Automorphisms of Green orders and their derived categories

Alexander Zimmermann^{∗}

Faculté de Mathématiques; LAMFA (CNRS FRE 2270); Université de Picardie 33, rue St Leu; 80039 Amiens Cedex; France

                  electronic mail: Alexander.Zimmermann@u-picardie.fr
               http://delambre.mathinfo.u-picardie.fr/alex/azim.html

October 2000; revised May 2001

Abstract

In an earlier paper Raphaël Rouquier and the author introduced the group of selfequivalences of a derived category. In the case of a Brauer tree algebra we determined a non trivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.

Introduction

The representation theory of a group G with cyclic p-Sylow subgroup over a big enough field k of characteristic p is completely understood. It is a classical result that the indecomposable factors of the group ring are Brauer tree algebras. Brauer tree algebras are defined by means of a combinatorial object, a finite tree with some additional data, and their representations are deduced by some combinatorial process.

In order to study many of the more subtle questions for representations of kG, it has proved desirable to replace the coefficient domain k by a complete discrete valuation ring R of characteristic 0 with residue field k and field of fractions K. In order to do so we have to replace Brauer tree algebras by another object. This object is called Green order and is introduced and studied by Roggenkamp in [10]. Green orders as well are defined by a finite tree together with some additional structure. The representation theory of Green orders is quite well understood and any indecomposable factor of RG is a Green order whenever G has cyclic p-Sylow subgroup. Moreover, for any Green order Λ we have that Λ ⊗_{R} k is a Brauer tree algebra and for any Brauer tree algebra A there is a Green order Λ so that Λ ⊗_{R} k ≃ A.

During the last decade parts of representation theory of groups were dominated by considerations of their derived categories. In an earlier work with Raphaël Rouquier the group of self-equivalences of the derived category of an algebra is studied [12]. We put special emphasis on the case of a Brauer tree algebra. There we get a close connection to Artin braid groups. In fact, let A be the Brauer tree algebra associated to a Brauer tree with n edges and no exceptional vertex. Then we prove that there is a homomorphism φ_{n} of the braid group B_{n+1} on n + 1 strings to the group T rP ic_{k}(A) of self-equivalences of standard type of the derived category of A. The homomorphism φ_{2} is injective and the image of φ_{2} is normal with cokernel of order 4 · | P ic_{k}(A) |. In a recent paper Khovanov and Seidel [3] announce that φ_{n} is injective for all n.

∗2000 AMS Mathematics Subject Classification: 16G30, 20C10, 18E30

In the present paper we continue this research and get a very similar result replacing Brauer tree algebras by Green orders. We shall prove that there is a group homomorphism ψ_{n} from the Artin braid group on n + 1 strings to the group of self-equivalences of the derived category of a Green order Λ_{n}. This Green order Λ_{n} is associated to a tree with n edges and with respect to some additional hypothesis corresponding to the absence of an exceptional vertex in the Brauer tree algebra setting. Then, Λ_{n} ⊗_{R} k =: A_{n} is a Brauer tree algebra without exceptional vertex. Moreover, denoting by π_{n} the group homomorphism T rP ic_{R}(Λ_{n}) ⟶ T rP ic_{k}(A_{n}) induced by − ⊗_{R} k, we get π_{n} ∘ ψ_{n} = ϕ_{n}.

We emphasize that the group ring ℤ̂_{p}𝔖_{p} of the symmetric group of degree p over the p-adic integers ℤ̂_{p} is an example of Λ_{p−1}.

We shall study the case n = 2 in greater detail since the braid group B_{3} on 3 strings has arithmetic relations coming from B_{3}/Z(B_{3}) ≃ P SL_{2}(ℤ). In this case we shall identify naturally defined normal subgroups T rP icent(Λ) and T rI_{R}(Λ) of T rP ic_{R}(Λ) with the level two congruence subgroup Γ(2) of P SL_{2}(ℤ) and its derived subgroup Γ(2)′. In fact, T rP icent(Λ) is the kernel of the natural homomorphism T rP ic_{R}(Λ) ⟶ Aut_{R}(Z(Λ)) induced by taking the action on degree 0 Hochschild cohomology. The group T rI_{R}(Λ) is the kernel of the group homomorphism T rP ic_{R}(Λ) ⟶ T rP ic_{K}(K ⊗_{R} Λ) induced by the functor − ⊗_{R} K. We should point out the fact that T rI_{R}(Λ) is constructed using an order and is invisible on the level of the Brauer tree algebra.

The paper is organized as follows. In the sections 1.1 and 1.2 we review the basic notations and results on derived equivalences of algebras [4] and put together the results of [12] as long as they are necessary for the present paper. In section 1.3 we review in some detail Roggenkamp’s theory of Green orders [10]. In section 2 we prove, under some assumptions on the combinatorial data of a Green order, that the identity is the only automorphism of a Green order fixing the indecomposable projective modules. Section 3 uses the results of section 2 to construct the homomorphism ψ_{n}. Finally section 4 gives the relation to the congruence subgroups.

