γ–bimodule, where Λ acts as multiplication by Λ · e_{3}^{Λ} on the left and Γ acts as multiplication by Γ · e_{3}^{Γ} on the right.

• In case Λ and Γ are Gorenstein orders, the homogeneous component X_{1} of X in degree 1 is a Λ–Γ–bimodule which induces a stable equivalence of Morita type.

Remark 5. We should remind the reader to Broué’s definition of a stable equivalence of Morita type. Let R be a commutative ring and let A and B be two R–algebras. If there are a finitely generated A ⊗_{R} B^{op}-module M and a finitely generated B ⊗_{R} A^{op}-module N such that

M ⊗_{B} N ≃ A ⊕ P_{A} and N ⊗_{A} M ≃ B ⊕ P_{B}

as A ⊗_{R} A^{op}-modules (or as B ⊗_{R} B^{op}-modules resp.) for a projective A ⊗_{R} A^{op}-module P_{A} and a projective B ⊗_{R} B^{op}-module P_{B}, then M is said to induce a stable equivalence of Morita type.

Theorem 4 follows from Proposition 1 and Lemma 1.

The proof of Proposition 1 will cover the two following subsections.

4.1. Restricting to the left. Certainly, the first column in X_{1} is isomorphic to Λi_{U} and the last column in X_{1} is isomorphic to Λi_{L}. On these both modules the differential is zero.

The third column gives rise to a a complex defined by the following pullback diagram.

e_{1}Λi_{1,2}  ⟵^{α}  P_{0}
↓ −f̃_{i_{1,2},e_{1}}  ↓ β

( 0 \overline{Ω} 0 0 )  ⟵ ( \begin{matrix}i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ i_{L}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ ker(π) \end{matrix} )

↓
Λi_{2,3} ⊗_{R} j_{1,3}Γj_{1,3}

The module in degree 1 is P_{0}, the module in degree 0 is Λi_{2,3} ⊗_{R} j_{1,3}Γ and the differential is given by the composition of the two right hand mappings.

We shall prove that this is isomorphic (in D^{b}(Λ)) to the natural mapping

g_{i_{1,2},e_{1}} : Λi_{1,2} ⟶ Λi_{2,3}.

Let

j_{1,3} : Λi_{1,2} ⟶ j_{1,3}Γj_{1,3}
λi_{1,2} ⟶ j_{1,3}

be the constant mapping. Then, we have a homomorphism

g_{i_{1,2},e_{2}} ⊗ j_{1,3} : Λi_{1,2} → Λi_{2,3} ⊗_{R} j_{1,3}Γj_{1,3}
λ ↦ g_{i_{1,2},e_{2}}(λ) ⊗ j_{1,3}

Since e_{2} · e_{3} = 0 and g_{i_{1,2},e_{2}}(Λi_{1,2}) ⊆ Λe_{2}, we get that

(g_{i_{1,2},e_{2}} ⊗ j_{1,3})(Λi_{2,3}) ⊆ kerπ̂ = ( \begin{matrix}i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ i_{L}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ ker(π) \end{matrix} )

Now, Roggenkamp’s classification of a Green order together with our choice of a first element in T_{Λ} ensure that i_{U}Λi_{2,3} = i_{U}Λi_{1,2} and that i_{1,2}Λi_{2,3} = e_{2}i_{1,2}Λi_{1,2}.

