Galois descent of equivalences between blocks of p-nilpotent groups
aa r X i v : . [ m a t h . G R ] F e b Galois descent of equivalences between blocks of p -nilpotentgroups ∗ Robert BoltjeDepartment of MathematicsUniversity of CaliforniaSanta Cruz, CA [email protected] Deniz YılmazInstitut f¨ur MathematikFriedrich-Schiller-Universit¨at Jena07737 [email protected]
February 25, 2021
Abstract
We give sufficient conditions on p -blocks of p -nilpotent groups over F p to be splen-didly Rickard equivalent and p -permutation equivalent to their Brauer correspon-dents. The paper also contains Galois descent results on p -permutation modules and p -permutation equivalences that hold for arbitrary groups. In [KL18], Kessar and Linckelmann proved that Brou´e’s Abelian Defect Group Conjecture(originally stated over splitting fields) holds for blocks with cyclic defect groups over arbitrary fields of characteristic p >
0, in particular over the prime field F p . More precisely,if G is a finite group and b is a block idempotent of F p G with cyclic defect group D , thenthere exists a splendid Rickard equivalence between F p Gb and its Brauer correspondentblock algebra F p H Br D ( b ), where H = N G ( D ) and Br D : ( F p G ) D → F p C G ( D ) is the Brauerhomomorphism, an F p -algebra homomorphism which is given by truncation.In this paper we investigate if a similar phenomenon holds for blocks of p -nilpotentgroups. In this case, a positive answer over a splitting field F of characteristic p > G was given by Rickard, see [R96], even without the assumption of abelian defect groups.There, he introduced and used the notion of an endosplit p -permutation resolution in orderto construct such splendid Rickard equivalences. We have two main results. The firstgives sufficient conditions under which there exists such a splendid Rickard equivalencebetween Brauer corresponding blocks of a p -nilpotent group over F p . The second gives ∗ MR Subject Classification:
Keywords: p -permutation modules, trivial sourcemodules, splendid Rickard equivalence, p -permutation equivalence, p -nilpotent groups, Galois descent. p -permutationequivalence , exists over F p .So let G be a p -nilpotent group , i.e., a finite group whose largest normal p ′ -subgroup N is a complement to a (and then each) Sylow p -subgroup of G . Moreover, let F be a finitesplitting field of G of characteristic p >
0. Let ˜ b be a block idempotent of F p G and let b be a block idempotent of F G which occurs in a primitive decomposition of ˜ b in Z ( F G ).Then b is contained in Z ( F N ). Let e be a block idempotent of F N that occurs in theprimitive decomposition of b in Z ( F N ). Adjoining the coefficients of e and of b to F p , oneobtains subfields F p [ b ] ⊆ F p [ e ] ⊆ F . Theorem A
Let G be a p -nilpotent group and let ˜ b be a block idempotent of F p G . Supposethat p is odd or ˜ b has abelian defect groups, and suppose that, with the above notation, F p [ b ] = F p [ e ] . Then there exists a splendid Rickard equivalence between the block algebra F p G ˜ b and its Brauer correspondent block algebra. Theorem A follows from the more precise statement in Proposition 5.15 and Re-mark 5.11. The proof uses Rickard’s original approach in [R96] involving endosplit p -permutation resolutions, a descent result in [KL18], and the classification of endopermu-tation modules over p -groups, see [T07] for a survey article on the latter.A weaker form of Brou´e’s Abelian Defect Group Conjecture states that if b is a blockidempotent of a group algebra F G with abelian defect groups then there exists a p -permutation equivalence between the block algebra F Gb and its Brauer correspondentblock algebra. The notion of a p -permutation equivalence was introduced in [BX08] andextended in [L09] and [BP20]. Theorem B
Let G be a p -nilpotent group with abelian Sylow p -subgroup and let ˜ b be ablock idempotent of F p G . Then there exists a p -permutation equivalence between F p G ˜ b andits Brauer correspondent block algebra. Theorem B follows from the more precise statement in Corollary 5.14 and Remark 5.11.The proof uses again Rickard’s construction and Galois descent arguments for the repre-sentation ring of trivial source modules, see Theorem 2.6 and Lemma 4.3.The paper is arranged as follows. In Section 2 we prove a Galois descent result on p -permutation modules, Theorem 2.6. The definition and basic properties of endosplit p -permutation resolutions are given in Section 3. In Section 4 we collect basic resultson the Galois group action on blocks and prove Lemma 4.3, a Galois descent result for p -permutation equivalences. Finally, in Section 5 we prove the main results.Our notation is standard. For any rings R and S we denote by R mod (resp. R mod S )the categories of finitely generated left R -modules (resp. ( R, S )-bimodules). For objects M and N in a module category or chain complex category we write M | N to indicate that M is isomorphic to a direct summand of N . If H and K are subgroups of a finite group G ,then g ∈ G/H (resp. g ∈ H \ G/K ) indicates that g runs through a set of representativesof the given cosets (resp. double cosets). 2 Galois descent of p -permutation modules Throughout this paper, G and H denote finite groups and F a finite field of characteristic p which is a splitting field for all subgroups of H and G . Moreover, Γ := Gal( F/ F p )denotes the Galois group of F over F p . For any subfield F ′ ⊆ F one has functors − F ′ : F G mod → F ′ G mod and F ⊗ F ′ − : F ′ G mod → F G mod defined by restriction and extension of scalars.