1 Recall some facts

1.1 Review on derived equivalences

Let A and B be R-algebras over a hereditary, commutative ring R and assume that A and B are projective as modules over R. We denote by D^{b}(A) the derived category of bounded complexes of A-modules. We follow the conventions of [4].

(1.1) Then Rickard [9], and under weaker hypotheses Keller [2], show that if there is an equivalence of triangulated categories D^{b}(A) ≃ D^{b}(B) then there is a bounded complex X in D^{b}(A ⊗_{R} B^{op}) so that X ⊗_{B}^{𝕃} − is an equivalence. Moreover, the inverse functor is given by left derived tensor product with a complex Y in D^{b}(B ⊗_{R} A^{op}). Equivalences given by tensor product by a bounded complex of bimodules are called of standard type. The complex X is called a two-sided tilting complex.

(1.2) A one-sided tilting complex is a bounded complex T of projective modules so that Hom_{D^{b}(A)}(T, T [i]) = 0 if i ≠ 0 and so that the rank one free A-module is contained in the smallest triangulated category which is closed under direct sums and direct summands and which contains T. The image of the rank one free module by tensor product with X is a one-sided tilting complex, or tilting complex for short. Given a one-sided tilting complex T, then there is a two-sided tilting complex X so that T is the image of the rank one free module by tensoring with X. This complex X is unique up to an automorphism of A.

(1.3) Suppose that R is a complete discrete valuation ring with residue field k. Let A be an R-order. Suppose that \overline{T} is a tilting complex in D^{b}(k ⊗_{R} A). Then, (cf Rickard [8]) up to isomorphism there is a unique tilting complex T in D^{b}(A) so that k ⊗_{R} T ≃ \overline{T}. In this case k ⊗_{R} End_{D^{b}(A)}(T) ≃ End_{D^{b}(k⊗_{R}A)}(\overline{T}).

(1.4) We define [12]

T rP ic_{R}(A) := { isomorphism classes of two-sided tilting complexes in D^{b}(A ⊗_{R} A^{op})}

It is clear by the definition that in case A and B are R-algebras which are projective as R-modules then,

D^{b}(A) ≃ D^{b}(B) ⟹ T rP ic_{R}(A) ≃ T rP ic_{R}(B) .

(1.5) We recall one of the main results of [12]. Denote by

B_{n} =< s_{1}, s_{2}, …, s_{n} | s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} ;

s_{i}s_{j} = s_{j}s_{i}∀i ∈ {1, …, n − 1}∀j ∈ {2, …, n} \ {i + 1, i − 1} >

the Artin braid group on n strings and let B̃_{3} be the extension of B_{3} =< s_{1}, s_{2} > generated by z, s_{1} and s_{2} with the relations z^{4} = (s_{1}s_{2})^{3} and zs_{1}z^{−1}s_{1}^{−1} = zs_{2}z^{−1}s_{2}^{−1} = 1. Let μ_{n}(R) be the set of n-th roots of unity of a ring R and let C_{m} be the cyclic group of order m.

Theorem 1 [12] Let k be a field and let A_{n} be a Brauer tree algebra over k associated to a tree with n edges and without exceptional vertex. Then, there is a group homomorphism

φ_{n} : B_{n+1} ⟶ T rP ic_{k}(A_{n})

so that each of the generators of B_{n+1} generate an infinite cyclic group in T rP ic_{k}(A_{n}). φ_{2} is injective and induces an isomorphism

T rP ic_{k}(A_{2}) ≃ B̃_{3} ×P ic_{k}(A_{2})

where P ic_{k}(A_{2}) ≃ C_{2} × (k^{×}/μ_{n}(k)) .

It is important to obtain the images of the generators of the braid group explicitly. Assume that the Brauer tree is a stem. Denote by P_{1}, P_{2}, …, P_{n} the projective indecomposable A-modules. Rearranging the numbering if necessary we may and will assume that Hom_{A}(P_{i}, P_{i−1}) ≠ 0 for any i ∈ {2, 3, …, n}. Then, for any i = 1, …, n the image of the standard braid group generator φ_{n}(s_{i}) is isomorphic to the complex

… ⟶ 0 ⟶ P_{i} ⊗_{k} Hom_{A}(P_{i}, A) ⟶ A ⟶ 0 ⟶ …

concentrated in degree 0 and −1.

We recall the result of [12] which implies that φ_{2} is injective. For any bounded complex Y = (Y, d) of projective A-modules denote by Y^{mod} the underlying A-module of Y forgetting the differential and the graduation. Now, let α : B_{3} ⟶ P SL_{2}(ℤ) and let w ∈ B_{3} with

α(w) = ( \begin{matrix}a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{matrix} ). It is proved in [12] that for every i ∈ \{1, 2\} the complex of smallest

dimension T_{i} which is homotopy equivalent to the complex φ_{2}(w) ⊗_{A} P_{i} has the property (T_{i})^{mod} ≃ P_{1}^{|a_{1,i}|} ⊕ P_{2}^{|a_{2,i}|}.

