Two-sided tilting complexes for Gorenstein orders

Alexander Zimmermann

1991 AMS Subject Classification: Primary: 16G30, 18E30, 20C05 Secondary: 16B50, 20C10

Abstract

Work of J. Rickard proves that the derived module categories of two rings A and B are equivalent as triangulated categories if and only if there is a particular object T, a so-called tilting complex, in the derived category of A such that B is the endomorphism ring of T. The functor inducing the equivalence however is not explicit by the knowledge of T. Suppose the derived categories of A and B are equivalent. If A and B are R-algebras and projective of finite type over the commutative ring R, then Rickard proves the existence of a so-called two-sided tilting complex X, which is an object in the derived category of bimodules. The left derived tensor product by X is then an equivalence between the derived categories of A and B. There is no general explicit construction known to derive X from the knowledge of T. In an earlier paper S. König and the author gave for a class of algebras a tilting complex T by a general procedure with prescribed endomorphism ring. Under some mild additional hypothesis we construct in the present paper an explicit two-sided tilting complex whose restriction to one side is any given one-sided tilting complex of the type described in the above cited paper. This provides two-sided tilting complexes for various cases of derived equivalences, making the functor inducing this equivalence explicit. In particular the perfect isometry induced by such a derived equivalences is determined.

Introduction

A well known process due to Grothendieck and Verdier [16] associates to any abelian category, e.g. category of modules over a ring A, a bounded derived category which is a so-called triangulated category. In case of the abelian category to be the category of modules over a ring A, the bounded derived category is denoted by D^{b}(A). For more details of this procedure we refer to [16] or [8]. The question of when two rings A and B induce equivalent bounded derived module categories D^{b}(A) and D^{b}(B) as triangulated categories is treated by Rickard in [10]. There he proves that D^{b}(A) is equivalent to D^{b}(B) as triangulated categories if and only if there is a particular object in D^{b}(A), a so-called tilting complex T, such that

End_{D^{b}(A)}(T) ≃ B .

This description is completely general, but the functor between D^{b}(A) and D^{b}(B) inducing the equivalence, called derived equivalence, remains nevertheless not at all easy to handle and is up to some extent even not constructable just from the knowledge of T. Rickard proves in [11] for algebras A and B over a commutative ring R which are flat as R-modules the existence of a complex X in D^{b}(A ⊗_{R} B^{op}) such that

X ⊗_{B}^{𝕃} − : D^{b}(B) ⟶ D^{b}(A)

is an equivalence of triangulated categories. The complex X is called two-sided tilting complex. One obtains the tilting complex T as image of the rank one free B-module:

T ≃ X ⊗_{B}^{𝕃} B .

B. Keller gives a significantly simpler construction of X in terms of differential graded algebras and without the hypothesis that B is flat over R in [6, 8]. To distinguish between tilting complexes and two-sided tilting complexes we call T a one-sided tilting complex or just tilting complex. By this description, the equivalence between D^{b}(A) and D^{b}(B) is as explicit as the knowledge X. However, a module theoretic construction of X out of a knowledge of T in general is not known so far, even though Keller constructs X as a certain projective resolution of T as differential graded module over a certain differential graded algebra [8].

In [7] a one-sided tilting complex T is constructed for a Gorenstein order Λ over a complete discrete valuation domain R with residue field of characteristic p such that the endomorphism ring T is a pullback of the endomorphism rings of the homology of the complex T over the endomorphism ring of the homology in the stable category. Recall that an R-algebra Λ is called an R-order if Λ is finitely generated projective as R-module such that K ⊗_{R} Λ is semisimple for the field of fractions K of R. An order Λ is Gorenstein if Ext_{Λ}^{1}(L, Λ) = 0 for any Λ-module L which is finitely generated projective as R-module. Gorenstein orders are well suited for the above problem. In fact, a lot of effort is undertaken in recent years to construct derived equivalences between blocks of group rings RG and RH for two finite groups G and H. Recall that the principal block of a group ring RG is the indecomposable ring direct factor of RG which acts on the trivial G-module. Broué conjectured [1], see also [8], that if the two groups G and H have isomorphic abelian Sylow p-subgroups and if moreover the normalizers of a Sylow p-subgroup of G and H are isomorphic, then the principal blocks of G and of H should have equivalent derived categories. Broué explains many for a long time conjectured and many known but not sufficiently explained phenomenons out of this derived equivalence. Now, principal blocks of group rings of finite groups are Gorenstein orders, they are even symmetric. Most of the complexes used so far in proving Broué’s conjecture satisfy the hypotheses of [7].

What we do in the present paper is to give an explicit two-sided tilting complex X such that the image of the rank one free End_{D^{b}(Λ)}(T)-module is the tilting complex T discussed in [7]. To do this we have to make an additional hypotheses which is not very restrictive if one is interested in Broué’s conjecture.

Twosided tilting complexes do not only provide the explicit equivalence between the derived categories but may be used also for other purposes. In fact, finding a one-sided tilting complex T of the type discussed in [7] together with verification of some hypothesis implies by our paper the explicit knowledge of a two-sided tilting complex X restriction of which to the left is isomorphic to T. The work of Marcus [9] reduces Broué’s conjecture to its validity between the principal blocks of finite simple groups with abelian Sylow p-subgroups and the principal block of the normalizer of a Sylow p-subgroup together with a technical condition to be verified on the two-sided tilting complex providing this derived equivalence. The two-sided tilting complex has to be known very explicitly for to be tested if it verifies these technical condition. Hence, our construction gives a tool for verifying these.

A perfect isometry [1] between two finite groups G and H is a special kind of isometry of the character ring of the characters belonging to the principal block of G and the character ring of the characters belonging to the principal block of H. As a further application we give explicitly the perfect isometry induced by the two-sided tilting complex X. Moreover, we are able to control up to a certain extent all perfect isometries who come from derived equivalences F : D^{b}(Γ) ⟶ D^{b}(Λ) such that F (Γ) ≃ T for T as in [7].

Besides the interest coming from Broué’s conjecture we feel that the construction itself deserves attention. The fact that two rings are derived equivalent gives many ring theoretic information, and controlling the equivalence explicitly should give even more.

At the very end we give examples where the theorem applies. Amongst them are algebras of semidihedral type in the sense of Erdmann [2].

Acknowledgement: I want to thank Lluis Puig for convincing me of the use of an abstract formulation of [18] which finally led to the present paper and I want to thank Bernhard Keller for numerous helpful remarks and discussions.

1 Recalling the one-sided situation

Our conventions composing mappings α : A ⟶ B and β : B ⟶ C are as follows. We write β ∘ α when we look at images of particular elements of A. We write αβ when we write mappings on the right, what we usually do when we discuss commutative diagrams.

Throughout, we fix a complete discrete valuation domain R with field of fractions K and a Gorenstein order Λ. We recall that an order Λ is an R-algebra which is finitely generated projective as R-module and K ⊗_{R} Λ is a semisimple K-algebra. A Λ-lattice is a Λ-module which is finitely generated projective as R-module. An R-order Λ is called a Gorenstein R-order if Ext_{Λ}^{1}(L, Λ) = 0 for any Λ-lattice L. A morphism U ⟶ V between two lattices U and V is called pure if its cokernel is a lattice. A sublattice U of a lattice V is a pure sublattice if the embedding is a pure homomorphism.

When we regard complexes we mean chain complexes T with differential d_{i} : T_{i} ⟶ T_{i−1} with d_{i} ∘ d_{i+1} = 0 for all i ∈ ℤ and we regard homology H_{∗}(T) unless otherwise stated. For further conventions and definitions concerning derived categories we refer to Verdier [16] and to [8].

In [7] we constructed a one-sided tilting complex T with endomorphism ring Γ as follows. Take a Λ-lattice L with projective cover

Q ⟶^{λ} L

and a projective cover

P ⟶^{π} ΩL

of ΩL := ker λ with C := ker π. Denote by φ the resulting homomorphism P ⟶ Q. We hence get a four term exact sequence

0 ⟶ C ⟶^{ι} P ⟶^{φ} Q ⟶^{λ} L ⟶ 0.

We choose a finitely generated projective Λ-module P̃ such that Q ⊕ P ⊕ P̃ is a progenerator for Λ.

Theorem 1 [7, 8] Let R, Λ, P, Q and P̃ be as above. Suppose that Hom_{Λ}(P ⊕ P̃, L) = 0. Then, the complex T

… ⟶ 0 ⟶ P ⊕ P ⊕ P̃ ⟶^{(φ,0,0)} Q ⟶ 0 ⟶ …

with homology concentrated in degrees 0 and 1 is a tilting complex. Denoting by \overline{End}_{Λ}(L) the quotient of the ring of Λ-linear endomorphisms of L modulo those endomorphisms factoring over a projective module, then End_{D^{b}(Λ)}(T) ≃ Γ occurs in the following pullback diagram.

Γ  ⟶ End_{Λ}(H_{0}(T))
↓  ↓
End_{Λ}(H_{1}(T)) ⟶  \overline{End}_{Λ}(L)

2 Exploring Γ

We fix L, Q, P, P̃, φ, π and ι as in Theorem 1.

Lemma 1 If Hom_{Λ}(P ⊕ P̃, L) = 0, then there is an idempotent e^{2} = e in the centre Z(K ⊗_{R} Λ) of K ⊗_{R} Λ such that L ≃ e · Q and e · (P ⊕ P̃) = 0.

