The perfect F-locality from the basic F-locality over a Frobenius P-category F
aa r X i v : . [ m a t h . G R ] M a r The perfect F -locality from the basic F -localityover a Frobenius P -category F Lluis Puig
Abstract:
Let p be a prime, P a finite p- group, F a Frobenius P- category and F sc the fullsubcategory of F over the set of F -selfcentralizing subgroups of P . Recently, we have understoodan easy way to obtain the perfect F sc -locality P sc from the basic F sc -locality L b ,sc : it dependson a suitable filtration of the basic F -locality L b and on a vanishing cohomology result, givenwith more generality in [11, Appendix].
1. Introduction p be a prime and P a finite p -group. After our introduction ofthe Frobenius P -categories F [7] and the question of Dave Benson [1] on theexistence of a suitable category P sc — called linking system in [2] and perfect F sc -locality in [8, Chap. 17] — extending the full subcategory F sc of F overthe set of F -selfcentralizing subgroups of P [8, Chap. 3], the existence andthe uniqueness of P sc has concentrate some effort.1.2. In [2] Carles Broto, Ran Levi and Bob Oliver formulate the existenceand the uniqueness of the category P sc in terms of the annulation of an obstruction -cohomology element and of the vanishing of a 2 -cohomologygroup , respectively. They actually state a sufficient condition for the va-nishing of the corresponding n -cohomology groups .1.3. In [3] Andrew Chermak has proved the existence and the uniquenessof P sc via his objective partial groups , but his proof depends on the so-called Classification of the finite simple groups and on some results by U. Meier-frankenfeld and B. Stellmacher. In [6] Bob Oliver, following some of Cher-mak’s methods, has also proved for n ≥ n -cohomologygroups mentioned above. In [5] George Glauberman and Justin Lynd removethe use of the Classification of the finite simple groups in [6] † .1.4. Independently, with direct methods which already employ the basic F -locality L b [8, Chap. 22], in [9] and [10] †† we prove the existence and theuniqueness of an extension P of F — called perfect F -locality in [8, Chap. 17]— which contains P sc as the full subcategory over the set of F -selfcentralizing subgroups of P [8, Chap. 3].1.5. But recently, we have understood an easier way to obtain P sc fromthe full subcategory L b ,sc of L b over the set of F -selfcentralizing subgroups † Although they need a partial classification for p =2. †† In [10] we give a full correction of the uniqueness argument for P sc in [9]. of P [8, Chap. 3]. Denoting by Z sc : L b ,sc → Ab the obvious contravariant functor from L b ,sc to the category Ab of finite Abelian groups, mapping any F -selfcentralizing subgroup Q of P on its center Z ( Q ) , it is easy to see thatwe have a quotient category ] L b ,sc = L b ,sc / Z sc and that the structural functor π sc : L b ,sc → F sc factorizes through a functor f π sc : ] L b ,sc → F sc . f π sc admits an essentially unique section functor f σ sc : F sc → ] L b ,sc , and then P sc is just the converse image in L b ,sc of the image f σ sc ( F sc ) of F sc in ] L b ,sc ; since in [9, Theorem 7.2] we prove that anyperfect F sc -locality P sc can be extended to a unique perfect F -locality P , this proves the existence of P . Moreover, in [9, 8.5 and Theorem 8.10] weprove that there is an F -locality functor σ from any perfect F -locality P to L b ; then, it is easy to check that σ induces a functor f σ sc : F sc → ] L b ,sc which is a section of f π sc , proving the uniqueness of P sc and therefore, by[9, Theorem 7.2], the uniqueness of P . sectionfuntor f σ sc mentioned above depend on a suitable filtration of L b and on a vanishing cohomology result ; this filtration has been already introduced in[9, 8.3 and Corollary 8.4], but it seems necessary to give here a completeaccount in Section 3. The vanishing cohomology result we need here is givenin [11, Appendix] in a more general framework. In Section 4 we give explicitproofs of all the results announced in 1.6 above and, in particular, an inde-pendent proof of the existence of the functor σ : P → L b mentioned above.
2. Definitions and quoted results Ab the category of Abelian groups and by iGr thecategory formed by the finite groups and by the injective group homomor-phisms. Recall that, for any category C , we denote by C ◦ the opposite cat-egory and, for any C -object A , by C A (or ( C ) A to avoid confusion) the cat-egory of “ C -morphisms to A ” [8, 1.7]; moreover, for any pair of objects A and B , C ( B, A ) denote the set of C -morphisms from A to B and we set C ( A ) = C ( A, A ) for short.2.2. For any finite subgroup G and any p -subgroup P of G , denote by F G,P and T G,P the respective categories where the objects are all the sub-groups of P and, for two of them Q and R , the respective sets of morphisms F G,P ( Q, R ) and T G,P ( Q, R ) are formed by the group homomorphisms from R to Q respectively induced by the conjugation by elements of G , and bythe set T G ( R, Q ) of such elements, the compositions being the obvious ones.2.3. For a finite p -group P , a Frobenius P -category F is a subcategoryof iGr containing F P = F P,P where the objects are all the subgroups of P and the morphisms fulfill the following three conditions [8, 2.8 and Proposi-tion 2.11]2.3.1 For any subgroup Q of P , the inclusion functor ( F ) Q → ( iGr ) Q isfull. F P ( P ) is a Sylow p -subgroup of F ( P ) . We say that a subgroup Q of P is fully centralized in F if for any F -morphism ξ : Q · C P ( Q ) → P we have ξ (cid:0) C P ( Q ) (cid:1) = C P (cid:0) ξ ( Q ) (cid:1) ; similarly, replacing in thiscondition the centralizer by the normalizer, we say that Q is fully normalized .2.3.3 For any subgroup Q of P fully centralized in F , any F -morphism ϕ : Q → P and any subgroup R of N P (cid:0) ϕ ( Q ) (cid:1) such that ϕ ( Q ) ⊂ R andthat F P ( Q ) contains the action of F R (cid:0) ϕ ( Q ) (cid:1) over Q via ϕ , there exists an F -morphism ζ : R → P fulfilling ζ (cid:0) ϕ ( u ) (cid:1) = u for any u ∈ Q .
We denote by ˜ F – called the exterior quotient of F — the quotient of F by the inner automorphisms of the F -objects [8, 1.3] and by ǫ F : F → ˜ F the canonical functor . Note that, with the notation above, if P is Sylow p -subgroup of G then F G,P is a
Frobenius P -category ; often, we write F G instead of F G,P . divisible ) F -locality † is a triple ( τ, L , π ) formed by a finite category L , a surjective functor π : L → F and a functor τ : T P → L from the transporter category T P = T P,P of P , fulfilling the following two conditions[8, 17.3 and 17.8]2.4.1
The composition π ◦ τ coincides with the composition of the canonicalfunctor defined by the conjugation κ P : T P → F P with the inclusion F P ⊂ F . We denote by ˜ κ P : T P → ˜ F P the composition of κ P with ǫ F P above.2.4.2 For any pair of subgroups Q and R of P ,
Ker( π R ) acts regularly onthe fibers of the following maps determined by ππ Q,R : L ( Q, R ) −→ F ( Q, R )Analogously, for any pair of subgroups Q and R of P , we denote by τ Q,R : T P ( Q, R ) −→ L ( Q, R ) 2 . . τ , and whenever R ⊂ Q we set i QR = τ Q,R (1) ; if R = Q then we write Q once for short.2.5. We say that an F -locality ( τ, L , π ) , or L for short, is coherent if itfulfills the following condition [8, 17.9](Q) For any pair of subgroups Q and R of P , any x ∈ L ( Q, R ) and any v ∈ R , we have x · τ R ( v ) = τ Q (cid:0) π Q,R ( x )( v ) (cid:1) · x . † Here we only consider divisible F -localities in the sense of [8, Chap. 17]. In this case,, if Q ′ and R ′ are subgroups of P , and we have the inclusions R ⊂ Q and R ′ ⊂ Q ′ , denoting by L ( Q ′ , Q ) R ′ ,R the set of y ∈ L ( Q ′ , Q ) suchthat (cid:0) π Q ′ ,Q ( y ) (cid:1) ( R ) ⊂ R ′ , we get a restriction map (possibly empty!) r Q ′ ,QR ′ ,R : L ( Q ′ , Q ) R ′ ,R −→ L ( R ′ , R ) 2 . . y · i QR = i Q ′ R ′ · r Q ′ ,QR ′ ,R ( y ) for any y ∈ L ( Q ′ , Q ) R ′ ,R . Note that, with thenotation above, if P is Sylow p -subgroup of G then T G = T G,P endowed withthe obvious functors τ G : T P −→ T G and π G : T G −→ F G = F G,P . . coherent F G -locality . Moreover, we say that a coherent F -locality ( τ, L , π ) is p -coherent (resp. ab-coherent ) when Ker( π Q ) is a finite p -group(resp. a finite abelian group) for any subgroup Q of P . F - hyperfocal subgroup is the subgroup H F of P generated by the union of the sets (cid:8) u − σ ( u ) (cid:9) u ∈ Q where Q runs over the setof subgroups of P and σ over the set of p ′ -elements of F ( Q ) . We say that an F -locality (ˆ τ , P , ˆ π ) is perfect if P is coherent and, for any subgroup Q of P fully centralized in F , the C F ( Q ) -hyperfocal subgroup H C F ( Q ) coincides withKer(ˆ π Q ) [8, 17.13]; actually, this is equivalent to say that P ( Q ) , endowedwith ˆ τ Q : T N P ( Q ) −→ P ( Q ) and ˆ π Q : P ( Q ) −→ F ( Q ) 2 . . , is an F -localizer of Q [8, 18.5 and Theorem 18.6], for any subgroup Q of P fully centralized in F .2.7. Further, for any F -locality ( τ, L , π ) we get a contravariant functorfrom L to the category Gr of finite groups [8, 17.8.2] Ker ( π ) : L −→ Gr . . Q of P to Ker( π Q ) and any L -morphism x : R → Q tothe group homomorphism Ker ( π ) x : Ker( π Q ) −→ Ker( π R ) 2 . . u · x = x · (cid:0) Ker ( π ) x ( u ) (cid:1) for any u ∈ Ker( π Q ) . If L is ab-coherent thenthe functor Ker ( π ) factorizes through the exterior quotient ˜ F , inducing afunctor g Ker ( π ) = ˜ k L : ˜ F −→ Gr . . Q of P , τ Q ( Q ) centralizes Ker( π Q ) and therefore, for any u ∈ Ker( π Q )and any v ∈ Q , we have τ Q ( v ) · u = u.τ Q ( v ) = τ Q ( v ) · (cid:0) Ker ( π ) τ Q ( v ) ( u ) (cid:1) . . , so that Ker ( π ) τ Q ( v ) = id Ker( π Q ) ; the same argument holds for w ∈ Ker( π Q ) . L ′ is a second F -locality with structural functors τ ′ and π ′ , wecall F -locality functor from L to L ′ any functor l : L → L ′ fulfilling τ ′ = l ◦ τ and π ′ ◦ l = π . . F -locality functors is obviously an F -locality functor.It is easily checked that any F -locality functor l : L → L ′ determines a naturalmap χ l : Ker ( π ) −→ Ker ( π ′ ) 2 . . h of Ker ( π ) determines a quotient F -locality L / h defined by the quotient sets( L / h )( Q, R ) = L ( Q, R ) / h ( R ) 2 . . , for any pair of subgroups Q and R of P , and by the corresponding inducedcomposition; moreover, L / h is coherent whenever L is it.2.9. We say that two F -locality functors l and ¯ l from L to L ′ are naturally F -isomorphic if we have a natural isomorphism λ : l ∼ = ¯ l fulfilling π ′ ∗ λ = id π and λ ∗ τ = id τ ′ ; in this case, λ Q belongs to Ker( π ′ Q ) for any subgroup Q of P and, since l ( i PQ ) = i ′ PQ = ¯ l ( i PQ ) , λ is uniquely determined by λ P ; indeed, wehave λ P · i ′ PQ = i ′ PQ · λ Q . . . Once again, the composition of a natural F -isomorphism with an F -localityfunctor or with another such a natural F -isomorphism is a natural F -iso-morphism.2.10. Moreover, from two F -localities ( τ, L , π ) and ( τ ′ , L ′ , π ) , we canconstruct a third F -locality L ′′ = L × F L ′ from the corresponding categorydefined by the pull-back of sets L ′′ ( Q, R ) = L ( Q, R ) × F ( Q,R ) L ′ ( Q, R ) 2 . . T P ( Q, R ) τ ′′ Q,R −−−−→ L ′′ ( Q, R ) π ′′ Q,R −−−−→ F ( Q, R ) 2 . . τ and τ ′ , and by π and π ′ . Note that we have obvious F -locality functors L ←− L × F L ′ −→ L ′ . . L × F L ′ is coherent if L and L ′ are so. 2.11. In order to define the basic F -locality , we have to consider the F -basic P × P -sets ; recall that an F -basic P × P -set Ω is a finite nonempty P × P -set Ω fulfilling the following three conditions [8, 21.4 and 21.5], whereΩ ◦ denotes the P × P -set obtained from Ω by exchanging both factors, forany subgroup Q of P we denote by ι PQ the corresponding inclusion map, andfor any ϕ ∈ F ( P, Q ) we set∆ ϕ ( Q ) = { ( ϕ ( u ) , u ) } u ∈ Q . . . We have Ω ◦ ∼ = Ω , { } × P acts freely on Ω and | Ω | / | P | 6≡ p . For any subgroup Q of P and any ϕ ∈ F ( P, Q ) we have a Q × P -setisomorphism Res ϕ × id P (Ω) ∼ = Res ι PQ × id P (Ω)2.11.4 Any P × P -orbit in Ω is isomorphic to ( P × P ) / ∆ ϕ ( Q ) for a suitablesubgroup Q of P and some ϕ ∈ F ( P, Q ) . Moreover, we say that an F -basic P × P -set Ω is thick if the multiplicityof the indecomposable P × P -set ( P × P ) (cid:14) ∆ ϕ ( Q ) is at least two for anysubgroup Q of P and any ϕ ∈ F ( P, Q ) [8, 21.4].2.12. The existence of a thick F -basic P × P -set is guaranteed by[8, Proposition 21.12]; we choose one of them Ω and denote by G the group of { } × P -set automorphisms of Res { }× P (Ω) ; it is clear that we have an injec-tive map from P × { } into G and we identify this image with the p -group P itself so that, from now on, P is contained in G and acts freely on Ω . Then,it follows from the conditions above that we have F G,P = F . . T G = T G,P becomes a coherent F -locality. Q of P , it is clear that the centralizer C G ( Q ) co-incides with the group of Q × P -set automorphisms of Res Q × P (Ω) ; moreover,since any Q × P -orbit in Ω is isomorphic to the Q × P -set ( Q × P ) (cid:14) ∆ η ( T ) , for a suitable subgroup T of P such that F ( Q, T ) = ∅ and some η ∈ F ( Q, T )(cf. condition 2.11.3), and since we have [8, 22.3]Aut Q × P (cid:0) ( Q × P ) (cid:14) ∆ η ( T ) (cid:1) ∼ = ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) . . , denoting by k η the multiplicity of ( Q × P ) (cid:14) ∆ η ( T ) in Ω and by S k η the cor-responding k ψ -symmetric group, we actually get obvious group isomorphisms C G ( Q ) ∼ = Y T ∈C P Y η ∈ O TQ ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) ≀ S k η . . ≀ denotes the wreath product , C P is a set of representatives for theset of P -conjugacy classes of subgroups T of P and, for any T ∈ C P , O TQ ⊂ F ( Q, T ) is a (possibly empty) set of representatives for the quotientset Q \F ( Q, T ) /N P ( T ) . For short, let us set O Q = G T ∈C P O TQ . . set of isomorphic classes of transitive Q × P -sets ;to avoid confusion, we note by ( T, η ) the element η in O TQ . the correspondencesending Q to the minimal normal subgroup S G ( Q ) of C G ( Q ) containing theimage of Y ( T,η ) ∈ O Q S k η for any isomorphism induces a functor S G : T G −→ iGr . . Ker ( π G ) (cf. 2.5.2) and therefore determinesa coherent F -locality L b = T G / S G (cf. 2.8) — called the basic F -locality [8, Chap. 22] — which, according to [9, Corollary 4.11], does not depend onthe choice of the thick F -basic P × P -set Ω . Moreover, denoting by τ b : T P −→ L b and π b : L b −→ F . . Q of P , isomorphisms in 2.13.2 induce a canonical isomorphism (cid:0)
Ker ( π b ) (cid:1) ( Q ) ∼ = Y ( T,η ) ∈ O Q ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1)(cid:17) . . ab : Gr → Ab denotes the obvious functor mapping any finite group H on its maximal Abelian quotient H/ [ H, H ] ; in particular, note that L b is p -coherent (cf. 2.5).2.15. Moreover, any L b -morphism x : R → Q can be lifted to an elementˆ x ∈ G fullfilling ˆ x ◦ R ◦ ˆ x − ⊂ Q in the group of bijections of Ω ; in particular,we also have ˆ x − ◦ C G ( Q ) ◦ ˆ x ⊂ C G ( R ) 2 . . C G ( Q ) and C G ( R ), it is clearthat the conjugation by ˆ x − sends the factor determined by T ∈ C P and by η ∈ O TQ in some factors determined by U ∈ C P and by θ ∈ O UR in such a waythat, setting ϕ = π b ( x ), there exists an injective R × P -set homomorphism f : ( R × P ) (cid:14) ∆ θ ( U ) −→ Res ϕ × id P (cid:0) ( Q × P ) (cid:14) ∆ η ( T ) (cid:1) . . ϕ ◦ θ ( U ) = (cid:0) ϕ ( R ) × P (cid:1) ∩ ( u,n ) ∆ η ( T ) 2 . . u ∈ Q and n ∈ P . L b -morphism x : R → Q determines the grouphomomorphism (cid:0) Ker ( π b ) (cid:1) ( x ) : (cid:0) Ker ( π b ) (cid:1) ( Q ) −→ (cid:0) Ker ( π b ) (cid:1) ( R )2 .