Let Q be a p -subgroup of G and let F ′ be a subfield of F . If M ∈ F ′ G mod is relatively Q -projective then F ⊗ F ′ M ∈ F G mod is relatively Q -projective. If N ∈ F G mod is relatively Q -projective then N F ′ ∈ F ′ G mod is relatively Q -projective. Proof
This follows immediately from the fact that restriction and extension of scalarscommute with Ind GQ .For each σ ∈ Γ one has a functor σ − : F G mod → F G mod (1)which assigns to M ∈ F G mod the
F G -module σ M whose underlying abelian group is equalto M and whose F G -module structure is given by restriction along the ring isomorphism σ − : F G → F G , P g ∈ G α g g P g ∈ G σ − ( α g ) g . For any F G -module homomorphism f one has σ f = f . Similarly one defines the functor σ − : F G mod
F H → F G mod
F H . For any M ∈ F G mod we set stab Γ ( M ) := { σ ∈ Γ | σ M ∼ = M } .We recall from [L18a, Definition 5.4.10] the definition of the Brauer construction func-tor − ( P ) : F G mod → F [ N G ( P ) /P ] mod , for any p -subgroup P of G , and we denote by − ◦ := Hom F ( − , F ) : F G mod → F G mod the functor of taking F -duals. The above functors extend to functors between appropriatecategories of chain complexes and they have the following properties. Let G , H and K be finite groups. Further, let M and N be F G -modules, U an ( F G, F H ) -bimodule, V an ( F H, F K ) -bimodule, L G a subgroup, P G a p -subgroup and σ, τ ∈ Γ . Moreover, let F ′ ⊆ F be a subfield and set ∆ := Gal( F/F ′ ) . Thenone has (a) τ ◦ σ M = τ ( σ M ) , σ ( M ⊕ N ) = σ M ⊕ σ N , and σ ( M ⊗ F N ) = σ M ⊗ F σ N . (b) Res GL ( σ M ) = σ (cid:0) Res GL M (cid:1) and Ind GL ( σ M ) = σ (cid:0) Ind GL M (cid:1) . (c) ( σ M ) ◦ = σ ( M ◦ ) , σ ( U ⊗ F H V ) = σ U ⊗ F H σ V , and ( σ M ) ( P ) = σ (( M )( P )) . (d) F ⊗ F ′ M F ′ ∼ = L σ ∈ ∆ σ M . roof The proofs of (a)–(c) are straightforward. For a proof of Part (d) see [KL18,Proposition 6.3].
Let F ′ be a subfield of F . An indecomposable F ′ G -module M and theindecomposable direct summands of F ⊗ F ′ G M have the same vertices. Similarly, anindecomposable F G -module N and the indecomposable direct summands of N F ′ have thesame vertices. Proof
This follows from Lemma 2.2(d) and Proposition 2.1.Feit attributes the following theorem to Brauer. [F82, Theorem 19.3] Let F ′ be a finite field, A a finite dimensional F ′ -algebra, K/F ′ a field extension, V an absolutely irreducible K ⊗ F ′ A -module such that tr V ( a ) ∈ F ′ for every a ∈ A . Then V has an A -form, i.e., there exists an absolutelyirreducible A -module W such that K ⊗ F ′ W ∼ = V as K ⊗ F ′ A -modules. Let V be an irreducible F G -module and let F ′ := F ∆ denote the fixedfield of ∆ := stab Γ ( V ) . Then there exists an (absolutely) irreducible F ′ G -module W with V ∼ = F ⊗ F ′ W . Proof
This follows from the above theorem noting that if σ V ∼ = V for some σ ∈ Γ then σ (tr V ( g )) = tr V ( g ) for all g ∈ G .Recall from [L18a, Section 5.11] that, for any field F ′ of characteristic p >
0, a p -permutation F ′ G -module M is a direct summand of a finitely generated permutation F ′ G -module. Equivalently, the restriction of M to any p -subgroup of G is a permutationmodule. Also equivalently, the sources of the indecomposable direct summands of M are trivial modules. We denote the Grothendieck group of p -permutation F ′ G -modules V with respect to split short exact sequences by T F ′ ( G ). It is a commutative ring withmultiplication induced by −⊗ F ′ − . The class of V in T F ′ ( G ) is denoted by [ V ]. The classesof indecomposable modules form a standard Z -basis of T F ′ ( G ). If E is a field extension of F then the ring homomorphism T F ( G ) → T E ( G ) of scalar extension is an isomorphism,see [BG07, Theorem 1.9]. The Galois conjugation functors in (1) induce an action of thegroup Γ on T F ( G ) via ring isomorphisms which stabilizes the standard basis. For a subfield F ′ of F the functors of scalar restriction and extension induce a group homomorphism T F ( G ) → T F ′ ( G ) and a ring homomorphism T F ′ ( G ) → T F ( G ) (2)which is injective by the Deuring-Noether Theorem and whose image is contained in thesubring T F ( G ) ∆ of ∆-fixed points, where ∆ := Gal( F/F ′ ). Note that also the abeliangroup T F ( G ) ∆ has a standard Z -basis, namely the ∆-orbit sums of the standard basis of T F ( G ). The goal of this section is the following theorem.4 .6 Theorem Let F ′ be a subfield of F and ∆ := Gal( F/F ′ ) . Then the ring homomor-phism T F ′ ( G ) → T F ( G ) ∆ in (2) induced by scalar extension is an isomorphism, mappingthe standard basis to the standard basis. Before proving the above theorem we need the following propositions.
