THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS

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1 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 1. Introduction Let O be a complete discrete valuation ring with algebraically closed residue field k of characteristic p > 0 and field of fractions K of characteristic 0, big enough for the groups considered. Let G be a finite group, OGe a block of G with a cyclic defect group D. Let ON G (D)e the block of N G (D) corresponding to OGe. In this work, we give a self-contained account of the theory of blocks with cyclic defect groups with aim the proof that the derived category of a block with cyclic defect is (in a strong sense) locally determined a special case of Broué s conjecture [Br1] : Theorem 1.1. The blocks OGe and ON G (D)e are splendidly Rickard equivalent. If O p (G) 1, then these blocks are in addition Morita equivalent. Some remarks. That the blocks should be not merely Rickard equivalent, but also splendidly Rickard equivalent, is Rickard s refinement of Broué s conjecture. When O p (G) 1, there need not be a splendid Morita equivalence! The block ON G (D)e has a very simple structure : it is Morita equivalent to OD E, where E = N G (D, e D )/C G (D) and (D, e D ) is an e-subpair (Proposition.15). Let us outline the organization of the proof of the theorem. In 3, we prove that a stable equivalence between OGe and OD E (the very local case) can always be lifted in a nice way to a Rickard equivalence. To elucidate part of the structure of OGe, we follow very closely Thompson s original approach [Th], translated to the setting of an abstract stable equivalence of Morita type using Broué s philosophy [Br3]. Ideas of Green [Gr], Alperin [Alp] and Linckelmann [Li4] enable us to understand well enough OGe and the stable equivalence to be left with a problem which is solved with (a simplification of) the ideas of [Rou]. Theorem 1.1 is then proved in 4 by induction on the order of G : it is enough to give a splendid Rickard equivalence between OGe and the corresponding block OHf for H = N G (R), where O p (G) < R D and [R : O p (G)] = p, since by induction OHf is splendidly Rickard equivalent to ON G (D)e. First, we consider the case where O p (G) = 1. By induction, we know that OHf is Morita equivalent to OD E. Now, Green s correspondence gives a stable equivalence between OGe and OHf, hence between OGe and OD E, which can be lifted to a Rickard equivalence by the results of 3. We have then constructed a splendid Rickard equivalence between OGe and OHf. Date: October The author thanks for its hospitality the Isaac Newton Institute for Mathematical Sciences, Cambridge, where this paper has been written. 1

2 Assume now O p (G) 1 but O p (G) is central in G. Then, the blocks have a unique simple module and we have already a Rickard equivalence for the blocks of the groups modulo O p (G). This equivalence turns out to be nice enough to be lifted to an equivalence between OGe and OHf. Finally when O p (G) 1 but O p (G) is not central in G, we extend the known Rickard equivalence between the blocks of C G (O p (G)) and C H (O p (G)), following Marcus [Mar]. In, we give various general results (i.e., not specific to blocks with cyclic defect) pertaining to 3 and 4. In the final part, we illustrate the constructions with an explicit study of P SL (p). As a final remark, let us say that the structure of blocks with cyclic defect has been determined originally by Dade [Da, Fe]. The existence of a Rickard equivalence is due to Rickard [Ri1] and Linckelmann [Li] (the Morita equivalence when O p (G) 1 goes back to Linckelmann and Puig [LiPu, Li1]). My thanks go to M. Broué for many a patient answer to my questions and to G. Robinson for useful discussions.. Miscellany : stable equivalences, Rickard equivalences and more.1. Notations. Let G be a finite group, e a block idempotent of OG (i.e., a primitive idempotent of the center of OG) and A = OGe the corresponding block of OG. By an A-module, we mean a finitely generated left A-module. The sign means O. For V an O-module, we put V = Hom(V, O). We write KA and ka for the algebras K A and k A. We denote by A the O-algebra opposite to A. Let H be another finite group, f a block idempotent of OH and B = OHf... Stable category, stable equivalences and invariants. We follow here [Br, Br3] for the main concepts associated with the stable category. We denote by A mod the stable category of A, whose objects are the A-modules. Let us recall that a relatively O-projective A-module is 1 an A-module of the form A O U for some O-module U. Given two A-modules V and W, the set of morphisms Hom(V, W ) in A mod is Hom(V, W ) modulo the submodule of O-projective morphisms, i.e., morphisms which factor through a relatively O-projective module. Let Ω A A be the kernel of the multiplication map A A A. Let Ω 1 A A be the cokernel of the map η A : A A A dual to the multiplication. These are (A A )-modules whose restrictions to A and A are projective. The functors Ω A A A and Ω 1 A A A induce inverse self-equivalences of A mod. Given an A-module V, we denote by ΩV an A-module without projective direct summand such that Ω A A A V ΩV in A mod such a module is unique up to isomorphism. Let f : V W be an injection between A-modules. There exists a morphism φ : W (A A ) A V such that we have a commutative diagram : V f φ η A 1 (A A ) A V W 1 a direct summand of which splits when restricted to O

3 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 3 Let δ be the induced map W/V Ω 1 A A A V. This gives rise to a standard distinguished triangle : V f W W/V δ Ω 1 A A A V. The category A mod becomes a triangulated category, with translation functor Ω A A A and distinguished triangles the triangles isomorphic to the standard distinguished triangles. We denote by R p (A) the kernel of the decomposition map R(KA) R(kA). We denote by R(A) the quotient of R(KA) by the submodule generated by the characters of the projective A- modules. Note that the restriction of the usual bilinear form on R(KA) R(KA) to R p (KA) R(KA) factors through R p (KA) R(A) and gives a perfect pairing between R p (KA) and R(A). Let M be an (A B )-module, projective as an A-module and as a B -module. The module M induces a stable equivalence of Morita type between A and B if M B M A projective modules and M A M B projective modules. Since we are dealing with blocks, Rickard has noted that it is enough to check one of the properties : Lemma.1. Let M be an (A B )-module, projective as an A-module and as a B -module. If A is not a matrix algebra over O and M B M A projective modules, then M induces a stable equivalence between A and B. Proof. Since A is not a matrix algebra over O, it follows that A is not a projective (A A )- module, hence M is not projective. The functors M B and M A between B mod and A mod are left and right adjoint. Consider the associated natural maps η : B M A M and ε : M A M B. By [Mac, IV.1, Theorem 1], the map 1 η : M B B M B M A M is a split injection. As M B M A M is isomorphic to M in (A B ) mod, the map 1 η is an isomorphism in (A B ) mod. Similarly, the map 1 ε : M B M A M M B B is an isomorphism in (A B ) mod. It follows that the composite (1 ε)(1 η) = 1 (εη) is an automorphism of M in (A B ) mod. Since B is a block, the algebra End B B (B) is local. If the composite εη is not invertible, it is in the radical of this algebra, hence gives an element in the radical of End B B (B). This can t happen, since after applying M B we get a non-zero isomorphism! Hence, εη is invertible and as a consequence, M A M B P for some module P. Now, M B (M A M) M M B P and (M B M ) A M M projective modules, hence M B P is projective. As P is isomorphic to a direct summand of M A M B P, we finally deduce that P is projective.