Proof. Set A := K ⊗_{R} Λ. Since Hom_{Λ}(P ⊕ P̃, L) = 0, for any primitive central idempotent f in A with f · (P ⊕ P̃) ≠ 0 one gets f · L = 0. In fact, take any α ∈ Hom_{A}(K ⊗_{R} (P ⊕ P̃), K ⊗_{R} L). Then α(P ⊕ P̃) is a Λ-lattice in K ⊗_{R} L, and hence there is a non zero element r in R, such that

r · α(P ⊕ P̃) ⊆ L.

Then,

r · α ∈ Hom_{Λ}(P ⊕ P̃, L) = 0

and therefore rα = 0, which yields α = 0.

Let E := {ϵ ∈ A | ϵ is a central primitive idempotent in A with ϵ · L ≠ 0}

Set e := ∑_{ϵ∈E} ϵ. Then, by the above, e · (P ⊕ P̃) = 0. We look at the short exact sequence

0 ⟶ ΩL ⟶ Q ⟶ L ⟶ 0

and we apply eΛ ⊗_{Λ} − to it. We get the exact sequence

… ⟶ eΩL ⟶ eQ ⟶ eL ⟶ 0

where eΩL = 0 by the above and eL = L by the definition of E. This concludes the proof of the lemma.

Lemma 2 L is free of rank 1 as End_{Λ}(L)-module.

Proof.

The following sequence is exact:

0 ⟶ ΩL ⟶ Q ⟶ L ⟶ 0

We apply Hom_{Λ}(−, L) to it and see

L ≃ Hom_{Λ}(Λ, L) ≃ ( \begin{matrix}Hom_{Λ}(Q, L) \\ Hom_{Λ}(P, L) \\ Hom_{Λ}(P̃, L) \end{matrix} ) = ( \begin{matrix}Hom_{Λ}(Q, L) \\ 0 \\ 0 \end{matrix} ) = ( \begin{matrix}Hom_{Λ}(L, L) \\ 0 \\ 0 \end{matrix} ) ≃ End_{Λ}(L)

as End_{Λ}(L)-module.

Assume that Hom_{Λ}(C, ΩL) = 0.

Since from the very beginning we assumed that Hom_{Λ}(P ⊕ P̃, L) = 0, Lemma 1 provides us with a central idempotent e in A = K ⊗_{R} Λ with e · Q = L and e · (P ⊕ P̃) = 0. Moreover, if we define e_{ΩL} the sum of all central primitive idempotents e_{i} of A with e_{i} · ΩL ≠ 0, and define

e_{C} := 1 − e − e_{ΩL}.

The above hypotheses ensure that

e · e_{ΩL} = e · e_{C} = e_{ΩL} · e_{C} = 0 .

We define P̃(c) by the following exact sequence.

0 ⟶ P̃(c) ⟶ P̃ ⟶ e_{ΩL}P̃ ⟶ 0

Analogously to Lemma 1 we have

Lemma 3 Assume that Hom_{Λ}(C, ΩL) = 0. Then, we have natural isomorphisms

• e_{ΩL} · P ≃ ΩL

• End_{Λ}(C) = Hom_{Λ}(C, P)

• Hom_{Λ}(P, Q) ≃ Hom_{Λ}(ΩL, ΩL)

• Hom_{Λ}(P̃, Q) ≃ Hom_{Λ}(e_{ΩL}P̃, ΩL)

• Hom_{Λ}(C, P̃(c)) ≃ Hom_{Λ}(C, P̃)

Proof.

0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0

is exact. Multiplying this short exact sequence by e_{ΩL}, i.e. applying e_{ΩL}Λ ⊗_{Λ} −, proves the first statement.

Applying Hom_{Λ}(C, −) to

0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0

yields the second isomorphism.

Applying Hom_{Λ}(P, −) to

0 ⟶ ΩL ⟶ Q ⟶ L ⟶ 0

yields Hom_{Λ}(P, Q) ≃ Hom_{Λ}(P, ΩL). Applying Hom_{Λ}(−, ΩL) to

0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0

proves the third statement.

Applying Hom_{Λ}(−, ΩL) to

0 ⟶ P̃(c) ⟶ P̃ ⟶ e_{ΩL}P̃ ⟶ 0

proves Hom_{Λ}(e_{ΩL}P̃, ΩL) ≃ Hom_{Λ}(P̃, ΩL). Applying Hom_{Λ}(P̃, −) to

0 ⟶ ΩL ⟶ Q ⟶ L ⟶ 0

gives the fourth statement.

Applying Hom_{Λ}(C, −) to

0 ⟶ P̃(c) ⟶ P̃ ⟶ e_{ΩL}P̃ ⟶ 0

proves the fifth statement.

3 Inverting T

We recall from [8, Lemma 5.2.5].

Lemma 4 Let S be a ring and let M and X be S-modules. Assume that X is a direct summand of M. Then, the functor

Hom_{S}(M, −) : S − Mod ⟶ End_{S}(M) − Mod

induces for any S-module Y an isomorphism

Hom_{S}(X, Y) ⟶ Hom_{End_{S}(M)}(Hom_{S}(M, X), Hom_{S}(M, Y))

Likewise, the functor

Hom_{S}(−, M) : S − Mod ⟶ End_{S}(M) − Mod

induces for any S-module Y an isomorphism

Hom_{S}(Y, X) ⟶ Hom_{End_{S}(M)}(Hom_{S}(X, M), Hom_{S}(Y, M))

Proof. The first part is proven in [8, Lemma 5.2.5].

We give a different proof due to B. Keller for the reader’s convenience.

The first statement follows from the counit of the following pair of adjunctions. The functors

Hom_{Λ}(M, −) : Λ − mod ⟶ End_{Λ}(M) − mod

and

M ⊗_{End_{Λ}(M)} − : End_{Λ}(M) − mod ⟶ Λ − mod

form an adjoint pair. The counit of this adjunction

η : M ⊗_{End_{Λ}(M)} Hom_{Λ}(M, −) ⟶ Id

has the property that its evaluation η_{M} on M is an isomorphism:

M ⊗_{End_{Λ}(M)} Hom_{Λ}(M, M) ≃ M

Since the two functors Hom_{Λ}(M, −) and M ⊗_{End_{Λ}(M)} − are additive, the same is true for any direct summand X. Hence,

Hom_{End_{S}(M)}(Hom_{S}(M, X), Hom_{S}(M, Y)) ≃ Hom_{Λ}(M ⊗_{End_{Λ}(M)} Hom_{Λ}(M, X), Y)
≃ Hom_{Λ}(X, Y)

For the second isomorphism we proceed as follows. Clearly, the functor Hom_{S}(−, M) induces a mapping

Θ : Hom_{S}(Y, X) ⟶ Hom_{End_{S}(M)}(Hom_{S}(X, M), Hom_{S}(Y, M))
α ⟶ (χ ⟶ αχ)

Take π ∈ Hom_{Λ}(M, X) the natural projection and denote by ι : X ⟶ M the natural embedding. Define

Ψ : Hom_{End_{S}(M)}(Hom_{S}(X, M), Hom_{S}(Y, M)) ⟶ Hom_{S}(Y, X)
ρ ⟶ ρ(ι)π

Now, for any α ∈ Hom_{S}(Y, X),

Ψ(Θ(α)) = Ψ(χ ⟶ αχ)
= αιπ
= α

Moreover,

ΘΨ(ρ)) = Θ(ρ(ι)π)
= (χ ⟶ ρ(ι)πχ)
= (χ ⟶ ρ(ιπχ))
= (χ ⟶ ρ(χ))
= ρ

This proves the lemma completely.

Take a Gorenstein order Λ and a Λ-module L as in Theorem 1. Then, choose a projective module P̃ to form a tilting complex T as in Theorem 1.

We are now able to prove that under certain circumstances there is an ’inverse’ to T which again is of the form described in Theorem 1.

A first step to this direction is in fact the observation that L has a natural structure of an End_{Λ}(L)-module. Moreover, End_{Λ}(L) is an epimorphic image of Γ as rings, as follows by the description of Γ as pullback and by the observation that H_{0}(T) ≃ L.

The projective cover of End_{Λ}(L) as Γ-module is computed as follows. Set C := kerϕ. Then,

End_{Λ}(H_{1}(T)) ≃ ( \begin{matrix}End_{Λ}(C) & Hom_{Λ}(C, P) & Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P, C) & End_{Λ}(P) & Hom_{Λ}(P, P̃) \\ Hom_{Λ}(P̃, C) & Hom_{Λ}(P̃, P) & End_{Λ}(P̃) \end{matrix} )

The pullback diagram

Γ  ⟶ End_{Λ}(H_{0}(T))
↓  ↓
End_{Λ}(H_{1}(T)) ⟶  \overline{End}_{Λ}(L)

gives us a morphism of the rank one free Γ-right module to End_{Λ}(L). We can even determine the projective cover of End_{Λ}(L) as Γ-module. Observe that

( \begin{matrix}Hom_{Λ}(P, C) & End_{Λ}(P) & Hom_{Λ}(P, P̃) \\ Hom_{Λ}(P̃, C) & Hom_{Λ}(P̃, P) & End_{Λ}(P̃) \end{matrix} )

maps as 0 to \overline{End}_{Λ}(L). In fact, only the component End_{Λ}(C) gives a contribution to the homomorphism, the mapping to \overline{End}_{Λ}(L) being induced by the exact sequence

0 ⟶ C ⟶ P ⟶ Q ⟶ L ⟶ 0.