Let V be an indecomposable p -permutation F G -module and let F ′ = F ∆ be the fixed field of ∆ := stab Γ ( V ) . Then there exists a unique indecomposable p -permutation F ′ G -module W such that V ∼ = F ⊗ F ′ W as F G -modules.
Proof
Let P be a vertex of the module V . Then the Brauer construction V ( P ) of V isprojective indecomposable as an F [ N G ( P ) /P ]-module and its inflation is the Green corre-spondent of V , see [L18a, Theorem 5.10.5]. The quotient module S := V ( P ) /J ( V ( P )) isabsolutely irreducible since F is a splitting field. Since the Green correspondence and tak-ing projective covers commutes with Galois conjugation, the stabilizer of the isomorphismclass of S in Γ is equal to ∆.By Corollary 2.5, there exists an irreducible F ′ [ N G ( P ) /P ]-module T such that S ∼ = F ⊗ F ′ T . Let W ′ be a projective indecomposable F ′ [ N G ( P ) /P ]-module such that W ′ /J ( W ′ ) ∼ = T , and let W ∈ F ′ G mod be the Green correspondent of the inflation of W ′ .We will show that V ∼ = F ⊗ F ′ W .First we claim that the projective F [ N G ( P ) /P ]-module F ⊗ F ′ W ′ is a projective coverof S . In fact, J ( F ⊗ F ′ F ′ [ N G ( P ) /P ]) = F ⊗ F ′ J ( F ′ [ N G ( P ) /P ]), see [L18a, Propositions1.16.14 and 1.16.18], so that J ( F ⊗ F ′ W ′ ) = F ⊗ F ′ J ( W ′ ). With this we obtain( F ⊗ F ′ W ′ ) /J ( F ⊗ F ′ W ′ ) = ( F ⊗ F ′ W ′ ) / ( F ⊗ F ′ J ( W ′ )) ∼ = F ⊗ F ′ ( W ′ /J ( W ′ )) ∼ = S , establishing the claim. Thus, F ⊗ F ′ W ′ ∼ = V ( P ) as F [ N G ( P ) /P ]-modules and also as F [ N G ( P )]-modules after inflation. Since W is the Green correspondent of W ′ , we have F ⊗ F ′ W | F ⊗ F ′ Ind GN G ( P ) ( W ′ ) ∼ = Ind GN G ( P ) ( F ⊗ F ′ W ′ ) ∼ = Ind GN G ( P ) ( V ( P )) . But since the modules V and V ( P ) are Green correspondents, the module V is the uniqueindecomposable direct summand of Ind GN G ( P ) ( V ( P )) with vertex P and has multiplicityone in Ind GN G ( P ) ( V ( P )). Now Corollary 2.3 implies F ⊗ F ′ W ∼ = V , as desired. Proof of Theorem 2.6.