1.2 Some subgroups of T rP ic

In [12] various subgroups of T rP ic_{R}(A) are defined. In case A is an order, we shall define one more interesting subgroup.

(1.6) Assume that R is an integral domain with field of fractions K and assume that A is an R-algebra, projective as R-module. Then, the functor K ⊗_{R} − : D^{b}(A) ⟶ D^{b}(K ⊗_{R} A) induces a group homomorphism

T rP ic_{R}(A) ⟶ T rP ic_{K}(K ⊗_{R} A) .

We denote by T rI_{R}(A) the kernel of this homomorphism.

Observe that the homology of each element in T rI_{R}(A) is R-torsion in non zero degrees. Moreover, in case A is an R-order, the homology in degree 0 contains a full A ⊗_{R} A^{op}-lattice in K ⊗_{R} A.

(1.7) J. Rickard proves in [9] that each self-equivalence of standard type of D^{b}(A) induces a graded automorphism of the Hochschild cohomology ring HH^{∗}(A) of A. Following [12] we denote by T rP icent(A) the kernel of the homomorphism

T rP ic_{R}(A) ⟶ Aut(HH^{0}(A)) .

1.3 Review on Green Orders

For the reader’s convenience we recall Roggenkamp’s notion of a ’Green order’. These orders are introduced to explain the structure of blocks with cyclic defect group. In fact, Green orders are a special class of rings with a very well prescribed ring theoretical structure. On the other hand Roggenkamp proves that every block of a group ring RG of a finite group G with cyclic defect group over a complete discrete valuation ring R of characteristic 0 is Morita equivalent to a Green order.

(1.8) Let R be a Dedekind domain with field of fractions K. Recall that an R–order is an R–algebra Λ which is finitely generated and projective as R–module and such that K ⊗_{R} Λ is a semisimple K–algebra.

(1.9) For the definition of a Green order we follow a suggestion of Luis Puig. 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 1.1 The indecomposable R–order Λ in the semisimple K-algebra K ⊗_{R} Λ is called a Green order if

• there is a set E of central idempotents of K ⊗_{R} Λ 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)}) ≃ Λ · π(t) · θ(t) .

(1.10) Let R be a complete discrete valuation ring of characteristic 0 with residue field of characteristic p and with big enough field of fractions. Let G be a finite group and let B(RG) be a block of RG with cyclic defect group. Roggenkamp proves in [10] that then, Λ is a Green order with the following additional property. There is at most one e_{0} ∈ E with End_{Λ}(Λπ(t)θ(t)) ≄ R for θ(t) = e_{0}. This e_{0} corresponds to the exceptional vertex of a Brauer tree algebra.

(1.11) Roggenkamp’s (equivalent) definition of a Green order goes another way. Roggenkamp has three conditions. The third asks for the existence of what people call ”Green’s walk around the Brauer tree” as a projective resolution of Λ · π(t)θ(t) for any t. This condition is exactly the second of our defining properties. Roggenkamp’s second condition states that the edges of the underlying tree correspond to the projective indecomposable modules, a fact which we ensure by asking that the primitive idempotents are the edges of a tree. Roggenkamp’s first condition asks that the vertices of the tree correspond to central idempotents in K ⊗_{R} Λ which add to the identity. We just reproduced this condition.

(1.12) If we abbreviate Λ · t = Λπ(t)θ(t) we have the following main result of [10].

Theorem 2 (Roggenkamp [10])

• Let Λ be a basic Green order. Then, the tree T as above is totally ordered by ω and the choice of a first element.

There is an R–torsion R–algebra \overline{Ω_{Λ}} and a family (f_{t} : tΛt ⟶ \overline{Ω_{Λ}}) of R–algebra homomorphisms with kernel being a principal ideal a_{t}tΛt in the radical of tΛt such that the first element of T can be chosen 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) ⟶^{·θ(t)} tΛt
↓ ·θ(t)′  ↓ f_{t}
t′Λt′  ⟶^{f_{t′}}  \overline{Ω_{Λ}}

• Moreover, let Λ be an R-order. Suppose that there is a complete set of primitive idempotents I of Λ and a set of central idempotents E in K⊗_{R}Λ such that ∑_{e∈E} e = 1, and so that the set T = {(i, e) ∈ I × E|i · e ≠ 0} is a tree which may be totally ordered by the choice of a lowest element t_{1} of T. If the Pierce decomposition has the properties (1), (2), (3) and (4) then Λ is a Green order.

