FOLDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS

Jeremy Rickard and Mary Schaps

Abstract. We give dual one-sided tilting complexes producing inverse equivalences of the derived category of a Brauer star algebra and a Brauer tree algebra of the same type, folded according to an additional combinatorial structure on the Brauer tree. We relate this to the two-sided two-term tilting complex of Rouquier in the case of a group block, showing that it induces the “completely folded” case for each one-sided complex.

§1. Introduction.

Let G be a tree with e edges. If one of the e + 1 vertices is designated as “exceptional” and assigned a multiplicity m, m ≥ 1, then f is called a Brauer tree of type (e, m). Brauer trees arise naturally in the modular theory of group representations in order to describe blocks of group algebras with cyclic defect group.

In [R1], the first author gave a tilting complex which would tilt an arbitrary group block to a Brauer star of type (e, m), in which the exceptional vertex in the center and all other vertices are terminal. Unlike the various recursively defined tiltings studied afterwards in [Z], [M], the original complex went from the given block to the algebra corresponding to the star in a single step. We will refer to this as the tree-to-star complex.

In [SZ1], [SZ2], [SZ3] the second author and her doctoral student considered the opposite problem: finding a combinatorial construction of a one-step star-to-tree tilting complex going from the Brauer star to an arbitrary algebra determined by a Brauer tree, both of type (e, m). The tilting complexes considered, designated

Typeset by AMS-TEX 2 JEREMY RICKARD AND MARY SCHAPS

two-restricted, were direct sums of indecomposables each involving no more than two projectives, and a complete classification for all such complexes was given. It turned out that the number of such complexes for a given tree is e times the product of the valences at the non-exceptional vertices, and they can be classified by imposing an additional structure called a “pointing” on the Brauer tree.

In this paper we will show that if the tree-to-star complex and the star-to-tree complex are “folded” according to the same pointing on the Brauer tree, then we have one-sided complexes T and T̂ which induce inverse equivalences between the derived categories. This implies the existence of a two-sided complex C such that C is homotopic to T and the dual C^{∗} is homotopic to T̂. We then show that in the case of a group block, with T and T̂ “completely folded”, the two-term sequence of Rouquier [Ro] will give the desired C.

We will give the precise definitions of the various ideas used here in §2. In §3 we will define the folding for the tree-to-star complex T̂ determined by a pointing and prove the duality of T and T̂. In §4 we will consider the construction of the two-sided complex in certain special cases.

§2. Definitions and previous results.

Each Brauer tree of type (e, m) determines a Brauer tree algebra, a finite dimensional algebra with e simple modules, in which each indecomposable module is uniserial or biserial. A complete description of this algebra in terms of composition series is given in [A]; a description in terms of quivers and relations is provided in [S]. Each simple module Σ_{i} corresponds to an edge, and the two series of composition factors in the indecomposable projective P_{i} are obtained by a counterclockwise circuit of the edges at the two ends. This circuit is made once unless the vertex is exceptional, in which case the circuit is made m times, returning to Σ_{i} in the end. If the vertex is terminal and not exceptional, then the circuit provides only two copies of Σ_{i}, and thus P_{i} is uniserial. If the vertex is terminal and exceptional,

FOLDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS 3

then we get m + 1 copies of Σ_{i} as one of the two series of composition factors. We will not need the full details in this paper, but it will be critical that there are no maps between indecomposable projectives which are not adjacent to a common vertex.

We recall the following definition from [R1].

Definition. Let R be a permutative Noetherian ring. A tilting complex for a finite dimensional R-algebra A is a complex T of projectives such that

(1) Hom_{D^{b}(A)}(T [n], T]) = 0 for n ≠ 0.

(2) The components of T generate the derived category D^{b}(A) as a triangulated

category. If T satisfies only condition (1), it will be called a partial tilting complex.

Remark. The definition of the standard triangles of D^{b}(A) in terms of mapping cones implies that if u : X → Y is a chain map between two elements of a subcategory D of D^{b}(A), then the mapping cone Cone(X → Y) also lies in D. Since all the complexes in D^{b}(A) can be built up recursively from the projective indecomposables, it suffices to show that we can obtain each projective indecomposable in order to establish (2).

In [R1] the first author gave a combinatorial construction of a tilting complex T which will tilt a given Brauer tree algebra to a Brauer star algebra.