Hence, since the above short exact sequence is the projective cover sequence for L as Λ-module, the projective cover of \overline{End}_{Λ}(L) as Γ-module is Q^{∗} defined as pullback as follows:

Q^{∗}  ⟶ End_{Λ}(L)
↓  ↓
Hom_{Λ}(C, C ⊕ P ⊕ P̃) ⟶ \overline{End}_{Λ}(L)

The kernel of Q^{∗} ⟶ End_{Λ}(L) is isomorphic to the kernel of Hom_{Λ}(C, C ⊕ P ⊕ P̃) ⟶ \overline{End}_{Λ}(L) using that Q^{∗} is defined as a pullback. Since again the latter is defined by pulling mappings along the exact sequence

0 ⟶ C ⟶ P ⟶ Q ⟶ L ⟶ 0,

already Hom_{Λ}(C, 0 ⊕ P ⊕ P̃) is in the kernel of the mapping in question. Moreover, using the fact that Λ is a Gorenstein order, we see that

\overline{End}_{Λ}(L) ≃ \overline{End}_{Λ}(C).

We shall need also for later use the following.

Lemma 5 Let Λ be a Gorenstein order, let U be a pure sublattice of a projective module V and denote by α the embedding. Then, for any lattice W the set of Λ-homomorphisms from U to W factoring through any projective module is α · Hom_{Λ}(V, W).

Proof. Let U ⟶ W be a homomorphism which factors through a projective module X. We form the pushout diagram as below:

0 ⟶ U ⟶ V ⟶ V/U ⟶ 0
↓ ↓ ∥
0 ⟶ X ⟶ P O ⟶ V/U ⟶ 0

Since U is pure in V, also X is pure in P O. Since X is projective, the Gorenstein property gives us that the lower short exact sequence splits. Hence there is a mapping P O ⟶ X such that X ⟶ P O ⟶ X is the identity on X. Hence,

(U ⟶ W) = (U ⟶ X ⟶ W)
= (U ⟶ X ⟶ P O ⟶ X ⟶ W)
= (U ⟶ V ⟶ P O ⟶ X ⟶ W)
= (U ⟶ V) ⟶ (P O ⟶ X ⟶ W)

and U ⟶ W already factors through the embedding of U into V.

We apply Lemma 5 to compute the kernel of the projective cover mapping. The endomorphisms of C factoring over any projective module are those factoring over the embedding C ⟶^{ι} P. Hence,

0 ⟶ ( ι · Hom_{Λ}(P, C), Hom_{Λ}(C, P), Hom_{Λ}(C, P̃) ) ⟶ Q^{∗} ⟶ End_{Λ}(L) ⟶ 0

is an exact sequence of Γ-modules.

We look for a projective Γ-module mapping onto this kernel. We have the short exact sequence

0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0

of Λ-modules. We apply Hom_{Λ}(−, C ⊕ P ⊕ P̃) to it and get the exact sequence

0 ⟶ Hom_{Λ}(ΩL, C ⊕ P ⊕ P̃) ⟶ Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ Hom_{Λ}(C, C ⊕ P ⊕ P̃) ⟶ Ext_{Λ}^{1}(ΩL, C)

of Γ-modules where

im( Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ Hom_{Λ}(C, C ⊕ P ⊕ P̃) ) = ι · Hom_{Λ}(P, C) ⊕ Hom_{Λ}(C, P ⊕ P̃).

Hence, the projective End_{Λ}(C ⊕ P ⊕ P̃)-module Hom_{Λ}(P, C ⊕ P ⊕ P̃) maps onto our kernel. In fact it is not only a projective End_{Λ}(C ⊕ P ⊕ P̃)-module but also a projective Γ-module: In fact, the pullback construction does only affect the End_{Λ}(C) entry in End_{Λ}(C ⊕ P ⊕ P̃).

We hence get a projective resolution of End_{Λ}(L) as Γ-module out of the following diagram.

0 ↓

Hom_{Λ}(ΩL, C ⊕ P ⊕ P̃)

↓

Hom_{Λ}(P, C ⊕ P ⊕ P̃)  →  Q^{∗}  → End_{Λ}(L) → 0
↓  ↓  p.b.  ↓
0 → ι · Hom_{Λ}(P, C) ⊕ Hom_{Λ}(C, P ⊕ P̃) → Hom_{Λ}(C, C ⊕ P ⊕ P̃) → \overline{End}_{Λ}(L) → 0

↓ We regard the complex T^{∗}

0 ⟶ Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ Q^{∗} ⟶ 0
⊕  ⊕
0 ⟶ Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ 0
⊕  ⊕
0 ⟶ Hom_{Λ}(P̃, C ⊕ P ⊕ P̃) ⟶ 0

of Γ-right modules with homology concentrated in degree 0 and 1.

We are ready to prove that T^{∗} verifies the hypotheses of Theorem 1. Lemma 6 1. H_{0}(T^{∗}) ≃ End_{Λ}(L) and H_{1}(T^{∗}) ≃ Hom_{Λ}(ΩL ⊕ P ⊕ P̃, C ⊕ P ⊕ P̃)

2. Q^{∗} ⊕ Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃) is a progenerator of Γ.

3. Hom_{Γ}( Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃), End_{Λ}(L) ) = 0.

Proof. Part 1 of the proposition is clear from the construction of T^{∗}.

Since the module in question is in fact free of rank 1, the second part follows also.

It is clear that Γ again is an order. We apply K ⊗_{R} − to Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃) and to End_{Λ}(L). Now, as a general fact, \overline{End}_{Λ}(L) is an R-torsion module and hence K ⊗_{R} \overline{End}_{Λ}(L) = 0. The idempotent e reappears in Γ as identity endomorphism on L. Since K ⊗_{R} \overline{End}_{Λ}(L) = 0,

K ⊗_{R} Γ ≃ ( K ⊗_{R} End_{Λ}(L) ) ⊕ ( K ⊗_{R} Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃) )

is a decomposition into two-sided ideals. Hence, also 3. follows.

Corollary 1 If Γ is again a Gorenstein order, then Λ^{op} ≃ End_{D^{b}(Γ)}(T^{∗}). Remark If Λ is symmetric, then also Γ is symmetric and symmetric orders are Gorenstein.

For the proof we use Lemma 4.

Hom_{Γ}( Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃), Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃) ) ≃ Hom_{Λ}(P ⊕ P̃, P ⊕ P̃)^{op} and Hom_{Γ}( Hom_{Λ}(P ⊕P̃, C ⊕P ⊕P̃), Hom_{Λ}(ΩL, C ⊕P ⊕P̃) ) ≃ Hom_{Λ}(ΩL, P ⊕P̃) ≃ Hom_{Λ}(Q, P ⊕P̃) Those Γ-linear mappings from Hom_{Λ}(P, C ⊕ P ⊕ P̃) to Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃) which factor through Q^{∗} are precisely those which factor through Hom_{Λ}(C, C ⊕ P ⊕ P̃) since End_{Λ}(L) lies in another component. Lemma 4 then gives us that

Hom_{Γ}(Hom_{Λ}(P, C ⊕ P ⊕ P̃), Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃) ≃ Hom_{Λ}(P ⊕ P̃, P)

and that

Hom_{Γ}(Hom_{Λ}(C, C ⊕ P ⊕ P̃), Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃) ≃ Hom_{Λ}(P ⊕ P̃, C).

Applying Hom_{Λ}(P ⊕ P̃, −) to the short exact sequence

0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0

gives us that

0 ⟶ Hom_{Λ}(P ⊕ P̃, C) ⟶ Hom_{Λ}(P ⊕ P̃, P) ⟶ Hom_{Λ}(P ⊕ P̃, ΩL) ⟶ 0

is exact. But,

Hom_{Λ}(P ⊕ P̃, ΩL) ≃ Hom_{Λ}(P ⊕ P̃, Q).

We have to compute the endomorphism ring over Γ of Hom_{Λ}(ΩL, C ⊕P ⊕P̃). But, by Lemma 3, we get e_{ΩL}P ≃ ΩL. Hence,

End_{Γ}(Hom_{Λ}(ΩL, C ⊕ P ⊕ P̃)) ≃ End_{Γ}(Hom_{Λ}(e_{ΩL}P, C ⊕ P ⊕ P̃))
≃ End_{Γ}(e_{ΩL}Hom_{Λ}(P, C ⊕ P ⊕ P̃))
≃ End_{Λ}(e_{ΩL}P)^{op}
≃ End_{Λ}(ΩL)^{op}

applying Lemma 4. Even more,

End_{Γ}(End_{Λ}(L)) ≃ End_{End_{Λ}(L)}(End_{Λ}(L)) ≃ End_{Λ}(L)^{op} .

Now, Γ is Gorenstein and hence the first three parts of Proposition 6 together justify that we may apply Theorem 1. Lemma 5 gives us that those Γ-linear endomorphisms of Hom_{Λ}(ΩL, C ⊕ P ⊕ P̃) factoring over a projective module are those factoring over Hom_{Λ}(P, C ⊕ P ⊕ P̃). These correspond then to those endomorphisms of ΩL factoring over a projective module and hence, the endomorphism ring of

T_{r} : 0 ⟶ Hom_{Γ}(P, C ⊕ P ⊕ P̃) ⟶ Q^{∗} ⟶ 0.

is isomorphic to the opposite of the endomorphism ring of Q.

