GLUING p-permutation MODULES 1. Introduction We give a local construction of the stable category of p-permutation modules : a p- permutation kg-module gives rise, via the Brauer functor, to a family of p-permutation modules for kn G Q)/Q, where Q runs over the non-trivial p-subgroups of G, together with certain isomorphisms. Conversely, the data of a compatible family of kn G Q)/Q-modules comes from a p-permutation kg-module, unique up to a unique isomorphism in the stable category. This should be the first half of a paper with a second part devoted to complexes of p- permutation modules. 2. The Brauer functor Let G be a finite group and k a field of characteristic p > 0. Let Q be a p-subgroup of G. We denote by Br Q the Brauer functor Br Q : kg mod kn G Q) mod. For V be a kg-module, ) Br Q V ) = V Q / Tr Q P V P. We write also V Q) for Br Q V ). For basic results about p-permutation modules and the Brauer functor, see [Br] and [Th, 27]. Restriction induces a fully faithful functor kn G Q)/Q mod kn G Q) mod and we will identify kn G Q)/Q mod with the full subcategory of kn G Q) mod of the modules with a trivial action of Q. We denote by kg perm the full subcategory of kg mod of p-permutation modules. This is the smallest full additive subcategory of kg mod closed under direct summands and containing the permutation modules. From now on, we will always consider the restriction of the Brauer functor Br Q : kg perm kn G Q) perm. Let Ω be a G-set. The composition γ Ω = γ Q Ω : kωq kω) Q kω)q) is an isomorphism. It induces an isomorphism of functors γ : k ) Q Br Q k ), i.e., there is a diagram of functors, commutative up to isomorphism : P <Q G sets k ) kg perm ) Q N G Q) sets k ) Br Q kn G Q) perm The following easy result describes the effect of the Brauer construction on a permutation module. Let H and L be two subgroups of G. We put T G L, H) = {g G L H g }. Date: August 1998. 1
2 Lemma 2.1. We have as N G L)-sets. G/H) L = g H\T G L,H)/N G L) N G L)/ N G L) H g ) 2.1. Tensor product and duality. Let us describe the effect of the functor Br Q on tensor products and duality. Let V and W be two p-permutation kg-modules. The inclusion V Q W Q V W ) Q induces a map α V,W : V Q) W Q) V W )Q). Restriction V ) Q = Hom kq V, k) V Q ) = Hom k V Q, k) induces a map β V : V Q) V Q). We put G = {x, x) x G} G G. Lemma 2.2. The maps α V,W and β V are isomorphisms and induce isomorphisms of functors from kg perm kg perm to kn G Q) perm and from kg perm to kn G Q) perm α : Res N GQ) N G Q) N G Q) Br Q Q Br Q Res G G G β : Br Q ) ) Br Q. Proof. In order to prove that α V,W and β V are isomorphisms, it is enough to consider permutation modules. So, let V = kω and W = kψ, where Ω and Ψ are G-sets. We have an isomorphism t Ω,Ψ : kω kψ kω Ψ) given by ω ψ ω ψ) for ω Ω and ψ Ψ. There is a commutative diagram : kω Q ) kψ Q ) t Ω Q,Ψ Q kω Ψ) Q kω) Q kψ) Q t Ω,Ψ kω Ψ)) Q kω)q) kψ)q) α V,W kω Ψ)Q) Hence, α V,W is an isomorphism. We have an isomorphism d Ω : kω kω) given by ω ω Ω a ) ω ω a ω for ω Ω. The composition kω Q kω) Q d Ω kω) ) Q kω) Q) is an isomorphism. Since the following diagram is commutative kω) Q d Ω kω) ) Q kω) Q) kω Q d Ω Q kω Q ) γω β V kω)q) we deduce that β V is an isomorphism. Let us now check that α V,W and β V induce natural transformations of functors.