(1.13) One should observe that since the elements in E are central and pairwise orthogonal, most of the entries in the Pierce decomposition are zero. The matrix coefficients fall naturally into matrix rings Ae for e ∈ E. Furthermore, by the first and the third points of the theorem, the matrix rings Ae are upper triangular n_{e} ×n_{e}–matrix rings over Ω_{θ(t)} := tΛt which does only depend on θ(t) and the lower diagonal entries is in a principal ideal a_{t} · tΛt, again depending only on θ(t). Here n_{e} = |θ^{−1}(e)|. In other words, for all t ∈ T,

H_{θ(t)} := Λ · θ(t) = ( \begin{matrix}Ω_{θ(t)} & Ω_{θ(t)} & … & … & Ω_{θ(t)} \\ (a_{θ(t)}) & Ω_{θ(t)} & … & … & Ω_{θ(t)} \\ ⋮ & ... & ... & & ⋮ \\ ⋮ & & ... & ... & ⋮ \\ (a_{θ(t)}) & … & … & (a_{θ(t)}) & Ω_{θ(t)} \end{matrix} )_{n_{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 [10]. 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.

(1.14) We call the set {(Ω_{e}, f_{e}, e)|e ∈ E} the structure data for the Green order A.

(1.15) As is done for Brauer tree algebras by J. Rickard [7], two Green orders Λ and Γ have equivalent derived categories if there is a bijection β between the sets E_{Λ} and E_{Γ} and such that

∀_{e_{Λ}∈E_{Λ}} Ω_{e_{Λ}} = Ω_{β(e_{Λ})} , \overline{Ω_{Λ}} = \overline{Ω_{Γ}} and (f_{Λ})_{e_{Λ}} = (f_{Γ})_{β(e_{Λ})}

This fact was proved by K.W.Roggenkamp and the author. The result was published in [14].

2 Automorphisms of Green orders

Let R be a complete discrete valuation domain of characteristic 0.

Even though we shall restrict to very special Green orders in our main application, recent discoveries [1] of Y. Drozd on polynomial functors seem to be enough motivation to keep the discussion more general.

The following result may be of independent interest.

Theorem 3 Let Λ be a basic Green order over R with structural data {(Ω_{e}, f_{e}, e)| e ∈ E}. Assume that Ω_{e} is a commutative maximal order in a field^{1} for all e ∈ E, and that ker(f_{e}) = rad Ω_{e} for all e ∈ E. Then, every ring automorphism of Λ which is linear over the centre of Λ and which maps each projective Λ–module to an isomorphic copy is inner.

Proof. We recall that a Green order Λ with data (f_{e}, Ω_{e})_{e∈E} can be embedded into an overorder H := ∏_{e∈E} H_{e} with H_{e} = Λ · e for any e ∈ E.

H_{e} = ( \begin{matrix}Ω_{e} & Ω_{e} & … & … & Ω_{e} \\ (a_{e}) & Ω_{e} & … & … & Ω_{e} \\ ⋮ & ... & ... & & ⋮ \\ ⋮ & & ... & ... & ⋮ \\ (a_{e}) & … & … & (a_{e}) & Ω_{e} \end{matrix} )_{n_{e}×n_{e}}

Our assumption implies that Ω_{e} is a complete discrete valuation ring and that a_{e} generates its radical for all e ∈ E. Then, the orders H_{e} are hereditary for all e ∈ E ([6, (39.14)]). By [6, Exercise 39.6], we see that the outer automorphism group of H_{e} is cyclic of order n_{e}. Conjugation by the element

1this implies that Ω_{e} is itself a complete discrete valuation domain

w_{e} := ( \begin{matrix}0 & 1 & 0 & … & 0 \\ ⋮ & ... & ... & ... & ⋮ \\ ⋮ & & ... & ... & 0 \\ 0 & & & ... & 1 \\ a_{e} & 0 & … & … & 0 \end{matrix} )_{n_{e}×n_{e}}

is an outer automorphism of H_{e} which is easily seen to be of order n_{e} in Aut(H_{e}). The conjugation by w_{e} is denoted by ω_{e}.

Let α be an automorphism of Λ which we assume to fix the central idempotents in E of K ⊗_{R} Λ. Then, α extends to an automorphism of

Ω := ∏_{e∈E} H_{e},

which, by abuse of notation, will be also denoted by α. By the above, there exist integers k_{e}; e ∈ E with 0 ≤ k_{e} < n_{e} and inner automorphisms γ_{e}; e ∈ E of H_{e} such that

α = ∏_{e∈E} ω_{e}^{k_{e}}γ_{e}

on Ω. We study the inner automorphisms γ_{e}.

Claim 2.1 Let Λ be a Green order with respect to the set of central idempotents E with structural data (Ω_{e}, a_{e}) such that all the Ω_{e} are commutative. Let u_{e} be a unit in H_{e} := Λ·e. Then, there is a unit u of Λ such that

u^{−1} · ∏_{e∈E} u_{e} ∈ centre(Ω) .

Proof. We shall construct u inductively. Since

H_{e} = ( \begin{matrix}Ω_{e} & Ω_{e} & … & … & Ω_{e} \\ (a_{e}) & Ω_{e} & … & … & Ω_{e} \\ ⋮ & ... & ... & & ⋮ \\ ⋮ & & ... & ... & ⋮ \\ (a_{e}) & … & … & (a_{e}) & Ω_{e} \end{matrix} )_{n_{e}×n_{e}}

and (a_{e}) is in the Jacobson radical of Ω_{e}, a unit u_{e} in H_{e} has to have the property that the main diagonal entries iu_{e}i of u_{e} are units in Ω_{e} for all e ∈ E and i ∈ I.