It suffices to show that every standard basis element of T F ( G ) ∆ comes via scalar extension from T F ∆ ( G ). So let V be an indecomposable p -permutation F G -module and set ∆ ′ := stab Γ ([ V ]). Then the ∆-orbit sum of [ V ], i.e., the class of L σ ∈ ∆ / (∆ ∩ ∆ ′ ) σ V is a standard basis element of T F ( G ) ∆ and every standard basis elementis of this form. By Proposition 2.7 there exists an indecomposable p -permutation F ∆ ′ G -module W ′ such that V ∼ = F ⊗ F ∆ ′ W ′ . By Lemma 2.2(d), we have M σ ∈ ∆ / (∆ ∩ ∆ ′ ) σ V ∼ = F ⊗ F ∆ ′ (cid:0) M σ ∈ ∆∆ ′ / ∆ ′ σ W ′ (cid:1) ∼ = F ⊗ F ∆∆ ′ ( W ′ F ∆∆ ′ ) ∼ = F ⊗ F ∆ W with W := F ∆ ⊗ F ∆∆ ′ ( W ′ F ∆∆ ′ ). 5 Endosplit p -permutation resolutions In this section, and only this section, F can be any field of characteristic p . The followingconcept is due to Rickard, see [R96, Section 7]. Let M be a finitely generated F G -module. An endosplit p -permutationresolution of M is a bounded chain complex X of p -permutation F G -modules with ho-mology concentrated in degree 0 such that H ( X ) ∼ = M and such that X ⊗ F X ◦ is splitas chain complex of F G -modules (with G acting diagonally and X ◦ denoting the F -dualof X ). Let X be an endosplit p -permutation resolution of a finitely generated F G -module M .(a) Every direct summand X ′ of X is again an endosplit p -permutation resolution of H ( X ′ ).(b) We can decompose X into a direct sum X = X ′ ⊕ X ′′ of chain complexes suchthat X ′′ is contractible and X ′ has no contractible non-zero direct summand. With X ,also X ′ is then an endo-split p -permutation resolution of M . If X ′′ = 0, we say that X is contractible-free .(c) Taking the 0-th homology induces an F -algebra isomorphism ρ : End K ( F G mod ) ( X ) ∼ = End F G ( M ) , (3)where K ( F G mod ) denotes the homotopy category of chain complexes in
F G mod , see [L18b,Proposition 7.11.2]. If N | M , then the projection map onto N yields an idempotent inEnd F G ( M ) and hence an idempotent in End K ( F G mod ) ( X ) via the isomorphism in (3). Thisidempotent lifts to an idempotent α in End Ch ( F G mod ) ( X ), where Ch ( F G mod ) denotes thecategory of chain complexes in
F G mod . It follows that the direct summand α ( X ) of X is anendosplit p -permutation resolution of N . The lifted idempotent is not unique up to conju-gation, but α ( X ) is unique up isomorphism and contractible direct summands. Therefore,if X is contractible-free, then α ( X ) is uniquely determined by N up to isomorphism in Ch ( F G mod ).(d) Suppose that M ∼ = F ⊗ F ′ M ′ for some subfield F ′ ⊆ F and some M ′ ∈ F ′ G mod .Then, M F ′ ∼ = M ′ [ F : F ′ ] in F ′ G mod and X F ′ ∈ Ch ( F ′ G mod ) is an endosplit p -permutationresolution of M ′ [ F : F ′ ] . By Part (c), also M ′ has an endosplit p -permutation resolution.Conversely, if M ′ has an endosplit p -permutation resolution X ′ ∈ Ch ( F ′ G mod ) then F ⊗ F ′ X ′ is an endosplit p -permutation resolution of F ⊗ F ′ M ′ ∼ = M . Let X V , X U , X V ′ and X U ′ be endosplit p -permutation resolutions of V, U, V ′ , U ′ ∈ F G mod , respectively, and assume that X V and X V ′ are contractible-free.Suppose further that X V ⊕ X U ∼ = X V ′ ⊕ X U ′ in Ch ( F G mod ) are endo-split p -permutationresolutions of V ⊕ U and V ′ ⊕ U ′ , respectively, and that V ∼ = V ′ in F G mod . Then U ∼ = U ′ in F G mod and X V ∼ = X V ′ in Ch ( F G mod ) . roof Taking 0-th homology of X V ⊕ X U and X V ′ ⊕ X U ′ yields V ⊕ U ∼ = V ′ ⊕ U ′ , andthe Krull-Schmidt Theorem implies U ∼ = U ′ . For the second statement let φ : X V ⊕ X U → X V ′ ⊕ X U ′ be an isomorphism in Ch ( F G mod ). Then φ ( X V ) and X V ′ are both directsummands of X V ′ ⊕ X U ′ and contractible-free endo-split p -permutation resolutions of V .Therefore, by [L18b, Proposition 7.11.2] (see also Remark 3.2(b)) they are isomorphic. p -permutation equivalences Since the Galois group Γ acts via F p -algebra automorphisms on the group algebra F G andalso on Z ( F G ), it permutes the block idempotents of