GLUING p-permutation MODULES 3 Let V 1, V 2, W 1 and W 2 be p-permutation kg-modules and f i HomV i, W i ). The following diagram is commutative : V 1 Q) V 2 Q) V Q 1 V Q 2 f 1 Q) f 2 Q) f 1 f 2 W Q 1 W Q 2 W 1 Q) W 2 Q) α V1,V 2 f 1 f 2 α W1,W 2 V 1 V 2 ) Q W 1 W 2 ) Q f 1 f 2 )Q) V 1 V 2 )Q) W 1 W 2 )Q) Hence, α V,W induces a morphism of functors : Res N GQ) N G Q) N G Q) The commutativity of the diagram : Br Q Q Br Q Res G G G. W1 Q) f 1 Q) V 1 Q) β W1 W1 ) Q f1 W Q 1 ) V Q W 1 Q) f 1 Q) f 1 V1 ) Q 1 ) β V1 V 1 Q) shows finally that β V induces a morphism of functors : Br Q ) ) Br Q. Proposition 2.3. There is a commutative diagram HomV, W ) HomV W, k) Br Q Br Q HomV W )Q), k) HomV Q), W Q)) βα 1 HomV Q) W Q), k) where the horizontal maps are isomorphisms provided by the adjoint pairs W, W ) and W Q), W Q)). Proof. More explicitely, the first horizontal map is the composition HomV, W ) W HomV W, W W trw ) ) HomV W, k) where trw ) : W W k is the trace map. Thanks to Lemma 2.2, we have a commutative diagram
4 HomV, W ) W HomV W, W W ) Br Q HomV W )Q), W W )Q)) Br Q HomV Q), W Q)) W Q) α 1 HomV Q) W Q), W Q) W Q)) W Q) HomV Q) W Q), W Q) W Q) ) We will be done if we prove that the image of trw ) : W W k under the morphism HomW W, k) βα 1 Br Q HomW Q) W Q), k) is trw Q)). It is enough to prove this for W a permutation module. Let W = kω, Ω a G-set. The claim follows from the commutativity of the diagram β kω kω) Q kω Q kω Q 1 d Ω Q kω Q kω Q ) 1 d Ω γ Ω γ 1 kω kω) ) Q kω kω) )Q) βα 1 Ω ) kωq) kωq) 2.2. Compatibilities. Let us define a category T G. Its objects are the non-trivial p-subgroups of G. Let P and Q be two non-trivial p-subgroups of G. Then, the set of maps between P and Q in T G is { P g Q g T G P, Q)}. The composition of maps is the product in G : Q h R ) P g Q ) = P hg) R. We put φ = g for φ = P g Q and φp ) = g P. We call a map φ = P g Q in Hom TG P, Q) normal if g P is normal in Q. Every normal map can be expressed uniquely as the composition of an isomorphism with a normal inclusion φ = φ φ where φ = P gg P and φ = g P 1 Q. An important property of the normal maps is that they generate the category T G, i.e., every map in T G is a composition of normal maps. For g G and H a subgroup of G, we denote by g : kh mod k g H mod the isomorphism of categories induced by the group isomorphism H g H, denote by g : H sets g H sets x g x. We also the isomorphism of categories induced by this group isomorphism. We have the obvious compatiblity with the previous isomorphism of categories. Let V be a p-permutation kh-module and φ Hom TG P, Q) invertible. Then, the isomorphism of Ng HQ)-modules φ V P ) φ V ) Q, v v, induces an isomorphism φ 0 V : φ V P )) φ V )Q).
