PICARD GROUPS FOR DERIVED MODULE CATEGORIES

RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

Abstract. We introduce in this paper a generalization of Picard groups to derived categories of algebras. We study general properties of those groups and we construct braid group actions on these groups for particular classes of algebras.

1. Introduction

Let k be a commutative ring and A a k-algebra. A bounded complex X of (A, A)- bimodules is invertible if there is a bounded complex Y of (A, A)-bimodules such that

X ⊗_{A}^{L} Y ≃ A in the derived category of (A, A) − bimodules
and Y ⊗_{A}^{L} X ≃ A in the derived category of (A, A) − bimodules.

We define the group TrPic(A) : its elements are isomorphism classes of invertible complexes in the derived category of (A, A)-bimodules. The product of the classes of X and X′ is the class of X ⊗_{A}^{L} X′. The inverse of the class of X is the class of Y where X ⊗_{A}^{L} Y ≃ A.

By Rickard’s theory [13], an equivalence between the derived categories of two k-algebras A and B which are projective as k-modules induces an isomorphism between TrPic(A) and TrPic(B). The subgroup of TrPic(A) given by complexes with homology concentrated in degree 0 is the usual Picard group Pic(A). As we shall see later, the group Pic(A) is not an invariant of the derived category.

The paper is organized as follows.

In a first section, we review and prove some results about standard derived equivalences. Flat central base change is dealt with in §2.4. Then we show that for commutative rings, standard derived equivalences come from Morita equivalences.

In a second part, we study various general properties of TrPic. Some of these are analogs of classical properties of Picard groups such as base change and Fröhlich’s localization sequence.

The third part of the paper is devoted to the study of Brauer tree algebras A with no exceptional vertex. Let n be the number of simple modules of A. We construct a morphism from Artin’s braid group on n + 1 strings to TrPic(A). When n = 2, we show this morphism is an isomorphism modulo some central subgroup : TrPic(A) is isomorphic

Date: October 1998, revised February 2002.

2 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

to a central extension of P SL_{2}(Z). This applies in particular when A is the group algebra of the symmetric group S_{3} over a field of characteristic 3.

The results in this paper have been exposed by the second author at the ICRA VII conference in August 1994 [19] and at the AMS Summer Research Institute “Cohomology, Representations and Actions on Finite Groups” in July 1996 in Seattle. In [18], Yekutieli has considered independently the group TrPic, in particular the case of local and commutative algebras, and given applications to dualizing complexes.

2. On Rickard’s tilting theory

2.1. Notations and terminology. Let us fix some conventions for the rest of the paper. Let k be a commutative ring and A a k-algebra. By an A-module, we always mean a left A-module. We denote by A^{∘} the opposite algebra of A. We denote by A-mod the category of finitely presented A-modules which is an abelian category if A is right coherent.

Let C = (C^{i}, d_{i}) be a complex of A-modules where we denote by d_{i} the differential C^{i} → C^{i+1}. For n an integer, we denote by C[n] the complex with C[n]^{i} = C^{n+i} and differential (−1)^{n}d.

We denote by D^{b}(A) the full subcategory of the derived category of A-modules consisting of objects with bounded homology. We identify the category of A-modules with the full subcategory of D^{b}(A) of complexes concentrated in degree 0. Unless otherwise specified, morphisms are taken in the derived category.

A complex of A-modules is perfect if it is quasi-isomorphic to a bounded complex of finitely generated projective A-modules. We denote by A-perf the full subcategory of D^{b}(A) of perfect complexes.

Given two k-modules M and N, we write M ⊗ N for M ⊗_{k} N.

For C = (C^{i}, d_{i}) and D = (D^{j}, δ_{j}) two bounded complexes, we denote by C ⊗ D the total complex associated with the double tensor complex. This has degree n term (C ⊗ D)^{n} = ⨁_{i+j=n} C^{i} ⊗ D^{j} and the differential in degree n is ∑_{i+j=n} d_{i} ⊗ 1 + (−1)^{i}1 ⊗ δ_{j}. Analogously we denote by Hom(C, D) the total complex of the double homomorphism complex. It has degree n term ∏_{i+n=j} Hom(C^{i}, D^{j}) and the differential ∂_{n} in degree n is ∂_{n}(f) = ∏_{i+j=n}(d_{i} ∘ f − (−1)^{n}f ∘ δ_{j}).

Note that given X in D^{b}(A^{∘}) and Y in D^{b}(A), then X ⊗_{A}^{L} Y is a complex with bounded homology when X is quasi-isomorphic to a bounded complex of flat A^{∘}-modules or Y is quasi-isomorphic to a bounded complex of flat A-modules.

A full triangulated subcategory of a triangulated category is called épaisse if it is closed under taking direct summands [12, §1].

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 3

The subcategory of a triangulated category generated by an object is the smallest épaisse full triangulated subcategory containing that object.

2.2. Standard derived equivalences.

2.2.1. Let B be a k-algebra.

The following theorem of Rickard gives the essentials of the Morita theory for derived categories [11, 13].

Theorem 2.1. The assertions (i)–(iii) are equivalent :

(i) The bounded derived categories D^{b}(A) and D^{b}(B) are equivalent as triangulated categories.

(ii) The categories of perfect complexes A-perf and B-perf are equivalent as triangulated

categories. (iii) There is a perfect complex T of A-modules such that

(a) B is isomorphic to End(T),

(b) Hom(T, T [i]) = 0 for i ≠ 0,

(c) A-perf is generated by T. If A and B are projective over k, then the assertions (i)–(iii) are equivalent to (iv) :

(iv) There is a bounded complex X of (A ⊗ B^{∘})-modules whose restrictions to A and to B^{∘} are perfect and a bounded complex Y of (B ⊗ A^{∘})-modules whose restrictions to B and to A^{∘} are perfect such that

X ⊗_{B}^{L} Y ≃ A in D^{b}(A ⊗ A^{∘}) and Y ⊗_{A}^{L} X ≃ B in D^{b}(B ⊗ B^{∘}).

If A and B are right coherent, then the assertions (i)–(iii) are equivalent to (v) :

(v) The bounded derived categories of finitely presented modules D^{b}(A-mod) and D^{b}(B-mod) are equivalent as triangulated categories.

When the assertions (i)–(iii) of the theorem are fulfilled, we say that A and B are derived equivalent. A complex T satisfying the conditions in (iii) is called a tilting complex for A. Complexes X and Y satisfying the conditions in (iv) are called two-sided tilting complexes, inverse to each other. The restriction of X to A is a tilting complex, as well as the restriction of Y to B. Similar statements hold of course restricting X to B^{∘} and Y to A^{∘}. It is clear from the definition that the functors Y ⊗_{A}^{L} − and X ⊗_{B}^{L} − are then inverse equivalences between D^{b}(A) and D^{b}(B), as well as between A-perf and B-perf. Equivalences between D^{b}(A) and D^{b}(B) of the form X ⊗_{B}^{L} − for a complex X ∈ D^{b}(A ⊗ B^{∘}) are called standard.

It is unknown whether every equivalence of derived categories is naturally isomorphic to a standard derived equivalence [13, §3]. It is only known that every equivalence between

4 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

D^{b}(A) and D^{b}(B) agrees with a standard equivalence on isomorphism classes of objects when A and B are projective over k [13, Corollary 3.5].

2.2.2. For algebras projective over k, two-sided tilting complexes can, up to isomorphism, be chosen to be bounded complexes of modules that are projective as A-modules and projective as B^{∘}-modules. This is a consequence of the following lemma (see also [14, Lemma 9.2.6]) :

Lemma 2.2. Assume A and B are projective over k. Let X be a bounded complex of (A ⊗ B^{∘})-modules such that the restrictions of X to A and to B^{∘} are perfect.

Then X is isomorphic to a bounded complex of (A ⊗ B^{∘})-modules all of whose terms are projective as (A ⊗ B^{∘})-modules except possibly the non-zero term in the smallest degree, which is projective as an A-module and projective as a B^{∘}-module.

Proof. Let us start with mentioning that the projectivity assumption of A and B over k ensures that the restriction to A of a projective (A ⊗ B^{∘})-module is projective and likewise the restriction to B^{∘}.

Let S be a bounded complex of projective A-modules isomorphic to Res_{A} X and let n be an integer such that the terms of S vanish in degrees less than n.

Let Y be a projective resolution of X : this is a right bounded complex of projective (A⊗B^{∘})-modules isomorphic to X. Let Z = τ_{≥n−1}Y : this is a bounded complex isomorphic to X, with zero terms in degrees less than n−1 and with projective terms in degrees greater than n − 1. We will now show that the degree n − 1 term of Z is projective as an A-module and as a B^{∘}-module.

Since Res_{A} Z is isomorphic to the bounded complex of projective A-modules S, there exists a morphism in the category of complexes of A-modules α : S → Res_{A} Z which is an isomorphism in D^{b}(A). Let D be the cone of α. Then D is an acyclic bounded complex all of whose terms are projective except possibly the non-zero term in the smallest degree. Such a complex is homotopy equivalent to zero. Indeed the largest degree non-zero differential is surjective with image a projective module, hence splits. So, D is homotopy equivalent to a smaller complex and we continue by induction. This shows that Res_{A} Z is a bounded complex of projective A-modules. The same argument shows that Res_{B^{∘}} Z is a bounded complex of projective B^{∘}-modules.

2.2.3. Assume A and B are projective over k. Let T be a tilting complex for A and f : B → End(T) an isomorphism. Then there exists a two-sided tilting complex X for A⊗B^{∘} with the following property: There is an isomorphism between T and the restriction Res_{A} X of X to A so that if we denote by ϕ : End(T) → End(Res_{A} X) the induced isomorphism,

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 5

then ϕf is right multiplication by B. Such a complex X associated with T is unique up to unique isomorphism in D^{b}(A ⊗ B^{∘}) (see [8] and [13]).

We denote by _{α}A_{1} the (A ⊗ A^{∘})-module equal to A as a right A-module and where the left action of a ∈ A is given by multiplication by α(a) : this is the restriction of the natural structure of (A ⊗ A^{∘})-module on A through the morphism α ⊗ 1 : A ⊗ A^{∘} → A ⊗ A^{∘}.

Let us consider the set of isomorphism classes of two-sided tilting complexes for A ⊗ B^{∘} whose restriction to B^{∘} is in a given isomorphism class. It is acted on simply transitively by Out(A) = Aut(A)/ Inn(A) as shown by the following proposition.

Proposition 2.3. Let X and X′ be two-sided tilting complexes for A⊗B^{∘}. The restrictions of X and X′ to B^{∘} are isomorphic if and only if there exists α ∈ Aut(A) such that

X′ ≃ _{α}A_{1} ⊗_{A} X.

Proof. Assume X and X′ have isomorphic restrictions to B^{∘}. Let Y be a two-sided tilting complex in D^{b}(B ⊗A^{∘}) inverse to X and let Y′ be a two-sided tilting complex in D^{b}(B ⊗A^{∘}) inverse to X′.