F G . [BKY20, Proposition 4.1] (a) Let b be a block idempotent of F G . Then ˜ b := Tr Γ ( b ) := P σ ∈ Γ / stab Γ ( b ) σ b is a block idempotent of F p G . (b) The map b ˜ b induces a bijection between the set of Γ -orbits of block idempotentsof F G and the set of block idempotents of F p G . (c) If b is a block idempotent of F G and ˜ b := Tr Γ ( b ) is the block idempotent of F p G associated to it as in (a) then ˜ b and b have the same defect groups. Let b be a block of F G with a defect group P and c be the block of F N G ( P ) which is in Brauer correspondence with b . For any σ ∈ Γ , the blocks σ b and σ c areagain in Brauer correspondence. In particular, the stabilizers of b and c in Γ are thesame. Moreover, the blocks ˜ b = Tr Γ ( b ) of F p G and ˜ c = Tr Γ ( c ) of F p N G ( P ) are Brauercorrespondents. Proof
The first assertion follows immediately from the fact that the action of Γ and theBrauer map Br P commute. We have σ ∈ stab Γ ( b ) ⇐⇒ σ ( b ) = b ⇐⇒ Br P ( σ ( b )) =Br P ( b ) ⇐⇒ σ ( c ) = c ⇐⇒ σ ∈ stab Γ ( c ). The last statement follows easily from theadditivity of the Brauer map.Let F ′ be a field of characteristic p and let b and c be central idempotents of F ′ G and F ′ H , respectively. As usual we identify F ′ [ G × H ] = F ′ G ⊗ F ′ F ′ H as F -algebras and weidentify ( F ′ Gb, F ′ Hc )-bimodules with left F ′ [ G × H ]( b ⊗ c ∗ )-modules, where c ∗ is definedby applying the F ′ -linear extension of h h − to c . We write T ∆ ( F ′ Gb, F ′ Hc ) for thesubgroup of T F ′ ( G × H ) spanned by indecomposable F ′ [ G × H ]( b ⊗ c ∗ )-modules whosevertices are twisted diagonal, i.e., of the form { ( φ ( y ) , y ) | y ∈ Q } for some isomorphism φ : Q → P between p -subgroups P and Q of G and H , respectively. Recall from [BP20] thata p -permutation equivalence between F ′ Gb and F ′ Hc is an element ω ∈ T ∆ ( F ′ Gb, F ′ Hc )such that ω · H ω ◦ = [ F ′ Gb ] in T ∆ ( F ′ Gb, F ′ Gb ) and ω ◦ · G ω = [ F ′ Hc ] in T ∆ ( F ′ Hc, F ′ Hc ).Here, · H is induced by − ⊗ F ′ H − , and ω ◦ ∈ T F ′ ( H × G ) is given by taking the F ′ -dual of ω . Note that if F ′ = F then stab Γ ( ω ) stab Γ ( b ) and stab Γ ( ω ) stab Γ ( c ). Let b and c be block idempotents of F G and
F H , respectively. Let ˜ b and ˜ c denote the block idempotents of F p G and F p H associated to b and c as in Proposi-tion 4.1(a). Moreover, let ω ∈ T ∆ ( F Gb, F Hc ) be a p -permutation equivalence between Gb and F Hc . Suppose that we have ∆ := stab Γ ( ω ) = stab Γ ( b ) = stab Γ ( c ) . Then thereexists a p -permutation equivalence between F p G ˜ b and F p H ˜ c . Proof