17 2 . . (cid:0) Ker ( π b ) (cid:1) ( Q ) and (cid:0) Ker ( π b ) (cid:1) ( R ), itmakes sense to introduce the projection in ab (cid:16) ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1)(cid:17) of the restric-tion of (cid:0) Ker ( π b ) (cid:1) ( x ) to 2.17 ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1)(cid:17) — noted (cid:0) Ker ( π b ) (cid:1) ( x ) ( U,θ )( T,η ) — for any ( T, η ) ∈ O Q and any ( U, θ ) ∈ O R ; according to 2.15 above, (cid:0) Ker ( π b ) (cid:1) ( x ) ( U,θ )( T,η ) = 0 forces∆ ϕ ◦ θ ( U ) = (cid:0) ϕ ( R ) × P (cid:1) ∩ ( u,n ) ∆ η ( T ) 2 . . u ∈ Q and n ∈ P . (cid:0)
Ker ( π b ) (cid:1) ( x ) ( U,θ )( T,η ) as follows. Consider the set of injective R × P -set homomorphisms as in 2.15.2above; it is clear that ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1) × ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) acts on this set by left-and right-hand composition and, denoting by I ( U,θ )( T,η ) ( ϕ ) a set of representativesfor the set of ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1) × ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) -orbits, for any f ∈ I ( U,θ )( T,η ) ( ϕ )consider the stabilizer ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) Im( f ) of Im( f ) in ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) , sothat we get an inclusion and an obvious group homomorphism ε f : ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) Im( f ) −→ ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) δ f : ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) Im( f ) −→ ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1) . . . fulfilling ¯ a · f = f · δ f (¯ a ) for any ¯ a ∈ ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1) Im( f ) . Then, denoting by ab c : iGr → Ab the contravariant functor mapping any finite group H onits maximal Abelian quotient H/ [ H, H ] and any injective group homomor-phism on the group homomorphism induced by the transfert , it follows from[8, Proposition 22.17] that for any (
T, η ) ∈ O Q and any ( U, θ ) ∈ O R fulfillingcondition 2.16.2 for suitable u ∈ Q and n ∈ P we have (cid:0) Ker ( π b ) (cid:1) ( x ) ( U,θ )( T,η ) = X f ∈ I ( U,θ )( T,η ) ( ϕ ) ab ( δ f ) ◦ ab c ( ε f ) 2 . . .
3. A filtration for the basic F -locality P be a finite p -group, F a Frobenius P -category and ( τ b , L b , π b )the corresponding basic F -locality ; we already know that the contravariant functor Ker ( π b ) : L b −→ Ab . . exterior quotient ˜ F of F (cf. 2.7), so that it definesa contravariant functor ˜ k L b = ˜ k b F : ˜ F −→ Ab . . F -morphism ˜ ϕ : R → Q onthe group homomorphism (cf. 2.17.2)˜ k b F ( ˜ ϕ ) = X ( T,η ) ∈ O Q X ( U,θ ) ∈ O R X f ∈ I ( U,θ )( T,η ) ( ϕ ) ab ( δ f ) ◦ ab c ( ε f ) 3 . . M ( T,η ) ∈ O Q ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1)(cid:17) to M ( U,θ ) ∈ O R ab (cid:16) ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1)(cid:17) , where weset I ( U,θ )( T,η ) ( ϕ ) = ∅ whenever condition 2.16.2 is not fulfillied for any u ∈ Q andany n ∈ P . k b F ( ˜ ϕ ) sends an elementof ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1)(cid:17) to a sum of terms indexed by elements ( U, θ ) in O R such that U is contained in a P -conjugated of T ; hence, for any subset N of C P which fulfills3.2.1 any U ∈ C P which is contained in a P -conjugated of T ∈ N belongsto N , setting O N Q = F T ∈N O TQ for any subgroup Q of P , it is quite clear that weget a contravariant subfunctor ˜ k N F : ˜ F −→ Ab of ˜ k b F sending Q to M ( T,η ) ∈ O N Q ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( T ) (cid:1)(cid:17) . . quotient F -locality L b / (˜ k N F ◦ ˜ π b ) (cf. 2.8)— denoted by ( τ b , N , L b , N , π b , N ) — of the basic F -locality L b above.3.3. It is quite clear that if M is another subset of C P fulfilling condi-tion 3.2.1 and containing N , we have a canonical functor l M , N F : L b , N → L b , M . From now on, we fix a proper subset N of C P fulfilling condition 3.2.1 and,in order to argue by induction on |C P − N | , we also fix a minimal element U in C P − N , setting M = N ∪ { U } . Hence, it makes sense to consider thequotient contravariant functor˜ k U F = Ker (¯ l M , N F ) = ˜ k M F / ˜ k N F : ˜ F −→ Ab . . U as we show in 3.4 and 3.5 below. More precisely, forany m ∈ N let us consider the subfunctor p m · id : Ab → Ab of the identity func-tor id Ab sending any Abelian group A to p m · A , setting s m = p m · id /p m +1 · id . for any m ≥ and any n ≥ , the n -th stable cohomology group — noted H n ∗ ( ˜ F , s m ◦ ˜ k U F ) (see [8, A3.17]) — of ˜ F over s m ◦ ˜ k U F vanish ; that is to say, thatthe differential subcomplex in [11, A2.2], where B = ˜ F and a = s m ◦ ˜ k U F , andwhere we only consider the elements which are stable by ˜ F -isomorphisms, isexact.3.4. This vanishing result will follow from Theorem 3.11 below andfrom [11, Theorem A5.5]; that is to say, with the terminology introducedin [11, 45.1], in Theorem 3.11 below we prove that, for any m ∈ N , the contravariant functor s m ◦ ˜ k U F above admits indeed a compatible complement .From definition 3.3.1 above it is clear that, for any subgroup Q of P ˜ k U F ( Q ) = M ( T,η ) ∈ O M Q − O N Q ab (cid:0) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:1) . . T, η ) ∈ O M Q − O N Q , we necessarily have T = U ; hence, weget ˜ k U F ( Q ) = M ( U,η ) ∈ O UQ ab (cid:0) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:1) . . O UQ ⊂ F ( Q, U ) is a set of representatives for Q \F ( Q, U ) /N P ( U ) ; moreprecisely, the group Q × N P ( U ) acts on F ( Q, U ) and if η, η ′ ∈ F ( Q, U ) are inthe same Q × N P ( U )-orbit then the conjugation by a suitable element ( u, n )in Q × N P ( U ) induces a group isomorphism ab (cid:0) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:1) ∼ = ab (cid:0) ¯ N Q × P (cid:0) ∆ η ′ ( U ) (cid:1)(cid:1) . . u, n ) fulfilling η ′ = ( u, n ) · η . Consequently, from 3.4.2 we get a canonical isomorphism˜ k U F ( Q ) ∼ = (cid:18) M η ∈F ( Q,U ) ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:17)(cid:19) Q × N P ( U ) . . . F -morphism ˜ ϕ : R → Q , from 3.1.3 above we stillget ˜ k U F ( ˜ ϕ ) = X ( U,η ) ∈ O UQ X ( U,θ ) ∈ O UR X f ∈ I ( U,θ )( U,η ) ( ϕ ) ab ( δ f ) ◦ ab c ( ε f ) 3 . . I ( U,θ )( U,η ) ( ϕ ) is empty unless for suitable u ∈ Q and n ∈ P we have ∆ ϕ ◦ θ ( U ) = ( u,n ) ∆ η ( U ) 3 . . n belongs to N P ( U ) and ϕ ◦ θ belongs to the classe of η in Q \F ( Q, U ) /N P ( U ) ; in this case we have an injective R × P -set homomor-phism f : ( R × P ) (cid:14) ∆ θ ( U ) −→ Res ϕ × id P (cid:0) ( Q × P ) (cid:14) ∆ η ( U ) (cid:1) . . ,
1) to the class of ( u, n ) . Then, denoting by ϕ θ : ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1) −→ ¯ N Q × P (cid:0) ∆ ϕ ◦ θ ( U ) (cid:1) . . ϕ × id P , and by κ η,ϕ ◦ θ ( u,n ) : ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1) ∼ = ¯ N Q × P (cid:0) ∆ ϕ ◦ θ ( U ) (cid:1) . . , the conjugation by ( u, n ) , it is quite clear that the image of ϕ θ stabilizes theimage of f and therefore that f is the unique element of I ( U,θ )( U,η ) ( ϕ ) , that δ f isan isomorphism in 2.17.1 and that we get [9, 8.8] ab ( δ f ) ◦ ab c ( ε f ) = ab c (cid:0) ( κ η,ϕ ◦ θ ( u,n ) ) − ◦ ϕ θ ) = ab c ( ϕ θ ) ◦ ab ( κ η,ϕ ◦ θ ( u,n ) ) 3 . . . Consequently, equality 3.5.1 becomes˜ k U F ( ˜ ϕ ) = X ( U,θ ) ∈ O UR ab c ( ϕ θ ) ◦ ab ( κ η,ϕ ◦ θ ( u,n ) ) 3 . . U, θ ) ∈ O UR , ( U, η ) belongs to O UQ and ( u, n ) fulfills 3.5.2.3.6. But, for our purposes, we need a better description as follows forthe functor ˜ k U F . It is quite clear that we have a functor n U F : F → Gr mappingany subgroup Q of P on the direct product of groups n U F ( Q ) = Y η ∈F ( Q,U ) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1) . . F -morphism ϕ : R → Q on the direct product of group homomor-phisms (cf. 3.5.4) n U F ( ϕ ) = Y θ ∈F ( R,U ) ϕ θ : Y θ ∈F ( R,U ) ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1) −→ Y η ∈F ( Q,U ) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1) . . u ∈ Q denoting by κ Q,u : Q ∼ = Q the conjugation by u , theaction of ( u, ∈ Q × N P ( U ) on n U F ( Q ) coincides with n U F ( κ Q,u ) . Similarly, asin 3.4.3 above, for any n ∈ N P ( U ) the action of (1 , n ) ∈ Q × N P ( U ) on n U F ( Q )induces obvious isomorphisms(1 , n ) η : ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1) ∼ = ¯ N Q × P (cid:0) ∆ η ◦ κ U,n − ( U ) (cid:1) . . η ∈ F ( Q, U ); moreover, for any θ ∈ F ( R, U ) , we obviously get(1 , n ) ϕ ◦ θ ◦ ϕ θ = ϕ θ ◦ κ U,n − . . . ab c ◦ n U F : F −→ Ab ◦ and ab ◦ n U F : F −→ Ab . . Q of P to( ab c ◦ n U F )( Q ) = M η ∈F ( Q,U ) ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:17) = ( ab ◦ n U F )( Q ) 3 . . Q × N P ( U ) acts on this Abelian p -group; then, it is quiteeasy to check that we have a subfunctor of ab c ◦ n U F and a quotient functorof ab ◦ n U F , respectively denoted by h ◦ ( ab c ◦ n U F ) : F −→ Ab ◦ and h ◦ ( ab ◦ n U F ) : F −→ Ab . . , sending any subgroup Q of P to the subgroup ( ab c ◦ n U F )( Q ) Q × N P ( U ) of fixed elements and to the quotient ( ab ◦ n U F )( Q ) Q × N P ( U ) of co-fixed elementsof M η ∈F ( Q,U ) ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:17) ; actually, it is easily checked that both fac-torize through the exterior quotient ˜ F (cf. 3.1) yielding respective functors˜ h ◦ ( ab c ◦ n U F ) : ˜ F −→ Ab ◦ and ˜ h ◦ ( ab ◦ n U F ) : ˜ F −→ Ab . . . In particular, it follows from 3.4.4 that for any subgroup Q of P we have˜ h ◦ ( ab c ◦ n U F )( Q ) ∼ = ˜ k U F ( Q ) 3 . . . F -morphism ˜ ϕ : R → Q we also havethe commutative diagram˜ h ◦ ( ab c ◦ n U F )( Q ) ∼ = ˜ k U F ( Q ) ˜ h ◦ ( ab c ◦ n U F )( ˜ ϕ ) y y ˜ k U F ( ˜ ϕ ) ˜ h ◦ ( ab c ◦ n U F )( R ) ∼ = ˜ k U F ( R ) 3 . . h ◦ ( ab c ◦ n U F ) and ˜ k U F from ˜ F to Ab ◦ are isomorphic. Indeed,˜ h ◦ ( ab c ◦ n U F )( ˜ ϕ ) sends any element a = X η ∈F ( Q,U ) a η ∈ ˜ h ◦ ( ab c ◦ n U F )( Q ) 3 . . , where a η belongs to ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:17) for any η ∈ F ( Q, U ) , to the element X θ ∈F ( R,U ) ab c ( ϕ θ )( a ϕ ◦ θ ) ∈ ˜ h ◦ ( ab c ◦ n U F )( R ) 3 . . Q of P we clearly have a canonicalgroup isomorphism tr Q : ( ab ◦ n U F )( Q ) Q × N P ( U ) ∼ = ( ab c ◦ n U F )( Q ) Q × N P ( U ) . . η ∈ F ( Q, U ) , maps the class in ( ab ◦ n U F )( Q ) Q × N P ( U ) of anelement a η of ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:17) on the element (cf. 3.6)tr Q × N P ( U ) N Q × P (∆ η ( U )) ( a η ) = X ( u,n ) ( u, n ) η · a η . . ab c ◦ n U F )( Q ) Q × N P ( U ) (cf. 3.7.2), where ( u, n ) runs over a set of represen-tatives for the quotient set (cid:0) Q × N P ( U ) (cid:1)(cid:14) N Q × P (∆ η ( U )) . u, n ) of Q × P belongs to N Q × P (∆ η ( U )) ifand only if we have n U = U and η ( n v ) = u η ( v ) for any v ∈ U ; in particular, u normalizes η ( U ) and, denoting by η ∗ : U ∼ = η ( U ) the isomorphism inducedby η , this element belongs to the converse image Q η of F Q (cid:0) η ( U ) (cid:1) ∩ (cid:0) η ∗ ◦ F P ( U ) ◦ η − ∗ (cid:1) . . N Q (cid:0) η ( U ) (cid:1) ; then, the conjugation by η − ∗ induces a group homomorphism ν η : Q η → F P ( U ) ; thus, setting∆ ν η ( Q η ) = { ( u, ν η ( u ) } u ∈ Q η ⊂ Q × F P ( U ) 3 . . , we get the exact sequence1 −→ { } × C P ( U ) −→ N Q × P (cid:0) ∆ η ( U ) (cid:1) −→ ∆ ν η ( Q η ) −→ . . a η the classe of a η ∈ ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:17) in( ab ◦ n U F )( Q ) Q × N P ( U ) and by Q η ⊂ Q a set of representatives for Q/Q η , since { } × C P ( U ) acts trivially on ab (cid:16) ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:17) , we still get tr Q (˜ a η ) = X u ∈ Q η X ν ∈F P ( U ) ( u, ν ) · a η = X ν ∈F P ( U ) (1 , ν ) · X u ∈ Q η ( ab ◦ n U F )( κ Q,u )( a η ) 3 . . . Finally, for any m ∈ N , for short we set˜ r U, ◦F ,m = s m ◦ ˜ h ◦ ( ab c ◦ n U F ) and ˜ r U,m F , ◦ = s m ◦ ˜ h ◦ ( ab ◦ n U F ) 3 . . Q of P , it is clear that tr Q induces a groupisomorphism tr mQ : ˜ r U,m F , ◦ ( Q ) ∼ = ˜ r U, ◦F ,m ( Q ) 3 . . . Theorem 3.11. † With the notation above, the functor ˜ r U, ◦F ,m : ˜ F → Ab ◦ admits a compatible complement (˜ r U, ◦F ,m ) c : ˜ F → Ab sending any ˜ F -morphism ˜ ϕ : R → Q to the group homomorphism (˜ r U, ◦F ,m ) c ( ˜ ϕ ) = tr mQ ◦ ˜ r U,m F , ◦ ( ˜ ϕ ) ◦ ( tr mR ) − . . . Proof:
It is clear that equalities 3.11.1 define a functor ˜
F → Ab sending anysubgroup Q of P to s m (cid:0) ( ab c ◦ n U F )( Q ) Q × N P ( U ) (cid:1) ; thus, it remains to prove that(˜ r U, ◦F ,m ) c fulfills the conditions A5.1.2 and A5.1.3 in [11]. With the notationin 3.7 above, assuming that a = X η ∈F ( Q,U ) a η belongs to p m · ˜ h ◦ ( ab c ◦ n U F )( Q ) anddenoting by ¯ a m its image in s m (cid:0) ( ab c ◦ n U F )( Q ) Q × N P ( U ) (cid:1) , for condition A5.1.3we have to compute (cid:0) (˜ r U, ◦F ,m ) c ( ˜ ϕ ) ◦ ˜ r U, ◦F ,m ( ˜ ϕ ) (cid:1) (¯ a m ) in ˜ r U, ◦F ,m ( Q ) ; clearly, thiselement is the image in s m (cid:0) ( ab c ◦ n U F )( Q ) Q × N P ( U ) (cid:1) of (cid:16) tr mQ ◦ (˜ h ◦ ( ab ◦ n U F ))( ˜ ϕ ) ◦ ( tr mR ) − (cid:17) ˜ h ◦ ( ab c ◦ n U F )( ˜ ϕ )( a ) (cid:1) = (cid:16) tr mQ ◦ ˜ h ◦ ( ab ◦ n U F )( ˜ ϕ ) ◦ ( tr mR ) − (cid:17)(cid:0) X θ ∈F ( R,U ) ab c ( ϕ θ )( a ϕ ◦ θ ) (cid:1) . . , which is equal to zero whenever F ( R, U ) is empty.Otherwise, for any element ( y, n ) in R × N P ( U ) , ( ϕ ( y ) , n ) belongs to Q × N P ( U ) and therefore, with the obvious action of R × N P ( U ) on F ( R, U ) , we have a ϕ ◦ ( y · θ · n − ) = ( ϕ ( y ) , n ) · a ϕ ◦ θ ; consequently, this element coincideswith (cf. 2.13 and 3.9) X θ ∈ O UR tr Q × N P ( U ) N Q × P (∆ ϕ ◦ θ ( U )) (cid:16)(cid:0) ab ( ϕ θ ) ◦ ab c ( ϕ θ ) (cid:1) ( a ϕ ◦ θ ) (cid:17) . . θ ∈ O UR we have (cid:0) ab ( ϕ θ ) ◦ ab c ( ϕ θ ) (cid:1) ( a ϕ ◦ θ ) = (cid:12)(cid:12) ¯ N Q × P (cid:0) ∆ ϕ ◦ θ ( U ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1)(cid:12)(cid:12) · a ϕ ◦ θ . . (cid:12)(cid:12) ¯ N Q × P (cid:0) ∆ ϕ ◦ θ ( U ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1)(cid:12)(cid:12) and the termtr Q × N P ( U ) N Q × P (∆ ϕ ◦ θ ( U )) (cid:16)(cid:0) ab ( ϕ θ ) ◦ ab c ( ϕ θ ) (cid:1) ( a ϕ ◦ θ ) (cid:17) . . p m +1 · ˜ h ◦ ( ab c ◦ n U F )( Q ) , or we have ϕ ( R θ ) = Q ϕ ◦ θ (cf. 3.10.2). † In [9, Proposition 8.9] the statement and the proof are far from correction η ∈ F ( Q, U ) and any element ( u, n ) in Q × N P ( U ) we stillhave a u · η · n − = ( u, n ) · a η ; consequently, for any η in the set ˜ ϕ ◦ F ( R, U ) , setting T Q × N P ( U ) ( η, ϕ ◦ θ ) = { ( u, n ) ∈ Q × N P ( U ) | η = ( u, n ) · ( ϕ ◦ θ ) } . . , in the second case we havetr Q × N P ( U ) N Q × P (∆ ϕ ◦ θ ( U )) (cid:16)(cid:0) ab ( ϕ θ ) ◦ ab c ( ϕ θ ) (cid:1) ( a ϕ ◦ θ ) (cid:17) = X η ∈ ˜ ϕ ◦F ( R,U ) |T Q × N P ( U ) ( η, ϕ ◦ θ ) | (cid:12)(cid:12) N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:12)(cid:12) · a η . . . Moreover, for any element u in the transporter T Q (cid:0) ϕ ( R ) , η ( U ) (cid:1) (cf. 2.2), thefollowing diagram R ϕ ′ ∼ = u − ϕ ( R ) θ ′ ↑ ∪ U η ∼ = η ( U ) 3 . . ϕ ′ ∈ ˜ ϕ and by θ ′ in the { } × N P ( U )-orbit of θ such that η = ϕ ′ ◦ θ ′ and therefore it is quite clear that T Q × N P ( U ) ( η, ϕ ◦ θ ) = T Q (cid:0) ϕ ( R ) , η ( U ) (cid:1) × N P ( U ) 3 . . . Finally, note that the map ϕ ( R ) \T Q (cid:0) ϕ ( R ) , η ( U ) (cid:1) −→ ϕ ( R ) \ Q/η ( U ) 3 . . u ∈ T Q (cid:0) ϕ ( R ) , η ( U ) (cid:1) to ϕ ( R ) uη ( U ) is injective and itsimage is the set of double classes of cardinal equal to | ϕ ( R ) | , so that p divides (cid:12)(cid:12) ϕ ( R ) \T Q (cid:0) ϕ ( R ) , η ( U ) (cid:1)(cid:12)(cid:12) ; in conclusion, p also divides the quotient |T Q × N P ( U ) ( η, ϕ ◦ θ ) | (cid:14)(cid:12)(cid:12) N Q × P (cid:0) ∆ η ( U ) (cid:1)(cid:12)(cid:12) · Consequently, in both cases we obtain (cid:0) (˜ r U, ◦F ,m ) c ( ˜ ϕ ) ◦ ˜ r U, ◦F ,m ( ˜ ϕ ) (cid:1) (¯ a m ) = 0 3 . . . In order to show condition A5.1.3 in [11], for any pair of ˜ F -morphisms˜ ϕ : R → Q and ˜ ψ : T → Q we have to prove that˜ r U, ◦F ,m ( ˜ ψ ) ◦ (˜ r U, ◦F ,m ) c ( ˜ ϕ ) = X w ∈ W (˜ r U, ◦F ,m ) c ( ˜ ψ w ) ◦ ˜ r U, ◦F ,m ( ˜ ϕ w ) 3 . . ϕ of ˜ ϕ and ψ of ˜ ψ , and a set ofrepresentatives W ⊂ Q for the set of double classes ϕ ( R ) \ Q/ψ ( T ) , for any w ∈ W we set S w = ϕ ( R ) w ∩ ψ ( T ) and denote by ϕ w : S w −→ R and ψ w : S w −→ T . . F -morphisms fulfilling ϕ (cid:0) ϕ w ( u ) (cid:1) = wuw − and ψ (cid:0) ψ w ( u ) (cid:1) = u for anyelement u ∈ S w . θ ∈ F ( R, U ) , let b θ be an element of p m · ab (cid:16) ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1)(cid:17) and denote by ¯ b θ the image of b θ in ( ab ◦ n U F )( R ) R × N P ( U ) (cf. 3.7.2); thus, tr mR (¯ b θ ) is an element of ˜ r ◦ m ( R ) (cf. 3.10.6) and we have to compute (cf. 3.11.1) (cid:0) ˜ r U, ◦F ,m ( ˜ ψ ) ◦ (˜ r U, ◦F ,m ) c ( ˜ ϕ ) (cid:1)(cid:0) tr mR (¯ b θ ) (cid:1) = (cid:0) ˜ r U, ◦F ,m ( ˜ ψ ) ◦ tr mQ ◦ ˜ r U,m F , ◦ ( ˜ ϕ ) (cid:1) (¯ b θ ) 3 . . r U, ◦F ,m ( T ) of the element (cf. 3.9.2) a = ˜ h ◦ ( ab c ◦ n U F )( ˜ ψ ) (cid:18) tr Q (cid:16) ˜ h ◦ ( ab ◦ n U F )( ˜ ϕ )( b θ ) (cid:17)(cid:19) = ( ab c ◦ n U F )( ψ ) (cid:16) tr Q × N P ( U ) N Q × P (∆ η ( U )) (cid:0) ab ( ϕ θ )( b θ ) (cid:1)(cid:17) . . h ◦ ( ab ◦ n U F )( ˜ ϕ )( b θ ) denotes the image of ˜ h ◦ ( ab ◦ n U F )( ˜ ϕ )( b θ ) in thequotient ( ab ◦ n U F )( Q ) Q × N P ( U ) and we set η = ϕ ◦ θ . Then, as in 3.10 above, denoting by Q η the converse image of the inter-section F Q (cid:0) η ( U ) (cid:1) ∩ (cid:0) η ∗ ◦ F P ( U ) ◦ η − ∗ (cid:1) in N Q (cid:0) η ( U ) (cid:1) and by Q η ⊂ Q a set ofrepresentatives for Q/Q η , we have (cf. 3.10.4)tr Q × N P ( U ) N Q × P (∆ η ( U )) (cid:0) ab ( ϕ θ )( b θ ) (cid:1) = X u ∈ Q η X ν ∈F P ( U ) ( u, ν ) · ab ( ϕ θ )( b θ ) 3 . . u ∈ Q η and any ν ∈ F P ( U ) , the element ( u, ν ) · ab ( ϕ θ )( b θ )belongs to p m · ab (cid:16) ¯ N Q × P (cid:0) ∆ u · η ◦ ν − ( U ) (cid:1)(cid:17) and therefore it follows from defini-tion 3.6.2 that the element (cf. 3.6) a u,ν = ( ab c ◦ n U F )( ψ ) (cid:0) ( u, ν ) · ab ( ϕ θ )( b θ ) (cid:1) = (1 , ν ) · ( ab c ◦ n U F )( κ Q,u − ◦ ψ ) (cid:0) ab ( ϕ θ )( b θ ) (cid:1) . . κ Q,u − ◦ ψ )( T ) contains η ( U ) , so that there is a unique ζ u ∈ F ( T, U ) fulfilling κ Q,u ◦ η = ψ ◦ ζ u ; in this case, setting ψ u = κ Q,u − ◦ ψ we get a u,ν = (1 , ν ) · ab c (cid:0) ( ψ u ) ζ u (cid:1)(cid:0) ab ( ϕ θ )( b θ ) (cid:1) . . Q η ⊂ Q η the subset of u ∈ Q η fulfilling this condition.In this situation, note that we have the two injective group homomor-phisms ¯ N Q × P (cid:0) ∆ η ( U ) (cid:1) ϕ θ ր տ ( ψ u ) ζu ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1) ¯ N T × P (cid:0) ∆ ζ u ( U ) (cid:1) . . ab c (cf. 2.17) is a Mackey complement of ab , for any u ∈ Q η weneed to consider the set of double classes X u = N ϕ ( R ) × P (cid:0) ∆ η ( U ) (cid:1) \ N Q × P (cid:0) ∆ η ( U ) (cid:1) /N ψ u ( T ) × P (cid:0) ∆ η ( U ) (cid:1) . . (cid:0) ϕ ( R ) ∩ Q η (cid:1)(cid:15) Q η (cid:14)(cid:0) ψ u ( T ) ∩ Q η (cid:1) ; hence,choosing a set X u ⊂ Q η of representatives for this last set of double classes,we get ab c (cid:0) ( ψ u ) ζ u (cid:1) ◦ ab ( ϕ θ ) = X x ∈ X u ab ( ψ η,u,x ) ◦ ab c ( ϕ θ,u,x ) 3 . . x ∈ X u we set S u,x = ϕ ( R ) x ∩ ψ ( T ) u and denote by ϕ θ,u,x : ¯ N S u,x × P (cid:0) ∆ η ( U ) (cid:1) −→ ¯ N R × P (cid:0) ∆ θ ( U ) (cid:1) ψ η,u,x : ¯ N S u,x × P (cid:0) ∆ η ( U ) (cid:1) −→ ¯ N T × P (cid:0) ∆ ζ u ( U ) (cid:1) . . F -morphisms fulfilling (cf. 3.10.2) ϕ θ (cid:0) ϕ θ,u,x ( s, n ) (cid:1) = (cid:0) x s, ˆ x n ) (cid:1) = ( x, ˆ x ) · ( s, n )( ψ u ) ζ u (cid:0) ψ η,u,x ( s, n ) (cid:1) = ( s, n ) 3 . . s, n ) ∈ N S u,x × P (cid:0) ∆ η ( U ) (cid:1) and for a choice of ˆ x ∈ Q η lifting ν η ( x ) (cf. 3.10.2); note that the element ( x, ˆ x ) ∈ Q η × P normalizes ∆ η ( U ) . Hence, from 3.11.15, 3.11.18, 3.11.19 and 3.11.21 we obtain a = X ν ∈F P ( U ) (1 , ν ) · X u ∈ ˆ Q η X x ∈ X u (cid:0) ab ( ψ η,u,x ) ◦ ab c ( ϕ θ,u,x ) (cid:1) ( b θ ) 3 . . . On the other hand, we have to prove that the element (cf. 3.11.12)¯ c = (cid:0) X w ∈ W (˜ r U, ◦F ,m ) c ( ˜ ψ w ) ◦ ˜ r U, ◦F ,m ( ˜ ϕ w ) (cid:1)(cid:0) tr mR (¯ b θ ) (cid:1) . . a . But, according to 3.9, tr mR (¯ b θ ) is the image in ˜ r U, ◦F ,m ( R ) oftr R × N P ( U ) N R × P (∆ θ ( U )) ( b θ ) = X y ∈ R θ X ν ∈F P ( U ) ( y, ν ) · b θ . . p m · ( ab c ◦ n U F )( R ) R × N P ( U ) (cf. 3.10.4) where, denoting by θ ∗ : U ∼ = θ ( U ) theisomorphism induced by θ and by R θ the converse image of the intersection F R (cid:0) θ ( U ) (cid:1) ∩ (cid:0) θ ∗ ◦F P ( U ) ◦ θ − ∗ (cid:1) in N R (cid:0) θ ( U ) (cid:1) , R θ ⊂ R is a set of representativesfor R/R θ . Note that, according to 3.10 we have R θ = ϕ − ( Q η ) . w ∈ W , the element ˜ r U, ◦F ,m ( ˜ ϕ w ) (cid:0) tr mR (¯ b θ ) (cid:1) is theimage in ˜ r U, ◦F ,m ( S w ) of the element (cf. 3.10.5) d w = X y ∈ R θ X ν ∈F P ( U ) ( ab c ◦ n U F )( ϕ w ) (cid:0) ( y, ν ) · b θ (cid:1) . . y ∈ R θ and any ν ∈ F P ( U ) , the element ( y, ν ) · b θ belongs to p m · ab (cid:16) ¯ N R × P (cid:0) ∆ y · θ ◦ ν − ( U ) (cid:1)(cid:17) and therefore it follows from defini-tion 3.6.2 that the element d w,y,ν = ( ab c ◦ n U F )( ϕ w ) (cid:0) ( y, ν ) · b θ (cid:1) . . ϕ w ( S w ) contains y θ ( U ) , so that there is a unique ξ w,y ∈ F ( S w , U ) fulfilling κ R,y ◦ θ = ϕ w ◦ ξ w,y , which forces the equality κ Q,w − ϕ ( y ) ◦ η = ι QS w ◦ ξ w,y ; in this case, we have d w,y,ν = (1 , ν ) · ab c (cid:0) ( κ R,y − ◦ ϕ w ) ξ w,y (cid:1) ( b θ ) 3 . . R θ,w ⊂ R θ the subset of y ∈ R θ fulfilling this condition;thus, we get d w = X ν ∈F P ( U ) (1 , ν ) · X y ∈ ˆ R θ,w ab c (cid:0) ( κ R,y − ◦ ϕ w ) ξ w,y (cid:1) ( b θ ) 3 . . . Moreover, for any y ∈ R θ fulfilling the above condition and any s ∈ S w , the product ϕ w ( s ) · y still fulfills this condition and we have ϕ w ◦ ξ w,ϕ w ( s ) · y = κ R,ϕ w ( s ) · y ◦ θ = κ R,ϕ w ( s ) ◦ κ R,y ◦ θ = κ R,ϕ w ( s ) ◦ ϕ w ◦ ξ w,y = ϕ w ◦ κ S w ,s ◦ ξ w,y . . , so that we get ξ w,ϕ w ( s ) · y = κ S w ,s ◦ ξ w,y ; in particular, ϕ w ( S w ) has an actionon ˆ R θ,w and, choosing a set of representatives ˆ Y θ,w ⊂ ˆ R θ,w for the set of ϕ w ( S w )-orbits, the element d w above is also equal to X ν ∈F P ( U ) (1 , ν ) · X y ∈ ˆ Y θ,w X s ∈ S θ,yw ab c (cid:0) ( κ R, ( ϕ w ( s ) · y ) − ◦ ϕ w ) ξ w,ϕw ( s ) · y (cid:1) ( b θ ) 3 . . y ∈ ˆ Y θ,w , setting S w,θ,y = ϕ − w (cid:0) ( R θ ) y (cid:1) , S θ,yw ⊂ S w is a set ofrepresentatives for S w /S w,θ,y ; but, it is quite clear that ab c (cid:0) ( κ R, ( ϕ w ( s ) · y ) − ◦ ϕ w ) ξ w,ϕw ( s ) · y (cid:1) = ab c (cid:0) ( κ R,y − ◦ ϕ w ◦ κ S w ,s − ) ξ w,ϕw ( s ) · y (cid:1) = ab (cid:0) ( κ S w ,s ) ξ w,y (cid:1) ◦ ab c (cid:0) ( κ R,y − ◦ ϕ w ) ξ w,y (cid:1) . . ϕ yw = κ R,y − ◦ ϕ w and denoting by ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1) ( b θ ) the im-age of ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1) ( b θ ) in the quotient ( ab ◦ n U F )( S w ) S w × N P ( U ) , accordingto 3.10.4 we easily