GLUING p-permutation MODULES 5 If H = G, then we have an isomorphism of kg-modules and an isomorphism of kn G Q)-modules ι φ,v : V φ V, v φv φ V = Br Q ι 1 φ,v ) φ 0 V : φ V P )) V Q). Let now P Q and φ = P 1 Q. The canonical map V Q V P V P ) factors through the inclusion V P ) Q V P ) to give a map V Q V P ) Q. Composing with the canonical map V P ) Q V P )Q), we get a map V Q V P )Q) which factors through the canonical map V Q V Q). The induced map V Q) V P )Q) is an isomorphism and we denote the inverse isomorphism by φ V φ V : V P )Q) V Q) Let us summarize the construction by the following commutative diagram : V P V P ) V Q V Q) V P ) Q V P )Q) For φ Hom TG P, Q) a normal map with P H, Q φh and V a p-permutation khmodule, we put φ 0 V = φ φ V Br Q φ 0 V ) : φ V P )) ) Q) φ V )Q). If V is a p-permutation kg-module, we put φ V = φ V Br Q φ V ) : φ V P )) ) Q) V Q). This gives an isomorphism of functors from kh perm to kn φh φp ), Q) perm φ 0 : Br Q φ Br P Res N φ H Q) N φh φp ),Q) Br Q φ. and an isomorphism of functors from kg perm to kn G φp ), Q) perm φ : Br Q φ Br P Res N GQ) N G φp ),Q) Br Q. When V = kω, Ω a G-set, we have a commutative diagram k φ Ω P Br Q φ γ Ω P )Q) γ Q φ Ω P k φ Ω P ) Q φ kω)p ))Q) φ V where k φ Ω kω Q γ Q Ω φ Ω : φ Ω P ) Q Ω Q, ω φ 1 ω. kω)q)
6 Note that this justifies the claim that φ V is an isomorphism for V a permutation module and consequently for V an arbitrary p-permutation module. We will now check a transitivity property of the isomorphisms constructed above. Lemma 2.4. Let φ Hom TG P, Q) and ψ Hom TG Q, R) such that φ, ψ and ψφ are normal maps and let V be a p-permutation kg-module. Then, the following diagram is commutative ψ φ V P )))Q) ) R) Br R ψ φ V ψ V Q)))R) ψ 0 φ V P )) ψφ V ψ φ) V P )))R) V R) ψ V Proof. Indeed, it is enough to check commutativity for V = kω a permutation module. follows from the commutativity of the following diagram It ψ φ Ω P )) Q) R ω ψ φ 1 ψ 1 ω ψ Ω Q )) R ω ω ω ψ φ 1 ω ψ φ) Ω P )) R Ω R ω ψ 1 ω The difference between φ and φ 0 is is given by the following lemma : Lemma 2.5. Let ψ Hom TG Q, R) be a normal map with ψ = 1, g G and V a p-permutation kg-module. Then, the following diagram is commutative Br R Br Q ι 1 g,v ) ψ V V Q))R) g V )Q))R) ψ g V V R) Br R ι 1 g,v ) g V )R) In particular, if φ Hom TG P, Q) is any normal map, then φ V = Br Q ι 1 φ,v ) φ 0 V. Proof. Again, it is enough to deal with V = kω a permutation module. reduces to the commutativity of ω g 1 ω Ω Q ) R ω ω g Ω) Q ) R ω ω Ω R g Ω) R ω g 1 ω Then, the lemma For the second part of the lemma, we take ψ = φ and g = φ in the commutative diagram. 3. A category of sheaves on p-subgroups complexes 3.1. Definition. Let F be a subcategory of T G. We define a category S F of sheaves on F. Its objects are families {V Q, [φ]} Q,φ where Q runs over the objects of F and φ over the normal maps of F. Here, V Q is a p-permutation kn G Q)/Q-module and for φ Hom F P, Q) normal, [φ] is an isomorphism of kn G φp ), Q)-modules [φ] : φ V P )Q) Res N GQ) N G φp ),Q) V Q.