For any σ ∈ Γ, the Galois conjugate σ ω is a p -permutation equivalence between F G σ b and F H σ c . Hence the sum P σ ∈ Γ / ∆ σ ω ∈ T ∆ ( F G ˜ b, F H ˜ c ) is a p -permutation equiv-alence between L σ ∈ Γ / ∆ F G σ b = F G ˜ b and L σ ∈ Γ / ∆ F H σ c = F H ˜ c . Note that the sum P σ ∈ Γ / ∆ σ ω ∈ T F ( G × H ) is fixed under Γ. By Theorem 2.6, there exists ˜ ω ∈ T F p ( G, H )such that P σ ∈ Γ / ∆ σ ω = F ⊗ F p ˜ ω . It follows that ˜ ω is a p -permutation equivalence between F p G ˜ b and F p H ˜ c . p -nilpotent groups Throughout this section we assume that G is a p -nilpotent group. Thus, G has a normal p ′ -subgroup N such that G/N is a p -group. We fix a block idempotent b of F G anddenote by ˜ b := Tr Γ ( b ) the corresponding block idempotent of F p G , see Proposition 4.1(a).Moreover, we fix a block idempotent e of F N such that be = 0. Then b = P g ∈ G/S g e ,where S := stab G ( e ), and the idempotent e is also a block idempotent of F S . Let Q be aSylow p -subgroup of S . Then Q is a defect group of the block idempotents e of F S , b of F G , and ˜ b of F p G . Finally, set ˜ e := Tr Γ ( e ), the block idempotent of F p N determined by e and set ˜ S := stab G (˜ e ). Then S ˜ S and ˜ b = P G/ ˜ S g ˜ e .The following diagram depicts the subgroups and block idempotents introduced above,together with additional subgroups, block idempotents and simple modules that will beintroduced later. • G, ˜ b, b • HN • ˜ S, ˜ e, e ′ , e V • S, ˜ e, e, e V , V • N, ˜ e, e, e V , V • H, ˜ c, c • ˜ T , ˜ f, f ′ • T, ˜ f, f, U • M, ˜ f, f, U ✏✏✏✏✏✏ ✏✏✏✏✏✏ ✏✏✏✏✏✏ ✏✏✏✏✏✏ • R • Q •{ } ✏✏✏✏✏✏ ✏✏✏✏✏✏ ✏✏✏✏✏✏ The group Γ × G acts on the block idempotents of F N . Set X := stab Γ × G ( e ). Sincestab G ( e ) = S we have k ( X ) := { g ∈ G | (1 , g ) ∈ X } = S . Similarly, k ( X ) := { σ ∈ Γ | ( σ, ∈ X } = stab Γ ( e ). Next we determine the images of X under the projection maps p : Γ × G → Γ and p : Γ × G → G . 8 .1 Lemma One has p ( X ) = ˜ S and S E ˜ S . Proof
Let g ∈ p ( X ). There exists σ ∈ Γ such that ( σ,g ) e = e . Therefore we have˜ e = Tr Γ ( e ) = Tr Γ ( ( σ,g ) e ) = Tr Γ ( σ ( g e )) = Tr Γ ( g e ) = g Tr Γ ( e ) = g ˜ e . This shows that g ∈ stab G (˜ e ) = ˜ S and hence that p ( X ) ˜ S .Now let ˜ s ∈ ˜ S . Then Tr Γ ( ˜ s e ) = ˜ s (Tr Γ ( e )) = ˜ s ˜ e = ˜ e . Since the blocks e and ˜ s e have the same Galois trace, they must be Γ-conjugate, andtherefore ˜ s ∈ p ( X ). This proves the first statement. The second statement holds, since k ( X ) is normal in p ( X ) in general, see [Bc10, p. 24].Next, set e ′ := P ˜ s ∈ ˜ S/S ˜ s e . Then e ′ is a block idempotent of F ˜ S and b = P g ∈ G/ ˜ S g e ′ . One has stab Γ ( e ′ ) = stab Γ ( b ) = p ( X ) . Moreover, ˜ S/S ∼ = stab Γ ( b ) / stab Γ ( e ) is cyclic. Proof
We have stab Γ ( e ′ ) stab Γ ( b ) since b = P g ∈ G/ ˜ S g e ′ . Next, let σ ∈ p ( X ). Thenthere exists ˜ s ∈ ˜ S such that ( σ, ˜ s ) ∈ X , and e ′ = X ˜ s ∈ ˜ S/S ˜ s e = X ˜ s ∈ ˜ S/S ˜ s ( ( σ, ˜ s ) e ) = σ ( X ˜ s ∈ ˜ S/S ˜ s ( ˜ s e )) = σ e ′ , where the last equation holds, because S E ˜ S . This shows that σ ∈ stab Γ ( e ′ ) and hence p ( X ) stab Γ ( e ′ ). Finally, let σ ∈ stab Γ ( b ). Then σ b = b implies that σ ( P g ∈ G/S g e ) = P g ∈ G/S g e . Therefore there exists g ∈ G such that ( σ,g ) e = e , i.e., σ ∈ p ( X ). The proofof the first statement is now complete. The second statement follows from the generalisomorphism p ( X ) /k ( X ) ∼ = p ( X ) /k ( X ), see [Bc10, p. 24], and since Γ is cyclic. One has Tr Γ ( e ′ ) = ˜ e . In particular, ˜ e is a block idempotent of F p ˜ S . Proof