Let e_{0} be a vertex in E.

Suppose we already constructed units v_{e} in H_{e} for those e ∈ E for which the distance of e to e_{0} is less or equal to a fixed number d and

• so that u_{e}^{−1} · v_{e} ∈ centre(H_{e})

• and so that the equation f_{e}(i · v_{e} · i) = f_{e′}(i · v_{e′} · i) holds for any i ∈ I and e′ ∈ E subject to the condition i · e ≠ 0 ≠ i · e′ and with distance from e′ to e_{0} is less than d.

Let e_{n} ∈ E be a vertex at distance d + 1 from e_{0}. Since T is a tree, there is a unique e_{l} ∈ E with distance d from e_{0}. Let j be the edge between e_{l} and e_{n}. Then,

f_{e_{n}}(j · u_{e_{n}} · j)^{−1} · f_{e_{l}}(j · v_{e_{l}} · j)

is a unit in \overline{Ω_{Λ}}. By Nakayama’s lemma there is a unit x_{e_{n},i} ∈ Ω_{e} so that

f_{e_{n}}(x_{e_{n},i}) = f_{e_{n}}(j · u_{e_{n}} · j)^{−1} · f_{e_{l}}(j · v_{e_{l}} · j) .

Now,

v_{e_{n}} := x_{e_{n},i} · u_{e_{n}}

has the property that

1. ) u_{e_{n}}^{−1} · v_{e_{n}} ∈ centre(H_{e_{n}}) and that

2. ) f_{e_{n}}(j · v_{e_{n}} · j) = f_{e_{l}}(j · v_{e_{l}} · j) in \overline{Ω_{Λ}}.

By induction we define this way an element u := ∏_{e∈E} v_{e}. By the previous two properties 1.) and 2.) we get u ∈ Λ. Moreover, the construction of modifying successively by entral elements implies that

u · ∏_{e∈E} u_{e} ∈ centre(Ω) .

By the second part of Theorem 2 we see that u ∈ Λ. Since u is in Λ and u is a unit in H, by a lemma of Roggenkamp and Scott (cf. [11, Lemma 3]) we conclude that u is actually a unit in Λ. This proves the claim.

Remark 2.2 Observe that this proof works in the same way for orders which are defined similarly to Green orders, with the difference that one allows the quotient \overline{Ω} attached to the edges of T to depend on the edge we are faced with. For Green orders of course it is not necessary to use an induction. An explicit formula can easily be given.

By Claim 2.1 conjugation by u induces the automorphism ∏_{e∈E} γ_{e} on K ⊗_{R} Λ. Hence, ∏_{e∈E} γ_{e} is actually an inner automorphism of Λ.

Modifying α by ∏_{e∈E} γ_{e} we may assume that γ_{e} = 1 for all e ∈ E.

But, α_{e} induces a cyclic permutation of order n_{e} of

H_{e}/ω_{e}H_{e} = ∏_{j=1}^{n_{e}} Ω_{e}/(a_{e}).

Hence, in case one α_{e} is not the identity, for an idempotent i ∈ I with i · e ≠ 0, we get α(i) ∉ Λ. Therefore, k_{e} = 0 for all e ∈ E.

The only possibility now is an automorphism which is not the identity on the central idempotents of E in K ⊗_{R} Λ. This automorphism, however, necessarily moves projective indecomposable Λ–modules. This proves Theorem 3.

3 Consequences for the self-equivalences of the derived category of a Green order

We shall derive some consequences of the result in section 2 for the group T rP ic_{R}(Λ).

Proposition 3.1 Let Λ be a basic Green order over R with structural data {(Ω_{e}, f_{e}, e)| e ∈ E}. We assume that Ω_{e} is a commutative maximal order in a field for all e ∈ E, and that ker(f_{e}) = rad Ω_{e} for all e ∈ E. Given X and Y in T rP icent(Λ), such that for each projective k ⊗_{R} Λ–module P we get X ⊗_{Λ} P ≃ Y ⊗_{Λ} P. Then, X ≃ Y.

Proof. We form Z := Y^{−1} ⊗_{Λ} X. Now, Z ⊗_{Λ} P ≃ P. Since R is a complete discrete valuation ring, also for each projective indecomposable Λ–module Q, we get Z ⊗_{Λ} Q ≃ Q. Then, Z is isomorphic to an invertible bimodule which is isomorphic to Λ as left Λ-module. Now, by [12], Z is isomorphic to _{1}Λ_{f} as Λ ⊗_{R} Λ^{op}-module, where the action from the right is multiplication twisted by an automorphism f fixing each projective Λ–module. Such automorphisms are inner by Theorem 3. Hence, Z ≃ Λ as Λ ⊗_{R} Λ^{∘}–modules.