We require the following two conditions to be satisfied : For φ Hom F Q, Q), we have GLUING p-permutation MODULES 7 [φ] = ι 1 φ,v Q. 1) Let φ Hom F P, Q) and ψ Hom F Q, R) such that φ, ψ and ψφ are normal maps. Then, the following diagram should be commutative ψ φ V P )Q) ) R) Br R ψ [φ] ψ V Q )R) 2) ψ 0 φ VP [ψφ] ψ φ) V P )R) For V = {V Q, [φ]} and V = {V Q, [φ] } two objects of S F, Hom SF V, V ) is the set of families Λ = {λ Q } Q, where Q runs over the objects of F. Here, λ Q Hom kng Q)V Q, V Q ). Furthermore, Λ should have the following property : for every normal map φ Hom F P, Q), the following diagram is commutative V R [ψ] φ V P )Q) Br Q φ λ P φ V P )Q) 3) [φ] V Q λ Q V Q [φ] Thanks to the results of 2.2, we have a functor where S = S TG. We can now state our main result : Br : kg perm S, V {V Q), φ V } Theorem 3.1. The functor Br induces an equivalence of categories kg perm /kg proj S. 3.2. Some properties of S F. Let us give a special case of the commutative diagram 2). Lemma 3.2. Let φ Hom F P, Q) and ψ Hom F Q, R) be two normal maps with ψ N G φp )). Then, the following diagram is commutative ψ φ V P )Q) ) R) Br R ψ [φ] ψ V Q )R) ψ φ VP [ψ φ] φ V P )R) Proof. Let ψ = ψφ and φ = P φ 1 ψ 1 φ)p. We have ψ φ = ψ φ. By 1), we have [φ ] = ι 1 φ,v P. We have also ψ φ ι 1 φ,v P = ι ψ, φ V P. The commutativity of the diagram 2) applied to ψ and φ gives the commutative diagram V R [ψ] φ V P )R) Br R ι ψ, φ VP ψ φ) V P )R) [ψφ] [ψ φ] V R
8 and we obtain the commutativity of the diagram of the lemma since ψ φ V P ψ 0 φ V P by Lemma 2.5. = Br R ι 1 ψ, φ V P ) The diagram 3) need be checked only on a generating set : Lemma 3.3. Let E be a set of normal maps in F such that every normal map of F is a product of elements of E and of inverses of invertible elements of E. Then, the commutativity of the diagram 3) for φ E implies the commutativity for every normal map φ of F Proof. Let φ Hom F P, Q) and ψ Hom F Q, R) such that ψ and ψφ are normal maps. Then, φ is a normal map and the following diagram is commutative ψ φ) V P )R) ψ 0 φ VP ψ φ V P )Q) ) R) [ψφ] Br R ψ [φ] ψ V Q )R) V R [ψ] Br R ψ φ) λ P Br R ψ Br Q φ λ P Br R ψ [φ] Br R ψ λ Q ψ φ V P )Q)) R) ψ V ψ 0 φ V Q )R) P [ψ] ψ φ) V P )R) [ψφ] The lemma follows. We now define a restriction functor from G to N G P )/P. Let P be an object in F. We assume P 1 Q Hom F P, Q) for every Q in F with P Q. Let FP ) be the subcategory of T NG P )/P whose objects are the Q/P where Q is in F and P Q, P Q and where Hom FP ) Q/P, R/P ) is given by the image in T NG P )/P Q/P, R/P ) of Hom F Q, R). Let V = {V Q, [φ]} be an object of S F. Let V Q/P = V Q. For φ Hom FP ) Q/P, R/P ), we put [φ ] = [φ] where φ Hom F Q, R) has image φ in Hom FP ) Q/P, R/P ). It follows from 1) and from the diagram 2)) that this is independent of the choice of φ. The restriction functor is Res F FP ) : S F S FP ), {V Q, [φ]} {V Q/P, [φ ]}. We denote by E P : T F kn G P )/P perm the functor sending V on V P. The commutative diagram in Lemma 3.2 says that objects can be glued locally : Lemma 3.4. For V in S F, we have an isomorphism {[ P 1 Q ]} Q : Br E P V) Res F FP ) V. This induces an isomorphism of functors Br E P Res F FP ). So, we have a diagram, commutative up to isomorphism : V R λ R S F Res F FP ) S FP ) Br E P kn G P )/P perm
GLUING p-permutation MODULES 9 Proof. Let V = {V Q, [φ]} in S F and V = Res F FP ) V. We have BrV P ) = {W Q/P, ψ VP } with W Q/P = V P Q). Let λ Q/P = [ P 1 Q ] : W Q/P V Q/P and Λ = {λ Q/P }. Let φ = P 1 Q in F and ψ Hom F Q, R) be two normal maps with ψ N G P ). We have a commutative diagram Lemma 3.2) ψ V P Q))R) ψ VP V P R) Br R ψ [φ] [ψ φ] ψ V Q )R) V R [ψ] This shows that Λ defines a map BrV P ) V. This induces an isomorphism between Br E P and Res F FP ). 3.3. Proof of Theorem 3.1. Let us first note that BrV ) = 0 if V is projective, hence Br induces indeed a functor Br : kg perm /kg proj S. Lemma 3.5. The functor Br is fully faithful. Proof. We have to prove that Br induces an isomorphism HomV, W ) HomBr V, Br W ) for V and W any p-permutation kg-modules. Thanks to Proposition 2.3, we have a commutative diagram : HomV, W ) Br HomBr V, Br W ) HomV W, k) Br HomBrV W ), Br k) So, it is enough to consider the case W = k. Since the modules kg/q), Q a p-subgroup of G, generate kg perm as an additive category closed under taking direct summands, we may assume V = kg/q). We may take Q 1 since otherwise V is projective. Now, HomkG/Q, k) HomkG/Q, k) is a one-dimensional vector space, generated by the unique map f between the G-sets G/Q and G/G. The map Br Q f) : V Q) = kg/q) Q k is induced by the unique map between the sets N G Q)/Q and N G Q)/N G Q). In particular, it is non-zero, hence Brf) 0. Let Λ HomBr kg/q, Br k). Since HomV Q), k) is one-dimensional, we have λ Q = α Br Q f) for some α k. So, Λ α Brf) vanishes on V Q). We assume now λ Q = 0. We will prove that λ P = 0 for all P. This is clear if P is not conjugate to a subgroup of Q, since then V P ) = 0. We will now prove the result for P Q by induction on [Q : P ]. Let P < Q and g T G P, Q). Let R be a p-subgroup of G such that P R Q g, P R. Then, N G P )/N G P ) Q g ) R. Now, λ P and λ R have the same restriction to N G P )/N G P ) Q g ) R. Consequently, λ P is zero on N G P )/N G P ) Q g ) R, by induction. By Lemma 2.1, we deduce that λ P = 0. The first part of following lemma is essentially due to Bouc [Bou1, Bou2] cf also [Li, Lemma 5.4]).
10 Lemma 3.6. i) Let f : V W be a morphism between two p-permutation kg-modules such that Br Q f) is injective for all Q 1. Then, there is a morphism σ : W V such that σf is a stable automorphism of V. In particular, if V has no projective direct summand, then f is a split injection. ii) Let Λ : V W be a morphism between two objects of S. Assume for all Q, there is a projective direct summand L Q of the kn G Q)/Q-module V Q such that the restriction of λ Q to L Q is injective and V Q /L Q has no projective direct summand. Then, for all Q, λ Q is a split injection. Proof. Let us prove part i) of the lemma. We can assume V has no projective direct summand. Assume G is a p-group. Let us first consider a morphism f : V = kg/q) W = kg/r) such that Br Q f) is injective, where Q and R are non-trivial p-subgroups of G. The module V Q) is a projective indecomposable kn G Q)/Q-module. Since W Q) 0, Q is contained in R, up to conjugacy. Without changing W, we can assume Q R. Assume Q R. Then, for any g T G Q, R), there is S such that P S Q g, P S. So, kn G Q)/N G Q) R g ) is not a projective kn G Q)/Q-module. By Lemma 2.1, V Q) is not isomorphic to a submodule of W Q) and we have reached a contradiction. If Q = R, then f becomes invertible in the quotient End kng Q)/QV Q)) of the local ring End kg V ), hence f is invertible. Let now f : V W be a morphism between two p-permutation modules such that Br Q f) is injective for Q non trivial. In order to prove that f is injective, we may assume V is indecomposable, i.e., V = kg/q) for some subgroup Q of G. We can assume Q 1, otherwise V is projective. Since V Q) is indecomposable, there is an indecomposable direct summand W of W such that if f f is the composition V W W, then Br Q f ) is injective. Now, the considerations above show that f is an isomorphism and we are done. Take now G an arbitrary finite group. Let U = ker f and let S be a Sylow p-subgroup of G. We know that the inclusion Res G S U Res G S V is projective. So, the inclusion U V is projective, hence U = 0 since we assume V has no projective direct summand. Finally, the short exact sequence 0 V W W/V 0 splits, since it splits by restriction to S. Let us come to part ii) of the lemma. We prove the result by inverse induction on the order of Q. When Q is a Sylow p-subgroup of G, then V Q = L Q, so λ Q is a split injection. Assume now Q is not a Sylow p-subgroup of G. Consider the restriction f : M Q W Q of λ Q, where V Q = L Q M Q. By induction, Br R/Q f) is injective, for all p-subgroups R with Q R, Q R. By part i) of the Lemma applied to N G Q)/Q, we deduce that f is a split injection. Hence, λ Q is a split injection. Lemma 3.7. Let G and H be two full subcategories of T G closed under inverse inclusion with G H. Let V, V S H. Then, the restriction map is surjective. Hom SH V, V ) Hom SG Res H G V, Res H G V ) Proof. It is enough to prove the lemma when H has one more object, Q, than G. Let λ Hom SG Res H G V, Res H G V ). Assume first there is g G such that Q g G and let ψ = Q g Q g Hom H Q, Q g ). Let λ R = λ R for R Q and λ Q = [ψ 1 ] λ Q g[ψ]. In order to prove that {λ R } gives a map between V and V extending λ), it is enough to check commutativity of the diagram 3) for the map ψ, thanks to Lemma 3.3. This is immediate.