The complexes X′ ⊗_{B}^{L} Y and X ⊗_{B}^{L} Y ≃ A have isomorphic restrictions to A^{∘}, hence they have both homology concentrated in degree 0. As a consequence, X′ ⊗_{B}^{L} Y is isomorphic to its degree 0 homology M = H^{0}(X′ ⊗_{B}^{L} Y) and M is free of rank 1 as an A^{∘}-module. The complexes X′ ⊗_{B}^{L} Y and X ⊗_{B}^{L} Y′ are two-sided tilting complexes for A ⊗ A^{∘}, inverse to each other. Consequently, M ⊗_{A} X ⊗_{B}^{L} Y′ ≃ A. It follows that X ⊗_{B}^{L} Y′ has homology concentrated in degree 0. Let N = H^{0}(X ⊗_{B}^{L} Y′). Then M ⊗_{A} N ≃ A. Since M is free of rank 1 as an A^{∘}-module, we deduce that N is free of rank 1 as an A^{∘}-module. Now, N ⊗_{A} M ≃ A. The bimodule M is invertible and free of rank 1 as an A^{∘}-module. By [3, Theorem 55.12], there exists an automorphism α of A so that M is isomorphic to _{α}A_{1} as an A ⊗ A^{∘}-module. The result follows.

2.2.4. Let us give now three results about compositions, products and sums of two-sided tilting complexes.

Standard equivalences can be composed [13, Proposition 4.1] :

Proposition 2.4. If C is a k-algebra, X a two-sided tilting complex for A ⊗ B^{∘} and X′ a two-sided tilting complex for B ⊗C^{∘}, then X ⊗_{B}^{L} X′ is a two-sided tilting complex for A⊗C^{∘}.

Tensor products of standard derived equivalences give standard derived equivalences [13, Lemma 4.3] :

6 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

Proposition 2.5. Let C and D be two k-algebras. Assume A and B are flat over k. Let X be a two-sided tilting complex for A ⊗ B^{∘} and let X′ be a two-sided tilting complex for C ⊗ D^{∘}. Then X ⊗^{L} X′ is a two-sided tilting complex for (A ⊗ C) ⊗ (B ⊗ D)^{∘}. In particular, X ⊗^{L} C is a two-sided tilting complex for (A ⊗ C) ⊗ (B ⊗ C)^{∘}.

The following result is clear :

Proposition 2.6. Let {A_{i}}_{i∈I} be a finite family of k-algebras, A = ∏_{i∈I} A_{i} and let e_{i} be the central idempotent of A such that e_{i}A = A_{i}.

Let T be a complex of A-modules. Then T is a tilting complex for A if and only if e_{i}T is a tilting complex for A_{i}, for every i ∈ I.

Let B be a k-algebra and X a two-sided tilting complex for A ⊗ B^{∘}. Let f_{i} be the central idempotent of B = End_{A}(X) given by the multiplication by e_{i} on X and B_{i} = f_{i}B. Then X_{i} = e_{i}X ≃ Xf_{i} is a two-sided tilting complex for A_{i} ⊗(B_{i})^{∘}, X = ⨁_{i} X_{i} and B = ∏_{i∈I} B_{i}.

2.3. Some invariants of a standard derived equivalence. Let X be a two-sided tilting complex for A ⊗ B^{∘}.

2.3.1. Hochschild cohomology. [13, Proposition 2.5]

Let i be an integer. Let f_{i} : HH^{i}(A) = Hom_{A⊗A^{∘}}(A, A[i]) → Hom(X, X[i]) be given by

(ϕ : A → A[i]) ↦ (ϕ ⊗_{A} 1_{X} : A ⊗_{A} X = X → A[i] ⊗_{A} X = X[i]).

Similarly, we have a map g_{i} : HH^{i}(B) → Hom(X, X[i]) given by

(ϕ : B → B[i]) ↦ (1_{X} ⊗_{B} ϕ : X ⊗_{B} B = X → X ⊗_{B} B[i] = X[i]).

Then f_{i} and g_{i} are isomorphisms and we put

HH^{i}(X) = f_{i}^{−1}g_{i} : HH^{i}(B) →^{∼} HH^{i}(A).

In particular, we have an isomorphism

HH^{0}(X) : ZB →^{∼} ZA.

2.3.2. Grothendieck groups. [6]

Recall that we denote the category of finitely generated projective A-modules by A-proj. Let K^{0}(A) be the Grothendieck group of A-proj. The natural embedding A-proj → K^{b}(A-perf) induces an isomorphism of the Grothendieck groups [17]. So, the equivalence X ⊗_{B}^{L} − : K^{b}(B-perf) → K^{b}(A-perf) induces an isomorphism K^{0}(B) →^{∼} K^{0}(A).

Let us assume now that A and B are right coherent. Let G^{0}(A) be the Grothendieck group of A-mod. The embedding A-mod → D^{b}(A-mod) induces an isomorphism of the Grothendieck groups [17]. So, the equivalence X ⊗_{B}^{L} − : D^{b}(B-mod) → D^{b}(A-mod) induces an isomorphism G^{0}(B) →^{∼} G^{0}(A).

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 7

2.4. Flat central base change. Proposition 2.5 solves trivially the problem of extending a standard derived equivalence through an extension of A coming from an extension of k. Base change with respect to the centers of A and B is more subtle, since the actions of the centers of A and B on a two-sided tilting complex for A ⊗ B^{∘} are the same only up to homotopy.

Let Z = ZA and let R be a flat commutative Z-algebra. Let X be a two-sided tilting complex for A ⊗ B^{∘}. We identify ZB with Z through the isomorphism HH^{0}(X) (cf §2.3.1). Let A′ = A ⊗_{Z} R and B′ = B ⊗_{Z} R. Let I be the ideal of R ⊗ R generated by the elements x ⊗ 1 − 1 ⊗ x for x ∈ R. We assume that A, B, A′ and B′ are flat over k.

Theorem 2.7. With the assumptions above, there is a pair (X′, f) associated with X unique up to unique isomorphism, where

• X′ is a bounded complex of A′-projective and B′^{∘}-projective (A′ ⊗ B′^{∘})-modules such that I is contained in the kernel of the canonical map R ⊗ R → End_{A′⊗B′^{∘}}(X′) and

• f : Res_{A′⊗B^{∘}} X′ →^{∼} A′ ⊗_{A} X an isomorphism.

Given such a pair, we have Res_{A⊗B′^{∘}} X′ ≃ X ⊗_{B} B′ and X′ is a two-sided tilting complex for A′ ⊗ B′^{∘}.

Let C be a k-algebra, Y be a two-sided tilting complex for B ⊗ C^{∘} and (Y′, g) a pair associated with Y. Let U = X ⊗_{B}^{L} Y. We identify Z with Z(C) via HH^{0}(U). Assume C and C ⊗_{Z} R projective over k and let

h = (f ⊗_{B}1_{Y})(1_{X′} ⊗_{B′} g) : Res_{A′⊗C^{∘}}(X′⊗_{B′}^{L} Y′) →^{∼} X′⊗_{B′}^{L} B′⊗_{B}^{L}Y →^{∼} A′⊗_{A}X⊗_{B}^{L}Y = A′⊗_{A}U.

Then (X′ ⊗_{B′}^{L} Y′, h) is a pair associated with U.

Lemma 2.8. Let M, N be two perfect complexes of A-modules. Then the canonical morphism

Hom_{A}(M, N) ⊗_{Z} R →^{∼} Hom_{A′}(A′ ⊗_{A} M, A′ ⊗_{A} N)

is an isomorphism.

Proof. Let E be the full subcategory of A-perf of complexes N such that the lemma holds for the pair (M, N), for all perfect complexes M.

Note that E is stable under translation. Let N_{1} → N → N_{2} ⇝ be a distinguished triangle in A-perf. We put M′ = A′ ⊗_{A} M, N′ = A′ ⊗_{A} N, etc... Thanks to the flatness of R over Z, we have a commutative diagram with exact rows

\xymatrix{Hom_{A}(M,N_{2}[−1])⊗_{Z} R \ar[r] \ar[d] & Hom_{A}(M,N_{1})⊗_{Z} R \ar[r] \ar[d] & Hom_{A}(M,N)⊗_{Z} R \ar[r] \ar[d] & Hom_{A}(M,N_{2})⊗_{Z} R \ar[r] \ar[d] & Hom_{A}(M,N_{1}[1])⊗_{Z} R \ar[d] \\ Hom_{A′} (M′,N′_{2}[−1]) \ar[r] & Hom_{A′} (M′,N′_{1}) \ar[r] & Hom_{A′} (M′,N′) \ar[r] & Hom_{A′} (M′,N′_{2}) \ar[r] & Hom_{A′} (M′,N′_{1}[1])}

8 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

The five lemma shows that, if the lemma holds for (M, N_{1}[i]) and (M, N_{2}[i]) for any i, it will hold for (M, N). Hence, E is a triangulated subcategory of A-perf. If N is a direct summand of N′ and the lemma holds for (M, N′), it holds for (M, N). Hence, E is an épaisse full triangulated subcategory of A-perf and in order to prove that it is equal to A-perf, it is enough to prove that it contains A.

The same arguments show that the full subcategory of A-perf of complexes M such that the lemma holds for Hom_{A}(M, A[i]) for any integer i is an épaisse full triangulated subcategory. So, all we have to do now is to prove the lemma for M = A and N = A[i]. When i ≠ 0, both terms in the lemma are zero, whereas for i = 0, the result is clear.

Proof of the theorem. Let T be the restriction of X to A and let T_{1} = A′ ⊗_{A} T. Consider the canonical morphism

ϕ_{1} : B′ = End_{A}(T) ⊗_{Z} R → End_{A′}(T_{1}).

By Lemma 2.8, ϕ_{1} is an isomorphism. Similarly, Hom_{A′}(T_{1}, T_{1}[i]) ≃ Hom_{A}(T, T [i])⊗_{Z} R = 0 for i ≠ 0.

Since A is in the subcategory of A-perf generated by T, it follows that A′ is in the subcategory of A′-perf generated by T_{1}. Hence T_{1} is a tilting complex for A′ and ϕ_{1} : B′ →^{∼} End_{A′}(T_{1}).

Similarly, the restriction T_{2} of X ⊗_{B} B′ to B′^{∘} is a tilting complex for B′^{∘} and we have a canonical isomorphism ϕ_{2} : A′^{∘} →^{∼} End_{B′}^{∘}(T_{2}).

For i ∈ {1, 2}, we denote by X_{i} a two-sided tilting complex for A′ ⊗ B′^{∘} associated with T_{i} as in §2.2.3. It comes with an isomorphism T_{i} →^{∼} Res_{A′} X_{i}.

We have a canonical isomorphism of B′^{∘}-modules :

g_{1} : Hom_{A′}(A′ ⊗_{A} X, X_{1}[i]) →^{∼} Hom_{A′}(Res_{A′} X_{1}, X_{1}[i]).

In particular, Hom_{A′}(Res_{A′}(A′ ⊗_{A} X), X_{1}[i]) = 0 for i ≠ 0. On the other hand, right multiplication by B on A′ ⊗_{A} X and X_{1} are compatible with the canonical isomorphism End_{A′}(A′ ⊗_{A} X) →^{∼} End_{A′}(X_{1}). So, g_{1} is an isomorphism of (B ⊗ B′^{∘})-modules and RHom_{A′}(A′ ⊗_{A} X, X_{1}) and RHom_{A′}(Res_{A′⊗B^{∘}} X_{1}, X_{1}) are isomorphic in D^{b}(B ⊗B′^{∘}). Since X_{1} is a two-sided tilting complex for A′ ⊗ B′^{∘}, we know that g comes from an isomorphism

f_{1} : Res_{A′⊗B^{∘}} X_{1} →^{∼} A′ ⊗_{A} X.