4 4 Assume we have a module M inducing a stable equivalence between A and B. Then, the functors M B and M A induce inverse equivalences of triangulated categories between the stable categories B mod and A mod. The maps R(KA) R(KB) and R(KB) R(KA) induced by M B and M A give rise to inverse isometries between R p (A) and R p (B) and inverse isomorphisms between R(A) and R(B). Lemma.. Let V be an indecomposable non relatively O-projective B-module. Let W be an A-module without projective direct summands such that M B V W relatively O-projective modules. Then, W is indecomposable. If V is free over O and V k is indecomposable, then, W k is indecomposable. Proof. Assume W = W 1 W, with W 1, W 0. Then, V relatively O-projective modules M A W 1 M A W. As M A W 1 and M A W contain both a non relatively O-projective direct summand, we get a contradiction. Assume now V k is indecomposable and W k has a non-zero projective direct summand. So, there is a projective A-module P and a surjective map W P k. As W is free over O, we have Ext 1 (W, P ) = 0, hence the surjection lifts to a map W P, which is forced to be surjective by Nakayama s lemma, hence which splits, contradicting the property of W to have no projective direct summand. The following very useful result comes from [Li4] : Lemma.3. The (A B )-module M is the direct sum of a non-projective indecomposable module and of a projective module. If M is indecomposable, then for any simple B-module V, the A-module M B V is indecomposable. Proof. Let M = M 1 M. Since M A M B projective modules, we have M A M 1 M A M B projective modules. As B is indecomposable (as a (B B )-module), there exists i {1, } such that M A M i is projective, so M B M A M i is projective. Now, (M B M ) A M i M i projective modules, hence M i is projective. Let us assume now that M is indecomposable. Denote by soc(ka) the largest semi-simple ka-submodule of ka. Recall that a ka-module V has no projective direct summand if and only if soc(ka)v = 0. We have soc(ka kb ) = soc(ka) soc(kb ). Since M has no projective direct summand, soc(ka kb )M = 0, hence soc(ka)(m B soc(kb)) = 0, which means that M B soc(kb) has no projective direct summand. But, if V is a simple B-module, it is a direct summand of soc(kb), so M B V has no projective direct summand : as M induces a stable equivalence, M B V is the direct sum of an indecomposable non-projective module and of a projective module and the lemma follows..3. Derived category and Rickard equivalences. The definition of a splendid Rickard equivalence follows [Ri4]. We denote by D b (A) the bounded derived category of A and by K b (A) the homotopy category of bounded complex of A-modules. Viewing a module as a complex concentrated in degree 0, we have a fully faithful functor from the category of A-modules to D b (A) and to K b (A).

5 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 5 Let C be a bounded complex of (A B )-modules, all of which are projective as A-modules and as B -modules. The complex C induces a Rickard equivalence between A and B (or is a Rickard complex) if C B C A complex homotopy equivalent to 0 C A C B complex homotopy equivalent to 0. Assume we have such a complex C. Then, the functors C B and C A induce inverse equivalences of triangulated categories between the categories D b (B) and D b (A) and between the categories K b (A) and K b (B). They give rise to inverse perfect isometries between R(KA) and R(KB), to inverse isomorphisms between the centers of A and B,... [Br1]. Suppose the complex C has homology only in one degree, isomorphic to M. If C is a Rickard complex, then M induces a Morita equivalence between A and B. The converse doesn t hold : if M induces a Morita equivalence between A and B, we deduce that C B C A R, where R has zero homology, but might be non homotopy equivalent to 0. Nevertheless, if C has only one non-zero term, M, then C induces a Rickard equivalence if and only if M induces a Morita equivalence. As for stable equivalences, Rickard [Ri4] has proved the following : Lemma.4. Let C be a bounded complex of (A B )-modules, all of which are projective as A-modules and as B -modules. If C B C is homotopy equivalent to A, then C induces a Rickard equivalence between A and B. (The proof is similar to the proof of Lemma.1) In D b (A B ), the complex C is isomorphic to a bounded complex of modules which are all projective except the degree n term N, for some integer n. The module M = Ω n N induces a stable equivalence between A and B [Ri3, Corollary 5.5]. Lemma.5. Let C be a bounded complex of (A B )-modules, all of whose terms are projective but the degree 0 term M, which is projective as an A-module and as a B -module. Assume M induces a stable equivalence between A and B. Then, A is a direct summand of C B C and B is a direct summand of C A C. Proof. The functors C B and C A between K b (B) and K b (A) are left and right adjoint and induce natural maps η : B C A C and ε : C A C B. Let η : B M A M and ε : M A M B be the natural maps induced by the functors M B and M A. The composite maps εη and ε η have the same image in End B B (B). As M induces a stable equivalence between A and B, the map ε η is invertible, hence εη is invertible. It follows that B is isomorphic to a direct summand of C A C. The second property is obtained by a similar proof. Assume A and B have a common defect group D. When the modules occurring in C are direct summands of permutation modules induced from D = {(x, x 1 ) x D} G H, the complex C is called splendid and the equivalence of derived categories it induces is called a splendid Rickard equivalence. Such an equivalence gives rise to derived equivalences for corresponding blocks of centralizers of p-subgroups of D in G and H ([Ri4], [Ha] ; cf also [Pu],[Li3]). Following a suggestion of M. Harris, we have