(1) In degree zero, for each edge i adjacent to the exceptional vertex, take a stalk complex P_{i}.

(2) For each edge not adjacent to the exceptional vertex, if i_{1}, i_{2}, …, i_{s} = j is a path connecting the edge j to the exceptional vertex, add the indecomposable complex

P_{i_{1}} → P_{i_{2}} → ⋯ → P_{i_{j}},

where each map is a uniquely determined nonzero homomorphism deter-

4 JEREMY RICKARD AND MARY SCHAPS

mined by the structure of the Brauer tree algebra. For this paper, we will refer to T̂ as a tree-to-star complex.

For a completely different purpose, connected with deformations, E. Zakay-Illouz studied certain combinatorial tilting complexes going in the opposite directions, from the Brauer star of type (e, m) to an arbitrary Brauer tree of type (e, m), [Za]. Definition. A partial tilting complex for the Brauer star algebra will be called two-restricted if it is a direct sum of shifts of the following elementary complexes, where the first nonzero term is in degree 0.

1) S_{i} : 0 → Q_{i} → 0

2) T_{jk} : 0 → Q_{j} →^{h_{jk}} Q_{k} → 0,

where the maps h_{jk} take the top of Q_{j} to the highest copy of the corresponding composition factor in Q_{k}.

The first section of the thesis determined necessary and sufficient conditions for a two-restricted complex to be a partial tilting complex. We will not need the full classification here. We will use only

(a) Each indecomposable projective Q_{i} occurs in the same degree in each elementary complex.

(b) After an appropriate cyclic renumbering of the vertices of the Brauer tree, each complex T_{jk} has j < k.

There are, in addition, a third set of conditions which we may summarize as non-overlap conditions. These are much easier to describe in terms of the full tilting complex, so we will postpone the discussion till we have given the connection between the Brauer tree and the tilting complex. Definition. Let G be a Brauer tree, identified with a planar diagram in which the edges are represented by line segments and the ordering of the edges at each vertex corresponds to a counterclockwise circuit of the vertex. A pointing of the

FOLDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS 5

tree consists of the choice of one sector at each non-exceptional vertex, indicated by placing a point in that sector. The resulting tree with this additional structure is called a pointed Brauer tree.

The main theorem of [Za], written up in [SZ2], states that the two-restricted tilting complexes of the Brauer star of type (e, m), modulo cyclic permutation of the star, are in one-to-one correspondence with the different pointed Brauer trees of type (e, m).

We give here the algorithm for computing a complex from the corresponding Brauer tree.

(1) Pick an arbitrary branch at the exceptional vertex as a starting point, and let the exceptional vertex be numbered as 0.

(2) Number all non-exceptional vertices by taking a Green’s walk around the tree in a counterclockwise direction, assigning a number to each vertex whenever the corresponding point is reached.

(3) Give each edge the same number as the vertex farthest from the exceptional vertex.

(4) Define a two-restricted complex T = ⨁_{i=1}^{e} R_{i}, by recursion on the distance from the exceptional vertex.

a) If edge i is adjacent to the exceptional vertex, then R_{i} is the stalk complex S_{i} in degree zero.

b) If i_{1}, i_{2}, …, i_{j} = i are the numbers assigned to the edges in a minimal path from the exceptional vertex to edge i, then we distinguish two cases: b.1) \underline{i_{j} < i_{j−1}}: In this case, where edge i is between the entering edge at vertex i_{j−1} and the point, we set

R_{i_{j}} = T_{i_{j}i_{j−1}}[n_{j}],

6 JEREMY RICKARD AND MARY SCHAPS

where n_{j} is the shift required to ensure that P_{j−1} is in the same degree in R_{i_{j−1}} and R_{i_{j}}. b.2) \underline{i_{j−1} < i_{j}}: In this case, when the edge i is between the point and the entering edge in the counterclockwise ordering, then we set R_{i_{j}} = T_{i_{j−1}i_{j}}[n_{j−1}], where again n_{j−1} is a shift which will line up the copies of P_{i_{j−1}} in R_{i_{j}} and R_{i_{j−1}}.

Remark. In all cases, we use only T_{jk} with j < k.

Example 1.