Similarly, X ⊗_{B} B′ and Res_{A⊗B′^{∘}} X_{2} are isomorphic.

We have isomorphisms in D^{b}(B ⊗ B′^{∘}) :

Res_{B⊗B′^{∘}} RHom_{A′}(X_{1}, X_{2}) ≃ RHom_{A′}(Res_{A′⊗B^{∘}} X_{1}, X_{2})
≃ RHom_{A′}(A′ ⊗_{A} X, X_{2})
≃ RHom_{A}(X, Res_{A⊗B′^{∘}} X_{2})
≃ RHom_{A}(X, X ⊗_{B} B′)
≃ RHom_{A}(X, X) ⊗_{B} B′
≃ Res_{B⊗B′^{∘}} B′.

Note that I is contained in the kernel of the canonical map R ⊗ R^{∘} → End_{A′⊗B′^{∘}}(X_{i}) for i ∈ {1, 2}. It follows that I is contained in the kernel of the canonical map R ⊗ R^{∘} → Hom_{A′}(X_{1}, X_{2}). Hence, the action of B′ ⊗ B′^{∘} on Hom_{A′}(X_{1}, X_{2}) factors through the canonical surjection

ψ : B′ ⊗ B′^{∘} → (B′ ⊗ B′^{∘})/I(B′ ⊗ B′^{∘}).

Similarly, the action of B′ ⊗ B′^{∘} on B′ factors through ψ. Now, the restriction of ψ to B ⊗ B′^{∘} is surjective. Hence, RHom_{A′}(X_{1}, X_{2}) ≃ Hom_{A′}(X_{1}, X_{2}) and B′ are isomorphic in D^{b}(B′ ⊗ B′^{∘}). Since X_{2} is a two-sided tilting complex, this shows that X_{1} and X_{2} are isomorphic and that (X_{1}, f_{1}) fulfills the requirements of the theorem.

Let now (X′, f) be as in the theorem. We have an isomorphism f_{1}^{−1}f : Res_{A′⊗B^{∘}} X′ →^{∼} Res_{A′⊗B^{∘}} X_{1}, hence we have isomorphisms in D^{b}(B ⊗ B′^{∘}) :

RHom_{A′}(Res_{A′⊗B^{∘}} X′, X_{1}) ≃ RHom_{A′}(Res_{A′⊗B^{∘}} X_{1}, X_{1}) ≃ Res_{B⊗B′^{∘}} B′.

Since I is contained in the kernel of the canonical maps R ⊗ R^{∘} → End_{A′⊗B′^{∘}}(X′) and R ⊗ R^{∘} → End_{A′⊗B′^{∘}}(X_{1}), we conclude as above that X′ and X_{1} are isomorphic.

The centralizer in B′ of B is the center of B′, hence the canonical map End_{B′⊗B′^{∘}}(B′) → End_{B′⊗B^{∘}}(Res_{B′⊗B^{∘}} B′) is an isomorphism. It follows that the canonical map End_{A′⊗B′^{∘}}(X_{1}) → End_{A′⊗B^{∘}}(Res_{A′⊗B^{∘}}(X_{1})) is an isomorphism. Consequently, there is a unique isomorphism i : X′ → X_{1} such that f = f_{1}i.

The last part of the theorem is clear.

Note that we have on our way proven the following result :

Proposition 2.9. Let A be a k-algebra, Z = ZA, R a flat commutative Z-algebra and A′ = A ⊗_{Z} R. Let T be a tilting complex for A. Then A′ ⊗_{A} T is a tilting complex for A′ with endomorphism ring End(T) ⊗_{Z} R.

2.5. Degenerate cases. We will see in this section that, for local or commutative algebras, Rickard’s theory gives nothing more than the usual Morita theory.

Lemma 2.10. Let A be an indecomposable k-algebra and T a tilting complex for A. If H^{i}(T) is projective for every i ∈ ℤ, then there is a progenerator P for A and an integer n such that T ≃ P [n].

Let B be a k-algebra and X a two-sided tilting complex for A ⊗ B^{∘}. Assume H^{i}(X) is a projective A-module for every i ∈ ℤ. Then there is an integer n such that M = H^{n}(X) induces a Morita equivalence between A and B and X ≃ M[−n].

Proof. We have T ≃ ⊕_{i∈ℤ}H^{i}(T)[−i]. The module M is finitely generated since T is perfect. Since T generates K^{b}(A-perf), it follows that ⨁_{i∈ℤ} H^{i}(T) is a progenerator for A. If T has non-zero homology in more than one degree, the indecomposability of A gives two distinct integers i and j such that Hom(H^{i}(T), H^{j}(T)) ≠ 0, hence such that Hom(T, T [j − i]) ≠ 0, which is impossible.

Let us come to the second part of the lemma. The assumption implies that Res_{A} X is a tilting complex for A with projective homology. Hence, by the first part of the lemma there is an integer n and an (A ⊗ B^{∘})-module M such that X is isomorphic to M[−n] and the restriction of M to A is a progenerator. Now, the canonical map B → End_{A} M is an isomorphism, hence M gives a Morita equivalence between A and B.

The following result is due to Roggenkamp and the second author for A local (cf [19]).

Theorem 2.11. Let A be an indecomposable k-algebra which is local or commutative.

Let T be a tilting complex for A. Then there is a progenerator P for A and an integer n such that T ≃ P [n].

Let B be a k-algebra and X a two-sided tilting complex for A⊗B^{∘}. Then there is an integer n such that M = H^{n}(X) induces a Morita equivalence between A and B and X ≃ M[−n].

In the local case, thanks to Lemma 2.10, the proposition follows from the following lemma (cf. also [15]) :

Lemma 2.12. Let A be a local ring. Let X be a bounded complex of projective A-modules such that Hom_{D^{b}(A)}(X[i], X) = 0 for i < 0. Then X is homotopy equivalent to a projective module translated in some degree.

Proof. Let us put X = (X^{n}, d^{n}). Replacing X by a complex which is homotopy equivalent to X, we may and will assume that the largest n such that X^{n} ≠ 0 satisfies H^{n}(X) ≠ 0 and that if m is minimal such that X^{m} ≠ 0, then d^{m} is not a split injection.

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 11

Let ψ_{1} : X^{m} → A be a surjection and ψ_{2} : A → X^{n} a split injection with splittings ζ_{1} and ζ_{2} such that d^{m}ζ_{1} is not split injective and ζ_{2}d^{n−1} is not surjective. We note that the existence of ψ_{1} and ψ_{2} follow from the fact that the indecomposable projective A-modules are free of rank 1 [1, Chap. II, §3, Exercice 3]. Let f = ψ_{2}ψ_{1} be the composition

f : X^{m} ⟶^{ψ_{1}} A ⟶^{ψ_{2}} X^{n}.

Let g_{m} : X → X^{m}[−m] be the morphism of complexes which is the identity in degree m and 0 in the other degrees and g′_{n} : X^{n}[−n] → X the morphism of complexes which is the identity in degree n and 0 in the other degrees. Let g be the composition

g : X[m − n] ⟶^{g_{m}[m−n]} X^{m}[−n] ⟶^{f[−n]} X^{n}[−n] ⟶^{g′_{n}} X.

Assume g is homotopy equivalent to zero. Then there are morphisms h ∈ Hom_{A}(X^{m}, X^{n−1}) and h′ ∈ Hom_{A}(X^{m+1}, X^{n}) such that

f = d^{n−1}h + h′d^{m}.

Therefore

1_{A} = ζ_{2}d^{n−1}hζ_{1} + ζ_{2}h′d^{m}ζ_{1}.

Since ζ_{2}d^{n−1} is not surjective, ζ_{2}d^{n−1}hζ_{1} is not invertible, hence lies in the radical of A. Similarly, d^{m}ζ_{1} is not split injective, hence ζ_{2}h′d^{m}ζ_{1} is in the radical of A. So, 1_{A} is in the radical of A. This is impossible.

It follows that g is not homotopy equivalent to zero and consequently Hom_{D^{b}(A)}(X[m − n], X) ≠ 0. This shows that m = n.

Proof of the theorem. We assume now that A is commutative. Let 𝔪 be a maximal ideal of A. We denote by A_{𝔪} the localization of A at 𝔪. By Proposition 2.9, T_{𝔪} = A_{𝔪} ⊗_{A} T is a tilting complex for A_{𝔪}. Since A_{𝔪} is local, it follows that H^{i}(T_{𝔪}) is finitely generated and projective for all integers i. Since H^{i}(T) ⊗_{A} A_{𝔪} is finitely generated and projective for all maximal ideals 𝔪 of A, we conclude that H^{i}(T) is a finitely generated projective A-module [1, Chap. I, §3, Proposition 12] for every i and the result follows from Lemma 2.10.

Together with Proposition 2.6, Theorem 2.11 has the following consequence :

Corollary 2.13. Let A be a local k-algebra or a commutative k-algebra and B a k-algebra derived equivalent to A. Then A and B are Morita equivalent.

Remark 1. There are nevertheless non-trivial equivalences of derived categories in commutative algebra and algebraic geometry involving non-affine varieties or some extra structures. An example is the Koszul duality between the exterior algebra Λ(V) of a vector space V and the symmetric algebra S(V^{∗}) of the dual vector space V^{∗}, where there is an equivalence

12 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

between the derived categories of bounded complexes of finitely generated graded modules. There are also derived equivalences of Mukai type involving in particular Calabi-Yau varieties.

2.6. Stable equivalences. We call a k-algebra A a Gorenstein k-algebra if A^{∗} = Hom_{k}(A, k) is a projective A-module. In this section, we assume that A is a finitely generated projective k-module and a Gorenstein k-algebra. All modules will be assumed to be finitely generated. Let B be a Gorenstein k-algebra finitely generated projective as a k-module.

The following proposition and corollary show that for such algebras, Rickard theory can be made more precise : equivalences of the derived categories can be lifted to equivalences of homotopy categories, which induce stable equivalences [13, Corollary 5.5] and [7].

Proposition 2.14. Let X be a two-sided tilting complex for A ⊗ B^{∘} and Y an inverse to X.

Then there exists a bounded complex C of (A ⊗ B^{∘})-modules and a bounded complex D of (B ⊗ A^{∘})-modules with the following properties :

1. X and C are isomorphic

2. Y and D are isomorphic

3. There is an integer n such that • C^{−i} = 0 and D^{i} = 0 for i > n,

• C^{−i} and D^{i} are projective for i < n,

• C^{−n} is projective as an A-module and projective as a B^{∘}-module,

• D^{n} is projective as a B-module and projective as a A^{∘}-module.