GLUING p-permutation MODULES 11 Assume now Q g / G for all g G. Let f : BrV Q ) BrV Q ) be the restriction of λ to S HQ) cf Lemma 3.4). By the fullness of Br applied to N G Q)/Q Lemma 3.5), there is a map λ Q : V Q V Q such that Brλ Q ) = f. Let λ R = λ R for R G. Since every map in H starting from Q is the composition of a map from Q to R, with Q a strict normal subgroup of R and of a map in G, Lemma 3.3 shows that {λ R } defines a map between V and V, extending λ. We now complete the proof of Theorem 3.1 by showing the essential surjectivity of Br. Let V S. We will prove by induction on the cardinality of {Q V Q 0} that V is in the image of Br. For Q T G, let L Q be a projective direct summand of V Q such that V Q /L Q has no projective direct summand. We denote by α Q : V Q L Q the canonical surjection. Let M = Ind G N G Q) L Q and M = Br M. We have MQ) L Q by Green s correspondence. α Q Let ζ : V Q LQ MQ). Let H = T G and G be the full subcategory of T G with objects the p-subgroups containing Q. Let ζ R = 0 for R G, R Q and ζ Q = ζ. Then, {ζ R } Hom SG Res H G V, Res H G M). By Lemma 3.7, there is χ Hom S V, M) extending {ζ R }. Let now V = Q T G /G IndG N G Q) L Q, V = Br V and λ : V V be the sum of the morphisms constructed for each Q above. Then, for all Q, the restriction of λ Q to L Q is injective. By Lemma 3.6, ii), we deduce that λ Q is a split injection for all Q. Let then W be the cokernel of λ. Take R with V R = 0. Then, V Q = 0 whenever R is contained up to G-conjugation in Q. So, V has no direct summand with vertex R, hence W R = 0. Let now Q be maximal such that V Q 0. Then, λ Q : V Q = L Q ) Ind G N G Q) L Q Q) is an isomorphism. So, WQ = 0. It follows that {Q W Q 0} is strictly contained in {Q V Q 0}. By induction, there is a p-permutation kg-module W without projective direct summand) such that W = Br W. By fulness of Br Lemma 3.5), the canonical morphism V W comes from a morphism f : V W. By Lemma 3.6, i), dualized, f is a split surjection. Hence, ker f is a p-permutation kg-module and we have an isomorphism V Brker f). This finishes the proof of Theorem 3.1. References [Bou1] S. Bouc, Résolutions de foncteurs de Mackey, in Groups, representations : cohomology, group actions and topology, Proc. of Symp. in Pure Math., p. 31 83, vol. 63, AMS, 1998. [Bou2] S. Bouc, Le complexe de chaînes d un G-complexe simplicial acyclique, preprint 1998). [Br] M. Broué, On Scott modules and p-permutation modules : an approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 1985), 401 408. [Li] M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, preprint 1997). [Th] J. Thévenaz, G-algebras and modular representation theory, Oxford University Press, 1995.