Hence,

End_{D^{b}(Γ)}(T^{∗}) ≃ ( \begin{matrix}End_{Λ}(Q)^{op} & Hom_{Λ}(P ⊕ P̃, Q) \\ Hom_{Λ}(Q, P ⊕ P̃) & End_{Λ}(P ⊕ P̃)^{op} \end{matrix} ) ≃ Λ^{op}

We recall a lemma of Rickard from [8, 18].

Lemma 7 (J. Rickard) Let R be a complete discrete valuation ring and let Λ and Γ be R-orders. Assume that we have a complex X in D^{b}(Λ⊗_{R}Γ^{op}) such that X ≃ T in D^{b}(Λ) and X ≃ T^{∗} ∈ D^{b}(Γ^{op}) where T is a tilting complex with endomorphism ring isomorphic to Γ and T^{∗} is a tilting complex with endomorphism ring isomorphic to Λ^{op}. Then, X is a two-sided tilting complex.

By Lemma 7 we are done if we have constructed a complex X in D^{b}(Λ⊗_{R}Γ^{op}) which is isomorphic to T in D^{b}(Λ) and to T^{∗} in D^{b}(Γ^{op}).

4 How to construct the two-sided tilting complex

We shall construct a complex X in D^{b}(Λ ⊗_{R} Γ^{op}) such that X ≃ T in D^{b}(Λ) and X ≃ T^{∗} in D^{b}(Γ^{op}) for our complexes T and T^{∗} from section 3. Lemma 7 then tells us that X is indeed a two-sided tilting complex.

The complex will be constructed by the following commutative diagram.

Ω̂L ↪ X_{0} ⟶ ⟶ End_{Λ}(L)
∥ ↓ P.B. ↓
H_{1}(X) ↪ X_{1} ⟶ ⟶ Ω̂L ↪ \overline{X_{0}} ⟶ ⟶ \overline{End}_{Λ}(L)
∥  ↓ P.B. ↓ α
H_{1}(X) ↪ \overline{X_{1}} ⟶ ⟶  \overline{Ω}

Here all sequences • ↪ • ⟶ ⟶ • are assumed to be exact. Ω̂L denotes the kernel of this map X_{0} ⟶ L as Λ ⊗_{R} Γ^{op}-module.

Remark 1 In general X_{0} is not projective neither as Λ nor as Γ^{op}-module.

What we have to do, is to define α, the Λ ⊗_{R} Γ^{op}-module \overline{Ω}, the module \overline{X_{1}} and its projection onto \overline{Ω}.

Our complex X will then be

… ⟶ 0 ⟶ X_{1} ⟶ X_{0} ⟶ 0 ⟶ …

where the non zero differential is induced by the mapping

Ω̂L ↪ X_{0}
∥
X_{1} ⟶ ⟶ Ω̂L

We begin with the parts which are easy to describe.

\overline{X_{0}} = ( \begin{matrix}End_{Λ}(Q) ⊗_{R} End_{Λ}(C) & End_{Λ}(Q) ⊗_{R} Hom_{Λ}(C, P) & End_{Λ}(Q) ⊗_{R} Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P, Q) ⊗_{R} End_{Λ}(C) & Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, P) & Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P̃, Q) ⊗_{R} End_{Λ}(C) & Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, P) & Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, P̃) \end{matrix} )

The mapping to \overline{End}_{Λ}(L) is defined as follows. Since there is a central idempotent e in K ⊗_{R} Λ with e·Q = L, each endomorphism ϕ of Q induces an endomorphism eϕ of L and hence the mapping \overline{X_{0}} ⟶ \overline{End}_{Λ}(L) is defined as

( \begin{matrix}ϕ ⊗ ψ & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{matrix} ) ⟶ \overline{ϕ · ψ}

denoting by the residue class in \overline{End}_{Λ}(L).

We see immediately that Ω̂L is

Ω̂L = ( \begin{matrix}Ỹ & End_{Λ}(Q) ⊗_{R} Hom_{Λ}(C, P) & End_{Λ}(Q) ⊗_{R} Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P, Q) ⊗_{R} End_{Λ}(C) & Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, P) & Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P̃, Q) ⊗_{R} End_{Λ}(C) & Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, P) & Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, P̃) \end{matrix} )

for Ỹ is defined via the exact sequence

0 ⟶ Ỹ ⟶ End_{Λ}(Q) ⊗_{R} End_{Λ}(C) ⟶ \overline{End}_{Λ}(L) ⟶ 0

We define

H_{1}(X) := ( \begin{matrix}Hom_{Λ}(ΩL, C) & Hom_{Λ}(ΩL, P) & Hom_{Λ}(ΩL, P̃) \\ Hom_{Λ}(P, C) & End_{Λ}(P) & Hom_{Λ}(P, P̃) \\ Hom_{Λ}(P̃, C) & Hom_{Λ}(P̃, P) & End_{Λ}(P̃) \end{matrix} )

Lemma 8 H_{1}(X) ≃ H_{1}(T) in Λ − mod and H_{1}(X) ≃ H_{1}(T^{∗}) in mod − Γ

Proof. Using that Hom_{Λ}(ΩL, C ⊕ P ⊕ P̃) ≃ Hom_{Λ}(Q, C ⊕ P ⊕ P̃) by Lemma 1, we recognize readily the direct summands.

5 Defining the remaining components

We shall assume from now on throughout that Hom_{Λ}(C, ΩL) = 0 Let P̃(c) := e_{C} P̃ ∩ P̃.

5.1 Defining \overline{X}_{1} and \overline{Ω} as R-modules

We define

\overline{X_{1}} := \overline{X_{1}}^{ΩL} ⊕ \overline{X_{1}}^{C}

with

\overline{X_{1}}^{ΩL} := ( \begin{matrix}0 & Hom_{Λ}(P, ΩL) & Hom_{Λ}(P, e_{ΩL}P̃) \\ 0 & Hom_{Λ}(P, ΩL) & Hom_{Λ}(P, e_{ΩL}P̃) \\ 0 & Hom_{Λ}(P̃, ΩL) & Hom_{Λ}((P̃, e_{ΩL}P̃) \end{matrix} )

and

\overline{X_{1}}^{C} := ( \begin{matrix}0 & 0 & 0 \\ Hom_{Λ}(C, P) & Hom_{Λ}(C, P) & Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P̃(c), P) & Hom_{Λ}(P̃(c), P) & Hom_{Λ}(P̃(c), P̃) \end{matrix} )

We shall discover a Λ ⊗_{R} Γ^{op}-module structure on \overline{X_{1}} in the sequel. However, want first define \overline{Ω}. To be able to define \overline{Ω} we discuss various pushout diagrams. First, we state a surely well known lemma.

Lemma 9 Let X ⟶^{χ} Z ⟵^{η} Y be the pushout of X ⟵^{α} W ⟶^{β} Y. Assume that kerα ∩ kerβ = 0. Then, α induces an isomorphism kerχ ≃ kerβ.

Proof. Z ≃ (X ⊕ Y)/{(α(w), −β(w))|w ∈ W }. And

kerχ = {x ∈ X|χ(x) ∈ {(α(w), −β(w))|w ∈ W }} = {x ∈ X|x = α(w) and w ∈ kerβ} = α(kerβ)

Hence, α is surjective as mapping kerβ ⟶ kerχ. Since kerα ∩ kerβ = 0, we get the statement.

We apply Hom_{Λ}(P, −) and Hom_{Λ}(−, P) to

0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0.

Since Hom_{Λ}(ΩL, C) = 0, the hypothesis of Lemma 9 are fulfilled and we obtain a commutative diagram, which is completed to a pushout diagram in form of Ω, as follows.

0  0
↓  ↓
Hom_{Λ}(P, C) = Hom_{Λ}(P, C)
↓  ↓
0 ⟶ Hom_{Λ}(ΩL, P) ⟶ Hom_{Λ}(P, P) ⟶ Hom_{Λ}(C, P) ⟶ 0
∥  ↓  ↓ σ
0 ⟶ Hom_{Λ}(ΩL, P) ⟶ Hom_{Λ}(P, ΩL) ⟶^{τ}  Ω  ⟶ 0
↓  ↓
0  0

Observe that by Lemma 9 this diagram is at once a pullback and a pushout diagram!

In the same way we introduce a commutative diagram associated to Hom_{Λ}(P̃, P̃) with respect to e_{C} and e_{ΩL} and complete it to a pushout diagram.

Hom_{Λ}(P̃, P̃) ⟶ Hom_{Λ}(P̃(c), P̃) ⟶ 0
↓  ↓
Hom_{Λ}(P̃, e_{ΩL}P̃) ⟶ Ω̃ ⟶ 0
↓  ↓
0  0

We shall do the same with the commutative diagram associated to Hom_{Λ}(P, P̃) with respect to e_{C} and e_{ΩL} as well as with Hom_{Λ}(P̃, P) with respect to the same central idempotents e_{C} and e_{ΩL} of A. Observe that e acts as 0 on each of these homomorphism sets.

Hom_{Λ}(P, P̃) ⟶ Hom_{Λ}(C, P̃) ⟶ 0
↓  ↓
Hom_{Λ}(P, e_{ΩL}P̃) ⟶ Ω̃_{P} ⟶ 0
↓  ↓
0  0

and

Hom_{Λ}(P̃, P) ⟶ Hom_{Λ}(P̃(c), P) ⟶ 0
↓  ↓
Hom_{Λ}(P̃, e_{ΩL}P) ⟶ Ω̃^{P} ⟶ 0
↓  ↓
0  0

are the corresponding pullback/pushout diagrams.