(3.1) Hypothesis: We will assume in the rest of this section 3 that the graph T is a stem and that the Green order Λ is basic. To avoid trivial cases we assume that |E| ≥ 3. We assume furthermore that

∀_{e,e′∈E} Ω_{e} = Ω_{e′} is a commutative maximal order in a field and f_{e} = f_{e′} .

(3.2) Let E = {e_{1}, e_{2}, …, e_{n}} and for any k ∈ {1, …, n−1} let i_{k} ∈ I be the idempotent with e_{k} · i_{k} ≠ 0 ≠ i_{k}e_{k+1}. Let P_{k} be the indecomposable projective left Λ-module Λ · i_{k} and let P_{k}^{∗} be the projective right Λ-module i_{k} · Λ. Observe that P_{k} is a right Ω_{k}-module and that P_{k}^{∗} is a left Ω_{k}–module.

Then, we define a complex

S_{k} := (… ⟶ 0 ⟶ P_{k} ⊗_{Ω_{k}} P_{k}^{∗} ⟶^{d_{k}} Λ ⟶ 0 ⟶ …)

of Λ ⊗_{Ω_{k}} Λ^{op}–modules with homology concentrated in degree −1 and 0. The differential d_{k} is defined by

d_{k}(α · i_{k} ⊗_{Ω_{k}} i_{k} · α′) = α · i_{k} · α′ for all α, α′ ∈ Λ

Clearly, d_{k} is a mapping of Λ ⊗_{Ω_{k}} Λ^{op}-modules. Define for technical reasons Ω_{0} := Ω_{1}.

Proposition 3.2 Suppose for a k ∈ {1, …, n − 1} that Ω_{k−1} = Ω_{k} = Ω_{k+1}. Then, the complex S_{k} is a two-sided tilting complex.

Proof. We show that S_{k} is a two-sided tilting complex. By a lemma of Rickard [4, Lemma 6.3.15] we have to show that the restriction to the left and to the right hand side is a one-sided tilting complex with endomorphism ring Λ. We get

S_{k} ⊗_{Λ} P_{l} ≃ P_{l} ∀l ∈ {1, …, n − 1} \ {k − 1, k, k + 1}

as is immediately verified. Moreover,

S_{k} ⊗_{Λ} P_{k} ≃ P_{k}[1]

and

S_{k} ⊗_{Λ} P_{k−1} ≃ … ⟶ 0 ⟶ P_{k} ⟶ P_{k−1} ⟶ 0 ⟶ …

when n − 1 ≥ k ≥ 2 as well as

S_{k} ⊗_{Λ} P_{k+1} ≃ … ⟶ 0 ⟶ P_{k} ⟶ P_{k+1} ⟶ 0 ⟶ …

when 1 ≤ k ≤ n − 2. Here the mappings are the homomorphisms with obvious maximal image. It is immediate to see that the triangulated category generated by these complexes contains the rank one free module Λ. The hypothesis on T implies that there are no morphisms between P_{k} and P_{l} for |k − l| ≥ 2. Moreover, any morphism from P_{k} to P_{k−1} factors through the maximal one. There is no nonzero homomorphism from P_{k−1} to P_{k} composing to 0 from the left with the differential since the differential and also any of the above homomorphism is given by multiplication with elements in Ω_{k}, which is an integral domain. The analogous statements hold for P_{k+1}. Altogether this shows that the restriction of S_{k} to the left is a one-sided tilting complex. It remains to see that the endomorphism ring of this tilting complex is Λ. This can be seen as it is carried out in [14]. In case Λ is Gorenstein, a shorter argument can be applied. It is clear that it is enough to show the statement for S_{k}^{∗} = Hom_{Λ}(S_{k}, Λ). Applying [4, Proposition 5.1.1] to the dual S_{k}^{∗}, we see that the endomorphism ring of the restriction of S_{k} to the left has endomorphism ring Λ.

Since the situation is symmetric with respect to the left and the right, we proved the statement.

(3.3) Assume Ω_{1} = Ω_{2} = … = Ω_{n}. Then, for all k = 1, …, n, tensor product with S_{k} induces an element in T rP ic_{R}(Λ). We denote by t_{k} the element in T rP ic_{R}(Λ) which is induced by tensor product by S_{k}. Furthermore we denote by ω the non zero element in the Picard group P ic_{R}(Λ) of Λ. Note that P ic_{R}(Λ) is of order 2.

In case Ω_{k−1}, Ω_{k} and Ω_{k+1} are possibly different, a slight modification of S_{k} gives a twosided tilting complex which does not necessarily induce an auto-equivalence of the derived category.

Theorem 4 Let Λ be a Green order over the complete discrete valuation domain R with structural data (Ω_{e}, f_{e})_{e∈E} associated to a stem T and such that Ω_{1} = Ω_{2} = … = Ω_{n} = R and f_{1} = f_{2} = … = f_{n} is the projection to k := R/rad R. Then, the tilting complexes {S_{1}, …, S_{n}} satisfy the braid relations

S_{i} ⊗_{Λ} S_{j} ≃ S_{j} ⊗_{Λ} S_{i} for |i − j| ≥ 2; i, j = 1, …, n.
S_{i} ⊗_{Λ} S_{i+1} ⊗_{Λ} S_{i} ≃ S_{i+1} ⊗_{Λ} S_{i} ⊗_{Λ} S_{i+1} for i = 1, …, n − 1.