4. C ⊗_{B} D is homotopy equivalent to A as a complex of (A, A)-bimodules

5. D ⊗_{A} C is homotopy equivalent to B as a complex of (B, B)-bimodules.

Proof. By Lemma 2.2, there is a bounded complex C of (A ⊗ B^{∘})-modules and an integer n such that C is isomorphic to X, C^{i} = 0 for i < −n, C^{i} is projective for i > −n and C^{−n} is projective as an A-module and projective as a B^{∘}-module.

Let D = Hom_{A}(C, A) : this is a bounded complex of (B ⊗ A^{∘})-modules isomorphic to Y and D^{i} = 0 for i > n, D^{i} is projective for i < n and D^{n} is projective as a B-module and projective as an A^{∘}-module.

All terms of the bounded complex C ⊗_{B} D are projective, except the degree 0 term, which is projective over k. Since C ⊗_{B} D has homology only in degree 0, it is homotopy equivalent to a bounded complex Z with no terms in positive degrees, whose terms in negative degrees are projective and whose degree 0 term is k-projective. Since Z has homology only in

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 13

degree 0 and this homology module H^{0}(Z) ≃ A is projective over k, the restriction to k of Z is homotopy equivalent to H^{0}(Z). Since A ⊗ A^{∘} is Gorenstein, an injection of a projective module inside a module splits if it splits when restricted to k. We deduce that Z is homotopy equivalent to H^{0}(Z) as a complex of (A ⊗ A^{∘})-modules.

Similarly, D ⊗_{A} C is homotopy equivalent to B.

Let M be an (A ⊗ B^{∘})-module, projective as an A-module and as a B^{∘}-module. Let N be a (B ⊗ A^{∘})-module, projective as a B-module and as an A^{∘}-module. We say that M induces a stable equivalence between A and B with inverse N if

M ⊗_{B} N ⊕ projective ≃ A ⊕ projective as (A, A) − bimodules and

N ⊗_{A} M ⊕ projective ≃ B ⊕ projective as (B, B) − bimodules.

Let Ω_{A⊗A^{∘}}A be the kernel of the multiplication map A ⊗ A^{∘} → A. This module Ω_{A⊗A^{∘}}A induces a stable equivalence of A. We denote by Ω_{A⊗A^{∘}}^{−1}A an indecomposable (A ⊗ A^{∘})- module which is an inverse of Ω_{A⊗A^{∘}}A.

For V an A-module, ε = ±1 and n a non-negative integer, we put

Ω_{A}^{εn}V = (Ω_{A⊗A^{∘}}^{ε}A)^{⊗_{A}n} ⊗_{A} V.

Corollary 2.15. Let X be a two-sided tilting complex for A ⊗ B^{∘} and Y an inverse to X. Let C and D be as in Proposition 2.14, M = Ω_{A⊗B^{∘}}^{−n}C^{−n} and N = Ω_{B⊗A^{∘}}^{n}D^{n}. Then M and N induce inverse stable equivalences between A and B. Furthermore, up to projective direct summands, the isomorphism classes of M and N are independent of the choice of C and D.

Proof. The complex C ⊗_{B} D is a bounded complex all of whose terms are projective but the degree 0 term, which is isomorphic to C^{−n} ⊗_{B} D^{n} ⊕ projective module. By Proposition 2.14 (4), we have C^{−n} ⊗_{B} D^{n} ≃ A ⊕ projective module. Since

(Ω_{A⊗B^{∘}}^{−n}C^{−n}) ⊗_{B} (Ω_{B⊗A^{∘}}^{n}D^{n}) ⊕ projective module ≃ C^{−n} ⊗_{B} D^{n} ⊕ projective module,

it follows that M ⊗_{B} N ⊕ projective module ≃ A ⊕ projective module. Similarly, N ⊗_{A} M ⊕ projective module ≃ B ⊕ projective module. So, M and N induce inverse stable equivalences between A and B.

Assume C_{1} and C_{2} are two complexes with the properties of C in Proposition 2.14 : they are quasi-isomorphic, C_{1}^{−i} = 0 for i > m, C_{2}^{−i} = 0 for i > n, C_{1}^{−i} is projective for i < m and C_{2}^{−i} is projective for i < n. Assume n ≥ m. Then C_{2}^{−n}[n] is isomorphic to the cone of a morphism E → C_{2} where E is a bounded complex of finitely generated projective modules. So, C_{2}^{−n}[n] is isomorphic to the cone of a morphism E → C_{1}. Since A⊗B^{∘} is Gorenstein, this morphism, a priori in the derived category, comes from a genuine morphism of complexes f.

14 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

The cone of f is homotopy equivalent to a bounded complex with C_{1}^{−m} ⊕ projective module in degree −m and zero or projective terms elsewhere. It follows that

C_{2}^{−n} ⊕ projective module ≃ Ω_{A⊗B^{∘}}^{−m+n}C_{1}^{−m} ⊕ projective module

and the unicity statement is proved.

3. Picard groups

3.1. Definitions.

Definition 3.1. We denote by TrPic(A) the group of isomorphism classes of two-sided tilting complexes for A ⊗ A^{∘} where the product of the classes of X and Y is given by the class of X ⊗_{A} Y.

That this is indeed a group follows from Proposition 2.4.

Note that a standard derived equivalence between two algebras A and B induces an isomorphism between TrPic(A) and TrPic(B).

By §2.3, we have canonical morphisms

TrPic(A) → Aut ZA

TrPic(A) → Aut G_{0}(A) (if A is right coherent)

TrPic(A) → Aut K_{0}(A).

The usual Picard group Pic(A) is the group of isomorphism classes of invertible (A, A)- bimodules. Hence, we have a canonical injection

Pic(A) → TrPic(A).

Note that Pic(A) is not a normal subgroup of TrPic(A) nor an invariant of D^{b}(A) . For example, two Brauer tree algebras with the same numerical invariants are standardly derived equivalent [11] but they have non-isomorphic Picard groups if the trees have non-isomorphic automorphism groups.

We denote by TrPic^{0}(A) the subgroup of TrPic(A) given by those elements X whose induced automorphism of ZA fixes the idempotents, i.e., such that eX ≃ Xe for every idempotent e of ZA. Recall that the k-algebra A is indecomposable if Z(A) has no nontrivial idempotent.

Thanks to Proposition 2.6, we have :

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 15

Lemma 3.2. Let {A_{i}}_{i∈I} be a finite family of indecomposable k-algebras, A = ∏_{i∈I} A_{i} and let e_{i} be the central idempotent of A such that e_{i}A = A_{i}. The map X ↦ {e_{i}Xe_{i}}_{i} induces an isomorphism

TrPic^{0}(A) →^{∼} ∏_{i} TrPic A_{i}.

We denote by Sh(A) the subgroup of TrPic(A) generated by A[1]. It is clear that Sh(A) is central in TrPic(A). The group Sh(A) is an infinite cyclic group and the direct product Pic(A) × Sh(A) is a subgroup of TrPic(A).

Theorem 2.11 has the following consequence :

Proposition 3.3. If A is a matrix algebra over an indecomposable commutative k-algebra or over a local k-algebra, then TrPic(A) = Pic(A) × Sh(A).

3.2. Base change. In this section, we assume A is flat over k.

Let R be a commutative k-algebra. Then Proposition 2.5 gives a canonical morphism

TrPic(A) → TrPic(A ⊗ R).

The next two lemmas help reducing the study of TrPic(A) to the case of algebras over fields.

Lemma 3.4. Assume k is a local ring with maximal ideal 𝔪. Then the kernel of the canonical map

TrPic(A) →^{ϕ} TrPic(A ⊗ k/𝔪)

is contained in Out(A).

Proof. Let T be a bounded complex of finitely generated projective A-modules such that T ⊗ k/𝔪 is homotopy equivalent to its 0-homology. Then T is homotopy equivalent to its 0-homology : this is a consequence of the following fact. Let f be a morphism between two finitely generated projective A-modules. By Nakayama’s lemma f is a surjection if and only if f ⊗ 1_{k/𝔪} = 1_{A/𝔪A} ⊗_{A} f is a surjection. Similarly, f is a split injection if and only if f ⊗ 1_{k/𝔪} = 1_{A/𝔪A} ⊗_{A} f is a split injection.

If X is in the kernel of the map ϕ of the lemma, then the restriction of X to A is isomorphic to a projective A-module N and X is in Pic(A). Since N ⊗ k/𝔪 is a free A ⊗ k/𝔪-module of rank 1, it follows that N is a free A-module of rank 1. Hence, X is actually in Out(A) by Proposition 2.3.

For 𝔪 a maximal ideal of k, we put A_{𝔪} = A ⊗ k_{𝔪}.

16 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

Let H be a set of maximal ideals of k such that, given a maximal ideal 𝔪 of k outside H, the restriction to A_{𝔪} of a two-sided tilting complex for A_{𝔪} ⊗ A_{𝔪}^{∘} has finitely generated and projective homology.

Lemma 3.5. Assume A is indecomposable. Then the kernel of the canonical map TrPic(A)/ Sh(A) → ∏_{𝔪∈H} TrPic(A_{𝔪})/ Sh(A_{𝔪}),

is contained in Pic(A).

Proof. Let X be a two-sided tilting complex such that X ⊗ k_{𝔪} ≃ A_{𝔪}[n_{𝔪}] as an (A_{𝔪} ⊗ A_{𝔪}^{∘})- module for every 𝔪 ∈ H, where n_{𝔪} is an integer. Then H^{i}(X) ⊗ k_{𝔪} is a finitely generated projective A_{𝔪}-module for 𝔪 ∈ H. It is also finitely generated projective for 𝔪 ∉ H by assumption. So, X has finitely generated projective homology. Now, Lemma 2.10 says that X is in Pic(A) × Sh(A).

Let Z = ZA be the centre of A, let R be a flat commutative Z-algebra and assume A and A ⊗_{Z} R are flat over k. Then Theorem 2.7 gives a canonical morphism

TrPic(A) → TrPic(A ⊗_{Z} R).

3.3. A localization sequence. Assume k is a Dedekind domain with field of fractions K. Recall that an algebra B over a field K is separable if B ⊗_{K} L is semisimple for any field extension L of K. The k-algebra A is a hereditary order if it is a finitely generated projective k-module, if every left ideal of A is a projective A-module and if A ⊗ K is separable.

When k is a discrete valuation ring, the following result follows from the classification of tilting complexes by S. König and the second author [9].

Lemma 3.6. Let A be an hereditary order. Then the restriction to A of a two-sided tilting complex for A ⊗ A^{∘} has projective homology. If A is in addition indecomposable, then TrPic(A) = Pic(A) × Sh(A).

Proof. Let X be a two-sided tilting complex for A ⊗ A^{∘}. As A is hereditary, every indecomposable direct summand of the restriction T of X to A has non-zero homology in at most one degree, and this homology group is a projective A-module or a torsion module. Since End_{A}(T) ≃ A is torsion free, it follows that there is no indecomposable direct summand of T whose non-zero homology group is a torsion module. Hence, H^{i}(T) is projective for every i and the second part of the lemma follows from Lemma 2.10.

Assume A is an indecomposable k-algebra, finitely generated and projective as a k-module. We assume also that A ⊗ K is separable. Let H be the set of maximal ideals 𝔪 of k for which A_{𝔪} is not a maximal order. It is known that H is finite [2, §29A].

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 17

Let TrPicent(A) be the kernel of the canonical morphism TrPic(A) → Aut ZA.