6 6 Lemma.6. Assume H H are two subgroups of G. Let f be a block idempotent of H. Assume e, f and f have defect D. Let C be a splendid Rickard complex for OGe (OHf) and C a splendid Rickard complex for OHf (OH f ). Then, C OHf C is a splendid Rickard complex for OGe (OH f ). Proof. Composing Rickard equivalences gives a Rickard equivalence. The point is to check that the modules in C OHf C have the required properties. Let M = Ind G H Q O OG OQ OH be the trivial module induced from a diagonal subgroup Q to G H, where Q D. Let N = Ind H H R O, where R D. We have M OH N OG OQ OH OR OH Ind G H Q R ResH H Q R IndH H H O. By Mackey s formula, we obtain M OH N (h 1,h ) Q R \H H / H h Q\H/R Ind G H (Q h R) O Ind G H ( H) (h 1,h ) (Q R ) O It follows that M OH N is a direct sum of direct summands of permutations modules induced from D to G H. Homotopy equivalence is detected by a Sylow p-subgroup : Lemma.7. Let G be a subgroup of G of index prime to p. Let g : X Y be a morphism between two bounded complexes of OG-modules. Then, g is a homotopy equivalence if and only if Res G G g is a homotopy equivalence. Proof. Let Z be the cone of g. Then, g is a homotopy equivalence if and only Z is homotopy equivalent to 0. Assume Z is not homotopy equivalent to 0. Up to homotopy, we α can assume Z = Z r 1 Z r 0, where α is not a split surjection. So, α gives a non-zero element in Ext 1 OG(Z r, ker α). As [G : G ] is prime to p, the restriction map Ext 1 OG(Z r, ker α) Ext 1 OG (ResG G Z r, Res G G ker α) is injective, hence the restriction to G of α doesn t split, i.e., the restriction to G of Z is not homotopy equivalent to 0 and the restriction to G of g is not a homotopy equivalence. The following lemma due to A. Marcus [Mar] shows how to solve (certain) extension problems for Rickard complexes : Lemma.8. Let G and H be normal subgroups of G and H with G/G = H/H = E. Assume e and f are block idempotents of OG and OH. Let C be a Rickard complex for eog and foh. Assume C extends to a complex C of OL-modules, where L = {(g, h) G H (gg, hh ) E}. If E has order prime to p or if C has only one non-zero term, then, C = Ind G H L C is a Rickard complex for eog and foh. Proof. We have Res G H G H C = ResG H G H by Mackey s formula. Similarly, Consequently, IndG H L Res G H G H C Ind G H G H C. C Ind G H G H C Res G G G G C OH C C OH OH OH C C OH C OG OG

7 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 7 is homotopy equivalent to eog OG OG eog. In particular, C OH C has homology only in degree 0, M, with Res G G G G M eog. The natural map g : eog C OH C induces a map g 0 = H 0 (g) : eog M. As the restriction of M to OG is isomorphic to eog, the OG -submodule of M generated by g 0 (e), the unit of M = End K b (OH )(C), must be isomorphic to eog, hence g 0 is an isomorphism. If the index of G G in G G is prime to p, it follows that g is a homotopy equivalence, since its restriction to G G is a homotopy equivalence (Lemma.7). Similarly, one proves that C OG C is homotopy equivalent to foh and the lemma follows. Let us finally consider normal p-subgroups. Let R be a normal p-subgroup of G. We put Ḡ = G/R. For M an OG-module, we put M = OḠ OG M and for φ : M N a morphism of OG-modules, we put φ = 1 φ : M N. Lemma.9. Let φ : M N be a morphism between OG-modules. Then, φ is surjective if and only if φ is surjective. Let N be an OG-module whose restriction to OR is projective. Then, every projective direct summand of N lifts to a projective direct summand of N, if M is a projective cover of N, then M is a projective cover of N. Proof. Assume φ is surjective. Then, N = φ(m)+rad(or)n, hence N = φ(m) by Nakayama s lemma and φ is surjective. Note that given a projective indecomposable OG-module N, the projective OḠ-module N is indecomposable, and we obtain a bijection between the set of isomorphism classes of projective indecomposable OG-modules and the set of isomorphism classes of projective indecomposable OḠ-modules. Let M be a projective OG-module and f : N M a surjection. Let M 0 be the kernel of the canonical surjective map M M. Then, M 0 is a direct summand of Ind G R Res G R M 0. In particular, Ext 1 OG(N, M 0 ) Ext 1 OG(N, Ind G R Res G R M 0 ) Ext 1 OR(Res G R N, Res G R M 0 ). Since Res G R N is projective, this last Ext-group is zero, hence the morphism f lifts to a morphism φ : N M. By the first part of the lemma, this morphism is surjective, hence split surjective, since M is projective. If φ : M N is a projective cover of N, then φ is a surjective map. Assume there is a direct summand M of M such that the restriction of φ to M is surjective. Using the second part of the lemma, we can lift M to a direct summand M of M, and the first part of the lemma shows that the restriction of φ to M is surjective. So, M = M and M = M, i.e., M is a projective cover of N. Homotopy equivalence between OG-modules can sometimes be controlled by OḠ : Lemma.10. Let n be an integer and φ : X Y be a morphism between two bounded complexes of OG-modules whose components are direct sums of indecomposable modules with trivial source and vertices Q such that Q R = n. Then, φ is a homotopy equivalence if and only if φ is a homotopy equivalence. Proof. Let P be a Sylow p-subgroup of G. Let Q be a subgroup of P. We have Res G P Ind G Q O Ind P P Q O g g P \G/Q and (P Q g ) R = Q R. Since Ind P P Q g O is indecomposable (it has a unique simple quotient), every direct summand of Res G P Ind G Q O has a vertex Q such that Q R = Q R.

8 8 Hence, the assumptions of the lemma hold if we restrict the modules from G to P and replace G by P. By Lemma.7, we can then assume that G = P. As in the proof of Lemma.7, we are reduced to the following problem. Let φ : M N be a morphism between OG-modules whose components are direct sums of indecomposable modules with trivial source and vertices Q such that Q R = n. Assume φ is a split surjection. Then, we have to show that φ is a split surjection. By Lemma.9, we know already that φ is a surjection. We may of course assume that N is indecomposable. But, this implies that N has a unique simple quotient, hence that N has a unique simple quotient and is therefore indecomposable. If M = M 1 M, then the restriction of φ to M 1 or to M is a split surjection : so, we can assume M, hence M, are indecomposable as well. Then, φ is a split surjection if and only if it is an isomorphism, i.e., if and only if M and N have the same rank. Let Q and Q be subgroups of G such that M Ind G Q O and N Ind G Q O. We have rank M = [G : RQ] and rank N = [G : RQ ]. Since φ is an isomorphism, we have RQ = RQ. But, by assumption, R Q = R Q, hence Q = Q and φ is an isomorphism. Under good circumstances, normal p-subgroups can be factored out, in order to check that a complex induces a Rickard equivalence : Lemma.11. Let R be a common normal p-subgroup of G and H and C a bounded complex of (OGe (OHf) )-modules, each of which is a direct sum of indecomposable modules with trivial source and vertices Q such that Q (1 H ) = Q (G 1) = 1 and R R (R 1)Q = (1 R )Q. Let ē and f be the images of e and f through the canonical morphisms OG OḠ and OH O H and C = OḠē OG C OH O H f. Then, C is a Rickard complex for OGe and OHf if and only C is a Rickard complex for OḠē and O H f. Proof. Let Q be a p-subgroup of G H such that Q (1 H ) = Q (G 1) = 1 and R R (R 1)Q = (1 R )Q. By Mackey s formula, we have Res G H R R IndG H Q O Ind R R (R R ) Q O g g R R \G H /Q and Q = (R R ) Q g satisfies Q (1 H ) = Q (G 1) = 1 and R R = (R 1)Q = (1 R )Q. Let N = Ind R R Q O. We have O OR N OR O Hom OR (OR) (N, O) O. We have Res R R R 1 N Ind R 1 1 O since R R = (R 1)Q and (R 1) Q = 1. It follows that O OR N O. So, the canonical surjective map O OR N O OR N OR O is an isomorphism. Similarly, the canonical surjective map N OR O O OR N OR O is an isomorphism. So, if M is a direct summand of Ind G H Q O, the canonical maps give isomorphisms of O(G H )-modules Consequently, we have M OR O O OR M O OR M OR O. OḠ OG C OH C OG OḠ OḠ OG C OH O H OH C OG OḠ C O H C