Proof. The statement follows from Theorem 1 and the remark following it in connection with Proposition 3.1.

(3.4) The assumption on the structure of the Green order has the following group theoretical interpretation. By the classical theory of blocks of group rings of finite groups with cyclic defect group, the multiplicity μ can be calculated from the number of edges e of the graph and the size p^{d} of the defect group.

μ = \frac{p^{d} − 1}{e}

with e divides p−1. Hence, μ = 1 is equivalent to e = p−1 and d = 1. Suppose that R is an unramified extension of the p–adic integers with residue field k and let i be an idempotent in the centre of RG. Suppose that kG·i is a Brauer tree algebra with exceptional multiplicity 1.

Roggenkamp proves [10] in this case that RG·i is a Green order with structure data satisfying the hypotheses of the theorem.

(3.5) Moreover, for any perfect field k and any Brauer tree algebra A over k without exceptional vertex there is a complete discrete valuation ring R with residue field k (cf. [13, II § 5 Théorème 3]) and a Green order Λ over R satisfying the assumptions in Theorem 3 such that k ⊗_{R} Λ ≃ A.

Recall that we denote by B̃_{3} the following extension by the braid group on 3 strings.

B̃_{3} :=< s, t, z | z^{4} = (st)^{3}; zs = sz; zt = tz; sts = tst >

Obviously, one has exact sequences of groups 1 ⟶< z >⟶ B̃_{3} ⟶ P SL_{2}(ℤ) ⟶ 1 and 1 ⟶ B_{3} ⟶ B̃_{3} ⟶ C_{4} ⟶ 1 given by identifying s, t with the usual braid generators.

We get the analogous result to Theorem 1 replacing Brauer tree algebras by Green orders.

Theorem 5 Let Λ be a Green order over the complete discrete valuation domain R such that there are exactly 2 non isomorphic simple A-modules. Suppose the structural data (Ω_{e}, f_{e})_{e∈{1,2,3}} have the property Ω_{1} = Ω_{2} = Ω_{3} = R and f_{1} = f_{2} = f_{3} is the projection to R/rad R. Then,

T rP ic_{R}(Λ) ≃ B̃_{3} ×C_{2}.

Proof. Again, this follows immediately from Theorem 1, Theorem 3 and Proposition 3.1.

(3.6) Let R be an integrally closed extension of the 3-adic integers and let 𝔖_{3} be the symmetric group on 3 letters. Then R𝔖_{3} satisfies the hypotheses of Theorem 5. Y. Drozd studied in [1] right continuous quadratic functors. These can be seen as modules over an order A. The completion of this order A at the prime 2 satisfies the hypotheses of the theorem.

4 Some subgroups of T rP icℤ̂_{3}(ℤ̂_{3}𝔖_{3})

Let R be a complete discrete valuation domain with field of fractions K and residue field k. Let Λ be a Green order over R such that there are exactly 2 non isomorphic simple Λmodules. Suppose Λ has that the structural data (Ω_{e}, f_{e})_{e∈{1,2,3}} where Ω_{1} = Ω_{2} = Ω_{3} = R and f_{1} = f_{2} = f_{3} is the natural projection R ⟶ k.

Recall

T rP icent(Λ) := ker(T rP ic_{R}(Λ) ⟶ Aut(Z(Λ)))
T rI_{R}(Λ) := ker(T rP ic_{R}(Λ) ⟶ T rP ic_{K}(K ⊗_{R} Λ))

Theorem 6 There is an exact sequence

1 ⟶ C_{∞} ⟶ T rP icent(Λ) ⟶ Γ(2) ⟶ 1

where Γ(2) is the congruence subgroup of level 2 of P SL_{2}(ℤ) and C_{∞} is the infinite cyclic group generated by shift in degrees < [1] >. Moreover, the above sequence induces an isomorphism

T rI_{R}(Λ) ≃ Γ(2)′

Proof. The automorphism group of the centre of Λ is a symmetric group of order 6. In fact the three irreducible characters may be permuted arbitrarily. Call the two onedimensional characters of Λ the trivial and the sign character. Of course, this is motivated by the case of the group ring of the symmetric group on three letters.

Recall (cf for example [5]) that in P SL_{2}(ℤ) there are two normal subgroups of index 6, the derived group P SL_{2}(ℤ)′ and the congruence subgroup Γ(2) of level 2. The quotient by the derived subgroup is cyclic, the quotient by the congruence subgroup of level 2 is

P SL_{2}(ℤ)/Γ(2) ≃ P SL_{2}(2) = GL_{2}(2) ≃ 𝔖_{3} .