The following theorem generalizes Fröhlich’s localization sequence for Picard groups [3, Theorem 55.25].

Theorem 3.7. There is an exact sequence :

1 → TrPicent(ZA) →^{f} TrPicent(A) →^{g} ∏_{𝔪∈H} TrPicent(A_{𝔪})/ Sh(A_{𝔪})

Here, g is the product of the canonical maps

TrPicent(A) → TrPicent(A_{𝔪}) → TrPicent(A_{𝔪})/ Sh(A_{𝔪})

and f : TrPicent(ZA) → TrPicent(A) is given by X ↦ Res_{ZA^{∘}}^{ZA⊗ZA^{∘}} X ⊗_{ZA} A, where A is viewed as a ZA ⊗ (A ⊗ A^{∘})-module by the action (z ⊗ (a_{1} ⊗ a_{2})) · a = za_{1}aa_{2} for a_{1}, a ∈ A, a_{2} ∈ A^{∘} and z ∈ ZA.

Proof. Fröhlich’s theorem [3, Theorem 55.25] says that the restriction of f to a map Picent(ZA) → Picent(A) is well-defined in the sense that X ⊗_{ZA} A is an invertible (A, A)- bimodule when X is an invertible (ZA, ZA)-bimodule. Fröhlich’s theorem states moreover that the sequence

1 → Picent(ZA) →^{f} Picent(A) →^{g} ∏_{𝔪∈H} Picent(A_{𝔪})

is exact.

Recall from Proposition 3.3 that

TrPicent(ZA) = Picent(ZA) × Sh(ZA).

It follows that f is well-defined and injective and that gf = 0.

When 𝔪 ∉ H, the order A_{𝔪} is maximal, hence hereditary [2, §26B]. It follows from Lemmas 3.6 and 3.5 that ker g is contained in Picent(A)×Sh(A), hence that ker g = im f.

Remark 2. 1. In general, g will not be surjective : for example, if A is indecomposable and commutative but there is a maximal ideal 𝔪 of k such that k_{𝔪} ⊗ A is not indecomposable, then g is not surjective. Examples for such rings are group rings ZG for an abelian group G over the integers Z.

2. We don’t know any example of an element in

TrPicent(ZG) − Picent(ZG) × Sh(ZG)

for a finite group G.

18 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

3.4. Stable Picard groups. We assume here A is a finitely generated Gorenstein kalgebra, projective as a k-module.

We say that an A-module M is projective-free if it has no projective direct summand. Given a finitely generated A-module M, there is a projective-free A-module N, unique up to isomorphism, such that M ≃ N ⊕ projective module. We call N the projective-free part of M.

Definition 3.8. We denote by StPic(A) the group of isomorphism classes of projective-free (A ⊗ A^{∘})-modules inducing a self-stable equivalence of A. The product of the classes of M and N is the class of the projective-free part of M ⊗_{A} N.

A stable equivalence induced by a bimodule between two algebras A and B gives rise to an isomorphism between StPic(A) and StPic(B).

We have a natural inclusion Pic(A) → StPic(A). Corollary 2.15 gives a canonical map TrPic(A) → StPic(A) and the following diagram is commutative :

\xymatrix{Pic(A) \ar[dr] \ar[rr] &  & TrPic(A) \ar@{-->}[dl] \\  & StPic(A)}

The bimodule Ω_{A⊗A^{∘}}A defines a central element of StPic(A) : this is the image of A[−1] ∈ TrPic(A).

Let Out_{0}(A) be the subgroup of Out(A) of those automorphisms fixing the isomorphism classes of non-projective indecomposable modules. The group Out_{0}(A) is invariant under stable equivalence [10].

4. Brauer tree algebras

4.1. Definition. Let Γ be a finite connected tree with a cyclic ordering of the edges adjacent to a given vertex and with a particular vertex v, the exceptional vertex, and a positive integer m, the multiplicity of the exceptional vertex. Let k be a field.

To this data (Γ, v, m) one associates a finite dimensional symmetric k-algebra, called a Brauer tree algebra, characterized up to Morita equivalence by the following properties :

The isomorphism classes of simple modules are parametrized by the edges of Γ. Denote by P_{j} a projective cover of a simple module S_{j} corresponding to an edge j. Then rad(P_{j})/ soc(P_{j}) is the direct sum of two uniserial modules U_{a} and U_{b} where a and b are the vertices of j. For c ∈ {a, b}, let j = j_{0}, j_{1}, …, j_{r} be the cyclic ordering of the r + 1 edges

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 19

around c. Then the composition factors of U_{c}, starting from the top, are

S_{j_{1}}, S_{j_{2}}, …, S_{j_{r}}, S_{j_{0}}, S_{j_{1}}, …, S_{j_{r}}

where the number of composition factors is m(r + 1) − 1 if c is the exceptional vertex and r otherwise. Note that when m = 1, the choice of an exceptional vertex is irrelevant.

Associated to a Brauer tree algebra are two numerical invariants: the number of edges of the tree Γ and the multiplicity of the exceptional vertex.

By [13, Theorem 4.2], two Brauer tree algebras with the same numerical invariants are derived equivalent. So, a Brauer tree algebra associated with (Γ, v, m) is derived equivalent to a Brauer tree algebra associated with a line with the same number of edges as Γ and with an exceptional vertex at an end having multiplicity m. Hence, to study TrPic for a Brauer tree algebra reduces to this last case.

4.2. Some elements of TrPic. We now restrict ourselves to the case of Brauer tree algebras where the multiplicity m is 1.

Let A be a basic Brauer tree algebra associated to a line with n edges numbered 1, …, n such that i is adjacent to i + 1, and with no exceptional vertex, so that the multiplicity m is 1. We assume n > 1.

The Loewy series of the projective indecomposable modules are as follows :

P_{1} = \begin{matrix}S_{1} \\ S_{2} \\ S_{1} \end{matrix} , P_{n} = \begin{matrix}S_{n} \\ S_{n−1} \\ S_{n} \end{matrix} and P_{i} = \begin{matrix}S_{i} \\ S_{i−1} S_{i+1} \\ S_{i} \end{matrix} for i ≠ 1, n.

The dimensions of the vector spaces of homomorphisms between projective modules is given by :

dim_{k} Hom_{A}(P_{i}, P_{j}) = \begin{cases}0 & if |i − j| > 1, \\ 1 & if |i − j| = 1, \\ 2 & if i = j. \end{cases}

By [16, Lemma 2] a projective cover of the (A⊗A^{∘})-module A is given by ⨁_{1≤i≤n} P_{i} ⊗P_{i}^{∗} →^{f} A, where P_{i}^{∗} is the A^{∘}-module Hom_{k}(P_{i}, k).

Let

X_{i} = (0 → P_{i} ⊗ P_{i}^{∗} →^{f} A → 0),

where A is in degree 0. The isomorphism class of this complex does not depend on the choice of f.

Theorem 4.1. The complex X_{i} is a two-sided tilting complex, hence defines an element t_{i} of TrPic(A).

20 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

Proof. To prove that X_{i} is a two-sided tilting complex, we follow the method of [16, Theorem 6]. We have

X_{i} ⊗_{A} X_{i}^{∗} = (0 → P_{i} ⊗ P_{i}^{∗} ⟶^{1⊗_{A}f^{∗}+f} P_{i} ⊗ P_{i}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} ⊕ A ⟶^{f⊗_{A}1−f^{∗}} P_{i} ⊗ P_{i}^{∗} → 0).

The map

f ⊗_{A} 1 : ⨁_{j} P_{j} ⊗ P_{j}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} → P_{i} ⊗ P_{i}^{∗}

is surjective. Now, the projective A-modules ⨁_{j≠i} P_{j} ⊗ P_{j}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} and P_{i} ⊗ P_{i}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} have no common non-zero direct summand. By [16, Lemma 1], this implies that the restriction of f ⊗_{A} 1 to P_{i} ⊗ P_{i}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} remains surjective. Since P_{i} ⊗ P_{i}^{∗} is projective, the map

f ⊗_{A} 1 − f^{∗} : P_{i} ⊗ P_{i}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} ⊕ A → P_{i} ⊗ P_{i}^{∗}

is a split surjection. By duality, the map

1 ⊗_{A} f^{∗} + f : P_{i} ⊗ P_{i}^{∗} → P_{i} ⊗ P_{i}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} ⊕ A

is a split injection. Hence, X_{i} ⊗_{A} X_{i}^{∗} is homotopy equivalent to a module V which satisfies

P_{i} ⊗ P_{i}^{∗} ⊕ P_{i} ⊗ P_{i}^{∗} ⊕ V ≃ P_{i} ⊗ P_{i}^{∗} ⊗_{A} P_{i} ⊗ P_{i}^{∗} ⊕ A.

As P_{i} ⊗_{A} P_{i}^{∗} ≃ Hom_{A}(P_{i}, P_{i}) has dimension 2, we obtain V ≃ A and finally X_{i} ⊗_{A} X_{i}^{∗} is homotopy equivalent to A.

The action of the functor

F_{i} := X_{i} ⊗_{A} −

on simple modules is easily described. One has

F_{i}(S_{i}) ≃ Ω(S_{i})[1] and F_{i}(S_{j}) ≃ S_{j} for j ≠ i,

where Ω(S_{i}) is the kernel of a surjective map P_{i} → S_{i}.

Let us now describe the action of F_{i} on projective modules.

Given i, j with |i − j| = 1, we denote by P_{i} → P_{j} a non-zero map from P_{i} to P_{j} — such a map is unique up to a scalar. From now on, complexes with zero terms in positive degrees will be written as C^{r} → ⋯ → C^{0}, where C^{0} is in degree 0.

Lemma 4.2. We have

F_{i}(P_{j}) ≃ \{ \begin{matrix}& P_{j} & if |i − j| > 1 \\ P_{i} → & 0 & if i = j \\ P_{i} → & P_{j} & if |i − j| = 1 \end{matrix}

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 21

Proof. When |i − j| > 1, we have P_{i}^{∗} ⊗_{A} P_{j} ≃ Hom_{A}(P_{i}, P_{j}) = 0, hence F_{i}(P_{j}) ≃ P_{j}.

The morphism

f ⊗_{A} 1 : ⨁_{j≠i} P_{j} ⊗ P_{j}^{∗} ⊗_{A} P_{i} → P_{i}

is not surjective. Since P_{i} is projective indecomposable, the morphism

f ⊗_{A} 1 : P_{i} ⊗ P_{i}^{∗} ⊗_{A} P_{i} → P_{i}

is surjective by [16, Lemma 1] and therefore X_{i} ⊗_{A} P_{i} has homology concentrated in degree −1. As P_{i}^{∗} ⊗_{A} P_{i} is two-dimensional, we obtain F_{i}(P_{i}) ≃ P_{i}[1].

The last case is clear.

4.3. Determination of the automorphism group. Now, we need to understand the automorphisms of A.

Lemma 4.3. Let B be a Brauer tree algebra with n edges and multiplicity m = 1. Then Out_{0}(B) ≃ k^{×}/μ_{n}(k), where μ_{n}(k) is the group of n-th roots of unity of k.