The lower horizontal mapping however is (0, f_{e_{2}} ·f_{e_{1}}, 0, 0)^{tr}. Since j_{1,3} is the identity element in j_{1,3}Γj_{1,3}, the square

e_{1}Λi_{1,2} ⟵  Λi_{1,2}
↓  ↓ −(g_{i_{1,2},e_{2}} ⊗ j_{1,3})

( 0 \overline{Ω} 0 0 ) ⟵  ker π̂

is commutative. The universal property of the pullback yields a unique mapping

ρ : Λi_{2,3} ⟶ P_{0}

such that αρ(λ) = λe_{1} and βρ(λ) = (λe_{2}) ⊗ j_{1,3} for any λ ∈ Λi_{1,2}. If we define the mapping

1 ⊗ j_{1,3} : Λi_{2,3} → Λi_{2,3} ⊗ j_{1,3}Γj_{1,3}
λ ↦ λ ⊗ j_{1,3}

we get a commutative diagram

e_{1}Λi_{1,2}  ⟵^{α}  P_{0}  ⟵^{ρ}  Λi_{1,2}
↓ −f_{i_{1,2}e_{1}}  ↓ β  ∥

( 0 \overline{Ω} 0 0 )  ⟵ ( \begin{matrix}i_{U}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ i_{1,2}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ i_{L}Λi_{2,3} ⊗ j_{1,3}Γj_{1,3} \\ ker(π) \end{matrix} ) ⟵^{−g_{i_{1,2}e_{2}} ⊗j_{1,3}}  e_{2}Λi_{1,2}

↓  ↓ −g_{i_{1,2}e_{2}} ⊗ j_{1,3}
Λi_{2,3} ⊗_{R} j_{1,3}Γj_{1,3}  ⟵^{·1⊗j_{1,3}}  Λi_{2,3}

By Remark 4 one gets that

ker(β) ≃ ker(f̃_{i_{1,2}e_{1}}) ≃ ker(g_{i_{1,2}e_{2}}) .

Moreover, the cokernel of αι is Ω_{3} by definition of π̂. Clearly, by construction of our Green order, coker(Λi_{1,2} ⟶ Λi_{2,3}) = Ω_{3} and 1 ⊗ j_{1,3} provides this isomorphism.

Hence, the complex, which is given by the composite of the two middle vertical mappings is isomorphic to the complex which is given by the two right most vertical mappings. This is what we claimed.

The second column gives a complex arising from the following pullback diagram:

e_{1}Λi_{1,2} ⊕ e_{2}Λi_{2,3} = ( \begin{matrix}i_{U}Λi_{2,3} \\ e_{1}i_{1,2}Λi_{1,2} ⊕ i_{1,2}Λi_{2,3} \\ i_{L}Λi_{1,2} \\ e_{2}i_{2,3}Λi_{2,3} \end{matrix} ) ⟵  P_{1}

↓ χ  ↓

( 0 \overline{Ω} 0 \overline{Ω} )  ⟵^{ψ} Λi_{2,3} ⊗_{R} j_{1,3}Γj_{1,2}

The mapping P_{1} ⟶ Λi_{2,3} ⊗_{R} j_{1,3}Γj_{1,2} is surjective, the mapping to the artinian quotient being so, and Λi_{2,3} ⊗_{R} j_{1,3}Γj_{1,2} is a projective Λ–module, such that the two term complex formed by this mapping is isomorphic in D^{b}(Λ) to its homology. This homology is isomorphic to the kernel of (the restriction of) χ, using the universal property of the pullback. We claim that this kernel is isomorphic to Λi_{1,2}.

Observe that the lowest entry in the column giving the mapping χ is just the structure giving mapping

f_{i_{2,3},e_{2}} : i_{2,3}Λi_{2,3}e_{2} ⟶ \overline{Ω}

in Roggenkamp’s theorem.

Moreover, the defining permutation ω of T_{Λ} is such that ω(i_{1,2}e_{2}) = i_{2,3}e_{2}. Hence, we have the defining map

g_{(i_{1,2},e_{2})} : Λi_{1,2} ⟶ Λi_{2,3}

of a Green order with image being Λi_{1,2}e_{2}.