obtain X ν ∈F P ( U ) (1 , ν ) · X s ∈ S θ,yw (cid:16) ab c (cid:0) ( κ R, ( ϕ w ( s ) · y ) − ◦ ϕ w ) ξ w,ϕw ( s ) · y (cid:1)(cid:17) ( b θ )= tr mS w (cid:16) ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1) ( b θ ) (cid:17) . . . Consequently, it follows from definition 3.11.1 that we have (cf. 3.11.25)¯ c = X w ∈ W (cid:0) tr mT ◦ ˜ r m ◦ ( ˜ ψ w ) (cid:1)(cid:16) X y ∈ ˆ Y θ,w ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1) ( b θ ) (cid:17) . . r m ◦ ( T ) of the element X w ∈ W X y ∈ ˆ Y θ,w tr T × N P ( U ) N T × P (∆ ζw,y ( U )) (cid:18)(cid:16) ab (cid:0) ( ψ w ) ξ w,y (cid:1) ◦ ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1)(cid:17) ( b θ ) (cid:19) k X w ∈ W X y ∈ ˆ Y θ,w X ν ∈F P ( U ) (1 , ν ) · X z ∈ Z w,y (cid:18)(cid:16) ab (cid:0) ( z ψ w ) ξ w,y (cid:1) ◦ ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1)(cid:17) ( b θ ) (cid:19) k X ν ∈F P ( U ) (1 , ν ) · X w ∈ W X y ∈ ˆ Y θ,w X z ∈ Z w,y (cid:18)(cid:16) ab (cid:0) ( z ψ w ) ξ w,y (cid:1) ◦ ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1)(cid:17) ( b θ ) (cid:19) . . p m · ( ab ◦ n U F )( T ) T × N P ( U ) where, for any w ∈ W and any y ∈ ˆ Y θ,w , set-ting ζ w,y = ψ w ◦ ξ w,y and denoting by ( ζ w,y ) ∗ : U ∼ = ζ w,y ( U ) the isomor-phism induced by ζ w,y and by T w,y the converse image of the intersection F T (cid:0) ζ w,y ( U ) (cid:1) ∩ (cid:0) ( ζ w,y ) ∗ ◦ F P ( U ) ◦ ( ζ w,y ) − ∗ (cid:1) in N T (cid:0) ζ w,y ( U ) (cid:1) , we choose asabove a set of representatives Z w,y for T /T w,y and, for any z ∈ Z w,y , weset z ψ w = κ T,z ◦ ψ w . Finally, we claim that this element ¯ c coincides with a in 3.11.24 above;that is to say, considering the sets X = G u ∈ ˆ Q η { u } × X u and Z = G w ∈ W { w } × G y ∈ ˆ Y θ,w { y } × Z w,y . . , in p m · ( ab ◦ n U F )( T ) T × N P ( U ) we have to prove the equality X ( u,x ) ∈ X X ν ∈F P ( U ) (1 , ν ) · (cid:0) ab ( ψ η,u,x ) ◦ ab c ( ϕ θ,u,x ) (cid:1) ( b θ )= X ( w,y,z ) ∈ Z X ν ∈F P ( U ) (1 , ν ) · (cid:18)(cid:16) ab (cid:0) ( z ψ w ) ξ w,y (cid:1) ◦ ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1)(cid:17) ( b θ ) (cid:19) . . X and Z such that the cor-responding terms in both sums coincide with each other.Indeed, for any w ∈ W , any y ∈ ˆ Y θ,w and any z ∈ Z w,y let us considerthe element ϕ ( y ) − wψ ( z ) − of Q ; this element certainly belongs to Q η u − for some u in Q η so that we have ϕ ( y ) − wψ ( z ) − = vu − for some v in Q η ;but, since y belongs to ˆ Y w , ϕ w ( S w ) contains y θ ( U ) and therefore w S w con-tains ϕ ( y ) η ( U ) (cf. 3.11.13); thus, ψ ( z ) w − ϕ ( y ) η ( U ) is contained in ψ ( T ) and,since Q η ⊂ N Q (cid:0) η ( U ) (cid:1) (cf. 3.9), u η ( U ) = ψ ( z ) w − ϕ ( y ) v η ( U ) is also containedin ψ ( T ) , so that u belongs to ˆ Q η (cf. 3.11.18). Moreover, the double classof v in (cid:0) ϕ ( R ) ∩ Q η (cid:1)(cid:15) Q η (cid:14)(cid:0) ψ u ( T ) ∩ Q η (cid:1) determines an element x in X u suchthat we have v = ϕ ( r ) xu − ψ ( t ) u for some r ∈ R θ and some t ∈ T fulfilling ψ ( t ) u ∈ Q η , so that we get ϕ ( y ) − wψ ( z ) − = ϕ ( r ) xu − ψ ( t ) 3 . . . Thus, we obtain a map from Z to X sending ( w, y, z ) to ( u, x ) . Moreover, with the same notation, setting q = ψ ( tz ) − u = w − ϕ ( yr ) x . . , it is clear that the automorphism κ Q,q of Q (cf. 3.6) maps S u,x onto S w inducing a group isomorphism χ : S u,x ∼ = S w ; hence, since we have (cf. 3.10.3) κ Q,q ◦ η = κ Q,w − ϕ ( y ) ◦ η ◦ ν η (cid:0) ϕ ( r ) x (cid:1) = ι QS w ◦ ξ w,y ◦ ν η (cid:0) ϕ ( r ) x (cid:1) . . , we get the group isomorphism (cf. 3.5.4) χ η : ¯ N S u,x × P (cid:0) ∆ η ( U ) (cid:1) ∼ = ¯ N S w × P (cid:0) ∆ ξ w,y ◦ ν η ( ϕ ( r ) x ) ( U )) 3 . . . Then, we claim that (cid:0) , ν η ( ϕ ( r ) x ) − (cid:1) · ab c ( ϕ θ,u,x ) = ab c ( χ η ) ◦ ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1)(cid:0) , ν η ( ϕ ( r ) xψ u ( t ) (cid:1) · ab ( ψ η,u,x ) = ab (cid:0) ( z ψ w ) ξ w,y (cid:1) ◦ ab ( χ η ) 3 . . . Indeed, for any ( s, n ) ∈ N S u,x × P (cid:0) ∆ η ( U ) (cid:1) it is easily checked that wehave (cf. 3.11.23) (cid:0) ϕ θ ◦ ν η ( ϕ ( r ) x ) ◦ ( ϕ yw ) ξ w,y ◦ ν η ( ϕ ( r ) x ) ◦ χ η (cid:1) ( s, n )= (cid:0) ϕ θ ◦ ν η ( ϕ ( r ) x ) ◦ ( ϕ yw ) ξ w,y ◦ ν η ( ϕ ( r ) x ) (cid:1) ( q s, n )= ( ϕ ( r ) x s, n ) = ( ϕ ( r ) x, id P ) · ( s, n )= ( ϕ ( r ) , ν η ( x ) − ) · (cid:0) ϕ θ ◦ ϕ θ,u,x (cid:1) ( s, n )= (cid:0) ϕ θ ◦ ν η ( ϕ ( r ) x ) ◦ ϕ θ,u,x (cid:1) ( s, n ) 3 . . κ R,r ◦ θ = θ ◦ ν θ ( r ) = θ ◦ ν η (cid:0) ϕ ( r ) (cid:1) ; thus,since the homomorphism ϕ θ ◦ ν η ( ϕ ( r ) x ) is injective, we get( ϕ yw ) ξ w,y ◦ ν η ( ϕ ( r ) x ) ◦ χ η = ϕ θ,u,x . . ϕ yw ) ξ w,y ◦ χ η = (cid:0) , ν η ( ϕ ( r ) x ) (cid:1) · ϕ θ,u,x . . . Similarly, since q = (cid:0) ψ ( z ) − u (cid:1) ψ u ( t ) − , we have (cf. 3.11.23) (cid:0) ( ψ u ) z ζ w,y ◦ ν η ( ϕ ( r ) x ) ◦ ( z ψ w ) ξ w,y ◦ ν η ( ϕ ( r ) x ) ◦ χ η (cid:1) ( s, n )= (cid:0) ( ψ u ) z ζ w,y ◦ ν η ( ϕ ( r ) x ) ◦ ( z ψ w ) ξ w,y (cid:1) ( q s, n )= ( ψ u ( t − ) s, n )= (cid:0) ψ u ( t − ) , id P (cid:1) · (cid:0) ( ψ u ) ζ u ◦ ψ η,u,x (cid:1) ( s, n )= (cid:0) ( ψ u ) ζ tu ◦ (cid:0) t − , id P (cid:1) · ψ η,u,x (cid:1) ( s, n ) 3 . . ζ u ∈ F ( T, U ) is the unique element fulfilling η = ψ u ◦ ζ u ; but, it is easily checked that ζ tu = z ζ w,y ◦ ν η ( ϕ ( r ) x ) ; thus, sincethe homomorphism ( ψ u ) ζ tu is injective, we get( z ψ w ) ξ w,y ◦ ν η ( ϕ ( r ) x ) ◦ χ η = (cid:0) t − , id P (cid:1) · ψ η,u,x . . z ψ w ) ξ w,y ◦ χ η = (cid:0) t − , ν η ( ϕ ( r ) x ) (cid:1) · ψ η,u,x . . ψ u ( t ) ∈ Q η , in 3.10.3 above (cid:0) t, ν η ( ψ u ( t ) (cid:1) is the image of anelement of N T × P (cid:0) ∆ ζ u ( U ) (cid:1) ; hence, it acts trivially over ab (cid:16) ¯ N T × P (cid:0) ∆ ζ u ( U ) (cid:1)(cid:17) and therefore we obtain ab (cid:0) ( z ψ w ) ξ w,y (cid:1) ◦ ab (cid:0) χ η (cid:1) = (cid:0) , ν η ( ϕ ( r ) xψ u ( t ) (cid:1) · ab (cid:0) ψ η,u,x (cid:1) . . . Finally, since ab c ( χ η ) = ab ( χ η ) − , the composition of both equalitiesin 3.11.43 yields (cid:0) , ν η ( ψ u ( t ) (cid:1) · ab ( ψ η,u,x ) ◦ ab c ( ϕ θ u,x )= ab (cid:0) ( z ψ w ) ξ w,y (cid:1) ◦ ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1) . . X ν ∈F P ( U ) (cid:16) ab ( ψ η,u,x ) ◦ ab c ( ϕ θ u,x ) (cid:17) ( b θ )= X ν ∈F P ( U ) (cid:16) ab (cid:0) ( z ψ w ) ξ w,y (cid:1) ◦ ab c (cid:0) ( ϕ yw ) ξ w,y (cid:1)(cid:17) ( b θ ) 3 . . . u ∈ ˆ Q η and any x ∈ X u let us consider the element w ∈ W determined by the double class of xu − in ϕ ( R ) \ Q/ψ ( T ) , so that wehave xu − = ϕ ( y ) − wψ ( z ) 3 . . y ∈ R and z ∈ T ; then, with the notation above, we claim that ϕ w ( S w ) contains y θ ( U ) or, equivalently, that w S w = ϕ ( R ) ∩ w ψ ( T ) contains ϕ ( y ) η ( U ) . Indeed, since θ ( U ) is contained in R , it is clear that ϕ ( y ) η ( U ) iscontained in ϕ ( R ) ; it remains to prove that η ( U ) is contained in ϕ ( y ) − w ψ ( T )or, equivalently, in xu − ψ ( T ) ; but, x normalizes η ( U ) and η = ψ u ◦ ζ u , sothat η ( U ) is contained in ψ u ( T ) ; this proves the claim.Consequently, from the very definitions of ˆ R θ,w , of ˆ Y θ,w and of ˆ S θ, ˆ yw above, we actually have y = ϕ w ( s )ˆ yr for a unique ˆ y ∈ ˆ Y θ,w , a unique s ∈ ˆ S θ, ˆ yw and a unique r ∈ R θ ; now, the equality 3.11.40 becomes ϕ ( r ) xu − = ϕ (ˆ y ) − ( ϕ ◦ ϕ w )( s − ) wψ ( z )= ϕ (ˆ y ) − ws − ψ ( z ) 3 . . s ∈ S w ⊂ ψ ( T ) , there exist a unique ˆ z ∈ Z w, ˆ y and a unique t ∈ T w, ˆ y fulfilling s − ψ ( z ) = ψ (ˆ zt − ) , so that equality 3.11.55 becomes ϕ ( r ) xu − ψ ( t ) = ϕ (ˆ y ) − wψ (ˆ z ) 3 . . . Thus, we obtain a map from X to Z sending ( u, x ) to ( w, ˆ y, ˆ z ) which is clearlythe inverse of the map from Z to X defined above. We are done.3.12. For the next result, we borrow the notation from A5 in [11]. Recallthat in 3.10.5 above, for any m ∈ N we actually define the functors˜ r U, ◦F ,m : ˜ F −→ O - mod ◦ and ˜ r U,m F , ◦ : ˜ F −→ O - mod . . . Corollary 3.13.
Let ˜ G be a subcategory of ˜ F having the same objects, onlyhaving ˜ G -isomorphisms and containing all the ˜ F P -isomorphisms. Then, withthe notation above, for any m ∈ N and any n ≥ we have H n ˜ G ( ˜ F , ˜ r U, ◦F ,m ) = { } . Proof:
It is an immediate consequence of Theorems 3.11 above and TheoremA5.5 in [11].
4. Existence and uniqueness of the perfect F -locality P be a finite p -group, F a Frobenius P -categoryand ( τ b , L b , π b ) the corresponding basic F -locality . Recall that we have a contravariant functor [8, Proposition 13.14] c f F : F −→ Ab . . Q of P fully centralized in F on C P ( Q ) /F C F ( Q ) ,where F C F ( Q ) denotes the C F ( Q ) -focal subgroup of C P ( Q ) [8, 13.1], and any F -morphism ϕ : R → Q between subgroups of P fully centralized in F , onthe group homomorphism C P ( Q ) /F C F ( Q ) −→ C P ( R ) /F C F ( R ) . . F -morphism [8, 2.8.2] ζ : ϕ ( R ) · C P ( Q ) −→ R · C P ( R ) 4 . . ζ (cid:0) ϕ ( v ) (cid:1) = v for any v ∈ R .
Actually, it is easily checked that this contravariant functor factorizes through the exterior quotient ˜ F inducing anew contravariant functor ˜ c f F : ˜ F −→ Ab . . Proposition 4.2.
The structural functor τ b : T P → L b induces a naturalmap ˆ τ b from ˜ c f F to ˜ k b F . Proof:
For any subgroup Q of P , the functor τ b induces a group homomor-phism τ b Q from N P ( Q ) to L b ( Q ) which clearly maps C P ( Q ) in (cid:0) Ker ( π b ) (cid:1) ( Q );we claim that this correspondence defines a natural map (cf. 3.1.2)ˆ τ b : ˜ c f F −→ ˜ k b F . . . First of all, we claim that τ b Q maps the C F ( Q ) -focal subgroup above onthe trivial subgroup of L b ( Q ); we may assume that Q is fully centralized in F and then we know that F C F ( Q ) is generated by the elements u − θ ( u ) where u runs over any subgroup T of C P ( Q ) and θ runs over F ( T.Q ) stabilizing T and acting trivially on Q [8, 13.1]; but, according to 2.12 above, θ can belifted to ˆ z ∈ N G ( T.Q ) normalizing T and centralizing Q ; hence, the element u − θ ( u ) = [ u, ˆ z − ] belongs to [ C G ( Q ) , C G ( Q )] and therefore it has indeed atrivial image in L b ( Q ) ; consequently, the canonical homomorphism C P ( Q ) ⊂ C G ( Q ) → Ker( π Q ) 4 . . τ b Q : ˜ c f F ( Q ) → Ker( π b Q ) . In order to prove the naturality of this correspondence, let x : R → Q be an L b -morphism between subgroups of P fully centralized in F and set ϕ = π b Q,R ( x ) ; it follows from [8, 2.8.2] that there exists an F -morphism ζ from ϕ ( R ) · C P ( Q ) to R · C P ( R ) fulfilling ζ (cid:0) ϕ ( v ) (cid:1) = v for any v ∈ R ; then, ζ can be lifted to an L b -morphism y : ϕ ( R ) · C P ( Q ) −→ R · C P ( R ) 4 . . (cid:0) π b R.C P ( R ) ,ϕ ( R ) · P ( Q ) ( y ) (cid:1)(cid:0) ϕ ( v ) (cid:1) = v for any v ∈ R ; in particular, bythe divisibility of L b , y induces an L b -isomorphism y R : ϕ ( R ) ∼ = R and then,setting xy R = z , the L b -morphism z : ϕ ( R ) → Q fulfills π b Q,ϕ ( R ) ( z ) = ι Qϕ ( R ) (cf. 2.4); consequently, we easily get the following commutative diagram˜ c f F ( Q ) ˆ τ b Q −→ Ker( π Q ) ˜ c f F ( ˜ ϕ ) y y ˜ k b F ( ˜ ϕ ) ˜ c f F ( R ) ˆ τ b R −→ Ker( π R ) 4 . . . We are done.4.3. The image ˆ τ b (˜ c f F ) of ˆ τ b is a subfunctor of ˜ k b F and therefore, by 2.8above, it determines a quotient F -locality f L b = L b (cid:14)(cid:0) ˆ τ b (˜ c f F ) ◦ ˜ π b (cid:1) of L b (cf. 2.3); we denote by f τ b : T P −→ f L b and f π b : f L b −→ F . . f π b admits an essen-tially unique section as proves the theorem below. First of all, we need thefollowing lemma. Lemma 4.4.