9 and THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 9 O H OH C OG C OH O H C O Ḡ C. The components of C OH C are direct sums of direct summands of modules Ind G H Q 1 O OH ( Ind G H Q O ) OG OQ1 OH OQ OG where Q i (G 1) = Q i (1 H ) = 1 and R R (R 1)Q i = (1 R )Q i for i {1, }. Let φ : G H G and ψ : G H H be the canonical projections. Then, we have isomorphisms Q i φ(q i ) and Q i ψ(q i ) and R is contained in φ(q i ) and ψ(q i ). We have Res H H ψ(q 1 ) ψ(q ) OH g ψ(q 1 ) ψ(q ) \H H / H Ind ψ(q 1) ψ(q ) (ψ(q 1 ) ψ(q ) ) ( H) g O. Note that, for Q = (ψ(q 1 ) ψ(q ) ) ( H) g, we have Q (R R ) = R (ψ(q 1 ) ψ(q ) ) g 1 = R. Hence, Ind G H Q 1 O OH ( Ind G H Q O ) Ind G G Q 1 Q ResH H ψ(q 1 ) ψ(q ) OH is a direct sum of modules Ind G G Q O, where Q (R R ) = R. Similarly, the components of C OG C have trivial source and vertices Q such that Q (R R ) = R. Let η : OGe C OH C and η : OHf C OG C be the natural maps induced by the functors C OHf and C OGe. Then, η = 1 OR η OR 1 : OḠē C O H C and η = 1 OR η OR 1 : O H f C C O Ḡ are the natural maps induced by the functors C O H f and C O Ḡē. By Lemma.10, η (resp. η ) is a homotopy equivalence if and only if η (resp. η ) is a homotopy equivalence..4. Some more lemmas. For a module V, we denote by P V a projective cover of V. Lemma.1. Let M be an (A B )-module. A projective cover of M is P M B W PW W where W runs over a complete set of representatives of isomorphism classes of simple B- modules. Proof. Let V be a simple A-module and W a simple B-module. We have an isomorphism of (A A )-modules Hom B (M, V W ) Hom O (M B W, V ) given by f (m B w f(m)(w)) and the lemma follows from the isomorphism Hom A B (M, V W ) Hom A (M B W, V ). The following well-known lemma solves the problem of lifting modules through cyclic p - extensions. Lemma.13. Let H be a normal subgroup of G with E = G/H a cyclic p -group. Let M be a G-stable OH-module. Then, there exists an OG-module M extending M and for any such module, we have Ind G H M Res G G G ( M OE).

10 10 Proof. Let g G generating G/H. Since M is G-stable, there exists φ End O (M) such that φ(g 1 hg(m)) = hφ(m) for all h H and m M. Let γ = φ e g e, where e = E, and R be the subring of End OH (M) generated by γ. Suppose there is α R such that α e = γ. Let ψ = α 1 φ. Then, ψ e acts on M as g e and ψ(g 1 hg(m)) = hψ(m) for h H and m M. It follows that we can extend the action of H on M to an action of G by letting g act as ψ. The existence of α follows from the fact that R is a finite algebra over the strictly henselian ring O, hence the etale extension R[X]/(X e γ) of R must be trivial : first, replacing R by one of its blocks, we can assume it is local. Then, the equation α e = γ has e distinct roots in the residue field of R. By Hensel s lemma, these solutions can be lifted to R and we are done. Let M be an OG-module extending M. Then, by Mackey s formula. Res G G G IndG G G H ( M O) Ind G H Res G H Let us recall some basic definitions of local block theory and some properties related to Brauer s first main theorem (see for example [Alp, IV]). Assume H is a subgroup of G. We say that the block OGe corresponds to OHf if OGe is a direct summand of Ind G G H H OHf. Let D be a defect group of OGe. When N G (D) H, there is a unique block idempotent f of OH such that OGe corresponds to OHf : OHf is called a Brauer correspondent of OGe. If R is a p-subgroup of D, e R is a block of ORC G (R) (equivalently, a block of OC G (R)) and OGe corresponds to ORC G (R)e R, then (R, e R ) is called an e-subpair. Assume furthermore R is normal in G and e R is G-stable. Then, e R = e, D RC G (R) is G-conjugate to a defect group of ORC G (R)e R and p [G : DC G (R)]. The next two lemmas deal with blocks of groups having a normal p-subgroup. Lemma.14. Let R be a normal p-subgroup of G, (R, f) an e-subpair and H its normalizer. Then, OGf induces a Morita equivalence between OGe and OHf. Proof. We have ef = f and (g 1 fg)f = 0 for g G H. The multiplication map fog OGe OGf fogf is an isomorphism and fogf = OHgg 1 fgf = OHf. g G/H The lemma is then a consequence of Lemma.4. Proposition.15. Assume a defect group D of OGe is normal in G and e is a block of OC G (D). Let E = G/DC G (D). Assume E is cyclic. Then, OGe is Morita equivalent to OD E. Proof. Let ē be the image of e in O(C G (D)/Z(D)). The canonical map from the center of OC G (D)e to the center of O(C G (D)/Z(D))ē is onto. Since O is complete, this forces ē to be a block of O(C G (D)/Z(D)). This block has defect zero, hence has a unique simple module. So, OC G (D)e has a unique simple module. Let H = D E and L = N G H ( D). The map C G (D) G H, x (x, 1) factors through L and gives an injection ϕ : C G (D) L/ D with cokernel isomorphic to E. The G-stable block e of OC G (D) has a unique simple module V, which is consequently G-stable. The action of C G (D) on V lifts to an action of L/ D on V (Lemma.13). Let P V be a projective cover of V as an O(L/ D)-module and Q the restriction of P V to L. M