The same procedure applied to Hom_{Λ}(P̃, C) yields the following. The sequence

0 ⟶ P̃(c) ⟶ P̃ ⟶ e_{ΩL}P̃ ⟶ 0

is exact. We apply Hom_{Λ}(−, C) to it and, using that Hom_{Λ}(e_{ΩL}P̃, C) = 0, we obtain

0 ⟶ Hom_{Λ}(P̃, C) ⟶ Hom_{Λ}(P̃(c), C) ⟶ Ext_{Λ}^{1}(e_{ΩL}P̃, C) ⟶ 0

is exact. Since Hom_{Λ}(P̃(c), ΩL) = 0, we have

Hom_{Λ}(P̃(c), C) ≃ Hom_{Λ}(P̃(c), P) .

Lemma 10 Let A be a ring and let

𝕄 : 0 ⟶ M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ 0

as well as

ℕ : 0 ⟶ N_{1} ⟶ N_{2} ⟶ N_{3} ⟶ 0

be short exact sequences of A-modules. Assume that

Ext_{A}^{1}(N_{2}, M_{1}) = Ext_{A}^{1}(N_{3}, M_{2}) = 0 and that Hom_{A}(N_{3}, M_{1}) = Hom_{A}(N_{1}, M_{3}) = 0.

Then, there is a short exact sequence

0 ⟶ Hom_{A}(N_{2}, M_{2}) ⟶ Hom_{A}(N_{1}, M_{2}) ⊕ Hom_{A}(N_{2}, M_{3}) ⟶ Ext_{A}^{1}(N_{3}, M_{1}) ⟶ 0

induced by the natural maps.

Proof. We apply Hom_{A}(−, M_{1}) to ℕ and obtain, using that Hom_{A}(N_{3}, M_{1}) = 0 and that Ext^{1}(N_{2}, M_{1}) = 0, a short exact sequence

0 ⟶ Hom_{A}(N_{2}, M_{1}) ⟶ Hom_{A}(N_{1}, M_{1}) ⟶ Ext_{A}^{1}(N_{3}, M_{1}) ⟶ 0

whence applying Hom(N_{2}, −) to 𝕄 we get, using that Ext_{A}^{1}(N_{2}, M_{1}) = 0 a short exact sequence

0 ⟶ Hom_{A}(N_{2}, M_{1}) ⟶ Hom_{A}(N_{2}, M_{2}) ⟶ Hom_{A}(N_{2}, M_{3}) ⟶ 0

Since Hom_{A}(N_{1}, M_{3}) = 0, we get Hom_{A}(N_{1}, M_{1}) ≃ Hom_{A}(N_{1}, M_{2}). We hence get a commutative diagrams with exact rows and columns.

0  0
↓  ↓
0 ⟶ Hom_{A}(N_{2}, M_{1}) ⟶ Hom_{A}(N_{1}, M_{1}) ⟶ Ext_{A}^{1}(N_{3}, M_{1}) ⟶ 0
↓  ↓
Hom_{A}(N_{2}, M_{2}) ⟶ Hom_{A}(N_{1}, M_{2}) ⟶  0
↓  ↓
Hom_{A}(N_{2}, M_{3})  0

↓ 0

The sequence

0 ⟶ Hom_{A}(N_{3}, M_{2}) ∩ Hom_{A}(N_{2}, M_{1}) ⟶ Hom_{A}(N_{2}, M_{2}) ⟶ Hom_{A}(N_{1}, M_{2}) ⊕ Hom_{A}(N_{2}, M_{3})

is exact; where the left term is identified with its image in Hom_{A}(N_{2}, M_{2}). An element in χ ∈ Hom_{A}(N_{3}, M_{2}) ∩ Hom_{A}(N_{2}, M_{1}) is a mapping N_{2} ⟶ M_{2} which factorizes via M_{1} and via N_{3}. The factorizing property via M_{1} implies that im χ ⊆ M_{1}. Since by the surjectivity of N_{2} ⟶ N_{3} the mapping χ : N_{2} ⟶ N_{3} ⟶ M_{2} hence is in fact a mapping N_{2} ⟶ N_{3} ⟶ M_{1}. The last part is zero since Hom_{A}(N_{3}, M_{1}) = 0. We get that Hom_{A}(N_{2}, M_{2}) ⟶ Hom_{A}(N_{1}, M_{2}) ⊕ Hom_{A}(N_{2}, M_{3}) is a monomorphism.

This induces a commutative diagram

0 ↓

0  0  ⟶  ker
↓  ↓  ↓
0 ⟶ Hom_{A}(N_{2}, M_{1}) ⟶  Hom_{A}(N_{1}, M_{1})  ⟶ Ext_{A}^{1}(N_{3}, M_{1}) ⟶ 0
↓  ↓  ↓
0 ⟶ Hom_{A}(N_{2}, M_{2}) ⟶ Hom_{A}(N_{1}, M_{2}) ⊕ Hom_{A}(N_{2}, M_{3}) ⟶  coker  ⟶ 0
↓  ↓  ↓
Hom_{A}(N_{2}, M_{3}) ⟶^{φ}  Hom_{A}(N_{2}, M_{3})  ⟶  cocoker  ⟶ 0
↓  ↓  ↓
0  0  0

with exact rows and columns. Since φ = id makes the diagram commutative, the snake lemma implies that cocoker = ker = 0 and that coker ≃ Ext_{A}^{1}(N_{3}, M_{1}). This proves the lemma.

We apply lemma 10 to determine Ω̃_{P} , Ω̃^{P} and Ω̃.

We shall use the sequences

ℙ̃ : 0 ⟶ P̃(c) ⟶ P̃ ⟶ e_{ΩL}P̃ ⟶ 0

and

ℙ : 0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0

The following identifications give the results below.

𝕄 = ℙ ℕ = ℙ̃ Ω̃_{P} = Ext_{Λ}^{1}(e_{Ω_{L}} P̃, C)
𝕄 = ℙ̃ ℕ = ℙ Ω̃^{P} = Ext_{Λ}^{1}(Ω_{L}, P̃(c))
𝕄 = ℙ̃ ℕ = ℙ̃ Ω̃ = Ext_{Λ}^{1}(e_{Ω_{L}} P̃, P̃(c))
𝕄 = ℙ ℕ = ℙ  Ω = Ext_{Λ}^{1}(Ω_{L}, C)

Summarizing, the sequences

0 ⟶ Hom_{Λ}(P̃, P̃) ⟶ Hom_{Λ}(P̃(c), P̃) ⊕ Hom_{Λ}(P̃, e_{ΩL}P̃) ⟶ Ext_{Λ}^{1}(e_{ΩL}P̃, P̃(c)) ⟶ 0

0 ⟶ Hom_{Λ}(P̃, P) ⟶ Hom_{Λ}(P̃(c), P) ⊕ Hom_{Λ}(P̃, ΩL) ⟶ Ext_{Λ}^{1}(e_{ΩL}P̃, C) ⟶ 0
0 ⟶ Hom_{Λ}(P, P̃) ⟶ Hom_{Λ}(C, P̃) ⊕ Hom_{Λ}(P, e_{ΩL}P̃) ⟶ Ext_{Λ}^{1}(ΩL, P̃(c)) ⟶ 0
0 ⟶ Hom_{Λ}(P, P) ⟶ Hom_{Λ}(C, P) ⊕ Hom_{Λ}(P, ΩL) ⟶ Ext_{Λ}^{1}(ΩL, C) ⟶ 0

are exact. These give the entries (3, 3), (2, 3), (3, 2) and (2, 2) of \overline{Ω}. Moreover, by the pullback property of the pushout diagrams we get that

0 ⟶ Hom_{Λ}(P̃, C) ⟶ Hom_{Λ}(P̃(c), P) ⟶ Ext_{Λ}^{1}(e_{ΩL}P̃, C) ⟶ 0
0 ⟶ Hom_{Λ}(ΩL, P̃) ⟶ Hom_{Λ}(P, e_{ΩL}P̃) ⟶ Ext^{1}(ΩL, P̃(c)) ⟶ 0
0 ⟶ Hom_{Λ}(ΩL, P) ⟶ Hom_{Λ}(P, ΩL) ⟶ Ext_{Λ}^{1}(ΩL, C) ⟶ 0
0 ⟶ Hom_{Λ}(P, C) ⟶ Hom_{Λ}(C, P) ⟶ Ext_{Λ}^{1}(ΩL, C) ⟶ 0

are exact. These sequences give the (1, 3), (3, 1), (2, 1) and (1, 2) entry of \overline{Ω}.

We see that

\overline{Ω} = ( \begin{matrix}0 & Ext_{Λ}^{1}(ΩL, C) & Ext_{Λ}^{1}(ΩL, P̃(c)) \\ Ext_{Λ}^{1}(ΩL, C) & Ext_{Λ}^{1}(ΩL, C) & Ext_{Λ}^{1}(ΩL, P̃(c)) \\ Ext_{Λ}^{1}(e_{ΩL}P̃, C) & Ext_{Λ}^{1}(e_{ΩL}P̃, C) & Ext_{Λ}^{1}(e_{ΩL}P̃, P̃(c)) \end{matrix} )

What we have to define is the module structure of \overline{X_{1}} and to verify that the mapping H_{1}(X) ⟶ \overline{X_{1}} and the mapping Ω̂L ⟶ \overline{Ω} are Λ ⊗_{R} Γ^{op}-module homomorphisms.