Let us determine the image J of T rP ic_{R}(Λ) in Aut_{R}(Z(Λ)). Denote by t_{1} the functor induced by tensor product with S_{1} and by t_{2} the functor induced by tensor product with S_{2}. Then, the image of the element t_{1} in Aut_{R}(Z(Λ)) permutes the trivial and the two-dimensional character and fixes the sign character. The image of the element t_{2} in Aut_{R}(Z(Λ)) permutes the sign and the two-dimensional character and fixes the trivial character. Hence, J is isomorphic to the symmetric group of order 6 and therefore,

T rP icent(Λ)/ < [1] > ≃ Γ(2) .

It is immediate to see that

T rP ic_{K}(K ⊗_{R} Λ) ≃ C_{∞} ≀ 𝔖_{3} .

Moreover, under this isomorphism the shift by n degrees [n] corresponds to the element (n, n, n; id) ∈ C_{∞} × 𝔖_{3}. Hence, we get a commutative diagram with exact rows and columns

1 ⟶  C_{∞}^{2}  ⟶ \frac{T rP ic_{K}(KΛ)}{<[1]>} ⟶ 𝔖_{3} ⟶ 1
↑ ψ  ↑  ↑ ϕ
1 ⟶  \frac{T rP icent(Λ)}{<[1]>}  ⟶ \frac{T rP ic_{R}(Λ)}{<[1]>} ⟶ 𝔖_{3} ⟶ 1
↑  ↑
1 ⟶ T rP icent(Λ) ∩ T rI_{R}(Λ) ⟶ T rI_{R}(Λ)
↑  ↑
1  1

where the upper row corresponds to the isomorphism T rP ic_{K}(K ⊗_{R} Λ) ≃ C_{∞} ≀ 𝔖_{3} and the second row corresponds to the definition of T rP icent(Λ) being the kernel of the epimorphism T rP ic(Λ) ⟶ Aut(Z(Λ)) ≃ 𝔖_{3}.

We claim that the right vertical morphism ϕ is injective, hence even an isomorphism. In fact, any R-linear automorphism of Z(Λ) is induced by a permutation of the characters of K ⊗_{R} Λ. Hence,

T rI_{R}(Λ) ⊆ T rP icent(Λ).

Recall that by Theorem 5 we get a monomorphism B_{3} ⟶^{φ_{2}} T rP ic_{R}(Λ). We call \overline{φ_{2}} the isomorphism P SL_{2}(ℤ) ⟶ T rP ic_{R}(Λ)/ < [1] > induced by φ_{2}. Recall moreover (cf for

example [5]) that Γ(2) is freely generated by the matrices ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} ). The

homomorphism φ_{2} maps t_{1}^{2} and t_{2}^{2} to these two matrices. It is readily verified that t_{1}^{2} and t_{2}^{2} are elements in T rP icent(Λ). Moreover, these two matrices induce shifts by degree two for the sign or the trivial character respectively in T rP ic_{K}(K ⊗_{R} Λ). Therefore, Γ(2) is mapped via \overline{φ_{2}}^{−1} to a subgroup of T rP icent(Λ)/ < [1] >. Moreover, the image of ψ is a subgroup of C_{∞}^{2} of index at most 4.

Automorphisms of Green orders 13

Now,

P SL_{2}(ℤ) = C_{3} ∗ C_{2}

with a generator a of order 3 and another generator b of order 2, the image of the element t_{1} in P SL_{2}(ℤ) by \overline{φ}_{2} is ab and the image of the element t_{2} is ba. Therefore,

P SL_{2}(ℤ)/ < \overline{φ_{2}}^{−1}(t_{1}^{2}), \overline{φ_{2}}^{−1}(t_{2}^{2}) >≃< a, b| b^{2}, (ab)^{2}, a^{3} >≃ 𝔖_{3} .

Hence, T rI_{R}(Λ) is normal in T rP icent(Λ)/ < [1] > with quotient being free abelian of rank 2. This shows at once that \overline{φ_{2}} : Γ(2) ≃ T rP icent(Λ)/ < [1] > and that T rI_{R}(Λ) is isomorphic to Γ(2)′ the derived group of the congruence subgroup of level 2 of the modular group (see [5]).

(4.1) By the above discussion it is clear that the commutator c of t_{1}^{2} with t_{2}^{2} is in T rI_{R}(Λ). One calculates immediately that \overline{φ_{2}}(c) = ( \begin{matrix}13 & 8 \\ 8 & 5 \end{matrix} ) . Observe that by [12] this means that the smallest complex representing the image under c of the first projective indecomposable module involves 13 copies of P_{1} as well as eight copies of P_{2} and the image of the second projective indecomposable involves 8 copies of P_{1} as well as 5 copies of P_{2}. It is not difficult to write down the corresponding tilting complex explicitly.

(4.2) It is known that most normal subgroups of P SL_{2}(ℤ) are free. In particular, since Aut(Z(Λ)) ≃ 𝔖_{3}, we get that T rP icent(Λ)/ < shift > is free on two generators and isomorphic to the congruence subgroup Γ(2) of P SL_{2}(ℤ). By the Nielsen-Schreier theorem subgroups of free groups are free. Hence T rI_{R}(Λ) is free.

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