0 → P_{1} → 0
0 → P_{1} → P_{3} → 0
0 → P_{2} → P_{3} → 0
0 → P_{3} → P_{4} → 0
0 → P_{1} → P_{8} → 0
0 → P_{5} → P_{8} → 0
0 → P_{5} → P_{6} → 0
0 → P_{7} → P_{8} → 0
0 → P_{11} → 0
0 → P_{10} → P_{11} → 0
0 →P_{9} → P_{10} → 0

FOLDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS 7

There are two particular pointings worthy of special mention. In the first, the point is always immediately after the entering vertex, and the resulting numbering is that given by the standard Green’s walk. We will call the resulting complex “completely unfolded”. In the second, the point alternates, first immediately after the entering vertex and then immediately before it. This gives a “completely folded” two-term lifting complex. Each of these special pointings has a dual version with the location of the point reversed.

§3. Folded tree-to-star complexes.

Given a pointed Brauer tree, we get folded star-to-tree tilting complex T which is unique up to cyclic permutation of the Brauer star. In this section, we want to build a folded tree-to-star complex such that T and T̂ will induce inverse equivalences.

We define the numbering of the vertices and edges of the Brauer star as (1) – (4) of the algorithm in §2.

ˆ5) We build the complex T̂ = ⨁ R̂_{i} by recursion on the distance from the

exceptional vertex. â) If edge i is adjacent to the exceptional vertex, then R̂_{i} is the stalk complex

with P_{i} in degree zero. b̂) If i_{1}, …, i_{j} = i are the numbers assigned to the edges in a minimal path leading to the exceptional vertex, assume we know by recursion that R_{i_{j−1}} contains one copy of P_{i_{j−1}}, then we distinguish two cases: b̂.1) \underline{i_{j−1} < i_{j}}: We set

R̂_{i_{j}} : P_{i_{j}} → R_{i_{j−1}}

where the map is induced by a nonzero map from P_{i_{j}} → P_{i_{j−1}}, which is unique up to isomorphism because P_{i_{j}} is not adjacent to the exceptional vertex. b̂.2) \underline{i_{j} < i_{j−1}} : We set

R̂_{i_{j}} : R̂_{i_{j−1}} → P_{i_{j}},

where the map is induced by a nonzero map from P_{i_{j−1}} → P_{i_{j}}, again unique up to isomorphism.

Note the duality: In T the smaller of two indices adjacent in the tree is in lower degree, whereas in T̂ it is in higher degree. For example, the complex T̂ for the previous example is:

0 → P_{1} → 0
0 → P_{3} → P_{1} → 0
0 → P_{3} → P_{1} ⊕ P_{2} → 0
0 → P_{4} → P_{3} → P_{1} → 0
0 → P_{8} → P_{1} → 0
0 → P_{8} → P_{1} ⊕ P_{5} → 0
0 → P_{8} ⊕ P_{6} → P_{1} ⊕ P_{5} → 0
0 → P_{8} → P_{1} ⊕ P_{7} → 0
0 → P_{11} → 0
0 → P_{11} → P_{10} → 0
0 → P_{11} → P_{10} → P_{9} → 0

In [SZ2], it was proven that the complex T is a tilting complex from the Brauer star to the Brauer tree algebra for the pointed Brauer tree with which we started.

Proposition 1. The endomorphism ring of the complex T̂ associated with a pointed Brauer tree is the Brauer star algebra.

FOLDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS 9

Proof. The various branches at the exceptional vertex determine a partition of 1, …, e into disjoint intervals. Let i_{1}, …, i_{r} be the distinct branches at the exceptional vertex, with R_{i_{k}} given by the stalk complex with P_{i_{k}} in degree zero. Let us define integers s_{k} and t_{k} such that the interval determined by the branch i_{k} is I_{k} = [s_{k}, …, t_{k}]. By the definition of the complex T̂, each R̂_{j} for j ∈ I_{k} contains a single copy of P_{i_{k}}, always in degree zero. We claim that we can define homomorphism f_{j} : R̂_{j} → R̂_{j+1} for j, j + 1 ∈ I_{k} which are the identity on P_{i_{k}}, and homomorphisms f_{t_{k}} : R̂_{t_{k}} → R̂_{s_{k+1}} which give the map with minimal quotient on the degree zero terms P_{i_{k}} → P_{i_{k+1}}. (Here we understand i_{r+1} = s_{1}.) \underline{j ≠ t_{k}}:

Case 1. j = i_{k}, or j lies on a path connecting j + 1 to i_{k}. Since none of the intermediate edges has a number assigned between j and j + 1, their points must be on the left as we go out, and thus we get

i_{k}. . . . .j.j_{1}.j_{2}. . . . .j_{ℓ}.j + 1

We conclude that j_{1} > j_{2} > ⋯ > j_{ℓ} > j + 1. Thus

R̂_{j}
R̂_{j+1} :  ↗
P_{j_{1}} → P_{j_{2}} → ⋯ → P_{j+1}.