11 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 11 Let M = Ind G H L Q and P = Res L C G (D) Z(D) Q. Then, P has vertex Z(D) and P = O(C G (D)/Z(D)) CG (D) P Z(D) O is a projective indecomposable O(C G (D)/Z(D))ē O- module, hence it induces a Morita equivalence between O(C G (D)/Z(D))ē and O. It follows from Lemma.11 that P induces a Morita equivalence between OC G (D)e and OZ(D) and from Lemma.8 that M induces a Morita equivalence between OGe and OH. What this proposition actually determines is a source algebra of the block. Note that instead of assuming E cyclic, one can assume the blocks are principal to get the same result. In general, a similar proof shows that OGe is Morita equivalent to a twisted group algebra O D Ê when D is normal in G, as proven by L. Puig [Thé, 45, theorem 1]. 3. Blocks stably equivalent to OD E Let G be a finite group, e a block of G with positive defect. Let E be a p -subgroup of the group of automorphisms of a non-trivial cyclic p-group D. Let A = OGe and B = OD E. Let M be an indecomposable (A B )-module which is projective as an A-module and as a B -module. This section is devoted to the proof of Theorem 3.1. Assume M induces a stable equivalence between A and B. Then, there exists a direct summand N of a projective cover p : P M M of M such that the complex 0 N p N M 0 induces a Rickard equivalence between A and B. If E = 1, then N = 0 or N = P M, i.e., M or ΩM induces a Morita equivalence between A and B. Let Ψ : R(KB) R(KA) and Θ : R(KA) R(KB) be the maps induced by the functors M B and M A Exceptional characters. An exceptional character of B is defined to be the character of an irreducible KB-module with a non-trivial D-action. A non-exceptional character of B is an irreducible character which is not exceptional. Let θ x be the sum of the exceptional characters of B. The simple B-modules are the simple OE-modules. They lift uniquely to O-free B-modules, with projective covers the indecomposable projective B-modules. If V is a one-dimensional O-free OE-module, then its projective cover as a B-module is Ind H E V, whose character is the character of V plus θ x. Recall also that the indecomposable kb-modules are uniserial. The set { θ} θ exceptional is a basis of R(B) (we denote by x the image of x R(KB) under the canonical morphism R(KB) R(B)). Assume B has at least two exceptional characters. An irreducible character ψ of A is called exceptional if there exists two exceptional characters θ, θ of B such that ψ, Ψ(θ) Ψ(θ ) 0. Note that θ θ R p (B), hence Ψ(θ) Ψ(θ ) =. So, the number of exceptional characters of A and B are the same. The rank of R p (B) is this number and R p (A) and R p (B) have the same rank. It follows that R p (A) is not generated by linear combinations of exceptional characters, hence A has at least one non-exceptional character. We denote by ψ x the sum of the exceptional characters of A. When B has a unique exceptional character θ x, then we pick an irreducible character ψ x = ψ θx of A such that ψ x 0 and we call it exceptional (actually, any irreducible character of A will do). For the non-exceptional characters, we have Lemma 3.. Let ψ be a non-exceptional irreducible character of A. Then, Θ(ψ) is a multiple of θ x.

12 1 Proof. Let θ, θ be two exceptional characters of B. We have Θ(ψ), θ θ = ψ, Ψ(θ θ ) = 0. We are now done since θ = θ x. 3.. Decomposition numbers. Lemma 3.3. Let V be a non-projective O-free A-module such that V k is indecomposable. Let W be a non-projective indecomposable B-module such that M A V W projective modules. Let ζ be the character of W. Then, (i) ζ is multiplicity free (ii) ζ contains at most one non-exceptional character (iii) if ζ contains a non-exceptional character, then it doesn t contain all exceptional characters. Proof. By Lemma., W k is indecomposable and non projective. Since W k is indecomposable, it is the quotient of an indecomposable projective kb-module. Hence, W is the quotient of an indecomposable projective B-module. It follows that the character ζ of W is multiplicity free and contains at most one non-exceptional character. Since W k is not projective, we know in addition that if ζ contains a non-exceptional character, then it doesn t contain all exceptional characters. Lemma 3.4. Let χ be a character of A and P an indecomposable projective A-module such that χ is contained in the character of P. Then, there exists an O-free A-module V with character χ and projective cover P. In particular, V k is indecomposable. Proof. Let L be a KA-submodule of K P such that K P/L has character χ. Let L = L P and V = P/L. Then, V has the required properties. As a consequence of Lemmas 3., 3.3 and 3.4, we obtain : Corollary 3.5. Let ψ be a non-exceptional character of A. Then, Θ(ψ) = ± θ x. If χ and χ are two characters, we say that χ is contained in χ if χ χ is a character. Proposition 3.6. Let P be an indecomposable projective A-module. Then, its character is the sum of two distinct non-exceptional characters or the sum of a non-exceptional character and of all exceptional characters. Proof. Let η be the character of P. Assume there are non-exceptional characters ψ and ψ such that ψ + ψ is strictly contained in η. By Lemma 3.4, there is an O-free A-module V with character ψ + ψ, such that V k is indecomposable. By Lemma 3.5, Θ(ψ) + Θ(ψ ) is zero or ± θ x. By Lemma 3.3, the second possibility can t arise. Now, assume Θ(ψ) + Θ(ψ ) = 0. Let W be a B-module such that M A V W projective modules, with W k indecomposable. Then, the character of W is 0 in R(B). By Lemma 3.3, this is impossible. We assume now that η contains at most one non-exceptional character. Let θ, θ be two distinct exceptional characters of B. We have η, Ψ(θ) Ψ(θ ) = Θ(η), θ θ = 0. Let ψ and ψ be the two distinct exceptional characters of A such that Ψ(θ) Ψ(θ ) = ±ψ ± ψ. Then, η, ψ = η, ψ, so ψ and ψ arise with the same multiplicity in η. It follows that there is a positive integer α such that χ = η αψ x is zero or a non-exceptional character. Assume χ = 0. Then the block A has a projective indecomposable module with character η different from η such that η, η 0. The character η is a non-zero multiple of ψ x plus a non-exceptional character ψ. But this implies that ψ = 0, which is impossible! Hence, χ is a non-exceptional character.