5.2 Defining the Λ ⊗_{R} Γ^{op}-module structure on \overline{X}_{1} and \overline{Ω}

For defining a left Λ-module structure on \overline{X_{1}}, we define the structure on \overline{X_{1}}^{ΩL} and on \overline{X_{1}}^{C} separately. Observe that

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

We begin with \overline{X_{1}}^{ΩL}.

Since

Hom_{Λ}(C, ΩL) = Hom_{Λ}(ΩL, L) = Hom_{Λ}(C, L) = 0,

any endomorphism of Q induces an endomorphism of ΩL which in turn induces an endomorphism of P. Two endomorphisms of P induced this way differ by an element in ι · Hom_{Λ}(P, C).

Therefore, the second column of \overline{X_{1}}^{ΩL} is isomorphic to

( \begin{matrix}End_{Λ}(ΩL) \\ Hom_{Λ}(P, ΩL) \\ Hom_{Λ}(P̃, ΩL) \end{matrix} )

which is certainly a Λ-module. It is an extension of ΩL and Ext^{1}(ΩL ⊕ P ⊕ P̃, ΩL).

The third column is as Λ left-module isomorphic to

( \begin{matrix}Hom_{Λ}(Q, e_{ΩL}P̃) \\ Hom_{Λ}(P, e_{ΩL}P̃) \\ Hom_{Λ}(P̃, e_{ΩL}P̃) \end{matrix} ) ≃ e_{ΩL}P̃

which as well is a Λ-left module.

The right module structure is seen as follows. We use again Lemma 3. Then,

( 0 Hom_{Λ}(P, ΩL) Hom_{Λ}(P, e_{ΩL}P̃) ) ≃ Hom_{Λ}(P, C ⊕ P ⊕ P̃) · e_{ΩL}

which is clearly a Γ-right module. Moreover,

( 0 Hom_{Λ}(P̃, e_{ΩL}P) Hom_{Λ}(P̃, e_{ΩL}P̃) ) ≃ Hom_{Λ}(P̃, C ⊕ P ⊕ P̃) · e_{ΩL}

which also is a Γ-right module.

We postpone for the moment the question if these left- and right module structure fit together to a bimodule structure.

We shall define the module structure on \overline{X_{1}}^{C}. The left Λ-structure on \overline{X_{1}}^{C} is defined by the following.

Lemma 3 gives us that Hom_{Λ}(C, P) is an End_{Λ}(P)-left module in a natural way. Also,

Hom_{Λ}(P, P̃) · Hom_{Λ}(P̃(c), P) ⊆ Hom_{Λ}(C, P).

Moreover, since C = e_{C}P ∩ P and P̃(c) = e_{C} P̃ ∩ P̃,

Hom_{Λ}(P̃, P)Hom_{Λ}(C, P) + End_{Λ}(P̃)Hom_{Λ}(P̃(c), P) ⊆ Hom_{Λ}(P̃(c), P)

This makes the matrix multiplication of

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

from the left on \overline{X_{1}}^{C} well defined.

The Γ right module structure on \overline{X_{1}}^{C} is defined as follows. By Lemma 3 one has Hom_{Λ}(C, P) ≃ Hom_{Λ}(C, C). Hence, for the second line in \overline{X_{1}}^{C} we get therefore

( Hom_{Λ}(C, P) Hom_{Λ}(C, P) Hom_{Λ}(C, P̃) ) ≃ Hom_{Λ}(C, C ⊕ P ⊕ P̃).

The third line in the matrix representation of \overline{X_{1}}^{C} is a Γ-right module in the following way. A Λ-linear mapping from P̃(c) ⟶ P has image in P and in e_{C}P as well. Therefore, it has image in C. this proves that

Hom_{Λ}(P̃(c), P) = Hom_{Λ}(P̃(c), C).

The third line of \overline{X_{1}}^{C} is therefore isomorphic to

Hom_{Λ}(P̃(c), C ⊕ P ⊕ P̃).

We shall now discuss why these left and right action are compatible. For proving this we form the module K ⊗_{R} \overline{X_{1}} and prove that this is now by extension of scalars a K ⊗_{R} Λ ⊗ Γ^{op} module. When we have proved this, again restriction to Λ ⊗ Γ^{op} of K ⊗_{R} \overline{X_{1}} gives us that the submodule \overline{X_{1}} is already a Λ ⊗ Γ^{op} module.

But,

K ⊗_{R} \overline{X_{1}}^{ΩL} ≃ K ⊗_{R} Hom_{Λ}(e_{ΩL}Λ, e_{ΩL}C ⊕ P ⊕ P̃)

and the operation of Λ ⊗_{R} Γ^{op} is the regular one. Analogously,

K ⊗_{R} \overline{X_{1}}^{C} ≃ K ⊗_{R} Hom_{Λ}(e_{C}Λ, e_{C}C ⊕ P ⊕ P̃)

with regular operation of Λ ⊗_{R} Γ^{op}.

This observation proves at the same time that the embedding

H_{1}(X) ↪ \overline{X_{1}}

is a Λ ⊗_{R} Γ^{op}-module homomorphism.

The Λ ⊗_{R} Γ^{op}-module structure is defined via the mapping \overline{X}_{1} ⟶ ⟶ \overline{Ω}_{1}.

5.3 Defining the mapping Ω̂L ⟶ \overline{Ω}

We shall have to define a homomorphism Ω̂L ⟶ \overline{Ω}.

Lemma 3 gives canonical isomorphisms

End_{Λ}(C) ≃ Hom_{Λ}(C, P)
Hom_{Λ}(P, Q) ≃ End_{Λ}(ΩL)
Hom_{Λ}(P̃, Q) ≃ Hom_{Λ}(e_{ΩL}P̃, ΩL)
Hom_{Λ}(C, P̃(c)) ≃ Hom_{Λ}(C, P̃)

Moreover, any endomorphism of Q induces a unique endomorphism of ΩL.

The following is the well known Baer construction.

Lemma 11 Let G and D be two Λ-modules. Then, there is a mapping

Hom_{Λ}(D, ΩL) ⊗_{R} Ext_{Λ}^{1}(ΩL, C) ⊗_{R} Hom_{Λ}(C, G) ⟶ Ext_{Λ}^{1}(D, G)

Proof. Let 0 ⟶ C ⟶ X ⟶ ΩL ⟶ 0 be an element in Ext_{Λ}^{1}(ΩL, C). Then, forming a pushout via an element γ ∈ Hom_{Λ}(C, G) and a pullback via δ ∈ Hom_{Λ}(D, ΩL) we get a commutative diagram

0 ⟶ C ⟶ X ⟶ ΩL ⟶ 0
↓ γ ↓ ∥
0 ⟶ G ⟶ X^{γ} ⟶ ΩL ⟶ 0
∥ ↑ ↑ δ
0 ⟶ G ⟶ X_{δ}^{γ} ⟶ D ⟶ 0

The lower sequence is the image in Ext_{Λ}^{1}(D, G). The diagram defines the mapping as stated and has the appropriate properties.

To define the mapping Ω̂L ⟶ \overline{Ω} we just apply lemma 11 and lemma 3. The image of the short exact sequence

0 ⟶ C ⟶ P ⟶ ΩL ⟶ 0

under the homomorphism set actions gives the desired mapping. The fact that the so defined mapping is Λ ⊗_{R} Γ^{op}-linear is then clear.

6 Looking at X one-sided

Recall the construction of X. The complex X is defined to be the composite of the mapping

X_{1} ⟶ ⟶ Ω̂L ↪ X_{0}

regarded as complex. Moreover, the second and the third line in the matrix

Ω̂L = ( \begin{matrix}Ỹ & End_{Λ}(Q) ⊗_{R} Hom_{Λ}(C, P) & End_{Λ}(Q) ⊗_{R} Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P, Q) ⊗_{R} End_{Λ}(C) & Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, P) & Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, P̃) \\ Hom_{Λ}(P̃, Q) ⊗_{R} End_{Λ}(C) & Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, P) & Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, P̃) \end{matrix} )

are mapped isomorphically to X_{0}. In fact, this was our observation that the pullback construction yielding Ω̂L as kernel, does only affect the position (1, 1) in the corresponding matrix rings. Since X_{1} maps surjectively to Ω̂L, the differential restricted to the second and third line of the matrix X_{1} are mapped surjectively to the lower lines of X_{0}, the two lower matrix lines of which are projective right Γ-modules. Hence, regarded as complex with two non zero entries, the complex restricted to the two lower lines is quasi-isomorphic to its homology. This is isomorphic to

Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃).

Completely analogously to the above, X restricted to the right has as direct summands P ⊕ P̃ as left Λ-modules.

Lemma 12 • Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃)[1] is a direct summand of X regarded as complex of right Γ-modules.

• (P ⊕ P̃)[1] is a direct summand of X regarded as complex of left Λ-modules.

We shall prove the following lemma.

Lemma 13 • X/(P ⊕P̃)[1] is isomorphic to 0 ⟶ P ⟶ Q ⟶ 0 as complex of left Λ-modules.

• X/Hom_{Λ}(P ⊕P̃, C ⊕P ⊕P̃)[1] is isomorphic to T_{r} : 0 ⟶ Hom_{Λ}(P, C ⊕P ⊕P̃) ⟶ Q^{∗} ⟶ 0 as complex of right Γ-modules.

Proof. We shall give a morphism of complexes of Γ-right modules

T_{r} ⟶ X/Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃)[1]

which we shall prove to be a quasi-isomorphism. Set

X_{r} := X/Hom_{Λ}(P ⊕ P̃, C ⊕ P ⊕ P̃)[1].