We let f_{j} be the canonical injection of R̂_{j} into R̂_{j+1}.

Case 2. j + 1 = i_{k}, or j + 1 lies on a path connecting j to i_{k}. By an argument dual to that in case 1, we have a path

i_{k}, …, j + 1, j_{1}, j_{2}, … j_{ℓ}, j,

with j_{1} < j_{2} < … j < j + 1. We then have

R̂_{j+1}
R̂_{j} :  ↘
P_{j} → P_{j_{ℓ}} → … P_{j_{2}} → P_{j_{1}}

and f_{j} : R_{j} → R_{j+1} is the surjection whose kernel is the subcomplex P_{j} → ⋯ → P_{j_{1}}. Case 3. The paths connecting j and j + 1 to i_{k} diverge at h ≠ j, j + 1. Case 3.a. \underline{h < j, j + 1}. Let j_{1}, j_{2}, …, j_{ℓ}, j and j′_{1}, …, j_{ℓ′}, j+1 be the two extensions. Then by the construction of the numbering of the pointed graph, we must have

h < j_{ℓ} < ⋯ < j_{2} < j_{1} < j < j + 1 < j′_{1} < j′_{2} < ⋯ < j′_{ℓ′}.

We thus have

P_{j}
↘
P_{j_{1}} → P_{j_{2}} … → P_{j_{ℓ}}
R̂_{j} :  ↘
R̂_{h}
R̂_{j+1} : P_{j′_{1}}  ⋯ →P_{j′_{1}} → R̂_{h}
↘
P_{j+1}

The homomorphism f_{j} : R̂_{j} → R̂_{j+1} can be defined by sending P_{j_{1}} → P_{j′_{1}} by the appropriate map at vertex h. Then f_{j} is well-defined on P_{j_{2}} since the composition P_{j_{1}} → P_{j_{2}} → P_{h} is zero, because j_{2} and h are not adjacent vertices.

Case 3.b. \underline{k > j, j + 1}. This is dual to 3.a.

\underline{j = t_{k}}: Map the copy of P_{i_{k}} in R̂_{t_{k}} to the copy of P_{i_{k+1}} in R̂_{j+1} by a homomorphism with minimal quotient. The composition of this map with any of the other maps in R̂_{t_{k}} or R̂_{s_{k+1}} is zero, so the f_{t_{k}} is well-defined.

It is clear that none of these maps is homotopic to zero, and that the result of one circuit starting at P_{i_{k}} is to map P_{i_{k}} to the maximal proper submodule with the same top. Thus the result of m circuits will be to map P_{i_{k}} to its socle. Now take any j ∈ I_{k}, let ε_{j}^{m} be the result of m circuits. We claim that ε_{j}^{m} is not homotopic to zero, and that f_{j} ∘ ε_{j}^{m} is homotopic to zero.

A full proof of this claim involves treating all of the cases in the definition of the f_{j} separately, so we will give only a sketch of the proof. Suppose that the common part of the paths from j and j + 1 to i_{k} is i_{k} = j_{0}, j_{1}, …, j_{h}. For any index i, let \overline{ε}_{i} : P_{i} → P_{i} be the non identity homomorphism with quotient of minimal length. Let s_{i} : P_{i} → P_{i} be the map of the projective P_{i} onto its socle. Thus s_{i_{k}} = \overline{ε}_{i}^{m}, and s_{i_{ℓ}} = \overline{ε}_{i_{ℓ}}, for ℓ ≠ 0.