By Lemma 5.2 and since k ( X ) = stab Γ ( e ), we haveTr Γ ( e ′ ) = X σ ∈ Γ /p ( X ) σ e ′ = X σ ∈ Γ /p ( X ) X ˜ s ∈ ˜ S/S σ ( ˜ s e ) = X σ ∈ Γ /p ( X ) X τ ∈ p ( X ) /k ( X ) σ ( τ e )= X σ ∈ Γ /k ( X ) σ e = Tr Γ ( e ) = ˜ e , as desired. The third equation holds, since the classes of ˜ s and τ correspond under theisomorphism p ( X ) /k ( X ) ∼ = p ( X ) /k ( X ) if and only if ( τ, ˜ s ) ∈ X , see [Bc10, p. 24].Let V denote the unique (up to isomorphism) simple F N e -module. By Theorem 2.6and since stab Γ ( V ) = stab Γ ( e ) = k ( X ), there exists a unique simple F p N -module e V suchthat M σ ∈ Γ /k ( X ) σ V ∼ = F ⊗ F p e V . (4)9ince ˜ e acts as identity on the above direct sum, e V is a simple F p N ˜ e -module. Since V is absolutely irreducible, it extends to a (unique up to isomorphism) simple F Se -modulewhich we denote again by V . Similarly, each σ V can be viewed as F S -module, so thatthe left hand side in (4) has an
F S ˜ e -module structure and is Γ-invariant. Again, byTheorem 2.6, the left hand side in (4) regarded as F S -module has an F p -form W ∈ F p S ˜ e mod . Restriction from S to N and the Deuring-Noether Theorem then imply thatRes SN ( W ) ∼ = e V . Thus, e V extends to a simple F S ˜ e -module and (4) is an isomorphism of F S ˜ e -modules. The F p S ˜ e -module e V extends to an F p ˜ S ˜ e -module. Proof
By the Fong-Reynolds correspondence, W := Ind ˜ SS V is the unique simple F ˜ Se ′ -module (up to isomorphism) and stab Γ ( W ) = stab Γ ( e ′ ) = p ( X ). By Theorem 2.6, thereexists a simple F p ˜ S -module f W such that L σ ∈ Γ /p ( X ) σ W ∼ = F ⊗ F p f W . Restriction to S impliesRes ˜ SS ( F ⊗ F p f W ) ∼ = Res ˜ SS ( M σ ∈ Γ /p ( X ) σ W ) ∼ = M σ ∈ Γ /p ( X ) σ (Res ˜ SS W ) ∼ = M σ ∈ Γ /p ( X ) σ (Res ˜ SS Ind ˜ SS V ) ∼ = M σ ∈ Γ /p ( X ) σ ( M ˜ s ∈ ˜ S/S ˜ s V ) ∼ = M σ ∈ Γ /k ( X ) σ V ∼ = F ⊗ F p e V , since L ˜ s ∈ ˜ S/S ˜ s V ∼ = L τ ∈ p ( X ) /k ( X ) τ V , which follows from the argument at the end of theproof of the previous proposition. This shows that Res ˜ SS f W = e V and the result follows. Proposition 5.4 extends the results of Michler [M73, Theorem 3.7] (z=1 inpart(e)).Now set H := N G ( Q ), which is again a p -nilpotent group, and set M := O p ′ ( H ), thelargest normal p ′ -subgroup of H . Then M = H ∩ N = C N ( Q ) . Let c denote the block idempotent of F H which is in Brauer correspondence with b .Then, by Lemmas 4.2 and 5.2, stab Γ ( c ) = stab Γ ( b ) = p ( X ) and ˜ c := Tr Γ ( c ) is the Brauercorrespondent of ˜ b .Further, let f denote the block idempotent of F M whose irreducible module is theGlaubermann correspondent of the Q -stable irreducible module V ∈ F N mod . Then f is QM = QC N ( Q ) = N S ( Q ) = N G ( Q ) ∩ S = H ∩ S =: T -stable and hence it remains a blockidempotent of F T . By [A76], the block idempotents e of F S and f of F T are Brauercorrespondents.
One has c = Tr HT ( f ) and stab H ( f ) = T . roof Since the block idempotents b and c are Brauer correspondents, we have c = Br Q ( b ) = Br Q (Tr GS ( e )) = Br Q (cid:0) X x ∈ Q \ G/S Tr QQ ∩ x S ( x e ) (cid:1) = X x ∈ Q \ G/S Br Q (cid:0) Tr QQ ∩ x S ( x e ) (cid:1) = X x ∈ Q \ GQ x S Br Q ( x e ) . The condition Q x S implies that x − Q S and hence x − Q = s Q for some s ∈ S since Q is a Sylow p -subgroup of S . This means that xs ∈ N G ( Q ) and so x ∈ N G ( Q ) S . Thereforethe above sum can be written as c = X x ∈ N G ( Q ) / ( N G ( Q ) ∩ S ) Br Q ( x e ) = X x ∈ H/T x Br Q ( e ) = Tr HT ( f ) , since f = Br Q ( e ). This proves the first assertion. The group stab H ( f ) has the group Q as a Sylow p -subgroup, since Q is a defect group of the bock c . This shows thatstab H ( f ) = QM = T , as desired.Let ˜ f := Tr Γ ( f ), ˜ T := stab H ( ˜ f ), f ′ := Tr ˜ TT ( f ) and Y := stab Γ × H ( f ). Since the blocks e and f are Brauer correspondents, we have k ( Y ) = stab Γ ( f ) = stab Γ ( e ) = k ( X ) . (5)Moreover, by Lemmas 5.1 and 5.2,stab Γ ( f ′ ) = p ( Y ) = stab Γ ( c ) = stab Γ ( b ) = p ( X ) = stab Γ ( e ′ ) , (6) p ( Y ) = ˜ T and k ( Y ) = T , and therefore˜