Set

T_{l} := T/(P ⊕ P̃)[1]

and

X_{l} := X/(P ⊕ P̃)[1]

as complexes of left Λ-modules. We will, analogously to the situation for Γ-right modules, give a quasi-isomorphism of complexes of left Λ-modules

T_{l} ⟶ X_{l}

Let us prove first the statement for the left Λ-structure.

P = ( \begin{matrix}Hom_{Λ}(Q, P) \\ Hom_{Λ}(P, P) \\ Hom_{Λ}(P̃, P) \end{matrix} ) ⟶ ( \begin{matrix}0 \\ Hom_{Λ}(C, P) \\ Hom_{Λ}(P̃(c), P) \end{matrix} )

in the obvious way. Moreover,

( \begin{matrix}Hom_{Λ}(Q, φ) \\ Hom_{Λ}(P, φ) \\ Hom_{Λ}(P̃, φ) \end{matrix} ) ⊗_{R} id_{C} : P = ( \begin{matrix}Hom_{Λ}(Q, P) \\ Hom_{Λ}(P, P) \\ Hom_{Λ}(P̃, P) \end{matrix} ) ⟶ ( \begin{matrix}Hom_{Λ}(Q, Q) ⊗_{R} Hom_{Λ}(C, C) \\ Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, C) \\ Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, C) \end{matrix} )

is Λ-linear from the left. Since

im( ( \begin{matrix}Hom_{Λ}(Q, φ) \\ Hom_{Λ}(P, φ) \\ Hom_{Λ}(P̃, φ) \end{matrix} ) ⊗_{R} id_{C} ) ⊆ ker(X_{0} ⟶ L),

we get a mapping

P ⟶ Ω̂L .

Moreover, the two mappings P ⟶ Ω̂L . and P ⟶ \overline{X_{1}} coincide in \overline{Ω}, and hence the pullback property ensures a unique morphism P ⟶ X_{1} which makes the corresponding diagrams

\begin{matrix}P & ⟶ & X_{1} \\ ∥ & & ↓ \\ P & ⟶ & \overline{X_{1}} \end{matrix} and \begin{matrix}P & ⟶ & X_{1} \\ ∥ & & ↓ \\ P & ⟶ & Ω̂L \end{matrix}

commutative. Define a homomorphism Q ⟶ X_{0} by means of

1_{Q} ⊗_{R} id_{C} : ( \begin{matrix}Hom_{Λ}(Q, Q) \\ Hom_{Λ}(P, Q) \\ Hom_{Λ}(P̃, Q) \end{matrix} ) ⟶ ( \begin{matrix}Hom_{Λ}(Q, Q) ⊗_{R} Hom_{Λ}(C, C) \\ Hom_{Λ}(P, Q) ⊗_{R} Hom_{Λ}(C, C) \\ Hom_{Λ}(P̃, Q) ⊗_{R} Hom_{Λ}(C, C) \end{matrix} )

ϕ ⟶ ϕ ⊗ id_{C}

and Q ⟶ L = End_{Λ}(L) in the obvious way. Since the two mappings coincide in \underline{End}_{Λ}(L), this defines a mapping Q ⟶ X_{0}. The mapping P ⟶ Ω̂L yields the diagram

P ⟶^{φ} Q
↓ χ  ↓
X_{1} ⟶ X_{0}

commutative. Hence, we get a morphism of complexes of left Λ-modules. By construction this mapping of complexes induces an isomorphism on the level of the degree 0 homology, namely the identity on L.

We shall prove that χ|_{C} is an isomorphism. This also is almost already done by the construction. In fact, χ|_{C} has image in ker(X_{1} ⟶ X_{0}) which is H_{1}(X). In turn, the first column there is C.

χ|_{C} : ( \begin{matrix}Hom_{Λ}(Q, C) \\ Hom_{Λ}(P, C) \\ Hom_{Λ}(P̃, C) \end{matrix} ) ⟶ ( \begin{matrix}0 \\ Hom_{Λ}(P, C) \\ Hom_{Λ}(P̃, C) \end{matrix} )

which is the identity mapping.

This completes the proof of the first part of the Lemma.

We come to the statement on the right Γ structure. We shall define first a complex morphism

T_{r} ⟶ X_{r}

which we then prove to be a quasi-isomorphism.

Multiplying by e_{ΩL} gives a morphism

Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ (0 Hom_{Λ}(P, ΩL) Hom_{Λ}(P, e_{ΩL}P̃) ) ⊆ \overline{X_{1}}.

Moreover

id_{Q} ⊗ Hom_{Λ}(ι, C ⊕ P ⊕ P̃) : Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ End_{Λ}(Q) ⊗_{R} Hom_{Λ}(C, C ⊕ P ⊕ P̃) gives a morphism to \overline{X_{0}}. Since

Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ Hom_{Λ}(C, C ⊕ P ⊕ P̃) ⟶ Ext_{Λ}^{1}(L, C)

is exact,

im( id_{Q} ⊗ Hom_{Λ}(ι, C ⊕ P ⊕ P̃) ) ⊆ Ω̂L .

Since the two morphisms coincide in \overline{Ω}, we define this way a morphism

Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ X_{1} .

Moreover, id_{Q} ⊗Hom_{Λ}(ι, C ⊕P ⊕P̃) factorizes via Q^{∗} in the following way. We define the morphism

Q^{∗} ⟶ ( \begin{matrix}Hom_{Λ}(Q, Q) \\ Hom_{Λ}(P, Q) \\ Hom_{Λ}(P̃, Q) \end{matrix} ) ⊗_{R} Hom_{Λ}(C, C ⊕ P ⊕ P̃)

q ⟶ ( \begin{matrix}id_{Q} \\ 0 \\ 0 \end{matrix} ) ⊗ q̂

where Q^{∗} ∋ q ⟶ q̂ ∈ Hom_{Λ}(C, C ⊕ P ⊕ P̃) is the mapping given by the defining property of Q^{∗} as pullback. We have a natural map Q^{∗} ⟶ End_{Λ}(L) also by the defining property of Q^{∗} as pullback. Since these two mappings coincide in \overline{End}_{Λ}(L), this defines a mapping Q^{∗} ⟶ X_{0}.

which makes the diagram

Hom_{Λ}(P, C ⊕ P ⊕ P̃) ⟶ Q^{∗}
↓ ξ  ↓
X_{1}  ⟶ X_{0}

commutative. By construction this mapping induces an isomorphism on the level of the degree 0 homology. Again, we shall show that ξ|_{Hom_{Λ}(ΩL,C⊕P ⊕P̃)} is an isomorphism. The argument there is completely analogous to the one we discussed for the Λ-structure.

This completes the proof of Lemma 13.

7 The main theorem

We are now ready to formulate the principal theorem.

Theorem 2 Under the hypotheses of Theorem 1 suppose in addition that Hom_{Λ}(C, ΩL) = 0 and that Γ is a Gorenstein order. Then the complex

… ⟶ 0 ⟶ X_{1} ⟶ X_{0} ⟶ 0 ⟶ …

is a two-sided tilting complex in D^{b}(Λ ⊗ Γ^{op}). The restrictions of X to Λ and to Γ^{op} are both one sided tilting complexes of the type described in Theorem 1.

Proof. By Lemma 13 and Lemma 12 the complex X restricts as complex of Λ-modules to T and as complex of Γ-modules to T^{∗}. Lemma 6 assures that T^{∗} is a tilting complex over Γ^{op} and Corollary 1 proves that it has endomorphism ring Λ^{op}. Theorem 1 assures that T is a tilting complex over Λ with endomorphism ring Γ. Lemma 7 proves that X then is a two-sided tilting complex as claimed.

As a corollary it is possible to compute the perfect isometry ([1, 8]) induced by the complex X. In fact, the only thing we have to do is to separate K ⊗_{R} P ≃ K ⊗_{R} ΩL ⊕ K ⊗_{R} C. To simplify the notation we write KΛ instead of K ⊗_{R} Λ etc. Then,

K ⊗_{R} Γ ≃ End_{KΛ}(KL ⊕ KC ⊕ KΩL ⊕ C ⊕ KP̃)
≃ End_{KΛ}(KL ⊕ (KC)^{2} ⊕ KΩL ⊕ KP̃)

and

K ⊗_{R} Λ ≃ End_{KΛ}(KL ⊕ KC ⊕ (KΩL)^{2} ⊕ KP̃)

Both expressions simplify considerably using that

Hom_{Λ}(L, ΩL ⊕ C ⊕ P̃) = Hom_{Λ}(C, ΩL) = 0.

Corollary 2 The perfect isometry induced by the complex X in theorem 2 is up to a global sign change

−[KL] + [( \begin{matrix}Hom_{KΛ}((KΩL)^{2}, KΩL) & 0 & Hom_{KΛ}((KΩL)^{2}, KP̃) \\ 0 & Hom_{KΛ}(KC, KC^{2}) & Hom_{KΛ}(KC, KP̃) \\ Hom_{KΛ}(KP̃, KΩL) & Hom_{KΛ}(KP̃, KC^{2}) & End_{KΛ}(KP̃) \end{matrix} )]

where the rings KΛ and KΓ act via the above matrices. Moreover, any perfect isometry induced by a derived equivalence whose one-sided tilting complex equals T of theorem 1 differs from the above only by an automorphism of Γ.