One can show that the map from R̂_{j_{h}} to R̂_{j_{h}} which is ε̄_{i_{k}}^{m} on P_{i_{k}}, (−1)^{h}s_{j_{h}}^{m} on P_{j_{h}} and zero elsewhere is homotopic to zero by a homotopy T̃. The homotopy T̃ is a direct sum of homomorphisms h_{j_{i}j_{i+1}} for j_{i} > j_{i+1} and h_{j_{i+1}j_{i}} for j_{i+1} > j_{i}. This demonstrates that ε_{j}^{m} is not homotopic to zero, since it is homotopic to the well-defined map ŝ_{j_{n}} : R̂_{j} → R̂_{j} which is equal to (−1)^{h+1}s_{j_{n}} on P_{j_{n}} and zero elsewhere. If j′_{1} is the first edge in the direction of j, and j′′_{1} is the first edge in the direction of j + 1, then the map from R̂_{j′_{1}} to R̂_{j′′_{1}} which is ε̄_{i_{k}}^{m} on P_{i_{k}} and zero elsewhere is homotopic to zero by an extension T̃′ of T̃ which produces a zero map at P_{j_{h}}. We describe the map T_{0} giving the extension from T̃ to T̃′.

\underline{j_{h} = j}.

P_{j_{h}}
T_{0}=(−1)^{h+1}h_{j_{h}j\begin{matrix}′ & ′ \\ 1 \end{matrix}} ↙ ↓ ^{(−1)^{h+1}s_{h}}
P_{j′′_{1}} → P_{j_{h}}

\underline{j_{h} = j + 1} is dual. \underline{j_{h} < j′_{1} < j < j + 1 < j_{1}”}.

P_{j′′_{1}}  → P_{j_{h}}
0 ↓ ^{T_{0}} ↙ ↓ ^{(−1)^{h+1}s_{h}}
P_{j′′_{1}}  → P_{j_{h}}

\underline{j′_{1} < j < j + 1 < j′′_{1} < j_{h}} is dual. The same homotopy T̃′ show that R̂_{j} ⟶^{f_{j}∘ε_{j}^{m}} R̂_{j+1}

is homotopic to zero.

We have actually shown that End_{B}(T̂^{·}) is isomorphic to the opposite algebra A^{0} of A but since A →^{∼} A^{0}, we have the desired result. Proposition 2. The complexes T^{·} and T̂^{·} determined by a given pointed Brauer tree determine inverse equivalences of categories. Proof. In the previous proposition, we established a correspondence between the projectives Q_{1}, …, Q_{e} of b and the components R̂_{1}, …, R̂_{e} of the tilting complex T̂. Knowing, from theoretical considerations, that an inverse equivalence exists, we will determine the inverses to the indecomposable projectives P_{1}, …, P_{e} of B using mapping cones.

Each branch at the exceptional vertex is sufficient to generate all the projectives appearing in that branch, so we will assume as in the proof of Proposition 1 that we are concerned with the branch at i_{k}, represented by an interval [s_{k}, …, t_{k}]. The individual projectives are generated recursively according to their distance from the

FOLDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS 13

exceptional vertex with Q_{i_{k}} corresponding to P_{i_{k}}. If i_{k} = j_{0}, j_{1}, …, j_{h} is a minimal path, then we consider two cases. \underline{j_{h−1} < j_{h}} : R̂_{j_{h}} : P_{j_{h}} → R̂_{j_{h−1}}

Suppose that in R̂_{j_{h−1}}, the projective P_{j_{h−1}} occurs in degree −n_{h−1}. Then R̂_{j_{h}} = Cone(P_{j_{h−1}}[n_{h−1}] → R̂_{j_{h−1}}).

Applying the inverse functor F^{−1} : (K^{b}(B) → K^{b}(A) and using the fact that F^{−1}(R̂_{j}) = Q_{j}, we get

Q_{j_{h}} →^{∼} Cone(F^{−1}(P_{j_{h−1}})[n_{h−1}] →^{u} Q_{j_{h−1}}).

We conclude that F^{−1}[P_{j_{h−1}}[n_{h−1}]] = T_{j_{h−1}j_{h}} and the map u is the identity on Q_{j_{h−1}}. After adjusting the shift, which commutes with the functor F^{−1}, we conclude that

F^{−1}(P_{j_{h−1}}) →^{∼} T_{j_{h−1}j_{h}}[−n_{h−1}].

Thus in the tilting complex T, Q_{j_{h−1}} occurs in degree n_{h−1}.

The case j_{h} < j_{h−1} is precisely dual.

§4. Two-term tilting complexes.