13 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 13 Now, Θ(η) = 0 = ± θ x + αθ(ψ x ) by Lemma 3.5. This implies α = 1 and Θ(ψ x ) = ± θ x and we are done The Brauer tree and its walk. Lemma 3.7. Let L be a simple B-module. Then, the module M B L has a unique simple quotient V and the correspondence L V induces a bijection between the sets of isomorphism classes of simple B-modules and simple A-modules. Proof. Let V be a simple A-module and W = M A V. Then, W is indecomposable (Lemma.3), hence it has a unique simple submodule L V. So, we have a map h from the set of isomorphism classes of simple A-modules to the set of isomorphisms classes of simple B-modules given by V L V. Let now L be a simple B-module and U = M B L. Let V be a simple A-module which is a quotient of the indecomposable module U. Then, Hom(L, M A V ) Hom(U, V ) 0. It follows that the map h is surjective. Let now V 1 and V be two simple A-modules such that W 1 = M A V 1 and W = M A V have the same simple submodule L. Since an injective hull of L is uniserial, there is an injection W i W j for some i, j with {i, j} = {1, }. Such an injection between modules with no projective direct summand is not an O-projective morphism, hence Hom(W i, W j ) 0, so Hom(V i, V j ) 0 and V i and V j are isomorphic. This proves the injectivity of h. Since h is bijective, given a simple B-module L, the module M B L has a unique simple quotient. The set {Ω i k} 0 i e 1 is a complete set of simple kb-modules (up to isomorphism). By Lemma.3, the module M B Ω i k is indecomposable. Hence, M B Ω i O is indecomposable as well. On the other hand, M B commutes with Heller translation up to projective modules, i.e., M B Ω i O Ω i S projective modules, where S = M B O. It follows that M B Ω i O Ω i S. Proposition 3.8. The character of Ω i S is a non-exceptional character or the sum of the exceptional characters. Proof. Let χ be the character of Ω i S. When i is even, Ω i S has a unique simple quotient, hence Ω i S is a quotient of a projective indecomposable A-module P i. In particular, χ is contained in the character of P i. Assume now i is odd. Then, we have an exact sequence 0 Ω i S P i 1 Ω i 1 S 0. (1) Again, we see that χ is contained in the character of a projective indecomposable module. We have χ = ±Ψ(θ x ) since the character of Ω i O is non-exceptional or equal to θ x. Hence, χ = ± ψ x and we get the conclusion from Proposition 3.6. Let us now define the Brauer tree T of A. The set of vertices is {ψ} ψ non exceptional {ψ x }. The vertices ψ and ψ are incident if ψ + ψ is the character of an indecomposable projective module. This defines a graph whose number of edges is the number of isomorphism classes of simple A-modules. By Proposition 3.6, this is a tree. The vertex corresponding to ψ x is called exceptional.

14 14 Let v i be the vertex corresponding to the character of Ω i S. Then, there is an edge l i connecting v i and v i+1, due to the exact sequence (1). The set {l i } is the set of all edges of T. Note that Ω e O O, where e is the order of E. It follows that v e+i = v i and l e+i = l i. For 0 i e 1, let v i {v i, v i+1 } be the further vertex of l i from the exceptional vertex. Let I be the set of non-negative integers i e 1 such that v i = v i+1. Note that {v i} is the set of non-exceptional vertices of T Construction of the complex. By Lemma.1, a projective cover of the (A B )- module M is P Ω i S P Ω i O. Let 0 i e 1 N = i I P Ω i S P Ω i O and let C = 0 N φ M 0 where M is in degree 0 and φ is the restriction to N of a surjection 0 i e 1 P Ω i S P Ω i O M. For 0 i, j e 1, we have { P Ω j O B Ω i O if i = j, O 0 otherwise. Hence, we have an isomorphism in D b (A) { C B Ω i 0 Ω i+1 S 0 0 if i I, O 0 0 Ω i S 0 otherwise where Ω i S is in degree 0 and Ω i+1 S in degree 1. In particular, the (Lefschetz) character of C B Ω i O is ε i v i where ε i = 1 if i I and ε i = 1 otherwise. Lemma 3.9. We have Hom D b (A)(C B Ω i k, C B Ω j k[ 1]) = 0 for all i, j. Proof. Let us recall first that Hom B (Ω n k, k) = 0 unless n 0(mod e). Put now T = Hom D b (A)(C B Ω i k, C B Ω j k[ 1]). Since Hom(Ω i S k, Ω j+1 S k) Hom(Ω i k, Ω j+1 k) = 0, we deduce that T = 0 unless i / I and j I, in which case T Hom(Ω i S k, Ω j+1 S k). Then, we have to prove that there are no k-projective morphisms from Ω i S k to Ω j+1 S k. As k-projective morphisms Ω i S k Ω j+1 S k lift to O-projective morphisms Ω i S Ω j+1 S, we are done, since Hom(Ω i S, Ω j+1 S) = 0 (the character of Ω i S is v i and the character of Ω j+1 S is v j, hence, these are distinct since i j). Corollary The complex C A C is homotopy equivalent to its 0-homology. Proof. Let L, L be two simple B-modules. We have Hom D b (B B )(C A C, L L [ 1]) Hom D b (A)(C B L, C B L [ 1]) = 0 by Lemma 3.9. Hence, C A C has no homology in degree 1. Since the degree 1 component of C A C is projective, C A C is homotopy equivalent to a complex with no component in degree 1. Since C A C is self-dual, it is homotopy equivalent to its 0-homology.

15 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 15 By Lemma.5, we have H 0 (C A C) B Q, where Q is a projective (B B )-module. Now, Hom D b (KB (KB) )(K (C A C), K (Ω i O (Ω j O) )) Hom D b (KA)(K (C B Ω i O), K (C B Ω j O)) = δ ij K, hence Hom(K Q, K (Ω i O (Ω j O) )) = 0 for all i, j. This implies Q = 0. So, we have proven that C A C is homotopy equivalent to B (seen as a complex concentrated in degree 0). By Lemma.4, we conclude that C is a Rickard complex. This completes the proof of the first part of Theorem 3.1. Note that, if E = 1, then a projective cover of M is indecomposable : it follows that C has homology only in one degree and it is isomorphic to M or ΩM[1], from which we derive the second part of Theorem Local study Let G be a finite group, e a block of G with a non-normal cyclic defect group D. Let Q be the subgroup of D containing R = O p (G) as a subgroup of index p. Let H = N G (Q) and f the block of H corresponding to e. Theorem 1.1 will follow from the following more precise result : Theorem 4.1. Let M be an indecomposable direct summand of the (OGe (OHf) )-module eogf with vertex D. Then, there is a direct summand N of the (OGe (OHf) )-module OGe OR foh such that the complex C = 0 N m M 0 induces a splendid Rickard equivalence between OGe and OHf. Here, m is the restriction of the multiplication map OGe OR foh eogf. If R 1, then C has homology only in one degree. Note that the (OGe (OHf) )-module eogf has, up to isomorphism, a unique indecomposable direct summand with vertex D (this can easily be deduced from the forthcoming proof of the theorem). Let us check that a complex C as defined in the theorem is splendid. Note that eogf is a direct summand of Ind G H D O, hence is isomorphic to a direct sum of modules with trivial source and vertex contained contained in D. Since OGe OR foh is isomorphic to a direct summand of Ind G H R O, it follows that N is isomorphic to a direct sum of modules with trivial source and vertex contained in R. Hence, C is splendid. Let us prove by induction on the order of G that Theorem 1.1 follows from Theorem 4.1. By induction, we know that Theorem 1.1 holds for OHf : there is a splendid Rickard equivalence between OHf and ON G (D)e given by a complex having homology only in one degree. Now, Theorem 4.1 gives a splendid Rickard equivalence between OGe and OHf. Hence, composing the two equivalences, we get a splendid Rickard equivalence between OGe and ON G (D)e, by Lemma.6. Furthermore, when O p (G) 1, the Rickard complex has homology only in one degree. Hence, Theorem 1.1 holds for G and, by induction, the proof of Theorem 1.1 is complete. We assume now that Theorem 4.1 holds for all finite groups of order strictly less than the order of G. The rest of this section is devoted to proving that the theorem holds then for G.