In fact, the perfect isometry is just the character induced by −[H_{0}(X)]+[H_{1}(X)] in the Grothendieck group of K ⊗_{R} Λ ⊗_{R} Γ^{op}. The second part of the corollary follows from joint work with R. Rouquier [14] (see also [17, Theorem 4])

Remark The perfect isometry changes the characters coming from the Λ-module ΩL to those coming from the Γ-module Hom_{Λ}(C ⊕ P ⊕ P̃, ΩL), the characters coming from the Λ-module C to the Γ-module Hom_{Λ}(C ⊕ P ⊕ P̃, C), associates to the character coming from L to the virtual character associated to the Γ-module −(End_{Λ}(L)) and leaves the rest of the characters unchanged.

8 Examples

8.1 Blocs with cyclic defect groups:

A block B of a group ring RG for a finite group G over a complete discrete valuation ring R with cyclic defect group is a Green order. A construction which is very similar to the above was carried out by the author in this special case in [18] (see also [8, section 6.4]). The complex carried out there is isomorphic to the one constructed above if one specializes to this special case.

8.2 Graph orders:

M. Kauer informed the author that a suitable generalization of the concept of Green orders, so called ’Graph orders’ [13], are discussed in his doctoral dissertation [5] and the classification of the derived equivalence classes of these is done by tilting complexes satisfying our hypotheses.

8.3 Algebras of semidihedral type:

We shall illustrate now, how one can apply theorem 2 even in case of finite dimensional algebras. Let R be a complete discrete valuation ring with field of fractions K of characteristic 0 and with algebraically closed residue field k of characteristic 2.

Type SD(3H)^{s} is equivalent to SD(3C_{2,I})^{s}: Th. Holm gave a tilting complex T over a finite dimensional k-algebra, named A := SD(3H)^{s} in [4] with endomorphism ring being a k-algebra B := SD(3C_{2,I})^{s}. The algebra A has three projective indecomposable modules P_{0}, P_{1} and P_{2} and K. Erdmann shows [2, 3] that a block with semidihedral defect group has a restricted structure as algebra and two of the possible algebras are A and B for certain parameters s.

Suppose that A is the image of a symmetric Gorenstein order, e.g. a block of a group ring kG.

Then, in this case, there is a symmetric R-order Λ, e.g. the block of RG, with R/rad R = k, such that Λ ⊗_{R} k ≃ A. In [2] it is proved that then, the decomposition matrix of A is of the form

D = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{matrix} )_{(∗)}

where (∗) means that the second lowest line of the matrix has to be taken 2^{n−2} − 1 times; where the parameter s equals 2^{n−2} which is one fourth the order of the defect group. Let K be the field of fractions of R. Following [4] we have

T : … ⟶ P_{0} ⊕ P_{1} ⊕ P_{1} ⟶^{(0,0,δ)} P_{2} ⟶ 0 ⟶ …

for a certain mapping δ. This is a complex as discussed in [7]. Then, by the composition series of the module P_{2} as described in [3] or directly by the quiver in [2], one gets coker δ ≃ S_{2} is a simple module.

Let T̂ be the unique lifting of T to Λ (the existence and unicity is proved in [12]); i.e. the unique tilting complex T̂ in D^{b}(Λ) with T̂ ⊗_{R} k ≃ T. Set L := H_{0}(T̂). Since K ⊗_{R} T is again a tilting complex over K ⊗_{R} Λ, expressing the images of the projective indecomposable Λ-modules in K ⊗_{R} Λ by means of their columns in the decomposition matrix, one gets that the complex is spliced together from the two exact sequences of K ⊗_{R} Λ-modules

0 ⟶ ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 1 \\ 1 \\ 1 \\ 0 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix} )_{(∗)} ⟶ 0

and

0 ⟶ ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 1 \\ 0 \\ 1 \\ 1 \\ 1 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{matrix} )_{(∗)} ⟶ 0 .

One sees that

K ⊗_{R} C ≃ ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix} )_{(∗)} and K ⊗_{R} ΩL ≃ ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{matrix} )_{(∗)}

which obviously do not have any common simple direct summand. Hence,

Hom_{Λ}(C, ΩL) = 0.

We have to prove that H_{0}(T) is torsion free. We have seen that coker δ is simple. Let L_{t} be the torsion submodule of L. Then, by vanishing of T or_{1}^{R}(k, R^{dim_{K}(K⊗_{R}L)}), we get that k ⊗_{R} L_{t} is a non zero submodule of k ⊗_{R} L. Now, k ⊗_{R} H_{0}(T) ≃ coker δ. Therefore, either L_{t} = L or L_{t} = 0. Since

K ⊗_{R} L ≃ ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix} )_{(∗)}

which is non zero, hence, L_{t} = 0 and L is torsion free. Therefore, the hypotheses of theorem 2 are satisfied. Theorem 2 gives us a two-sided tilting complex X in D^{b}(Λ ⊗_{R} Γ^{op}) with Γ being the endomorphism ring of T̂. Moreover, one knows by [12] that k ⊗_{R} Γ ≃ B. By [11], one gets that k ⊗_{R} X is a two-sided tilting complex in D^{b}(A ⊗_{R} B^{op}).

Type SD(3H)^{k} is equivalent to SD(3C_{2,II})^{k}: The very same argument applies to the tilting complex T given by TH. Holm between the forenamed types of algebras. The tilting complex he uses is

T : … ⟶ 0 ⟶ P_{1} ⊕ P_{0} ⊕ P_{0} ⟶ P_{2} ⟶ 0 ⟶ …

with simple homology in degree 0 over the algebra C := SD(3C_{2,II})^{k}. For the decomposition matrix Erdmann give two possibilities depending on the parameters, namely:

D = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \end{matrix} )_{(∗)} or D = ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{matrix} )_{(∗)}

If there is a symmetric order Λ with k ⊗_{R} Λ ≃ C, then our theorem applies. In fact, then there is a unique tilting complex T̂ with k ⊗_{R} T̂ ≃ T and endomorphism ring being an R-order reducing to B. Tensored with K the tilting complex is a splicing of the short exact sequences

0 ⟶ ( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}1 \\ 0 \\ 1 \\ 0 \\ 1 \\ 1 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \\ 1 \\ 0 \end{matrix} )_{(∗)} ⟶ 0

and

0 ⟶ ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \\ 1 \\ 0 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 1 \\ 1 \\ 1 \\ 0 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix} )_{(∗)} ⟶ 0

or in the second case

0 ⟶ ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1 \end{matrix} )_{(∗)} ⟶ 0

and

0 ⟶ ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}1 \\ 1 \\ 0 \\ 1 \\ 0 \\ 1 \end{matrix} )_{(∗)} ⟶ ( \begin{matrix}1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} )_{(∗)} ⟶ 0

One observes that the second case does not produce a tilting complex. Hence, the second decomposition matrix does not occur in the case SD(3C_{2,II}). The rest of the argument is completely analogous to the afore discussed case.

Existence of a lifting to an order: The question if there is an order Λ such that Λ ⊗_{R} k is Morita equivalent to A can be treated using theorem 1.

The algebras A = SD(3H)^{s} and E := SD(3D)^{s} of Erdmann’s list in [2] are derived equivalent, choosing parameters for E such that blocks theoretically could occur, as shown in [4] by giving a tilting complex over A with endomorphism ring E. We know, that the algebra E is Morita equivalent to a principal block of a group ring, namely B_{0}(P SL_{3}(q)) over an algebraically closed field of characteristic 2, the principal block of the projective special linear group of degree 3 over 𝔽_{q}, the field with q elements with q ≡ 3 (mod 4) (see the remark at the end of [3]). Then, s = 2^{n−2} where the Sylow-2 subgroup of P SL_{3}(q) is semidihedral and has order 4s.

We can give a tilting complex T over E with endomorphism ring being isomorphic to A.

Let P_{0}, P_{1}, P_{2} be the three projective indecomposable modules of D. We apply theorem 1 to L being the top of P_{0}. Identify P_{1} with the projective indecomposable corresponding to the vertex 2 in A and P_{2} with the projective indecomposable corresponding to the vertex 1 in A. Then, the main result in [7] gives that End_{D^{b}(E)}(T) ≃ A. Moreover L is simple. Using that the decomposition matrix is, according to [2]

D = ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )_{(∗)}

and hence the lifting T̂ of T to the corresponding order has a non torsion part in the degree 0 homology, we see that H_{0}(T̂) is a lattice. The hypothesis of theorem 2 is not satisfied. Nevertheless, theorem 1 shows that there is an order Λ with Λ ⊗_{R} k ≃ A for s = 2^{n}.

We even do not have to verify that End_{D^{b}(E)}(T) ≃ A. The complex Th. Holm gives is a two term complex S with H_{0}(S) being simple. We may assume that A and E are basic. Then, the k-dimension of H_{0}(S) is one. Hence, the existence of a twosided tilting complex X being isomorphic to S if restricted to the left implies the existence of a two term tilting complex T over E with endomorphism ring being A and simple homology in degree 0. Since homology being the simple with projective cover P_{1} or P_{2} do not lead to a T with sufficiently big k-dimension in degree 1, the complex we give above is the correct one.

Remark It is not clear to the author if there is always an R-order Λ reducing to A. As mentioned in [15] this seems to be an open problem in general.

References

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[13] K. W. Roggenkamp, Generalized Brauer Tree Orders, Colloq. Math. 71 (1996) 225-242.

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Address of the author:

Alexander Zimmermann LAMFA Faculté de Mathématiques Université de Picardie Jules Verne 33, rue Saint Leu 80039 Amiens CEDEX France

e-mail: Alexander.Zimmermann@u-picardie.fr