The Λ–module homomorphism

Λi_{1,2} \underrightarrow{\binom{·e_{1}}{e_{2}g_{(i_{1,2},e_{2})}}} e_{1}Λi_{1,2} ⊕ e_{2}Λi_{2,3}

is injective. In fact, since g_{(i_{1,2},e_{2})}(Λi_{1,2}) = e_{2}Λi_{1,2}, an element λ in the kernel has to be in the kernel of the mapping (·e_{1}, ·e_{2}) which is impossible since

Λi_{1,2} ⊆ Λi_{1,2}e_{1} ⊕ Λi_{1,2}e_{2}.

On the other hand, the composite

Λi_{1,2} \underrightarrow{\binom{·e_{1}}{e_{2}g_{(i_{1,2},e_{2})}}} e_{1}Λi_{1,2} ⊕ e_{2}Λi_{2,3} ⟶^{χ} ( 0 \overline{Ω} 0 \overline{Ω} )

is zero. In fact, in the lowest entry of the module on the right the composition yields 0: We have a commutative diagram

Λi_{1,2} ⟶^{g_{(i_{1,2},e_{2})}} Λi_{2,3}  ⟶^{·e_{2}} Λi_{2,3}e_{2}
↓ ·e_{3}  ↓ f̃_{i_{2,3},e_{2}}

Λi_{2,3}e_{3} ⟶^{f̃_{i_{2,3},e_{3}}}  ( 0 0 0 \overline{Ω} )

Now,

χ ∘ g_{(i_{1,2},e_{2})} = f_{i_{2,3},e_{2}} ∘ ·e_{2} ∘ g_{(i_{1,2},e_{2})} = f_{i_{2,3},e_{3}} ∘ ·e_{3} ∘ g_{(i_{1,2},e_{2})}

and g_{(i_{1,2},e_{2})} has image in Λe_{2}. Since e_{2}e_{3} = 0, we obtain the result.

The second entry is dealt by the following argument.

We have to deal with

i_{1,2}Λi_{1,2} \underrightarrow{\binom{·e_{1}}{e_{2}g_{(i_{1,2},e_{2})}}} i_{1,2}Λi_{1,2}e_{1} ⊕ e_{2}i_{1,2}Λi_{2,3} = i_{1,2}Λi_{1,2}e_{1} ⊕ e_{2}i_{2,3}Λi_{2,3} ⟶^{(−f_{i_{1,2}e_{1}} ,f_{i_{2,3}e_{2}} )} \overline{Ω}

We know that f_{i_{2,3}e_{2}} = f_{i_{1,2}e_{2}} by the second property of a Green order in Roggenkamp’s theorem. Hence, since we started with an element in Λi_{1,2}, the composite is 0.

Now, since i_{1,2}e_{2} < i_{2,3}e_{2} in T_{Λ}, we get that ( 0 \overline{Ω} 0 0 ) is a Λ-submodule of ( 0 \overline{Ω} 0 \overline{Ω} ) and hence

we get a short exact sequence of Λ-modules

0 ⟶ ( 0 \overline{Ω} 0 0 ) ⟶ ( 0 \overline{Ω} 0 \overline{Ω} ) ⟶^{ν} ( 0 0 0 \overline{Ω} ) ⟶ 0

Clearly,

ker χ ⊆ ker νχ .

Moreover,

νχ|_{Λi_{1,2}e_{1}} = 0 and νχ|_{Λi_{2,3}e_{2}} = f̃_{i_{2,3}e_{2}} .

Now,

ker f̃_{i_{2,3}e_{2}} = ker g_{i_{2,3}e_{3}} = im g_{i_{1,2}e_{2}} = (Λi_{1,2}e_{2}) .