16 O p (G) = 1. Let us first consider the case where R = 1, i.e., Q has order p. Following Alperin, we have : Lemma 4.. The module eogf induces a stable equivalence between OGe and OHf. Proof. We have Res G G H H OG ResG G H H IndG G G O Ind H H O. ( G) (g 1,g ) (H H ) (g 1,g ) H H \G G / G Now, (Q Q ) ( G) (g 1,g ) 1 if and only if g 1 g 1 H. Since Q Q is the maximal elementary abelian( subgroup of D D and ) is normal in H H, it follows that the OHf (OHf) -module f Ind H H O f is projective when g ( G) (g 1,g ) (H H ) 1 g 1 / H. Hence, OHf OH OG OH foh OHf projective modules. Since f is the Brauer correspondent of e, OHf is a direct summand of Res G G H H OGe. So, foge OG eogf fogef OHf OH OGe OH foh OHf projective modules. The result is now a consequence of Lemma.1. Let M be an indecomposable non-projective direct summand of eogf : by Lemma.3, it still induces a stable equivalence between OGe and OHf. By induction, Theorem 1.1 holds for OHf. Hence, there is an (OHf (OD E) )-module M inducing a Morita equivalence between OHf and OD E. So, the indecomposable module M 0 = M OHf M induces a stable equivalence between OGe and OD E. By Theorem 3.1, there exists a direct summand N 0 of a projective cover P 0 of M 0, such that the complex C 0 = 0 N 0 M 0 0 induces a Rickard equivalence between OGe and OD E. It follows that C 1 = C 0 OD E M 0 N 0 OD E M M 0 induces a Rickard equivalence between OGe and OHf. The module N 0 OD E M is a direct summand of P M = P 0 OD E M, a projective cover of M, and the map N 0 OD E M M is the restriction of a surjective morphism P M M. Now, the multiplication map m : OGe foh eogf is surjective and eogf = M projective modules, hence there is a direct summand N of OGe foh such that the complex C = 0 N m M 0 is isomorphic to C 1. So, the theorem holds when R = O p (G) 1. Let us now consider the case where R is non-trivial. Let Ḡ = G/R and ē be the image of e through the canonical morphism OG OḠ. Similarly, let H = H/R and f be the image of f through the canonical morphism OH O H. Note that the canonical map E = N G (D, e D )/C G (D) Aut(D/R) is injective, since it factors through the group of p -automorphisms of D. Hence, O H f and OḠē are blocks with defect D/R. Let (R, e R ) be an e-subpair. By Lemma.14, the (OGe ON G (R, e R )e R )-module eoge R induces a Morita equivalence. Hence, we can assume that G stabilizes e R, that is, that e = e R. Let C = C G (R)/R. Since Ḡ/ C G/C G (R) is a cyclic p -group, a simple O Cē-module extends in exactly [G : C G (R)] non-isomorphic ways to Ḡ and every simple OḠē-module is obtained in this way (Lemma.13). By induction, OḠē is derived equivalent to (D/R) E, hence has E simple modules. It follows that [G : C G (R)] = E, hence the canonical map N G (D, e D )/C G (D) G/C G (R) is an isomorphism. Note that O p (C G (R)) = R. Similarly, the canonical map N H (D, e D )/C H (D) H/C H (R) is an isomorphism and O p (C H (R)) = R.

17 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS O p (G) Z(G). By assumption, Theorem 4.1 holds for OḠē and O H f. The observation above shows that these two blocks have a unique simple module. Let M be an indecomposable direct summand of the O(G H )/ R-module eogf with vertex D/ R. Since ēoḡ f is the direct sum of an indecomposable non-projective module and of a projective module, it follows from Lemma.9 that the O(Ḡ H )-module M = M OR O is an indecomposable direct summand of ēoḡ f. Let f : N M be a projective cover of M. Then, N = N OR O is a projective cover of M (Lemma.9). Let M (resp. N) be the restriction of M (resp. N ) to G H. If M induces a Morita equivalence between OḠē and O H f, then let C be the complex with only one non-zero term, M, in degree 0. If the kernel of a surjective map f : N M induces a Morita equivalence between OḠē and O H f, then let C be the complex with N in degree 1, M in degree 0 and differential f. Then, it follows from Lemma.11 that C is a Rickard complex. Note that C has the form required O p (G) Z(G). Note that OC H (R)f is the Brauer correspondent of the block OC G (R)e. Let M be an indecomposable direct summand of the O(C G (R) C H (R) )/ R-module eoc G (R)f with vertex (D/R). Let L = N G H ( R). The restriction to L of the action of G H on OG leaves eoc G (R)f invariant. This gives a natural extension of the action of (C G (R) C H (R))/ R on eoc G (R)f to L/ R. Since (eoc G (R)f)/M is a sum of modules with vertices strictly contained in (D/R), the module M is L-stable. The isomorphisms G/C G (R) E and H/C H (R) E induce an isomorphism L/(C G (R) C H (R) ) E. So, by Lemma.13, there is an indecomposable summand M of Ind L/ R (H/R) O OC G(R) lifting M. Let N be a projective cover of M. This is an L-stable indecomposable O(C G (R) C H (R) )/ Rmodule, hence it lifts to a projective L/ R-module Ñ (Lemma.13). Since IndL/ R (C G (R) C H (R) )/ R N is a projective cover of Ind L/ R (C G (R) C H (R) )/ R M and M is isomorphic to a direct summand of the latter, one may choose Ñ to be a projective cover of M. We have a natural map giving rise to a surjective map Ind H/R 1 Res H/R 1 O O f : Ind L/ R 1 O Ind L/ R (H/R) O. So, we may choose Ñ to be a direct summand of IndL/ R 1 O with f(ñ) = M. Let N = Ind G H L Ñ and M = Ind G H L M. Then, N is a direct summand Res L/ R L Res L/ R L of OGe OR foh, M is a direct summand of eogf and m (N ) = M, where m is the multiplication map OGe OR foh eogf. If Res (C G(R) C H (R) )/ R C G (R) C H (R) M induces a Morita equivalence between OC G (R)e and OC H (R)f, then we define C to be M. Otherwise, let C be the complex with N in degree 1, M in degree 0 and differential m. Then, Lemma.8 says that C is a splendid Rickard complex between eog and foh. Note that C has homology only in one degree. Hence, the proof of Theorem 4.1 is complete. 5. An example : P SL (p) We make the constructions of 3 explicit for the group P SL (p). Let V 1 be the natural -dimensional representation of SL (p). Then, the simple ksl (p) modules are the symmetric powers S i (V 1 ), where 0 i p 1 [Alp, pp 14 16]. The center of SL (p) acts trivially on S i (V 1 ) when i is even and we denote by V i the module S i (V 1 ) induced