Proof. As explained in §4.1, the algebra B is derived equivalent to a basic Brauer tree algebra whose tree is a star. By §3.4, we are reduced to proving the lemma for such a B. Let B a basic star Brauer tree algebra with n edges and multiplicity 1.

Let us define an algebra A_{l} for l ≥ 0 with generators e_{i}, i ∈ Z/nZ, and t, with relations

t^{l+1} = 0, e_{i}^{2} = e_{i}, e_{i}e_{j} = 0 if i ≠ j, 1 = ∑_{i} e_{i} and te_{i} = e_{i+1}t.

It is well known that B ≃ A_{n} (cf [4]).

The algebra A_{l} is serial, hence indecomposable modules are determined (up to isomorphism) by their Loewy series. So Out_{0}(A_{l}) is the subgroup of Out(A_{l}) given by the automorphisms fixing the isomorphism classes of simple modules.

For x ∈ k^{×}, let α_{l}(x) be the automorphism of A_{l} given by α_{l}(x)(e_{i}) = e_{i} and α_{l}(x)(t) = xt. Then α_{l}(x) gives an element of Out_{0}(A_{l}). Assume x ∈ μ_{n}(k) and let y = ∑_{i} x^{i}e_{i} ∈ A_{l}. Then yty^{−1} = xt and ye_{i}y^{−1} = e_{i} for all i, hence α_{l}(x) is an inner automorphism.

We will prove by induction on l that for 1 ≤ l ≤ n, the morphism k^{×} → Out_{0}(A_{l}), x ↦ α_{l}(x) is surjective and has kernel μ_{n}(k).

Note that t is a generator of the Jacobson radical J(A_{l}) of A_{l} and that the map sending the generators t, e_{i} of A_{l} on the generators t, e_{i} of A_{l−1} induces an isomorphism A_{l}/J(A_{l})^{l} →^{∼} A_{l−1}.

An automorphism φ of A_{1} inducing the trivial automorphism of A_{0} ≃ A_{1}/J(A_{1}) has to fix the elements e_{i} : Indeed, we have φ(e_{i}) = e_{i} + t ∑_{j} φ_{ij}e_{j} where φ_{ij} ∈ k for such an

22 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

automorphism. Since φ(e_{i})φ(e_{j}) = 0 for i ≠ j, we get φ_{ij} = 0 if i ≠ j. As 1 = φ(1) = ∑_{i} φ(e_{i}), we also get φ_{ii} = 0. This implies φ(e_{i}) = e_{i}.

Let y = ∑_{i} a_{i}e_{i} + ∑_{i} b_{i}e_{i}t be an arbitrary invertible element of A_{1} (here, a_{i} ∈ k^{×} and b_{i} ∈ k). Then an elementary calculation shows

y^{−1} = ∑_{i} \frac{1}{a_{i}} e_{i} − ∑_{i} \frac{b_{i}}{a_{i−1}a_{i}} e_{i}t.

Hence, yty^{−1} = (∑_{i} c_{i}e_{i})t, where c_{i} = a_{i}/a_{i−1}. Note that ∏_{i} c_{i} = 1. It follows that α_{1}(x) is not inner, for x^{n} ≠ 1.

Assume the result holds for A_{l−1}, l ≥ 2. Let φ be an automorphism of A_{l} in Out_{0}(A_{l}). Then φ induces an automorphism of A_{l−1} in Out_{0}(A_{l−1}). By the induction hypothesis, we may assume that this induced automorphism is trivial, multiplying if necessary by some α_{l}(x) and by an inner automorphism. Then φ(e_{i}) = e_{i} + t^{l} ∑_{j} φ_{ij}e_{j} for some φ_{i,j} ∈ k. As φ(e_{i})φ(e_{j}) = 0 for i ≠ j, we get φ_{ij} = 0 if i ≠ j. Since 1 = φ(1) = ∑_{i} φ(e_{i}), we also get φ_{ii} = 0. This implies φ(e_{i}) = e_{i}. We have now φ(t) = t + t^{l} ∑_{i} φ_{i}e_{i} for some φ_{i} ∈ k. So,

φ(t)φ(e_{i}) = e_{i+1}t + φ_{i}t^{l}e_{i} and φ(e_{i+1})φ(t) = e_{i+1}t + t^{l}φ_{i+l}e_{i+1−l}.

As 1 < l ≤ n, we have e_{i+1−l} ≠ e_{i}, hence φ_{i} = 0 for 1 ≤ i ≤ n. Therefore, φ(t) = t and φ is trivial. Hence, the result is true for A_{l}.

It follows that Out_{0}(A_{n}) =< α_{n}(x) >_{x∈k^{×}}≃ k^{×}/μ_{n}(k).

Up to isomorphism, Ω_{A⊗A^{∘}}^{n}(A) has a unique non-zero and non-projective direct summand. We denote it by M. It induces a self-stable equivalence of Morita type. Since it is indecomposable, the module M ⊗_{A} V is indecomposable for any simple A-module V [10, Theorem 2.1]. We have then M ⊗_{A} V_{i} ≃ Ω^{n}V_{i} ≃ V_{n+1−i}. So, M ⊗_{A} − sends simple modules to simple modules. It now follows from [10, Theorem 2.1] that M induces a self-Morita equivalence. In other words, M is an invertible bimodule and we denote by ω the element (of order 2) of Out(A) it induces. Note that the image of ω in StPic(A) is central, hence ω is central in Out(A).

The image of ω in Aut G_{0}(A) corresponds to the non-trivial automorphism of the tree of A.

We denote by Γ the subgroup of TrPic(A) generated by t_{1}, …, t_{n} and by G the subgroup of TrPic(A) generated by t_{1}, …, t_{n}, ω and [1].

Proposition 4.4. We have

Out(A) =< ω > × Out_{0}(A).

The group Out_{0}(A) centralizes Γ and ωt_{i}ω = t_{n+1−i}. Furthermore, Γ ∩ Out(A) = 1 and G ∩ Out_{0}(A) = 1.

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 23

Proof. Since indecomposable A-modules are determined by their radical series, an automorphism of A which fixes the isomorphism classes of simple modules will fix the isomorphism classes of all modules. Hence, Out_{0}(A) is the kernel of the canonical map Out(A) → Aut G_{0}(A). This map factors actually through the group of automorphisms of the tree of A. The group of automorphisms of the tree has order 2 and is generated by the image of ω. It follows that Out(A) =< ω > × Out_{0}(A). Note that we have followed [10, Theorem 4.7].

Let us consider the complex

1A_{ω} ⊗_{A} ( ⨁_{1≤i≤n} P_{i} ⊗ P_{i}^{∗} →^{f} A) ⊗_{A} (_{ω}A_{1}) ≃ ( ⨁_{1≤i≤n}(_{1}A_{ω} ⊗_{A} P_{i}) ⊗ (_{1}A_{ω} ⊗_{A} P_{i})^{∗} →^{g} A).

This complex defines again a projective cover of A, hence is isomorphic, as a complex, to

the complex ⨁_{1≤i≤n} P_{i} ⊗ P_{i}^{∗} →^{f} A. As _{1}A_{ω} ⊗_{A} P_{i} ≃ P_{n+1−i}, the complex

1A_{ω} ⊗_{A} X_{i} ⊗_{A} (_{ω}A_{1}) ≃ (P_{n+1−i} ⊗ P_{n+1−i}^{∗} →^{g} A)

is isomorphic to X_{n+1−i}, hence, ωt_{i}ω = t_{n+1−i}.

Similarly, one proves that Out_{0}(A) centralizes each of the elements t_{i} of Γ.

Let us prove that Out(A) ∩ Γ = 1. Since the canonical map Out(A) → StPic(A) is injective, we can check this property in StPic(A). But, the image of t_{i} in StPic(A) is trivial, hence the property holds. Finally, the image of G in StPic(A) intersects trivially the image of Out_{0}(A), so we conclude that G ∩ Out_{0}(A) = 1.

Remark 3. By Propositions 2.3 and 4.4, when doing calculations inside G, it is enough to look at the action on projective indecomposable modules : for σ, σ′ ∈ G, we have σ = σ′ if and only if σ(P) ≃ σ′(P) for any indecomposable projective module P. Our main tool will then be Lemma 4.2.

4.4. Braid relations. Denote by B_{n+1} the Artin braid group on n + 1 strings, generated by σ_{1}, …, σ_{n} with the relations σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| > 1 and σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}. We put w_{0} = σ_{1}(σ_{2}σ_{1}) ⋯ (σ_{n−1} ⋯ σ_{1})(σ_{n} ⋯ σ_{1}).

Theorem 4.5. There is a surjective group morphism

B_{n+1} → Γ, σ_{i} ↦ t_{i}.

The image of w_{0} is ω[n].

Proof. The first braid relation t_{i}t_{j} = t_{j}t_{i} if |i − j| > 1 is immediate, since P_{i}^{∗} ⊗_{A} P_{j} ≃ Hom_{A}(P_{i}, P_{j}) = 0 for |i − j| > 1.

By Remark 3, we have only to check that F_{i}F_{i+1}F_{i}(P) ≃ F_{i+1}F_{i}F_{i+1}(P) for every projective indecomposable module P.

24 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

Since F_{i+1} preserves cones, the complex F_{i+1}(P_{i} → P_{i−1}) is isomorphic to the cone of a non-zero morphism (P_{i+1} → P_{i}) ⟶ P_{i−1}. Therefore this complex is isomorphic to a three terms complex P_{i+1} → P_{i} → P_{i−1}. Note that the notation follows the convention above and that such a complex is well defined up to isomorphism.

The complex F_{i}(P_{i+1} → P_{i}) is isomorphic to the cone of a non-zero morphism (P_{i} → P_{i+1}) ⟶ (P_{i} → 0) hence is isomorphic to P_{i+1} → 0.

Similarly, F_{i+1}(P_{i} → P_{i+1}) ≃ (P_{i} → 0). Now we have done the necessary computations to determine F_{i+1}F_{i}(P_{j}) for all j. Two

more are necessary to determine F_{i}F_{i+1}F_{i}(P_{j}) for all j.

The complex F_{i}(P_{i+1} → P_{i} → P_{i−1}) is isomorphic to the cone of a non-zero morphism (P_{i+1} → 0) ⟶ (P_{i} → P_{i−1}), hence to P_{i+1} → P_{i} → P_{i−1}.