Moreover,

Λi_{1,2}  ⟶^{e_{2}g_{i_{1,2}e_{2}}}  Λi_{2,3}e_{2}

↓ ·e_{2}  ↓ ( \begin{matrix}0 \\ f_{e_{2}} \\ 0 \\ f_{e_{2}} \end{matrix} )

Λi_{1,2}e_{2}  ⟶^{f̃_{i_{1,2}e_{2}}}  ( 0 \overline{Ω} 0 \overline{Ω} )

is commutative; the image of f̃_{i_{1,2}e_{2}} being ker ν. What remains now to prove is that

0 ⟶ Λi_{1,2} \underrightarrow{\binom{e_{1}}{e_{2}}} Λi_{1,2}e_{1} ⊕ Λi_{1,2}e_{2} ⟶^{(−f̃_{i_{1,2}e_{1}} ,f̃_{i_{1,2}e_{2}} )} ( 0 \overline{Ω} 0 0 ) ⟶ 0

is exact. But this follows from the definition.

Collecting the pieces, we get that

ker(χ) ≃ Λi_{1,2}.

4.2. Restricting to the right. The restriction to the right is treated completely analogously as the restriction to the left. We leave the verification to the reader.

Remark 6. It might be a interesting to try to see if the functor obtained by tensoring with our module X_{1} differs from the functor defined by P. Gabriel and Chr. Riedtmann in [3].

5. Appendix

We shall repeat a lemma which is well known to the experts^{1}, however, as far as it is known to the author, it was never written down.

Lemma 2. • Let R be a commutative ring and let Λ and Γ be R–algebras which are

projective as R–modules.

• Let X be a complex in D^{b}(Λ ⊗_{R} Γ^{op}) such that X in D^{b}(Λ) is isomorphic to a tilting complex T and X in D^{b}(Γ^{op}) is a isomorphic to a tilting complex S.

• Let m_{Λ} be the smallest natural number, such that there is a T′ ≃ T in K^{b}(P_{Λ}) with T_{k}′ = 0 for all k ≤ m_{Λ}

• and let n_{Λ} be the smallest natural number such that there is a T′′ ≃ T in K^{b}(P_{Λ}) with T_{k}′′ = 0 for all k ≥ n_{Λ}.

• Similarly, m_{Γ} and n_{Γ} are defined. Then, m_{Λ} = m_{Γ} =: m and for n = max(n_{Γ}, n_{Λ}) there is a X̃ ≃ X in D^{b}(Λ ⊗_{R} Γ^{op}) with

X̃_{k} being a projective Λ ⊗_{R} Γ^{op} module for k = n, n − 1, ..., m and X_{n+1} is a module which is projective if restricted to Λ and is projective when restricted to Γ^{op}.

Proof. Since we are dealing with projective modules, m_{Λ} = m_{Γ}.

Without loss of generality we may, and will assume that m = 0. We take a projective resolution as complex of bimodules of X:

… ⟶ P_{3} ⟶^{d_{3}} P_{2} ⟶^{d_{2}} P_{1} ⟶^{d_{1}} P_{0} ⟶ 0 ⟶ …

We truncate as follows:

X′ := (0 ⟶ ker(d_{n}) ⟶ P_{n} ⟶ … ⟶ P_{0} ⟶ 0)

1in fact, it was mentioned to the author by J.Rickard

and observe that, since both rings are projective as R–modules, P_{0}, …, P_{n} are both projective when restricted to either side. Clearly, X ≃ X′ in D^{b}(Λ⊗_{R}Γ^{op}) since the homology in the degree n + 1 is 0. Since n = max{n_{Λ}, n_{Γ}}, the complex (0 ⟶ ker(d_{n}) ⟶ P_{n} ⟶ 0) decomposes as

L ⟶^{(id,0)} L ⊕ L′ as complex of Λ–modules. Since P_{n} = L ⊕ L′ we conclude that L is projective. Likewise, ker(d_{n}) is projective as right Γ-module.

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Address of the author: Mathematisches Institut B Universität Stuttgart Pfaffenwaldring 57 70550 Stuttgart Germany

electronic mail: alex@machnix.mathematik.uni-stuttgart.de Current address: UFR de Mathématiques Université de Paris 7 2, place Jussieu F-75251 Paris CEDEX 05 France

electronic mail:

alex@mathp7.jussieu.fr