18 18 from SL (p) to P SL (p). Then, the simple modules in the principal block e of G = P SL (p) are the V i, 0 i p 3. There is only one other block in G, it has defect 0 and its simple module is the Steinberg module V p 1. For i < p 1, the dimension of V i is i+1 < p. Let B be the normalizer of a Sylow p-subgroup of G. By Proposition 4., the (OGe (OB) )-module OGe induces a stable equivalence between OGe and OB. For V a simple OGe-module, the module OGe V has a dimension strictly smaller than p, hence it cannot have a projective direct summand. By Proposition.3, this implies the indecomposabiblity of OGe as a (OGe (OB) )-module. Since indecomposable kb-modules are uniserial, the restriction of V i to B has a unique simple quotient W i, for 0 i p 3 (this is actually a special case of the general property of simple modules for groups with a (B, N)-pair to have a unique simple quotient when restricted to B). Furthermore, W i W i 1. Let U be the Sylow p-subgroup of B and T a complement to U in B. The group T is cyclic, with order p 1 and there are two conjugacy classes of non-trivial p-elements in B (hence also in G). Let x 1, x be representatives for these classes. Let T be a Coxeter torus of G, i.e., the centralizer of an element of order p+1 p+1. This is a cyclic group of order. Let η be the irreducible character of T which gives the simple module W 1. Let δ be a nontrivial irreducible character of T which occurs in the character of the restriction of V 1 to T (the restriction is then 1 + δ + δ 1 ). Let ε = ±1 with ε p (mod 4). The character table of the principal block of G is : x 1 x t T t T (p 1) j (δ j (t ) + δ j (t )) (p + 1) l 1 1 η l (t) + η l (t) 0 ( p+ε ) i 1 (ε + i 1 εp) (ε i εp) {η p 1 4 (t) if ε = 1 0 if ε = 1 { 0 if ε = 1 δ p+1 4 (t ) if ε = 1 where j {1,..., p+ε }, l {1,..., p ε 1} and i { 1, 1}. 4 4 The character table of B is : x 1 x t T 1 l 1 1 η l (t) ) 1 ( 1 + i 1 p) ( 1 i p) 0 ( p 1 i where l {0, p 1 1} and i { 1, 1}. The restrictions of the irreducible characters of the principal block of G to B are : Res G B 1 = 1, Res G B (p 1) j = ( p 1 )0 + ( ) p 1, 1 Res G B (p + 1) l = ( p 1 )0 + ( ) p l + 1 p 1 l. {( ( Res G p+ε ) p 1 ) B = if ε = 1, ( i i p 1 ) + 1 i q 1 if ε = 1. 4 The Brauer tree of OGe is

19 THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 19 1 P 0 P p 3 P 1 P p 5 (p 1) 1 (p + 1) 1 (p 1) (p + 1) (p ε) p +ε P p ε 4 ( p+ε 4 The arrows describe Green s walk on the tree (i.e., the sequence of vertices v 0, v 1,... ). A projective cover of the (OGe (OB) )-module OGe is P λ Q λ 0 λ p 3 where P λ is a projective cover of V λ and Q λ a projective cover of W λ. The module N constructed in 3.4 is N = P λ Q λ Restricting a surjective map 0 λ p 1 p++ε λ p 3 4 P λ Q λ OGe to N gives a complex C = 0 N OGe which induces a splendid Rickard equivalence between OB and OGe. Let I be the isometry R(KGe) R(KB) induced by this equivalence of derived categories. We have I(1) = 1, I((p + 1) l ) = 1 l, I((p 1) j ) = 1 p 1 j, I( ( p+ε )i ) = ε ( ) p 1 εi. References [Alp] J.-L. Alperin, Local representation theory, Cambridge University Press, [Br1] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque (1990), [Br] M. Broué, On representations of symmetric algebras : an introduction, preprint ETH Zürich, [Br3] M. Broué, Equivalences between block algebras : an introduction, in Finite dimensional algebras and related topics, pp 1 6, Kluwer Academic Press, [Da] E.C. Dade, Blocks with cyclic defect group, Ann. Math. 84 (1966), [Fe] W. Feit, The representation theory of finite groups, North-Holland, 198. [Gr] J.A. Green, Walking around the Brauer tree, J. Aust. Math. Soc. 17 (1974), [Ha] M.E. Harris, Splendid Rickard equivalences for arbitrary blocks of finite groups, preprint (1997). [Li1] M. Linckelmann, Variations sur les blocs à défaut cyclique, thèse (1988), Publ. Univ. Paris 7, vol. 34 (1996). [Li] M. Linckelmann, Derived equivalences for cyclic blocks over a p-adic ring, Math. Z. 07 (1991), [Li3] M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, preprint (1995). [Li4] M. Linckelmann, Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z. 3 (1996), [LiPu] M. Linckelmann et L. Puig, Structure des p -extensions des blocs nilpotents, Comptes Rendus Acad. Sci. Paris Ser. I, 304 (1987), [Mac] S. MacLane, Categories for the working mathematician, Springer Verlag, [Mar] A. Marcus, On equivalences between blocks of group algebras : reduction to simple components, J. of Alg. 184 (1996), ) ±1