We have F_{i}(P_{i+1} → P_{i+2}) ≃ P_{i} → P_{i+1} → P_{i+2}. Summarizing, we have :

( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i−1} ; P_{i} ; P_{i+1} ; P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} ) ⟶^{F_{i}}  ( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i} → P_{i−1} ; P_{i} → 0 ; P_{i} → P_{i+1} ; P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} ) ⟶^{F_{i+1}} ( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i+1} → P_{i} → P_{i−1} ; P_{i+1} → P_{i} → 0 ; P_{i} → 0 ; P_{i+1} → P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} ) ⟶^{F_{i}} ( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i+1} → P_{i} → P_{i−1} ; P_{i+1} → 0 → 0 ; P_{i} → 0 → 0 ; P_{i} → P_{i+1} → P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} )

and

( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i−1} ; P_{i} ; P_{i+1} ; P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} ) ⟶^{F_{i+1}}  ( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i−1} ; P_{i+1} → P_{i} ; P_{i+1} → 0 ; P_{i+1} → P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} ) ⟶^{F_{i}} ( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i} → P_{i−1} ; P_{i+1} → 0 ; P_{i} → P_{i+1} → 0 ; P_{i} → P_{i+1} → P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} ) ⟶^{F_{i+1}} ( P_{1} ; P_{2} ; ⋮ ; P_{i−2} ; P_{i+1} → P_{i} → P_{i−1} ; P_{i+1} → 0 → 0 ; P_{i} → 0 → 0 ; P_{i} → P_{i+1} → P_{i+2} ; P_{i+3} ; ⋮ ; P_{n} )

Hence, we have indeed F_{i}F_{i+1}F_{i}(P_{j}) ≃ F_{i+1}F_{i}F_{i+1}(P_{j}) for all j. By induction on i, we have

( P_{1} ; P_{2} ; P_{3} ; ⋮ ; P_{i} ; P_{i+1} ; P_{i+2} ; ⋮ ; P_{n} ) ⟶^{F_{i}F_{i−1}···F_{1}}  ( P_{i} → P_{i−1} → ⋯ → P_{2} → P_{1} → 0 ; P_{1} → 0 ; P_{2} → 0 ; ⋮ ; P_{i−1} → 0 ; P_{i} → P_{i+1} ; P_{i+2} ; ⋮ ; P_{n} )

and in particular,

( P_{1} ; P_{2} ; P_{3} ; ⋮ ; P_{n} ) ⟶^{F_{n}F_{n−1}···F_{1}}  ( P_{n} → P_{n−1} → ⋯ → P_{2} → P_{1} → 0 ; P_{1} → 0 ; P_{2} → 0 ; ⋮ ; P_{n−1} → 0 )

Now, by induction on i, we have

F_{i}F_{i−1} ⋯ F_{1}(P_{r} → P_{r−1} → ⋯ P_{2} → P_{1}) ≃ (P_{r} → P_{r−1} → ⋯ → P_{i+1})[i]

for i < r. In particular,

F_{r−1} ⋯ F_{1}(P_{r} → P_{r−1} → ⋯ P_{2} → P_{1}) ≃ P_{r}[r − 1].

We deduce that

F_{1}(F_{2}F_{1}) ⋯ (F_{n} ⋯ F_{1})(P_{i}) ≃ P_{n−i+1}[n].

In the next section, we will prove that for n = 2, the morphism B_{3} → Γ is bijective and in section 4.6 that TrPic(A) is generated by Γ and Pic(A) × Sh(A).

4.5. Faithfulness of the braid group action. We assume now that A is a Brauer tree algebra associated to a line with two edges and no exceptional vertex. An example of such an algebra is the group algebra of the symmetric group 𝔖_{3} over a field of characteristic 3.

Put ϕ = t_{1}ω. A tilting complex corresponding to ϕ acts as

( \begin{matrix}P_{1} \\ P_{2} \end{matrix} ) ⟶^{ϕ} ( \begin{matrix}P_{1} & → & P_{2} \\ P_{1} & → & 0 \end{matrix} )

As is shown in Theorem 4.5, we have ω = t_{1}t_{2}t_{1}[−2]. Note that ϕ^{3} = t_{1}t_{2}t_{1}ω = [2].

Theorem 4.6. The map

S = ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ) ↦ ϕ and T = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ↦ ω

induces an isomorphism χ : PSL_{2}(Z) →^{∼} G/ Sh(A). Hence the subgroup G of TrPic(A) is isomorphic to a central extension of PSL_{2}(Z).

26 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

Since PSL_{2}(Z) is generated by S and T with the relations S^{3} = T^{2} = 1, we have indeed a morphism PSL_{2}(Z) → G/ Sh(A). This morphism is surjective, since G is generated by ϕ, ω and [1].

Note that the morphism B_{3}/Z(B_{3}) → PSL_{2}(Z) given by

σ_{1} ↦ ST and σ_{2} ↦ T S

is an isomorphism. As a consequence, we have :

Corollary 4.7. The morphism B_{3} → TrPic(A) given by σ_{i} ↦ t_{i} is injective.

Let C be a bounded complex of projective modules. Then we have a decomposition C = C_{r} ⊕ C_{a} in the category of complexes, where C_{a} is homotopy equivalent to 0 and C_{r} has no non-zero direct summand which is homotopy equivalent to 0. We call C_{r} the reduced part of C. This is well defined up to isomorphism in the category of complexes.

For X a complex of k-modules, we denote by dim X the dimension of X, viewed as a k-module by forgetting the differential and the grading.

Let C ∈ TrPic(A) and C_{i} be the reduced part of C ⊗_{A} P_{i}. For {i, j} = {1, 2}, we denote by Cone(C_{i} → C_{j}) the reduced part of the cone of a non-zero morphism from C_{i} to C_{j}. Since Hom_{A}(C_{i}, C_{j}) ≃ Hom_{A}(P_{i}, P_{j}) is one-dimensional, the morphism is well defined up to a scalar in the homotopy category, hence Cone(C_{i} → C_{j}) is well defined up to isomorphism in the category of complexes.

We deduce Theorem 4.6 from the following more precise result :

Proposition 4.8. Let C ∈ G equal to χ( \begin{matrix}a & b \\ c & d \end{matrix} ) up to shift.

Then forgetting the differential and the grading, C_{1} is isomorphic to P_{1}^{|a|} ⊕ P_{2}^{|c|} and C_{2} is isomorphic to P_{1}^{|b|} ⊕ P_{2}^{|d|}.

Assume ab and cd are not both zero. Let C_{12} = Cone(C_{1} → C_{2}) and C_{21} = Cone(C_{2} → C_{1}).

If ab ≤ 0 and cd ≤ 0, then dim C_{12} = | dim C_{1} − dim C_{2}| and dim C_{21} = dim C_{1} + dim C_{2}. If ab ≥ 0 and cd ≥ 0, then dim C_{12} = dim C_{1} + dim C_{2} and dim C_{21} = | dim C_{1} − dim C_{2}|.

Remark 4. Note that, for example when ab ≤ 0, cd ≤ 0, |b| ≤ |a| and |d| ≤ |c|, the statement of the proposition is that every morphism C_{1} → C_{2} which is not homotopy equivalent to zero is surjective. It is an obvious fact that any morphism in the homotopy category can be represented by a monomorphism or an epimorphism in the category of complexes by adding a large enough complex which is homotopy equivalent to 0. The statement in the proposition is concerned with monomorphisms or epimorphisms between reduced complexes.

PICARD GROUPS FOR DERIVED MODULE CATEGORIES 27

Proof. Note first that an element x = ( \begin{matrix}a & b \\ c & d \end{matrix} ) of PSL_{2}(Z) is determined by |a|, |b|, |c|, |d| and by the signs of ab and cd. Note that if both ab and cd are non-zero, then these signs are equal.

The proposition is clear when ab = cd = 0, since then C is isomorphic, up to shift, to A or to M.

So, we assume (ab, cd) ≠ (0, 0). We will prove the proposition by induction on |a| + |b| + |c| + |d|.

Conjugating if necessary x by T, we may assume that ab ≤ 0 and cd ≤ 0. Let us assume that |b| ≤ |a| and |d| ≤ |c|. The other case can be dealt with the same proof as below, conjugating all matrices by ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ).

We have

x = ( \begin{matrix}−b & a + b \\ −d & c + d \end{matrix} ) S.

When b|a + b| = d|c + d| = 0, we have two cases:

If x = ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) then C ≃ X_{2}, up to a shift.

If x = ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ) then C ≃ X_{1} ⊗_{A} M, up to a shift.

In the first case, we have

\begin{aligned}C_{1} & ≃ & P_{2} → P_{1}, & C_{12} & ≃ & P_{1}[1], \\ C_{2} & ≃ & P_{2} → 0, & C_{21} & ≃ & P_{2} → P_{2} → P_{1} \end{aligned}

and we are done.

In the second case, we have

\begin{aligned}C_{1} & ≃ & P_{1} → P_{2}, & C_{12} & ≃ & P_{2}[1], \\ C_{2} & ≃ & P_{1} → 0, & C_{21} & ≃ & P_{1} → P_{1} → P_{2} \end{aligned}

and we are also done.

Assume now b|a + b| and d|c + d| are not both zero. Denote by C′ a two-sided tilting complex such that the class of C in TrPic(A) is the product of the class of C′ by ϕ. Let C′_{1} be the reduced part of C′ ⊗_{A} P_{1} and let C′_{2} be the reduced part of C′ ⊗_{A} P_{2}. The image of

C′ in G/ Sh(A) is equal to χ( \begin{matrix}−b & a + b \\ −d & c + d \end{matrix} ).

Note that our assumptions on the sign of ab and cd, on the absolute value of a compared to that of b and on the absolute value of c compared to that of d imply |a+b|+|c+d| < |a|+|c|. Hence by induction we have dim Cone(C′_{1} → C′_{2}) = dim C′_{1} + dim C′_{2} and dim Cone(C′_{2} → C′_{1}) = | dim C′_{1} − dim C′_{2}|. We have C_{1} ≃ Cone(C′_{1} → C′_{2}) and C_{2} ≃ C′_{1}[1]. Consider the canonical map Hom(C′_{1}[1], C′_{1}[1]) → Hom(C_{1}, C_{2}). Then, the morphism C_{1} → C_{2} which is the image of the identity morphism on C′_{1}[1] under this canonical map is not homotopy equivalent to zero. So, dim Cone(C_{1} → C_{2}) = dim C′_{2} = dim C_{1} − dim C_{2}.

We need to prove now that dim Cone(C_{2} → C_{1}) = dim C_{1} + dim C_{2}.

28 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

Let z ∈ Z(A) such that multiplication by z induces a non-zero but non-invertible endomorphism of P_{1} : such an element is obtained as follows. Since End_{A⊗A^{∘}}(A) ≃ Z(A) and the head and the socle of A as an (A ⊗ A^{∘})-module are isomorphic to S_{1} ⊗ S_{1}^{∗} ⊕ S_{2} ⊗ S_{2}^{∗}, we take for z an (A ⊗ A^{∘})-endomorphism of A with image isomorphic to S_{1} ⊗ S_{1}^{∗}.

Let z′ be the image of z by the automorphism of Z(A) induced by C′. Then under the isomorphism End(P_{1}) ≃ End(C′_{1}) induced by C′, the image of the endomorphism given by multiplication by z is the endomorphism given by multiplication by z′.

Multiplication by z′ on a projective module has image in the socle of this module. Hence, the morphism f : C′_{1}[1] → C′_{1}[1] given by multiplication by z′ extends to a morphism g : C′_{1}[1] → Cone(C′_{1} → C′_{2}). Now, the identity map C′_{1}[1] → C′_{1}[1] extends to a map h : Cone(C′_{1} → C′_{2}) → C′_{1}[1] and we have f = hg. As f is not zero, g is not zero either. The reduced part of the cone of g has dimension dim C′_{1} + dim Cone(C′_{1} → C′_{2}). Now, a non-zero morphism C_{2} → C_{1} is equal to g up to a scalar. Hence, its cone has dimension dim C_{2} + dim C_{1}. So, the second part of the proposition holds for x.