For any subgroup Q of P there is a group homomorphism µ Q : F ( Q ) → f L b ( Q ) fulfilling µ Q ◦ κ Q = f τ b Q . Proof:
Since we can choose an F -isomorphism θ : Q ∼ = Q ′ such that Q ′ isfully normalized in F and θ can be lifted to f L b ( Q ′ , Q ) , we may assume that Q is fully normalized in F . We apply [8, Lemma 18.8] to the groups F ( Q ) and f L b ( Q ) , to the normal p -subgroup Ker( f π b Q ) of f L b ( Q ) and to the group homomorphism id F ( Q ) . Weconsider the group homomorphism f τ b Q : N P ( Q ) → f L b ( Q ) and, for any sub-group R of N P ( Q ) and any α ∈ F ( Q ) such that α ◦ F R ( Q ) ◦ α − ⊂ F P ( Q ) , it follows from [8, Proposition 2.11] that there exists ζ ∈ F (cid:0) N P ( Q ) , Q · R (cid:1) extending α ; then, it follows from [8, 17.11.2] that there exists x ∈ f L b ( Q )fulfilling f τ b Q (cid:0) ζ ( v ) (cid:1) = x f τ b Q ( v ) 4 . . v ∈ R .
That is to say, condition 18.8.1 in [8, Lemma 18.8] is fulfilledand therefore this lemma proves the existence of µ Q as announced. Theorem 4.5.
With the notation above, the structural functor f π b admits aunique natural F -isomorphism class of F -locality functorial sections. Proof:
We consider the filtration of f L b induced by the filtration of the basic F -locality introduced in section 3 and then argue by induction. That is to say,recall that we denote by C P a set of representatives for the set of P -conjugacyclasses of subgroups U of P (cf. 2.13); now, for any subset N of C P fulfillingcondition 3.2.1, we consider the obvious functor (cf. 3.2)ˆ τ b (˜ c f F ) · ˜ k N F : ˜ F −→ Ab . . Q of P to ˆ τ b Q (cid:0) ˜ c f F ( Q ) (cid:1) . ˜ k N F ( Q ) , and the quotient F -lo-cality ] L b , N = L b (cid:14)(cid:0) ˆ τ b (˜ c f F ) · ˜ k N F ◦ ˜ π b (cid:1) with the structural functors g τ b , N : T P −→ ] L b , N and g π b , N : ] L b , N −→ F . . . Note that if N = ∅ then ] L b , N = f L b ; hence, arguing by induction on |C P −N | , it suffices to prove that g π b , N admits a unique natural F -isomorphism classof F -locality functorial sections.Moreover, if N = C P then ˜ k N F = ˜ k b F ; therefore ] L b , N = F and g π b , N = id F , so that we may assume that N 6 = C P ; then, we fix a minimal element U in C P − N , setting M = N ∪ { U } and ˜ k U F = ˜ k M F / ˜ k N F . If U = P then M 6 = C P and, as a matter of fact, we have ˆ τ b (˜ c f F ) ∩ ˜ k M F = { } , so that (cid:0) ˆ τ b (˜ c f F ) · ˜ k M F (cid:1)(cid:14)(cid:0) ˆ τ b (˜ c f F ) · ˜ k N F (cid:1) ∼ = ˜ k U F . . m ∈ N we simply denote by ˜ l U,m F the converse imageof p m · ˜ k U F in ˆ τ b (˜ c f F ) · ˜ k M F ; set ^ L b ,U,m = L b / ˜ l U,m F and, coherently, denote by ^ π b ,U,m and ^ τ b ,U,m the corresponding structural functors. Note that, by 3.8and 3.10.5 above we get ˜ l U,m F / ˜ l U,m +1 F ∼ = ˜ r U, ◦F ,m . . n ∈ N we still get H n ∗ ( ˜ F , ˜ l U,m F / ˜ l U,m +1 F ) = { } . . . If U = P then M = C P , so that in this case ˜ k M F = ˜ k b F and, denoting by d P : ˜ F → Ab the functor mapping P on Z ( P ) and any other subgroup of P on { } , from 3.7 and 3.8 it is easily checked that˜ k b F (cid:14)(cid:0) ˆ τ b (˜ c f F ) · ˜ k C P −{ P } F (cid:1) ∼ = Y ˜ σ ∈ ˜ F ( P ) d P . ∆( d P ) 4 . . diagonal map ; but, similarly we have˜ r P, ◦F ,m ∼ = Y ˜ σ ∈ ˜ F ( P ) s m ◦ d P . . H n F P ( ˜ F , s m ◦ d P ) = { } ; moreover,since p does not divide | ˜ F ( P ) | , we still have Y ˜ σ ∈ ˜ F ( P ) d P . ∆( d P ) ∼ = Y ˜ σ ∈ ˜ F ( P ) −{ e id P } d P . . l P,m F = p m · ˜ k b F and ^ L b ,P,m = L b / ˜ l P,m F , we still get H n ∗ ( ˜ F , ˜ l P,m F / ˜ l P,m +1 F ) = { } . . . Further, we denote by C F a set of representatives, fully normalized in F , for the F -isomorphism classes of subgroups of P and, for any subgroup Q in C F , we choose a group homomorphism µ Q : F ( Q ) → f L b ( Q ) as in Lemma4.4 above and, for any m ∈ N , simply denote by µ m Q the corresponding grouphomomorphism from F ( Q ) to ^ L b ,U,m ( Q ) . For any F -morphism ϕ : R → Q denote by F ( Q ) ϕ and by ^ L b ,U,m ( Q ) ϕ the respective stabilizers of ϕ ( R ) in F ( Q ) and in ^ L b ,U,m ( Q ) ; it is clear that we have a group homomorphism a ϕ : F ( Q ) ϕ → F ( R ) fulfilling η ◦ ϕ = ϕ ◦ a ϕ ( η ) for any η ∈ F ( Q ) ϕ ; similarly,for any x m ∈ ^ L b ,U,m ( Q, R ) we have a group homomorphism a x m : ^ L b ,U,m ( Q ) ϕ −→ ^ L b ,U,m ( R ) 4 . . y m · x m = x m · a x m ( y m ) for any y m ∈ ^ L b ,U,m ( Q ) ϕ . For any subgroups Q and R in C F and any F -morphism ϕ : R → Q , F P ( Q ) and F P ( R ) are respective Sylow p -subgroups of F ( Q ) and F ( R )[8, Proposition 2.11]; therefore, there are α ∈ F ( Q ) such that F P ( Q ) α contains a Sylow F P ( Q ) αϕ p -subgroup of F ( Q ) ϕ and β ∈ F ( R ) such that a ϕ (cid:0) F P ( Q ) αϕ (cid:1) is contaioned in F P ( R ) β . Thus, we choose a set of representati-ves F Q,R for the set of double classes F ( Q ) \F ( Q, R ) / F ( R ) such that, for any ϕ in F Q,R , F P ( Q ) contains a Sylow p -subgroup of F ( Q ) ϕ and a ϕ (cid:0) F P ( Q ) ϕ (cid:1) is contaioned in F P ( R ) ; of course, we choose F Q,Q = { id Q } . With all this notation and arguing by induction on |C P − N | and on m , we will prove that there is a functorial section σ m : F −→ ^ L b ,U,m . . Q ∈ C F and any u ∈ Q , we have σ m (cid:0) κ Q ( u ) (cid:1) = ^ τ b ,U,m Q ( u ) , and that, for any groups Q and R in C F , and any F -morphism ϕ : Q → R F Q,R , we have the commutative diagram F ( Q ) ϕ µ mQ −→ ^ L b ,U,m ( Q ) ϕa ϕ y y a σm ( ϕ ) F ( R ) µ mR −→ ^ L b ,U,m ( R ) 4 . . . Since we have ^ π b ,U, = ] π b , M and |M| = |N | + 1 , by the induction hypothesiswe actually may assume that m = 0 , that ^ π b ,U,m − admits a functorial section σ m − : F −→ ^ L b ,U,m − . . ϕ ∈ F Q,R it follows from [8, Proposition 2.11], applied tothe inverse ϕ ∗ of the isomorphism ϕ ∗ : R ∼ = ϕ ( R ) induced by ϕ , that thereexists an F -morphism ζ : N P ( Q ) ϕ → N P ( R ) fulfilling ζ (cid:0) ϕ ( v ) (cid:1) = v for any v ∈ R , so that we easily get the following commutative diagram‘ N P ( Q ) ϕ κ Q −→ F ( Q ) ϕζ y y a ϕ N P ( R ) κ R −→ F ( R ) 4 . . Q = R and ϕ = κ Q ( u ) for some u ∈ Q , we may assume that ζ = κ N P ( Q ) ( u ) . In particular, since σ m − fulfills the corresponding commu-tative diagram 4.5.12, we still get the following commutative diagram N P ( Q ) ϕ ^ τ b ,U,m − Q −−−−−−→ ^ L b ,U,m − ( Q ) ϕζ y y a σm − ϕ ) N P ( R ) ^ τ b ,U,m − R −−−−−−→ ^ L b ,U,m − ( R ) 4 . . F -morphism ϕ in F Q,R , to choose a suitablelifting \ σ m − ( ϕ ) of σ m − ( ϕ ) in ^ L b ,U,m ( Q, R ) . We start by choosing a lifting \ σ m − ( ζ ) of σ m − ( ζ ) in the obvious stabilizer ^ L b ,U,m (cid:0) N P ( R ) , N P ( Q ) ϕ (cid:1) R,ϕ ( R ) ;thus, by the coherence of ^ L b ,U,m (cf. (Q)), for any u ∈ N P ( Q ) ϕ we have \ σ m − ( ζ ) · ^ τ b ,U,m N P ( Q ) ϕ ( u ) = ^ τ b ,U,m N P ( R ) (cid:0) ζ ( u ) (cid:1) · \ σ m − ( ζ ) 4 . . divisibility of ^ L b ,U,m (cf. 2.4), we find z ϕ ∈ ^ L b ,U,m (cid:0) R, ϕ ( R ) (cid:1) fulfilling \ σ m − ( ζ ) · ^ τ b ,U,m N P ( Q ) ϕ ,ϕ ( R ) (1) = ^ τ b ,U,m N P ( R ) ,R (1) · z ϕ . . σ m − ( ζ ) resrtricts to σ m − ( ϕ ∗ ) ∈ ^ L b ,U,m − (cid:0) R, ϕ ( R ) (cid:1) , so that itis easily checked that z ϕ lifts σ m − ( ϕ ∗ ) to ^ L b ,U,m (cid:0) R, ϕ ( R ) (cid:1) and therefore ^ σ m − ( ϕ ) = ^ τ b ,U,m Q · ϕ ( R ) (1) z − ϕ lifts σ m − ( ϕ ) to ^ L b ,U,m ( Q, R ) . u ∈ N P ( Q ) ϕ we get \ σ m − ( ζ ) · ^ τ b ,U,m N P ( Q ) ϕ ( u ) · . ^ τ b ,U,m N P ( Q ) ϕ ,ϕ ( R ) (1) = ^ τ b ,U,m N P ( R ) ,R (1) · z ϕ · ^ τ b ,U,m ϕ ( R ) ( u ) k ^ τ b ,U,m N P ( R ) (cid:0) ζ ( u ) (cid:1) · \ σ m − ( ζ ) · . ^ τ b ,U,m N P ( Q ) ϕ ,ϕ ( R ) (1)= ^ τ b ,U,m N P ( R ) ,R (1) · ^ τ b ,U,m R (cid:0) ζ ( u ) (cid:1) · z ϕ . . z ϕ · ^ τ b ,U,m ϕ ( R ) ( u ) = ^ τ b ,U,m R (cid:0) ζ ( u ) (cid:1) · z ϕ , so that ^ τ b ,U,m Q ( u ) · ^ σ m − ( ϕ ) = ^ σ m − ( ϕ ) · ^ τ b ,U,m R (cid:0) ζ ( u ) (cid:1) . . a ^ σ m − ( ϕ ) (cid:0) ^ τ b ,U,m Q ( u ) (cid:1) = ^ τ b ,U,m R (cid:0) ζ ( u ) (cid:1) . At this point, we will apply the uniqueness part of [8, Lemma 18.8] tothe groups F ( Q ) ϕ and ^ L b ,U,m ( R ) and to the composition of group homomor-phisms a σ m − ( ϕ ) ◦ µ m − Q : F ( Q ) ϕ −→ ^ L b ,U,m − ( Q ) ϕ −→ ^ L b ,U,m − ( R ) 4 . . , together with the composition of group homomorphisms ^ τ b ,U,m R ◦ ζ : N P ( Q ) ϕ −→ N P ( R ) −→ ^ L b ,U,m ( R ) 4 . . . Now, according to the commutative diagrams 4.5.12 for m − a ^ σ m − ( ϕ ) ◦ µ m Q : F ( Q ) ϕ −→ ^ L b ,U,m ( Q ) ϕ −→ ^ L b ,U,m ( R ) µ m R ◦ a ϕ : F ( Q ) ϕ −→ F ( R ) −→ ^ L b ,U,m ( R ) 4 . . k ϕ in the kernel of the canonical homomorphism from ^ L b ,U,m ( R ) to ^ L b ,U,m − ( R ) such that, denoting by int ^ L b ,U,m ( R ) ( k ϕ ) the conju-gation by k ϕ in ^ L b ,U,m ( R ) , we haveint ^ L b ,U,m ( R ) ( k ϕ ) ◦ a ^ σ m − ( ϕ ) ◦ µ m Q = µ m R ◦ a ϕ . . ^ L b ,U,m ( R ) ( k ϕ ) ◦ a ^ σ m − ( ϕ ) = a ^ σ m − ( ϕ ) · k − ϕ . . . \ σ m − ( ϕ ) = ^ σ m − ( ϕ ) · k − ϕ , lifting indeed σ m − ( ϕ ) to ^ L b ,U,m ( Q, R ) and, according to equalities 4.5.23 and 4.5.24, fulfilling the fol-lowing commutative diagram F ( Q ) ϕ µ mQ −→ ^ L b ,U,m ( Q ) ϕa ϕ y y a \ σm − ϕ ) F ( R ) µ mR −→ ^ L b ,U,m ( R ) 4 . . Q = R and ϕ = κ Q ( u ) for some u ∈ Q , this choice is com-patible wtih \ σ m − (cid:0) κ Q ( u ) (cid:1) = ^ τ b ,U,m Q ( u ) . In particular, considering the actionof F ( Q ) × F ( R ) , by composition on the left- and on the right-hand, on F ( Q, R ) and on ^ L b ,U,m ( Q, R ) via µ m Q and µ m R , we have the inclusion of sta-bilizers (cid:0) F ( Q ) × F ( R ) (cid:1) ϕ ⊂ (cid:0) F ( Q ) × F ( R ) (cid:1) σ m − ( ϕ ) . . α, β ) ∈ (cid:0) F ( Q ) × F ( R ) (cid:1) ϕ forces α ∈ F ( Q ) ϕ ;then, since α ◦ ϕ = ϕ ◦ a ϕ ( α ) , we get β = a ϕ ( α ) and the inclusion abovefollows from the commutativity of diagram 4.5.26.This allows us to choose a family of liftings { \ σ m − ( ϕ ) } ϕ , where ϕ runsover the set of F -morphisms, which is compatible with F -isomorphisms; pre-cisely, for any pair of subgroups Q and R in C F , and any ϕ ∈ F Q,R , wechoose a lifting \ σ m − ( ϕ ) of σ m − ( ϕ ) in ^ L b ,U,m ( Q, R ) as above. Then, anysubgroup Q of P determines a unique ˆ Q in C F which is F -isomorphic to Q and we choose an F -isomorphism ω Q : ˆ Q ∼ = Q and a lifting x Q ∈ ^ L b ,U,m ( Q, ˆ Q )of‘ ω Q ; in particular, we choose ω ˆ Q = id ˆ Q and x ˆ Q = ^ τ b ,U,m ˆ Q (1) . Thus, any F -morphism ϕ : R → Q determines subgroups ˆ Q and ˆ R in C F and an elementˆ ϕ in F ˆ Q, ˆ R in such a way that there are α ϕ ∈ F ( ˆ Q ) and β ϕ ∈ F ( ˆ R ) fulfilling ϕ = ω Q ◦ α ϕ ◦ ˆ ϕ ◦ β − ϕ ◦ ω − R . . \ σ m − ( ϕ ) = x Q · µ m ˆ Q ( α ϕ ) · \ σ m − ( ˆ ϕ ) · µ m ˆ R ( β ϕ ) − · x − R . . Q = R and ϕ = κ Q ( u ) for some u ∈ Q , we actually get \ σ m − (cid:0) κ Q ( u ) (cid:1) = ^ τ b ,U,m Q ( u ) . This definition does not depend on the choice of( α ϕ , β ϕ ) since for another choice ( α ′ , β ′ ) we clearly have α ′ = α ϕ ◦ α ′′ and β ′ = β ϕ ◦ β ′′ for a suitable ( α ′′ , β ′′ ) in (cid:0) F ( ˆ Q ) × F ( ˆ R ) (cid:1) ˆ ϕ and it suffices toapply inclusion 4.5.26.0 Moreover, for any pair of F -isomorphisms ζ : Q ∼ = Q ′ and ξ : R ∼ = R ′ , considering ϕ ′ = ζ ◦ ϕ ◦ ξ − we claim that \ σ m − ( ϕ ′ ) = \ σ m − ( ζ ) · \ σ m − ( ϕ ) · \ σ m − ( ξ ) − . . Q ′ also determines ˆ Q in C F and therefore, if we have ζ = ω Q ◦ α ζ ◦ ω − Q ′ then we obtain \ σ m − ( ζ ) = x Q ′ · µ m ˆ Q ( α ζ ) · x − Q ; similarly, ifwe have ξ = ω R ◦ β ξ ◦ ω − R ′ we also obtain \ σ m − ( ξ ) − = x R · µ m ˆ R ( β ξ ) − · x − R ′ ;further, ϕ ′ also determines ˆ ϕ in F ˆ Q, ˆ R ; consequently, we get \ σ m − ( ζ ) · \ σ m − ( ϕ ) · \ σ m − ( ξ ) − = ( x Q ′ · µ m ˆ Q ( α ζ ) · x − Q ) · \ σ m − ( ϕ ) · ( x R · µ m ˆ R ( β ξ ) − · x − R ′ )= x Q ′ · µ m ˆ Q ( α ζ ◦ α ϕ ) · ˆ ϕ · µ m ˆ R ( β − ϕ ◦ β − ξ ) · x − R ′ = \ σ m − ( ϕ ′ ) 4 . . . Recall that we have the exact sequence of contravariant functors from˜ F to Ab (cf. 2.7 and 2.8)0 −→ ˜ l U,m − F / ˜ l U,m F −→ g Ker ( ^ ¯ π b ,U,m ) −→ g Ker ( ^ ¯ π b ,U,m − ) −→ . . F -morphism ψ : T → R we clearly have \ σ m − ( ϕ ) · \ σ m − ( ψ ) = \ σ m − ( ϕ ◦ ψ ) · γ m ψ,ϕ . . γ m ψ,ϕ in (˜ l U,m − F / ˜ l U,m F )( T ) . That is to say, borrowing notation andterminology from [8, A2.8], we get a correspondence sending any F -chain q : ∆ → F to the element γ m q (0 • , q (1 • in (˜ l U,m − F / ˜ l U,m F ) (cid:0) q (0) (cid:1) and, setting C n (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) = Y ˜ q ∈ Fct (∆ n , ˜ F ) (˜ l U,m − F / ˜ l U,m F ) (cid:0) ˜ q (0) (cid:1) . . n ∈ N , we claim that this correspondence determines an stable ele-ment γ m of C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) [8, A3.17].Indeed, for another F− isomorphic F -chain q ′ : ∆ → F and a natural F -isomorphism ν : q ∼ = q ′ , setting T = q (0) , T ′ = q ′ (0) , R = q (1) , R ′ = q ′ (1) , Q = q (2) , Q ′ = q ′ (2) ψ = q (0 • , ϕ = q (1 • , ψ ′ = q ′ (0 • , ϕ ′ = q ′ (1 • ν = η , ν = ξ and ν = ζ . . , from 4.5.30 we have \ σ m − ( ϕ ′ ) = \ σ m − ( ζ ) · \ σ m − ( ϕ ) · \ σ m − ( ξ ) − \ σ m − ( ψ ′ ) = \ σ m − ( ξ ) · \ σ m − ( ψ ) · \ σ m − ( η ) − \ σ m − ( ϕ ′ ◦ ψ ′ ) = \ σ m − ( ζ ) · \ σ m − ( ϕ ◦ ψ ) · \ σ m − ( η ) − . . \ σ m − ( ϕ ′ ◦ ψ ′ ) · γ m ϕ ′ ,ψ ′ = \ σ m − ( ϕ ′ ) · \ σ m − ( ψ ′ )= (cid:0) \ σ m − ( ζ ) · \ σ m − ( ϕ ) · \ σ m − ( ξ ) − (cid:1) · (cid:0) \ σ m − ( ξ ) · \ σ m − ( ψ ) · \ σ m − ( η ) − (cid:1) = \ σ m − ( ζ ) · (cid:0) \ σ m − ( ϕ ◦ ψ ) · γ m ϕ,ψ (cid:1) · \ σ m − ( η ) − = \ σ m − ( ϕ ′ ◦ ψ ′ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( \ σ m − ( η ) − ) (cid:1) ( γ m ϕ,ψ ) 4 . . , so that, by the divisibility of ^ L b ,U,m , we have γ m ϕ ′ ,ψ ′ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( \ σ m − ( η ) − ) (cid:1) ( γ m ϕ,ψ ) 4 . . γ m sending ( ϕ, ψ ) to γ mϕ,ψ is stable and,in particular, that γ mϕ,ψ only depends on the corresponding ˜ F -morphisms ˜ ϕ and ˜ ψ ; thus we set γ m ˜ ϕ, ˜ ψ = γ m ϕ,ψ . On the other hand, considering the usual differential map d : C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) −→ C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) . . , we claim that d ( γ m ) = 0 ; indeed, for a third F -morphism ε : W → T we get (cid:0) \ σ m − ( ϕ ) · \ σ m − ( ψ ) (cid:1) · \ σ m − ( ε ) = ( \ σ m − ( ϕ ◦ ψ ) · γ m ˜ ϕ, ˜ ψ ) · \ σ m − ( ε )= (cid:0) \ σ m − ( ϕ ◦ ψ ) · \ σ m − ( ε ) (cid:1) · (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ )= \ σ m − ( ϕ ◦ ψ ◦ ε ) · γ m ˜ ϕ ◦ ˜ ψ, ˜ ε · (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ ) \ σ m − ( ϕ ) · (cid:0) \ σ m − ( ψ ) · \ σ m − ( ε ) (cid:1) = \ σ m − ( ϕ ) · (cid:0) \ σ m − ( ψ ◦ ε ) · γ m ˜ ψ, ˜ ε (cid:1) = \ σ m − ( ϕ ◦ ψ ◦ ε ) · γ m ˜ ϕ, ˜ ψ ◦ ˜ ε · γ m ˜ ψ, ˜ ε . . divisibility of ^ L b ,U,m forces γ m ˜ ϕ ◦ ˜ ψ, ˜ ε · (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ ) = γ m ˜ ϕ, ˜ ψ ◦ ˜ ε · γ m ˜ ψ, ˜ ε . . ^ π b ,U,m W ) is abelian, with the additive notation we obtain0 = (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ ) − γ m ˜ ϕ, ˜ ψ ◦ ˜ ε + γ m ˜ ϕ ◦ ˜ ψ, ˜ ε − γ m ˜ ψ, ˜ ε . . , proving our claim.At this point, it follows from equalities 4.5.5 and 4.5.9 that γ m = d ( β m )for some stable element β m = ( β m ˜ r ) ˜ r ∈ Fct (∆ , ˜ F X ) in C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) ; thatis to say, with the notation above we get γ m ˜ ϕ, ˜ ψ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( β m ˜ ϕ ) · ( β m ˜ ϕ ◦ ˜ ψ ) − · β m ˜ ψ . . F -morphism with the obvious ˜ F -chain ∆ → ˜ F ;hence, from equality 4.5.32 we obtain (cid:0) \ σ m − ( ϕ ) · ( β m ˜ ϕ ) − (cid:1) · (cid:0) \ σ m − ( ψ ) · ( β m ˜ ψ ) − (cid:1) = (cid:0) ( \ σ m − ( ϕ ) · \ σ m − ( ψ ) (cid:1) · (cid:16) β m ˜ ψ · (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( β m ˜ ϕ ) (cid:17) − = \ σ m − ( ϕ ◦ ψ ) · ( β m ˜ ϕ ◦ ˜ ψ ) − . . , which amounts to saying that the correspondence σ m sending ϕ ∈ F ( Q, R )to \ σ m − ( ϕ ) · ( β m ˜ ϕ ) − ∈ ^ L b ,U,m ( Q, R ) defines a functorial section of ^ π b ,U,m ;note that, if Q = R and ϕ = κ Q ( u ) for some u ∈ Q , we have ˜ ϕ = e id Q and β m ˜ ϕ = 1 , so that σ m (cid:0) κ Q ( u ) (cid:1) = ^ τ b ,U,m Q ( u ) . It remains to prove that this functorial section fulfills the commutativity of the corresponding diagram4.5.12; since we already have the commutativity of diagram 4.5.25, it sufficesto get the commutativity of the following diagram F ( R ) µ mR −→ ^ L b ,U,m ( R ) id F ( R ) y y a ( βm ˜ ϕ ) − F ( R ) µ mR −→ ^ L b ,U,m ( R ) 4 . . β m is stable and therfore ( β m ˜ ϕ ) − fixes theimage of µ mR . We can modify this correspondence in order to get an F -locality functo-rial section ; indeed, for any F P -morphism κ Q,R ( u ): R → Q where u belongsto T P ( Q, R ) , the ^ L b ,U,m ( Q, R )-morphisms σ m (cid:0) κ Q,R ( u ) (cid:1) and ^ τ b ,U,mQ,R ( u ) bothlift κ Q,R ( u ) ∈ F ( Q, R ) ; thus, the divisibility of ^ L b ,U,m guarantees the exis-tence and the uniqueness of δ κ Q,R ( u ) ∈ Ker( ^ π b ,U,m R ) fulfilling ^ τ b ,U,mQ,R ( u ) = σ m (cid:0) κ Q,R ( u ) (cid:1) .δ κ Q,R ( u ) . . σ m (cid:0) κ Q ( w ) (cid:1) = ^ τ b ,U,m Q ( w ) for any w ∈ Q , it is quite clearthat δ κ Q,R ( u ) only depends on the class of κ Q,R ( u ) in ˜ F ( Q, R )For a second F P -morphism κ R,T ( v ) : T → R , setting ξ = κ R,T ( u ) and η = κ R,T ( v ) we get σ m ( ξ ◦ η ) · δ ˜ ξ ◦ ˜ η = ^ τ b ,U,mQ,T ( uv ) = ^ τ b ,U,mQ,R ( u ) · ^ τ b ,U,mR,T ( v )= σ m ( ξ ) · δ ˜ ξ · σ m ( η ) .δ ˜ η = σ m ( ξ ◦ η ) · (cid:0)g Ker ( ^ π b ,U,m )(˜ η ) (cid:1) ( δ ˜ ξ ) · δ ˜ η . . divisibility of ^ L b ,U,m forces δ ˜ ξ ◦ ˜ η = (cid:0)g Ker ( ^ π b ,U,m )(˜ η ) (cid:1) ( δ ˜ ξ ) · δ ˜ η . . ^ π b ,U,m T ) is abelian, with the additive notation we obtain0 = (cid:0)g Ker ( ^ π b ,U,m )(˜ η ) (cid:1) ( δ ˜ ξ ) − δ ˜ ξ ◦ ˜ η + δ ˜ η . . . That is to say, denoting by i : ˜ F P ⊂ ˜ F the obvious inclusion functor ,the correspondence δ sending any ˜ F P -morphism ˜ ξ : R → Q to δ ˜ ξ defines a1 -cocycle in C (cid:0) ˜ F P , g Ker ( ^ π b ,U,m ) ◦ i (cid:1) ; but, since the category ˜ F P has a finalobject, we actually have [8, Corollary A4.8] H (cid:0) ˜ F P , g Ker ( ^ π b ,U,m ) ◦ i (cid:1) = { } . . δ = d ( w ) for some element w = ( w Q ) Q ⊂ P in C (cid:0) ˜ F P , g Ker ( ^ π b ,U,m ) ◦ i (cid:1) = C (cid:0) ˜ F , g Ker ( ^ π b ,U,m )) (cid:1) . . . In conclusion, equality 4.5.45 becomes ^ τ b ,U,mQ,R ( u ) = σ m (cid:0) κ Q,R ( u ) (cid:1) · (cid:0)g Ker ( ^ π b ,U,m )( ^ κ Q,R ( u ) (cid:1) ( w Q ) · w − R = w Q · σ m (cid:0) κ Q,R ( u ) (cid:1) · w − R . . ϕ ∈ F ( Q, R ) to w Q · σ m (cid:0) ϕ ) · w − R defines a F -locality functorial section of ^ π b ,U,m . From now on, we still denoteby σ m this F -locality functorial section of ^ π b ,U,m . Let σ ′ m : F → ^ L b ,U,m be another F -locality functorial section of ^ π b ,U,m ;arguing by induction on |C P − N | and on m , and up to natural F -isomor-phisms, we clearly may assume that σ ′ m also lifts σ m − ; in this case, forany F -morphism ϕ : R → Q , we have σ ′ m ( ϕ ) = σ m ( ϕ ) · ε m ϕ for some ε m in (˜ l U,m − F / ˜ l U,m F )( R ) ; that is to say, as above we get a correspondence sendingany F -chain q : ∆ → F to ε m q (0 • , in (˜ l U,m − F / ˜ l U,m F ) (cid:0) q (0) (cid:1) and we claim thatthis correspondence determines an F P -stable element ε m of C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) [8, A3.17].Indeed, for another F P -isomorphic F -chain q ′ : ∆ → F and a natural F P -isomorphism ν : q ∼ = q ′ , as in 4.5.34 above setting R = q (0) , R ′ = q ′ (0) , Q = q (1) , Q ′ = q ′ (1) ϕ = q (0 • , ϕ ′ = q ′ (0 • ν = κ R ′ ,R ( v ) and ν = κ Q ′ ,Q ( u ) 4 . . , σ ′ m ( ϕ ′ ) = κ Q ′ ,Q ( u ) · σ m ( ϕ ) · ε m ϕ · κ R ′ ,R ( v ) − = σ m ( ϕ ′ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( ^ κ R ′ ,R ( v ) − ) (cid:1) ( ε m ϕ ) σ ′ m ( ϕ ′ ) = σ m ( ϕ ′ ) · ε m ϕ ′ . . ^ L b ,U,m forces ε m ϕ ′ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( ^ κ R ′ ,R ( v ) − ) (cid:1) ( ε m ϕ ) 4 . . ε m sending ϕ to ε m ϕ is F P -stable and, inparticular, that ε m ϕ only depends on the corresponding ˜ F -morphism ˜ ϕ , thuswe set ε m ˜ ϕ = ε m ϕ , . On the other hand, considering the usual differential map d : C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) −→ C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) . . , we claim that d ( ε m ) = 0 ; indeed, for a second F -morphism ψ : T → R weget σ ′ m ( ϕ ) · σ ′ m ( ψ ) = σ m ( ϕ ) · ε m ϕ · σ m ( ψ ) · ε m ψ = σ m ( ϕ ◦ ψ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( ε m ˜ ϕ ) · ε m ψ σ ′ m ( ϕ ) · σ ′ m ( ψ ) = σ m ( ϕ ◦ ψ ) · ε m ϕ ◦ ψ . . divisibility of ^ L b ,U,m forces (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( ε m ˜ ϕ ) · ε m ˜ ψ = ε m ˜ ϕ ◦ ˜ ψ . . ^ π b ,U,m T ) is Abelian, with the additive notation we obtain0 = (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( ε m ˜ ϕ ) − ε m ˜ ϕ ◦ ˜ ψ + ε m ˜ ψ . . , proving our claim.At this point, it follows from equalities 4.5.5 and 4.5.9 that ε m = d ( y )for some stable element y = ( y Q ) Q ⊂ P in C (cid:0) ˜ F , ˜ l U,m − F / ˜ l U,m F (cid:1) ; that is to say,with the notation above we get ε m ˜ ϕ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ϕ ) (cid:1) ( y Q ) · y − R . . σ ′ m ( ϕ ) = σ m ( ϕ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ϕ ) (cid:1) ( y Q ) · y − R = y Q · σ m ( ϕ ) · y − R . . , which amounts to saying that σ ′ m is naturally F -isomorphic to σ m . We aredone5
Corollary 4.6.
There exists a perfect F -locality P . Proof:
Denote by ¯ P the converse image in L b of the image of F in f L b bya section of f π b ; since ˆ τ ( c f F ) is contained in the image of τ b , we still havea functor τ b : T P → ¯ P ; thus, together with the restriction of π b to ¯ P , ¯ P becomes an F -locality and, since L b is coherent , ¯ P is coherent too.We claim that ¯ P sc is a perfect F sc -locality ; indeed, for any F -selfcentrali-zing subgroup Q of P fully normalized in F , since C P ( Q ) /F C F ( Q ) = Z ( Q )we have a group extension (cf. 4.3)1 −→ Z ( Q ) −→ ¯ P ( Q ) −→ F ( Q ) −→ . . τ b Q : N P ( Q ) → ¯ P ( Q ) ; con-sequently, it follows from [8, 18.5] that ¯ P ( Q ) is the F -localizer of Q ; thus,by the very definition in [8, 17.4 and 17.13], ¯ P sc is a perfect F sc -locality .But, in [8, Ch. 20] we prove that any perfect F sc -locality can be extendedto a unique perfect F -locality P . We are done.4.7. The uniqueness of the perfect F -locality is an easy consequence ofthe following theorem; the proof of this result follows the same pattern thanthe proof of Theorem 4.5, but we firstly need the following lemmas. Lemma 4.8.
Let ( τ, P , π ) be a perfect F -locality and ˆ ϕ : Q → P be a P -morphism such that π ˆ ϕ ( Q ) is fully normalized in F . Then there is a P -morphism ˆ ζ : N P ( Q ) → P such that ˆ ϕ = ˆ ζ · τ N P ( Q ) ,Q (1) . Proof:
Denoting by ϕ the image of ˆ ϕ in F ( P, Q ) , it follows from [8, 2.8.2]that there is an F -morphism ζ : N P ( Q ) → P extending ϕ ; then, lifting ζ to ˆ ζ in P (cid:0) P, N P ( R ) (cid:1) , it is clear that the P -morphisms ˆ ζ · τ N P ( Q ) ,Q (1) and ˆ ϕ havethe same image ϕ in F ( P, Q ) and therefore, by the very definition of P in[8, 17.13], there is z ∈ C P ( Q ) such that ˆ ζ · τ N P ( Q ) ,Q (1) · τ Q ( z ) = ˆ ϕ ; bur, it isclear that τ N P ( Q ) ,Q (1) · τ Q ( z ) = τ N P ( Q ) ,Q ( z ) = τ N P ( Q ) ( z ) · τ N P ( Q ) ,Q (1) 4 . . ζ · τ N P ( Q ) ( z ) extends ˆ ϕ in P . We are done
Lemma 4.9.