Let us now suppose that we have a two term two-sided tilting complex C, as in [Rou], with the following properties. Let A be a basic Brauer star algebra, and let B be a Brauer tree algebra. Let M be a B − A bimodule, which is projective both as a left B-module and as a right A-module, such that

(∗)

M ⊗_{A} M^{∗} →^{∼} B ⊕ projective B-bimodules
M ⊗_{B} M^{∗} →^{∼} A ⊕ projective A-bimodules,

Suppose that C : 0 → N → M → 0 is a complex of B − A bimodules with N a projective B − A bimodule and, satisfying (∗) in degree zero. Suppose further that C ⊗_{A} C^{∗} is homotopy equivalent to B (as a stalk complex) and C^{∗} ⊗_{B} C is

14 JEREMY RICKARD AND MARY SCHAPS

homotopy equivalent to A. Then C as a complex of projective B-modules is a onesided tilting complex from B to A and C^{∗} as a complex of B modules is a one-sided tilting complex from A to B, giving inverse equivalences.

Proposition 3. If C is a two-term, two-sided tilting complex of B − A bimodules, then C is homotopic to a tree-to-star, completely folded tilting complex T̂, and C^{∗} is homotopic to a two-restricted completely folded star-to-tree complex T obtained from the same pointing of the Brauer tree.

Proof. Since, by Proposition 2, the complexes T and T̂ obtained from a given Brauer tree give inverse equivalences, it suffices to show that C^{∗} is homotopic to a tworestricted one-sided tilting complex. For this, the only information we need about our complex is that it is a two-term complex of projectives, so we will consider the more general case

D : 0 → V →^{u} W → 0

with

V = ⨁_{j=1}^{s} V_{j}, V_{j} →^{∼} P_{i_{j}}, V_{j} = ⟨v_{j}⟩,

W = ⨁_{ℓ=1}^{t} W_{j}, W_{ℓ} →^{∼} P_{k_{ℓ}}, W_{ℓ} = ⟨w_{ℓ}⟩.

Choose an element x of the Brauer star algebra A which generates the radical, with xn̂ ≠ 0, x^{n̂+1} = 0, for n̂ = em + 1.

If u = 0, then D decomposes into a direct sum of stalk complexes, which is surely two-restricted, so we may assume that u ≠ 0. We first want to prove, by induction on S, that D is a direct sum of stalk complexes and complexes of the form P_{i} → P_{k}. For s = 0 we have direct sum of stalk complexes, so we assume s > 0 and that every D with smaller s decomposes.

FOLDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS 15

We have

u(V) ⊈ x^{n̂+1}W = {0},
u(V) ⊆ x^{0}W = W.

Therefore, there is a unique integer g, ˆ0 ≤ g ≤ n̂, such that

u(V) ⊈ x^{g+1}W, u(V) ⊆ x^{g}W.

There is some j such that u(V_{j}) ∉ x^{g+1}W, and some ℓ such that

π_{ℓ}(u(v_{j})) = cx^{g}w_{ℓ}, c ∈ k,

where π_{ℓ} is the projection of W on W_{ℓ}. Since u(v_{j}) ∈ x^{g}W, there is an element w ∈ W such that u(v_{j}) = x^{g}(w), and we can make a change of coordinates substituting w for w_{ℓ} so that u(v_{j}) = x^{g}w_{ℓ}, since u(v_{j}) = v(w) implies that ⟨w⟩ →^{∼} P_{k_{ℓ}} →^{∼} w_{ℓ} and π_{ℓ}(w) = cw_{ℓ} with c ≠ 0.

We now make a change of coordinates in V such that π_{ℓ}(v_{i}) = 0 for every i ≠ j, by adding appropriate linear combinations of x^{t}v_{j} to v_{i}. It is then possible to split off V_{j} → W_{ℓ}, and get a complex D′ with smaller s.