Finally, we know by induction that C′_{1} is isomorphic to P_{1}^{|b|} ⊕ P_{2}^{|d|} and C′_{2} is isomorphic to P_{1}^{|a+b|} ⊕ P_{2}^{|c+d|}, when the differential and the grading are omitted. As dim C_{1} = dim C′_{1} + dim C′_{2} and C_{1} = Cone(C′_{1} → C′_{2}), we deduce that C_{1} is isomorphic to P_{1}^{|a|} ⊕ P_{2}^{|c|} when the differential and the grading are omitted. Since C_{2} ≃ C′_{1}[1], when omitting the differential and the grading, C_{2} becomes isomorphic to P_{1}^{|b|} ⊕ P_{2}^{|d|}. So, the first part of the proposition holds for x.

4.6. Transitivity of the braid group action. In this section we prove that, for A a Brauer tree algebra with two edges and no exceptional vertex, the group TrPic(A) is generated, by t_{1}, t_{2} and Pic(A) × Sh(A), and we deduce the structure of TrPic(A).

Let us start with some general properties on the image of simple modules by a derived equivalence.

Let A be a finite dimensional algebra over a field k.

For X a bounded complex with non-zero homology, we denote by lb(X) the smallest integer i with H^{i}(X) ≠ 0. Similarly, we denote by rb(X) the largest integer i with H^{i}(X) ≠ 0. We define the amplitude of X as Λ(X) = {lb(X), lb(X) + 1, …, rb(X)}. Finally, the length ℓ(X) of X is the cardinality of its amplitude.

Lemma 4.9. Let U ⟶ V ⟶ W ⇝ be a distinguished triangle in D^{b}(A).

Then Λ(V) ⊆ Λ(U) ∪ Λ(W).

If lb(U) ≠ lb(W) + 1 and rb(U) ≠ rb(W) + 1, then we have Λ(V) = Λ(U) ∪ Λ(W).

Proof. This follows immediately from the long exact sequence

⋯ → H^{i}(U) → H^{i}(V) → H^{i}(W) → H^{i+1}(U) → · · · .

Let C be a two-sided tilting complex in D^{b}(A ⊗ A^{∘}). Replacing C by an isomorphic complex, we may and will assume that C^{i} = 0 for i ∉ Λ(C). To avoid trivialities, we assume furthermore that Λ(C) has more than one element, or in other words that C is not a shifted module.

Denote by F : D^{b}(A) → D^{b}(A) the functor C ⊗_{A} −.

The following lemma is clear :

Lemma 4.10. If M is an A-module, then Λ(F (M)) ⊆ Λ(C).

Lemma 4.11. We have Λ(C) = ⋃_{V} Λ(F (V)) where V runs over the simple A-modules.

Proof. Since, as an A-module, C ≃ F (A), and as A has a composition series of simple modules, Lemma 4.9 gives the inclusion Λ(C) ⊆ ⋃_{V} Λ(F (V)). The reverse inclusion follows from Lemma 4.10.

The next lemma is crucial — when ℓ(C) = 2, this had been pointed out to us by J. Rickard.

Lemma 4.12. If V is simple, then Λ(F (V)) ≠ Λ(C).

Proof. Let V be a simple module with Λ(F (V)) = Λ(C). Let T be the restriction of C^{∗} to A. Then the complex of k-modules Hom_{A}(T, V) ≃ C ⊗_{A} V has amplitude Λ(C) = Λ(T^{∗}). Let m = lb(T) and n = rb(T). There is a non-zero morphism T → V [−n], hence a non-zero morphism f : P [−n] → T, which is injective, where P is a projective cover of V. There is also a non-zero morphism T → V [−m]. This means that T^{m} has a direct summand isomorphic to P whose intersection with H^{m}(T) is non-zero. So, there is a non-zero morphism g : T → P [−m] which is surjective. Now, the morphism f[n − m] ∘ g : T → T [n − m] is non-zero :

0 → T^{m} → T^{m+1} → ⋯
↓
P
↓
⋯ → T^{n−1} → T^{n} → 0

But, T being a tilting complex, that is not possible unless m = n, which has been excluded, so we get a contradiction.

From now, A is a Brauer tree algebra with two edges and no exceptional vertex.

30 RAPHAËL ROUQUIER AND ALEXANDER ZIMMERMANN

The algebra A has two simple modules S_{1} and S_{2} and we may assume the indexing is chosen so that

(1) lb(F (S_{1})) > lb(F (S_{2})) and rb(F (S_{1})) > rb(F (S_{2}))

since by Lemmas 4.11 and 4.12, there is no inclusion between the sets Λ(F (S_{1})) and Λ(F (S_{2})). Note that, we have then lb(C) = lb(F (S_{2})) and rb(C) = rb(F (S_{1})).

Denote by X = P_{1} → P_{2} a complex with P_{2} in degree 0 and where the differential P_{1} → P_{2} is non-zero. We have H^{0}(X) ≃ S_{2} and H^{−1}(X) ≃ S_{1}, hence we have a distinguished triangle

S_{1}[1] → X → S_{2} ⇝

and applying F, a distinguished triangle

F (S_{1})[1] → F (X) → F (S_{2}) ⇝

By Lemma 4.9, this implies Λ(F (X)) ⊆ {lb(F (S_{2})), …, rb(F (S_{1})) − 1}, using (1). In particular, ℓ(F (X)) < ℓ(C).

Let L be the kernel of a surjective map P_{2} → S_{2}. We have an exact sequence

0 → S_{2} → L → S_{1} → 0,

hence a distinguished triangle

F (S_{2}) → F (L) → F (S_{1}) ⇝

By Lemma 4.9, we obtain Λ(F (L)) = Λ(C). The distinguished triangle

F (L) → F (P_{2}) → F (S_{2}) ⇝

shows that lb(F (P_{2})) = lb(C). The distinguished triangle

F (S_{1}) → F (P_{1}) → F (L) ⇝

shows that rb(F (P_{1})) = rb(C).

If rb(F (P_{2})) < rb(C), then Λ(F (X) ⊕ F (P_{2})) is strictly contained in Λ(C). We use the notation of §4.2 for the complex X_{2}. Let C′ = C ⊗_{A} X_{2}^{∗}[1]. Then C′ ⊗_{A} P_{1} ≃ F (X) and C′ ⊗_{A} P_{2} ≃ F (P_{2}). Consequently, Λ(C′) is strictly contained in Λ(C).

If rb(F (P_{2})) = rb(C), then Λ(F (X)[−1] ⊕ F (P_{1})) ⊆ Λ(C) and ℓ(F (X)) + ℓ(F (P_{1})) < ℓ(F (P_{2})) + ℓ(F (P_{1})). Let C′ = C ⊗_{A} X_{1}[−1]. Then C′ ⊗_{A} P_{1} ≃ F (P_{1}) and C′ ⊗_{A} P_{2} ≃ F (X)[−1]. So, Λ(C′) ⊆ Λ(C) and ℓ(C′ ⊗_{A} P_{1}) + ℓ(C′ ⊗_{A} P_{2}) < ℓ(C ⊗_{A} P_{1}) + ℓ(C ⊗_{A} P_{2}).

It follows by induction first on ℓ(C), then on ℓ(C ⊗_{A} P_{1}) + ℓ(C ⊗_{A} P_{2}), that, modulo the subgroup generated by t_{1} and t_{2}, every element of TrPic(A) is in Pic(A) × Sh(A).

Denote by B̃_{3} the extension of B_{3} =< σ_{1}, σ_{2} > generated by z, σ_{1} and σ_{2} with the

relations z^{4} = (σ_{1}σ_{2})^{3} and zσ_{1}z^{−1}σ_{1}^{−1} = zσ_{2}z^{−1}σ_{2}^{−1} = 1. We have an injective morphism

B̃_{3} → TrPic(A) given by σ_{i} ↦ t_{i} and z ↦ [1].

We have completed our description of TrPic :

Theorem 4.13. Let A be a Brauer tree algebra over a field k, with two edges and without

exceptional vertex. Then

TrPic(A) ≃ B̃_{3} × (k^{×}/{±1}).

Remark 5. The results of §4 have a counterpart for Green orders as defined by

Roggenkamp. Details and proofs are given in [20].

Let O be a complete discrete valuation ring with residue field k and A a Brauer tree

algebra over k with n edges and no exceptional vertex. For Green-orders Λ over O, such

that Λ ⊗_{O} k ≃ A, one can construct a morphism B_{n+1} ⟶ TrPic(Λ) lifting the morphism

B_{n+1} → TrPic(A) constructed in §4.4. Moreover, one proves that when n = 2, there is

an isomorphism B̃_{3} ≃ TrPic(Λ). The canonical map TrPic(Λ) → TrPic(A) will not be

surjective in general.

References

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[2] C.W. Curtis and I. Reiner, “Methods of representation theory”, volume I, John Wiley Interscience, 1981.

[3] C.W. Curtis and I. Reiner, “Methods of representation theory”, volume II, John Wiley Interscience, 1987.

[4] P. Gabriel and Chr. Riedtmann, Group representations without groups, Comm. Math. Helv. 54 (1978), 240-287.

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

[6] A. Grothendieck, Groupe des classes des catégories abéliennes et triangulées, complexes parfaits, in SGA 5, Lecture Notes in Math. 589, pp. 351–371, 1977.

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[9] S. König and A. Zimmermann, Tilting hereditary orders, Comm. in Alg. 24 (1996), 1897–1913.

[10] M. Linckelmann, Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z. 223 (1996), 87–100.

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[12] J. Rickard, Derived categories and stable equivalence, J. Pure and Appl. Alg. 61 (1989), 303–317.

[13] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37–48.

[14] J. Rickard, Triangulated categories in the modular representation theory of finite groups, in “Derived categories for group rings”, Lecture Notes in Math. 1685, pp. 177–198, Springer Verlag, 1998.

[15] J. Rickard, letter to the second author from 21. October 1993.

[16] R. Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, proceedings of “Groups 1993, Galway–Saint-Andrews conference” volume 2, 512–523, London Math. Soc. series 212, Cambridge University Press, 1995.

[17] J. L. Verdier, Catégories dérivées, état 0, in SGA4\frac{1}{2}, Lecture Notes in Math. 569, pp. 262–308, Springer Verlag, 1977.

[18] A. Yekutieli, Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. London Math. Soc. 60 (1999) 723-746.

[19] A. Zimmermann, Derived equivalences of orders, Proceedings of the ICRA VII, Mexico, eds : Bautista, de la Pena, Martinez, Can. Math. Soc. Conf. Proc. 18 (1996) 721–749.

[20] A. Zimmermann, Automorphisms of Green orders and their derived categories, to appear in Algebras and Representation theory.

Raphaël Rouquier : UFR de Mathématiques, Université Paris 7 Denis Diderot et UMR 7586 du CNRS, 2 place Jussieu, F–75251 Paris CEDEX 05, France

  E-mail address: Raphael.Rouquier@math.jussieu.fr

Alexander Zimmermann : Faculté de Mathématiques, Université de Picardie Jules Verne et LAMFA UMR 6140 du CNRS, 33 rue Saint Leu, 80039 Amiens CEDEX, France

  E-mail address: Alexander.Zimmermann@u-picardie.fr