Let ( τ, P , π ) be a perfect F -locality. For any subgroup Q of P there is a group homomorphism ˆ µ Q : P ( Q ) → L b ( Q ) fulfilling ˆ µ Q ◦ τ Q = τ b Q . Proof:
Since we can choose an F -isomorphism θ : Q ∼ = Q ′ such that Q ′ isfully normalized in F and θ can be lifted to P ( Q ′ , Q ) and to L b ( Q ′ , Q ) , wemay assume that Q is fully normalized in F . P ( Q ) and L b ( Q ) , to the normal p -subgroup Ker( π b Q ) of L b ( Q ) and to the group homomorphism τ Q from P ( Q ) to F ( Q ) ∼ = L b ( Q ) / Ker( π b Q ) . We consider the group homomorphism τ b Q : N P ( Q ) → L b ( Q ) and, for any subgroup R of N P ( Q ) and any ˆ α ∈ P ( Q )such that ˆ α · τ Q ( R ) · ˆ α − ⊂ τ Q (cid:0) N P ( Q ) (cid:1) , it follows from [8, 2.10.1] that thereexists ζ ∈ F (cid:0) N P ( Q ) , Q · R (cid:1) extending the image of ˆ α in F ( Q ) ; then, it followsfrom [8, 17.11.2] that there exists x ∈ L b ( Q ) fulfilling τ b Q (cid:0) ζ ( v ) (cid:1) = x τ b Q ( v ) 4 . . v ∈ Q.R .
That is to say, condition 18.8.1 in [8, Lemma 18.8] is fulfilledand therefore this lemma proves the existence of ˆ µ Q as announced.. Theorem 4.10.
For any perfect F -locality P there exists a unique natural F -isomorphism class of F -locality functors to L b . Proof:
Let P be a perfect F -locality with the structural functors τ : T P −→ P and π : P −→ F . . Q of P and R of Q we set i QR = τ Q,R (1) . We considerthe filtration of the basic F -locality introduced in section 3 and then argue byinduction. That is to say, recall that we denote by C P a set of representativesfor the set of P -conjugacy classes of subgroups U of P (cf. 2.13); now, for anysubset N of C P fulfilling condition 3.2.1, we have the functor ˜ k N F : ˜ F → Ab (cf. 3.2) and we consider the quotient F -locality L b , N = L b / (˜ k N F ◦ ˜ π b ) withthe structural functors τ b , N : T P −→ L b , N and π b , N : L b , N −→ F . . . Note that if N = ∅ then L b , N = L b ; hence, arguing by induction on |C P −N | , it suffices to prove the existence of a unique natural F -isomorphism class of F -locality functors from P to L b , N .Moreover, if N = C P then ˜ k N F = ˜ k b F and therefore L b , N = F , so that wemay assume that N 6 = C P ; in this situation, we fix a minimal element U in C P − N , setting M = N ∪ { U } and ˜ k U F = ˜ k M F / ˜ k N F ; for any m ∈ N we simplydenote by ˜ l U,m F the converse image of p m · ˜ k U F in ˜ k M F ; set L b ,U,m = L b / ˜ l U,m F and,coherently, denote by π b , U , m and τ b , U , m the corresponding structural functors.Note that, by 3.8 and 3.10.5 above we get˜ l U,m F / ˜ l U,m +1 F ∼ = ˜ r U, ◦F ,m . . n ∈ N we still get H n ∗ ( ˜ F , ˜ l U,m F / ˜ l U,m +1 F ) = { } . . . C F a set of representatives, fully normalizedin F , for the F -isomorphism classes of subgroups of P and, for any subgroup Q in C F , we choose a group homomorphism ˆ µ Q : P ( Q ) → L b ( Q ) as in Lemma4.9 above and, for any m ∈ N , simply denote by ˆ µ m Q the corresponding grouphomomorphism from P ( Q ) to L b ,U,m ( Q ) . For any F -morphism ϕ : R → Q denote by P ( Q ) ϕ and by L b ,U,m ( Q ) ϕ the respective stabilizers of ϕ ( R ) in P ( Q )and in L b ,U,m ( Q ) . As above, for any ˆ ϕ ∈ P ( Q, R ) and any x m ∈ L b ,U,m ( Q, R )we have group homomorphisms a ˆ ϕ : P ( Q ) ϕ −→ P ( R ) and a x m : L b ,U,m ( Q ) ϕ −→ L b ,U,m ( R ) 4 . . · For any subgroups Q and R in C F , we choose as above a set of representati-ves P Q,R for the set of double classes P ( Q ) \P ( Q, R ) / P ( R ) such that, for anyˆ ϕ in P Q,R , denoting by ϕ its image in F ( Q, R ) , F P ( Q ) contains a Sylow p -subgroup of F ( Q ) ϕ and a ϕ (cid:0) F P ( Q ) ϕ (cid:1) is contaioned in F P ( R ) ; of course, wechoose P Q,Q = { τ Q (1) } . With all this notation and arguing by induction on |C P − N | and on m , we will prove that there is a functor λ m : P −→ L b ,U,m . . Q ∈ C F and any u ∈ Q , we have λ m (cid:0) τ Q ( u ) (cid:1) = τ b ,U,m Q ( u ) , and that, for any groups Q and R in C F , and any ˆ ϕ in P Q,R , denoting by ϕ its image in F ( Q, R ) , we have the commutative diagram P ( Q ) ϕ ˆ µ mQ −→ L b ,U,m ( Q ) ϕa ˆ ϕ y y a λm ( ˆ ϕ ) P ( R ) ˆ µ mR −→ L b ,U,m ( R ) 4 . . . Since we have π b ,U, = π b , M and |M| = |N | + 1 , by the induction hypothesiswe actually may assume that m = 0 and that we have a functor λ m − : P −→ L b ,U,m − . . ϕ ∈ P Q,R , denoting by ϕ its image in F ( Q, R ) , it follows from [8, Proposition 2.11], applied to the inverse ϕ ∗ of the iso-morphism ϕ ∗ : R ∼ = ϕ ( R ) induced by ϕ , that there exists an F -morphism ζ : N P ( Q ) ϕ → N P ( R ) fulfilling ζ (cid:0) ϕ ( v ) (cid:1) = v for any v ∈ R , so that we easilyget the following commutative diagram‘ N P ( Q ) ϕ τ Q −→ P ( Q ) ϕζ y y a ˆ ϕ N P ( R ) τ R −→ P ( R ) 4 . . Q = R and ˆ ϕ = τ Q ( u ) for some u ∈ Q , we may assume that ζ = κ N P ( Q ) ( u ) . In particular, since λ m − fulfills the corresponding commu-tative diagram 4.10.7, we still get the following commutative diagram N P ( Q ) ϕ τ b ,U,m − Q −−−−−−→ L b ,U,m − ( Q ) ϕζ y y a λm −
1( ˆ ϕ ) N P ( R ) τ b ,U,m − R −−−−−−→ L b ,U,m − ( R ) 4 . . \ λ m − ( ˆ ϕ ) of λ m − ( ˆ ϕ ) in L b ,U,m ( Q, R ) . Choosing a lifting ˆ ζ of ζ in the obviousstabilizer P (cid:0) N P ( R ) , N P ( Q ) ϕ (cid:1) R,ϕ ( R ) , we start by choosing a lifting \ λ m − (ˆ ζ ) of λ m − (ˆ ζ ) in L b ,U,m (cid:0) N P ( R ) , N P ( Q ) ϕ (cid:1) R,ϕ ( R ) ; thus, by the coherence of L b ,U,m (cf. (Q)), for any u ∈ N P ( Q ) ϕ we have \ λ m − (ˆ ζ ) · τ b ,U,m N P ( Q ) ϕ ( u ) = τ b ,U,m N P ( R ) (cid:0) ζ ( u ) (cid:1) · \ λ m − (ˆ ζ ) 4 . . divisibility of L b ,U,m (cf. 2.4), we find z ˆ ϕ ∈ L b ,U,m (cid:0) R, ϕ ( R ) (cid:1) fulfilling \ λ m − (ˆ ζ ) · τ b ,U,m N P ( Q ) ϕ ,ϕ ( R ) (1) = τ b ,U,m N P ( R ) ,R (1) · z ˆ ϕ . . ϕ ∗ : ϕ ( R ) ∼ = R the P -isomorphism determined by ˆ ϕ ,λ m − (ˆ ζ ) restricts to λ m − ( ˆ ϕ ∗ ) and it is easily checked that z ˆ ϕ lifts λ m − ( ˆ ϕ ∗ )to L b ,U,m (cid:0) R, ϕ ( R ) (cid:1) ; consequently, ^ λ m − ( ˆ ϕ ) = τ b ,U,m Q · ˆ ϕ ( R ) (1) · z − ϕ lifts λ m − ( ˆ ϕ )to L b ,U,m ( Q, R ) . .Then, from 4.10.11 and 4.10.12 above, for any u ∈ N P ( Q ) ϕ we get \ λ m − (ˆ ζ ) · τ b ,U,m N P ( Q ) ϕ ( u ) · .τ b ,U,m N P ( Q ) ϕ ,ϕ ( R ) (1) = τ b ,U,m N P ( R ) ,R (1) · z ˆ ϕ · τ b ,U,m ϕ ( R ) ( u ) k τ b ,U,m N P ( R ) (cid:0) ζ ( u ) (cid:1) · \ λ m − (ˆ ζ ) · .τ b ,U,m N P ( Q ) ϕ ,ϕ ( R ) (1)= τ b ,U,m N P ( R ) ,R (1) · τ b ,U,m R (cid:0) ζ ( u ) (cid:1) · z ˆ ϕ . . z ˆ ϕ · τ b ,U,m ϕ ( R ) ( u ) = τ b ,U,m R (cid:0) ζ ( u ) (cid:1) · z ˆ ϕ , so that τ b ,U,m Q ( u ) · ^ λ m − ( ˆ ϕ ) = λ m − ( ˆ ϕ ) · τ b ,U,m R (cid:0) ζ ( u ) (cid:1) . . a ^ λ m − ( ˆ ϕ ) (cid:0) τ b ,U,m Q ( u ) (cid:1) = τ b ,U,m R (cid:0) ζ ( u ) (cid:1) . P ( Q ) ϕ and L b ,U,m ( R ) , to the kernel of the canonical homo-morphism from L b ,U,m ( R ) to L b ,U,m − ( R ) , and to the composition of grouphomomorphisms a λ m − ( ˆ ϕ ) ◦ ˆ µ m − Q : P ( Q ) ϕ −→ L b ,U,m − ( Q ) ϕ −→ L b ,U,m − ( R ) 4 . . , together with the composition of group homomorphisms τ b ,U,m R ◦ ζ : N P ( Q ) ϕ −→ N P ( R ) −→ L b ,U,m ( R ) 4 . . . Now, according to the commutative diagrams 4.10.7 for m − a ^ λ m − ( ˆ ϕ ) ◦ ˆ µ m Q : P ( Q ) ϕ −→ L b ,U,m ( Q ) ϕ −→ L b ,U,m ( R )ˆ µ m R ◦ a ˆ ϕ : P ( Q ) ϕ −→ P ( R ) −→ L b ,U,m ( R ) 4 . . , both fulfill the conclusion of [8, Lemma 18.8]; consequently, according tothis lemma, there is k ˆ ϕ in the kernel of the canonical homomorphism from L b ,U,m ( R ) to L b ,U,m − ( R ) such that, denoting by int L b ,U,m ( R ) ( k ˆ ϕ ) the conju-gation by k ˆ ϕ in L b ,U,m ( R ) , we haveint L b ,U,m ( R ) ( k ˆ ϕ ) ◦ a ^ λ m − ( ˆ ϕ ) ◦ ˆ µ m Q = ˆ µ m R ◦ a ˆ ϕ . . L b ,U,m ( R ) ( k ˆ ϕ ) ◦ a ^ λ m − ( ˆ ϕ ) = a ^ λ m − ( ˆ ϕ ) · k − ϕ . . . Finally, we choose \ λ m − ( ˆ ϕ ) = ^ λ m − ( ˆ ϕ ) · k − ϕ , lifting indeed σ m − ( ϕ ) to L b ,U,m ( Q, R ) and, according to equalities 4.10.18 and 4.10.19, fulfilling thefollowing commutative diagram P ( Q ) ϕ ˆ µ mQ −→ L b ,U,m ( Q ) ϕa ˆ ϕ y y a \ λm −
1( ˆ ϕ ) P ( R ) ˆ µ mR −→ L b ,U,m ( R ) 4 . . Q = R and ˆ ϕ = τ Q ( u ) for some u ∈ Q , this choice is com-patible wtih \ λ m − (cid:0) τ Q ( u ) (cid:1) = τ b ,U,m Q ( u ) . In particular, considering the action of P ( Q ) × P ( R ) , by composition on the left- and on the right-hand, on P ( Q, R )and on L b ,U,m ( Q, R ) via ˆ µ m Q and ˆ µ m R , we have the inclusion of stabilizers (cid:0) P ( Q ) × P ( R ) (cid:1) ˆ ϕ ⊂ (cid:0) P ( Q ) × P ( R ) (cid:1) λ m − ( ˆ ϕ ) . . α, ˆ β ) ∈ (cid:0) P ( Q ) × P ( R ) (cid:1) ˆ ϕ forces ˆ α ∈ P ( Q ) ϕ ;then, since ˆ α · ˆ ϕ = ˆ ϕ · a ˆ ϕ (ˆ α ) , we get ˆ β = a ˆ ϕ (ˆ α ) by the divisibility of P , andthe inclusion above follows from the commutativity of diagram 4.10.20.This allows us to choose a family of liftings { \ λ m − ( ˆ ϕ ) } ˆ ϕ , where ˆ ϕ runsover the set of P -morphisms, which is compatible with P -isomorphisms; pre-cisely, for any pair of subgroups Q and R in C F , and any ˆ ϕ ∈ P Q,R , wechoose as above a lifting \ λ m − ( ˆ ϕ ) of λ m − ( ˆ ϕ ) in L b ,U,m ( Q, R ) . Then, anysubgroup Q of P determines a unique ¯ Q in C F which is F -isomorphic to Q and we choose a P -isomorphism ˆ ω Q : ¯ Q ∼ = Q and a lifting x Q ∈ L b ,U,m ( Q, ˆ Q )of the image ω Q ∈ F ( Q, ¯ Q ) of ˆ ω Q ; in particular, we choose ω ¯ Q = τ ¯ Q (1) and x ¯ Q = τ b ,U,m ¯ Q (1) . Thus, any P -morphism ˆ ϕ : R → Q determines subgroups¯ Q and ¯ R in C F and an element ¯ˆ ϕ in P ¯ Q, ¯ R in such a way that there areˆ α ˆ ϕ ∈ P ( ¯ Q ) and ˆ β ˆ ϕ ∈ P ( ¯ R ) fulfillingˆ ϕ = ˆ ω Q · ˆ α ˆ ϕ · ¯ˆ ϕ · ˆ β − ϕ · ˆ ω − R . . \ λ m − ( ˆ ϕ ) = x Q · ˆ µ m ¯ Q (ˆ α ˆ ϕ ) · \ λ m − ( ¯ˆ ϕ ) · ˆ µ m ˆ R ( ˆ β ˆ ϕ ) − · x − R . . Q = R and ˆ ϕ = τ Q ( u ) for some u ∈ Q , we actually get \ λ m − (cid:0) τ Q ( u ) (cid:1) = τ b ,U,m Q ( u ) . This definition does not depend on the choice of(ˆ α ˆ ϕ , ˆ β ˆ ϕ ) since for another choice (ˆ α ′ , ˆ β ′ ) we clearly have ˆ α ′ = ˆ α ˆ ϕ · ˆ α ′′ andˆ β ′ = ˆ β ˆ ϕ · ˆ β ′′ for a suitable (ˆ α ′′ , ˆ β ′′ ) in (cid:0) P ( ¯ Q ) × P ( ¯ R ) (cid:1) ˆ ϕ and it suffices to applyinclusion 4.10.21.Moreover, for any pair of P -isomorphisms ˆ ζ : Q ∼ = Q ′ and ˆ ξ : R ∼ = R ′ , considering ˆ ϕ ′ = ˆ ζ · ˆ ϕ · ˆ ξ − we claim that \ λ m − ( ˆ ϕ ′ ) = \ λ m − (ˆ ζ ) · \ λ m − ( ˆ ϕ ) · \ λ m − ( ˆ ξ ) − . . Q ′ also determines ¯ Q in C F and therefore, if we haveˆ ζ = ˆ ω Q · ˆ α ˆ ζ · ˆ ω − Q ′ then we obtain \ λ m − (ˆ ζ ) = x Q ′ · ˆ µ m ¯ Q (ˆ α ˆ ζ ) · x − Q ; similarly, if wehave ˆ ξ = ˆ ω R · ˆ β ˆ ξ · ˆ ω − R ′ we also obtain \ λ m − ( ˆ ξ ) − = x R · ˆ µ m ˆ R ( ˆ β ˆ ξ ) − · x − R ′ ; further,ˆ ϕ ′ also determines ¯ˆ ϕ in P ¯ Q, ¯ R ; consequently, we get \ λ m − (ˆ ζ ) · \ λ m − ( ˆ ϕ ) · \ λ m − ( ˆ ξ ) − = ( x Q ′ · ˆ µ m ¯ Q (ˆ α ˆ ζ ) · x − Q ) · \ λ m − ( ˆ ϕ ) · ( x R · ˆ µ m ˆ R ( ˆ β ˆ ξ ) − · x − R ′ )= x Q ′ · ˆ µ m ˆ Q (ˆ α ˆ ζ · ˆ α ˆ ϕ ) · ¯ˆ ϕ · ˆ µ m ˆ R ( ˆ β − ϕ · ˆ β − ξ ) · x − R ′ = \ λ m − ( ˆ ϕ ′ ) 4 . . . contravariant functors from˜ F to Ab (cf. 2.7 and 2.8)0 −→ ˜ l U,m − F / ˜ l U,m F −→ g Ker ( π b ,U,m ) −→ g Ker ( π b ,U,m − ) −→ . . P -morphism ˆ ψ : T → R we clearly have \ λ m − ( ˆ ϕ ) · \ λ m − ( ˆ ψ ) = \ λ m − ( ˆ ϕ · ˆ ψ ) · γ m ˆ ψ, ˆ ϕ . . γ m ψ,ϕ in (˜ l U,m − F / ˜ l U,m F )( T ) . That is to say, borrowing notation andterminology from [8, A2.8], we get a correspondence sending any P -chain q : ∆ → P to the element γ m q (0 • , q (1 • in (˜ l U,m − F / ˜ l U,m F ) (cid:0) q (0) (cid:1) and, setting C n (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) = Y ˜ q ∈ Fct (∆ n , ˜ P ) (˜ l U,m − F / ˜ l U,m F ) (cid:0) ˜ q (0) (cid:1) . . n ∈ N , we claim that this correspondence determines an stable ele-ment γ m of C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) [8, A3.17]; note that ˜ P ∼ = ˜ F . Indeed, for another P− isomorphic P -chain q ′ : ∆ → P and a natural P -isomorphism ν : q ∼ = q ′ , setting T = q (0) , T ′ = q ′ (0) , R = q (1) , R ′ = q ′ (1) , Q = q (2) , Q ′ = q ′ (2)ˆ ψ = q (0 • , ˆ ϕ = q (1 • , ˆ ψ ′ = q ′ (0 • , ˆ ϕ ′ = q ′ (1 • ν = ˆ η , ν = ˆ ξ and ν = ˆ ζ . . , from 4.5.30 we have \ λ m − ( ˆ ϕ ′ ) = \ λ m − (ˆ ζ ) · \ λ m − ( ˆ ϕ ) · \ λ m − ( ˆ ξ ) − \ λ m − ( ˆ ψ ′ ) = \ λ m − ( ˆ ξ ) · \ λ m − ( ˆ ψ ) · \ λ m − (ˆ η ) − \ λ m − ( ˆ ϕ ′ · . ˆ ψ ′ ) = \ λ m − (ˆ ζ ) · \ λ m − ( ˆ ϕ · ˆ ψ ) · \ λ m − (ˆ η ) − . . \ λ m − ( ˆ ϕ ′ · ˆ ψ ′ ) · γ m ϕ ′ ,ψ ′ = \ λ m − ( ˆ ϕ ′ ) · \ λ m − ( ˆ ψ ′ )= (cid:0) \ λ m − (ˆ ζ ) · \ λ m − ( ˆ ϕ ) · \ λ m − ( ˆ ξ ) − (cid:1) · (cid:0) \ λ m − ( ˆ ξ ) · \ λ m − ( ˆ ψ ) · \ λ m − (ˆ η ) − (cid:1) = \ λ m − (ˆ ζ ) · (cid:0) \ λ m − ( ˆ ϕ · ˆ ψ ) · γ m ˆ ϕ, ˆ ψ (cid:1) · \ λ m − (ˆ η ) − = \ λ m − ( ˆ ϕ ′ · ˆ ψ ′ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( \ λ m − (ˆ η ) − ) (cid:1) ( γ m ˆ ϕ, ˆ ψ ) 4 . . , so that, by the divisibility of L b ,U,m , we have γ m ˆ ϕ ′ , ˆ ψ ′ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( \ λ m − (ˆ η ) − ) (cid:1) ( γ m ˆ ϕ, ˆ ψ ) 4 . . γ m sending ( ˆ ϕ, ˆ ψ ) to γ m ˆ ϕ, ˆ ψ is stable and,in particular, that γ m ˆ ϕ, ˆ ψ only depends on the corresponding ˜ P -morphisms˜ˆ ϕ and ˜ˆ ψ ; thus we set γ m ˜ ϕ, ˜ ψ = γ m ˆ ϕ, ˆ ψ where ϕ and ψ are the correponding F -morphisms.On the other hand, considering the usual differential map d : C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) −→ C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) . . , we claim that d ( γ m ) = 0 ; indeed, for a third F -morphism ε : W → T we get (cid:0) \ λ m − ( ˆ ϕ ) · \ λ m − ( ˆ ψ ) (cid:1) · \ λ m − (ˆ ε ) = ( \ λ m − ( ˆ ϕ · ˆ ψ ) · γ m ˜ ϕ, ˜ ψ ) · \ λ m − (ˆ ε )= (cid:0) \ λ m − ( ˆ ϕ · ˆ ψ ) · \ λ m − (ˆ ε ) (cid:1) · (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ )= \ λ m − ( ˆ ϕ · ˆ ψ · ˆ ε ) · γ m ˜ ϕ · ˜ ψ, ˜ ε · (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ ) \ λ m − ( ˆ ϕ ) · (cid:0) \ λ m − ( ˆ ψ ) · \ λ m − (ˆ ε ) (cid:1) = \ λ m − ( ˆ ϕ ) · (cid:0) \ λ m − ( ˆ ψ · ˆ ε ) · γ m ˜ ψ, ˜ ε (cid:1) = \ λ m − ( ˆ ϕ · ˆ ψ · ˆ ε ) · γ m ˜ ϕ, ˜ ψ · ˜ ε · γ m ˜ ψ, ˜ ε . . divisibility of L b ,U,m forces γ m ˜ ϕ · ˜ ψ, ˜ ε · (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ ) = γ m ˜ ϕ, ˜ ψ · ˜ ε · γ m ˜ ψ, ˜ ε . . ^ π b ,U,m W ) is abelian, with the additive notation we obtain0 = (cid:0) (˜ l U,m − F / ˜ l U,m F )(˜ ε ) (cid:1) ( γ m ˜ ϕ, ˜ ψ ) − γ m ˜ ϕ, ˜ ψ · ˜ ε + γ m ˜ ϕ · ˜ ψ, ˜ ε − γ m ˜ ψ, ˜ ε . . , proving our claim.At this point, it follows from equality 4.10.4 that γ m = d ( β m ) for some stable element β m = ( β m ˜ r ) ˜ r ∈ Fct (∆ , ˜ P ) in C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) ; that is to say,with the notation above we get γ m ˜ ϕ, ˜ ψ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( β m ˜ ϕ ) · ( β m ˜ ϕ · ˜ ψ ) − · β m ˜ ψ . . (cid:0) \ λ m − ( ˆ ϕ ) · ( β m ˜ ϕ ) − (cid:1) · (cid:0) \ λ m − ( ˆ ψ ) · ( β m ˜ ψ ) − (cid:1) = (cid:0) ( \ λ m − ( ˆ ϕ ) · \ λ m − ( ˆ ψ ) (cid:1) · (cid:16) β m ˜ ψ · (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( β m ˜ ϕ ) (cid:17) − = \ λ m − ( ˆ ϕ · ˆ ψ ) · ( β m ˜ ϕ ◦ ˜ ψ ) − . . , λ m sending ˆ ϕ ∈ P ( Q, R )to \ λ m − ( ˆ ϕ ) · ( β m ˜ ϕ ) − ∈ L b ,U,m ( Q, R ) defines the announced functor; note that,if Q = R and ˆ ϕ = τ Q ( u ) for some u ∈ Q , we have ˜ ϕ = e id Q and β m ˜ ϕ = 1 , so that λ m (cid:0) τ Q ( u ) (cid:1) = τ b ,U,m Q ( u ) . It remains to prove that this functorial sec-tion fulfills the commutativity of the corresponding diagram 4.10.7; sincewe already have the commutativity of diagram 4.10.20, it suffices to get thecommutativity of the following diagram P ( R ) ˆ µ mR −→ L b ,U,m ( R ) id P ( R ) y y a ( βm ˜ ϕ ) − P ( R ) ˆ µ mR −→ L b ,U,m ( R ) 4 . . β m is stable and therfore ( β m ˜ ϕ ) − fixes theimage of ˆ µ mR . We can modify this correspondence in order to get an F -locality functor ;indeed, for any P -morphism τ Q,R ( u ): R → Q where u belongs to T P ( Q, R ) , the L b ,U,m ( Q, R )-morphisms λ m (cid:0) τ Q,R ( u ) (cid:1) and τ b ,U,mQ,R ( u ) , both lift κ Q,R ( u )in F ( Q, R ) ; thus, the divisibility of L b ,U,m guarantees the existence and theuniqueness of δ κ Q,R ( u ) ∈ Ker( π b ,U,m R ) fulfilling τ b ,U,mQ,R ( u ) = λ m (cid:0) τ Q,R ( u ) (cid:1) .δ κ Q,R ( u ) . . λ m (cid:0) τ Q ( w ) (cid:1) = τ b ,U,m Q ( w ) for any w ∈ Q , it is quite clearthat δ κ Q,R ( u ) only depends on the class of κ Q,R ( u ) in ˜ F ( Q, R )For a second P -morphism τ R,T ( v ) : T → R , setting ˆ ξ = τ R,T ( u ) andˆ η = τ R,T ( v ) we get λ m ( ˆ ξ · ˆ η ) · δ ˜ ξ ◦ ˜ η = τ b ,U,mQ,T ( uv ) = τ b ,U,mQ,R ( u ) · τ b ,U,mR,T ( v )= λ m ( ˆ ξ ) · δ ˜ ξ · λ m (ˆ η ) .δ ˜ η = λ m ( ˆ ξ · ˆ η ) · (cid:0)g Ker ( π b ,U,m )(˜ η ) (cid:1) ( δ ˜ ξ ) · δ ˜ η . . divisibility of L b ,U,m forces δ ˜ ξ ◦ ˜ η = (cid:0)g Ker ( π b ,U,m )(˜ η ) (cid:1) ( δ ˜ ξ ) · δ ˜ η . . π b ,U,m T ) is abelian, with the additive notation we obtain0 = (cid:0)g Ker ( π b ,U,m )(˜ η ) (cid:1) ( δ ˜ ξ ) − δ ˜ ξ · ˜ η + δ ˜ η . . . i : ˜ F P ⊂ ˜ F the obvious inclusion functor ,the correspondence δ sending any ˜ F P -morphism ˜ ξ : R → Q to δ ˜ ξ defines a1 -cocycle in C (cid:0) ˜ F P , g Ker ( π b ,U,m ) ◦ i (cid:1) ; but, since the category ˜ F P has a finalobject, we actually have [8, Corollary A4.8] H (cid:0) ˜ F P , g Ker ( π b ,U,m ) ◦ i (cid:1) = { } . . δ = d ( w ) for some element w = ( w Q ) Q ⊂ P in C (cid:0) ˜ F P , g Ker ( π b ,U,m ) ◦ i (cid:1) = C (cid:0) ˜ F , g Ker ( π b ,U,m )) (cid:1) . . . In conclusion, equality 4.10.40 becomes τ b ,U,mQ,R ( u ) = λ m (cid:0) τ Q,R ( u ) (cid:1) · (cid:0)g Ker ( π b ,U,m )( ^ τ Q,R ( u ) (cid:1) ( w Q ) · w − R = w Q · λ m (cid:0) τ Q,R ( u ) (cid:1) · w − R . . ϕ ∈ P ( Q, R ) to w Q · λ m (cid:0) ˆ ϕ ) · w − R defines a F -locality functor . From now on, we still denote by λ m this F -lo-cality functor .Let λ ′ m : P → L b ,U,m be another F -locality functor; arguing by inductionon |C P − N | and on m , and up to natural F -isomorphisms, we clearly mayassume that λ ′ m also lifts λ m − ; in this case, for any P -morphism ˆ ϕ : R → Q , we have λ ′ m ( ϕ ) = λ m ( ˆ ϕ ) · ε m ˆ ϕ for some ε m ˆ ϕ in (˜ l U,m − F / ˜ l U,m F )( R ) ; that is to say,as above we get a correspondence sending any P -chain q : ∆ → P to ε m q (0 • , in (˜ l U,m − F / ˜ l U,m F ) (cid:0) q (0) (cid:1) and we claim that this correspondence determines a P -stable element ε m of C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) [8, A3.17].Indeed, for another P− isomorphic P -chain q ′ : ∆ → P and a natural P -isomorphism ν : q ∼ = q ′ , as in 4.10.29 above setting R = q (0) , R ′ = q ′ (0) , Q = q (1) , Q ′ = q ′ (1)ˆ ϕ = q (0 • , ˆ ϕ ′ = q ′ (0 • ν = ˆ ξ and ν = ˆ ζ . . , from 4.10.24 we get λ ′ m ( ˆ ϕ ′ ) = ˆ ζ · λ m ( ϕ ) · ε m ϕ · ˆ ξ − = λ m ( ˆ ϕ ′ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( e ξ − ) (cid:1) ( ε m ˆ ϕ ) λ ′ m ( ˆ ϕ ′ ) = λ m ( ˆ ϕ ′ ) · ε m ˆ ϕ ′ . . L b ,U,m forces ε m ˆ ϕ ′ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( e ξ − ) (cid:1) ( ε m ˆ ϕ ) 4 . . ε m sending ˆ ϕ to ε m ˆ ϕ is P -stable and, inparticular, that ε m ˆ ϕ only depends on the corresponding ˜ F -morphism ˜ ϕ , thuswe set ε m ˜ ϕ = ε m ˆ ϕ . On the other hand, considering the usual differential map d : C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) −→ C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) . . , we claim that d ( ε m ) = 0 ; indeed, for a second P -morphism ˆ ψ : T → R weget λ ′ m ( ˆ ϕ ) · λ ′ m ( ˆ ψ ) = λ m ( ˆ ϕ ) · ε m ˜ ϕ · λ m ( ˜ ψ ) · ε m ˆ ψ = λ m ( ˆ ϕ · ˆ ψ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( ε m ˜ ϕ ) · ε m ˜ ψ λ ′ m ( ˆ ϕ ) · λ ′ m ( ˆ ψ ) = λ m ( ˆ ϕ · ˆ ψ ) · ε m ˜ ϕ · ˜ ψ . . divisibility of L b ,U,m forces (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( ε m ˜ ϕ ) · ε m ˜ ψ = ε m ˜ ϕ · ˜ ψ . . π b ,U,m T ) is Abelian, with the additive notation we obtain0 = (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ψ ) (cid:1) ( ε m ˜ ϕ ) − ε m ˜ ϕ · ˜ ψ + ε m ˜ ψ . . , proving our claim.At this point, it follows from equality 4.10.4 that ε m = d (ˆ ν ) for some stable element ˆ ν = (ˆ ν Q ) Q ⊂ P in C (cid:0) ˜ P , ˜ l U,m − F / ˜ l U,m F (cid:1) ; that is to say, with thenotation above we get ε m ˜ ϕ = (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ϕ ) (cid:1) (ˆ ν Q ) · ˆ ν − R . . λ ′ m ( ˆ ϕ ) = λ m ( ˆ ϕ ) · (cid:0) (˜ l U,m − F / ˜ l U,m F )( ˜ ϕ ) (cid:1) (ˆ ν Q ) · ˆ ν − R = ˆ ν Q · λ m ( ˆ ϕ ) · ˆ ν − R . . , which amounts to saying that λ ′ m is naturally F -isomorphic to λ m . We aredone
Corollary 4.11.
There exists a unique perfect F -locality P up to natural F -isomorphisms. Proof:
The existence has been proved in Corollary 4.6 above and the unique-ness is an easy consequence of Theorem 4.10.6
References [1] Dave Benson, personal letter 1994[2] Carles Broto, Ran Levi and Bob Oliver,
The homotopy theory of fusionsystems , Journal of Amer. Math. Soc. 16(2003), 779-856.[3] Andrew Chermak.
Fusion systems and localities , Acta Mathematica,211(2013), 47-139.[4] Stefan Jackowski and James McClure,
Homotopy decomposition of classi-fying spaces via elementary abelian subgroups , Topology, 31(1992), 113-132.[5] George Glauberman & Justin Lynd,
Control of fixed points and existenceand uniqueness of centric systems , arxiv.org/abs/1506.01307.[6] Bob Oliver.
Existence and Uniqueness of Linking Systems: Chermak’sproof via obstruction theory , Acta Mathematica, 211(2013), 141-175.[7] Llu´ıs Puig,
Brauer-Frobenius categories , Manuscript notes 1993[8] Llu´ıs Puig, “Frobenius categories versus Brauer blocks” , Progress in Math.274(2009), Birkh¨auser, Basel.[9] Llu´ıs Puig,