It remains to show that the summands P_{i} →^{h} P_{k} are of type T_{i_{k}}. If i = k, with h an isomorphism, it is homotopic to zero. If u factors through ε_{i}, then it was shown in [SZ1] that the complex cannot occur in a tilting complex. Thus we must have P_{i} →^{h_{ik}} P_{k} as desired. D is two-restricted and thus, by the main theorem in [SZ2], it corresponds to a pointed Brauer tree. Since it is two-term, it is completely folded. Proposition 2 then finishes the theorem. Example. G = P SL(2, p), as in [Rou]. Take the numbering starting at the nonexceptional vertex

P_{e} P_{e−1}
•  •  •  •  •  •
P_{1}  P_{2}  P[_{\frac{e+1}{2}}]

16 JEREMY RICKARD AND MARY SCHAPS

Our numbering is one greater than that in [Rou], so that each P_{i} has a simple top of dimension 2i − 1 (instead of 2i + 1). The module M in this case is OGê, where ê is the idempotent of the principal block and contains 2i − 1 copies of each P_{i}. The module M, as a B module, contains p copies of each P_{i}, since dim Q_{i}^{∗} = p. The B-module T̂ is ⨁_{i=1}^{e} R̂_{j}. Since each P_{i} occurs at most once in R̂_{j}, and the number of R̂_{j} in which P_{i} occurs in the distance of P_{i} from the non-exceptional vertex, we get:

T̂ : ⨁_{j=[\frac{e+1}{2}]+1}^{e} P_{j}^{2(e−j+1)} → \begin{matrix}[ & \frac{e+1}{2} & ] \\ ⨁ \\ j=1 \end{matrix} P_{i}^{2j−1}

C : ⨁_{j=[\frac{e+1}{2}]+1}^{e} P_{j}^{p} → ⨁_{j=1}^{e} P_{i}^{2j−1}.

Since we know that T̂ is homotopic to C, we see that, for each j = [\frac{e+1}{2}] + 1, …, e. We must split off from C 2j − 1 copies of P_{j} →^{id} P_{j}.

p − (2j − 1) = (2e + 1) − 2j + 1
= 2(e − j + 1).

We get just the right number of copies of P_{i} left for T̂.

§5. Applications.

Although two-sided tilting complexes exist whenever there is a one-sided tilting complex, in fact there are few known examples. We hope that having the matched pairs T and T̂ will help us work out a recursive procedure for constructing the two-sided complexes. We would be particularly interested in finding a two-sided complex with only one term which is not projective as a bimodule.

Should we be able to work out a procedure to construct the two-sided complexes C − C^{∗} for the T − T̂ pairs,then we would get a large number of elements in the derived Picard group of the Brauer star, by considering two different points for a fixed Brauer tree. If we have C − C^{∗} for one pointing and C′ − C′^{∗} for another

pointing, then C ⊗_{B} C′ will be a self equivalence of A. We conjecture that these

complexes generate the entire derived Picard group of A. If this is true, then any

two-sided tilting complex between A and B is a composition of two-sided tilting

complexes whose corresponding one-sided tilting complexes are the tree-to-star and

star-to-tree complexes associated with pointed Brauer trees.

References

[A] Alperin J., Local Representation Theory, vol II, Cambridge Studies in Mathematics, 1986.

[M] Membrillo F.H., Homological Properties of Finite Dimensional Algebras, Ph.D. Thesis, Oxford University, 1993, pp. 6-13, 78.

[R1] Rickard J., Derived categories and stable equivalence, J. of Pure and Applied Algebra 61 (1989), 303-317.

[R2] Rickard J., Conjectural calculation of a two-sided tilting complex for cyclic blocks, preprint (1990).

[R3] Rickard J., Derived equvialences as derived functors, J. London Math. Soc. 43 (1991), 37-48.

[R4] Rickard J., Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331-358.

[Rou] Rouquier R., The derived category of blocks with cyclic defect groups, LNM, vol. 1685, 1998, pp. 199-220.

[S] Schaps M., A modular version of Maschke’s theorem for groups with cyclic p-Sylow subgroup, J. of Algebra 163 (1994), 623-635.

[SZ1] Schaps M. and Zakay-Illouz E., Pointed Brauer trees, preprint.

[SZ2] Schaps M. and Zakay-Illouz E., Homogeneous deformations of Brauer tree algebras, preprint.

[SZ3] Schaps M. and Zakay-Illouz E., Combinatorial partial tilting complexes for the Brauer star algebras (to appearProc. Int. Conference on Representations of Algebra, Sao Paulo).

[Z] Zimmerman, A., A two-sided tilting complex for Green orders and Brauer tree algebras, J. of Alg. 187 (1997), no. 2, 446-473.

Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel

Department of Mathematics, University of Bristol, Bristol, England