T /T = p ( Y ) /k ( Y ) ∼ = p ( Y ) /k ( Y ) = p ( X ) /k ( X ) ∼ = p ( X ) /k ( X ) = ˜ S/S (7)which implies that ˜ T = H ∩ ˜ S . We recall Rickard’s construction of a splendid Rickard equivalence between
F Se and
F Q , i.e., a chain complex X of relatively ∆( Q )-projective p -permutation ( F Se, F Q )-bimodules such that X ⊗ F Q X ◦ ∼ = F Se and X ◦ ⊗ F S X ∼ = F Q in the homotopy categoriesof (
F Se, F Se )-bimodules and (
F Q, F Q )-bimodules, respectively, where
F Se and
F Q areconsidered as chain complex concentrated in degree 0. For more details we refer the readerto [R96].Set ∆ Q S := { ( nq, q ) : n ∈ N, q ∈ Q } S × Q and note that p : S × Q → S restricts toan isomorphism ∆ Q S ∼ → S . The module Res SQ V is a capped endopermutation F Q -module.In everything that follows, we suppose that there exists W ∈ F p Q mod such thatRes SQ V ∼ = F ⊗ F p Q W and W has an endosplit p -permutation resolution X W . (8)11hen the complex F ⊗ F p X W is an endosplit p -permutation resolution of Res SQ V . Bythe proof in [R96, Lemma 7.7], see also Remark (a) at the end of Section 7 in [R96],the induced complex Ind SQ ( F ⊗ F p X W ) ∼ = F ⊗ F p Ind SQ X W is an endosplit p -permutationresolution of Ind SQ Res SQ V as F S -modules. Since V | Ind SQ Res SQ V , there exists a directsummand Y V of F ⊗ F p Ind SQ X W such that Y V is an endosplit p -permutation resolution of V as an F S -module, and we may choose Y V to be contractible-free, see Remark 3.2(c)and (b). The induced chain complex Ind S × Q ∆ Q S Y V is then a splendid Rickard equivalencebetween F Se and
F Q , see [R96, Theorem 7.8] and its following Remark (a). (a) The ( F ˜ Se ′ , F Se )-bimodule F ˜ Se induces a Morita equivalence (the Fong-Reynoldscorrespondence) between F ˜ Se ′ and F Se . Hence the complex F ˜ Se ⊗ F S
Ind S × Q ∆ Q S Y V gives asplendid Rickard equivalence between F ˜ Se ′ and F Q .(b) For any
F S -module M , let M ⊗ F F Q be the (
F S, F Q )-bimodule, given by s ( m ⊗ x ) y := sm ⊗ qxy , for s = nq ∈ S , n ∈ N , m ∈ M , and x, y, q ∈ Q .It is straightforward to check that the map φ M : M ⊗ F F Q → F [ S × Q ] ⊗ F ∆ Q S Mv ⊗ q (1 , q − ) ⊗ v is an isomorphism of ( F S, F Q )-bimodules and that it is natural in M . Therefore, it yieldsan isomorphism Y V ⊗ F F Q ∼ = Ind S × Q ∆ Q S Y V of chain complexes of ( F Se, F Q )-bimodules.(c) Let U be the simple F M -module belonging to the block idempotent f . Since Q isnormal in H , we have T = Q × M . Thus, the unique extension of U to T (with Q actingtrivially on U ) is a p -permutation F T -module and plays the same role as the complex Y V .Similar as in (a), the bimodule F ˜ T f ⊗ F T
Ind T × Q ∆ Q T U induces a splendid Rickard equivalencebetween F ˜ T f ′ and F Q .(d) Altogether, the complex Z := F ˜ Se ⊗ F S
Ind S × Q ∆ Q S Y V ⊗ F Q
Ind Q × T (∆ Q T ) ◦ ( U ) ◦ ⊗ F T f F ˜ T induces a splendid Rickard equivalence between F ˜ Se ′ and F ˜ T f ′ . Here, (∆ Q T ) ◦ := { ( q, t ) ∈ Q × T | ( t, q ) ∈ ∆ Q T } . Set ω := X n ∈ Z ( − n [ Z n ] ∈ T ∆ ( F ˜ Se ′ , F ˜ T f ′ ) . By [BX08, Theorem 1.5], ω is a p -permutation equivalence between F ˜ Se ′ and F ˜ T f ′ .Moreover, the isomorphism in (b) implies that Z = F ˜ Se ⊗ F S
Ind S × Q ∆ Q S Y V ⊗ F Q
Ind Q × T (∆ Q T ) ◦ ( U ) ◦ ⊗ F T f F ˜ T ∼ = F ˜ Se ⊗ F S ( Y V ⊗ F F Q ) ⊗ F Q ( F Q ⊗ F U ◦ ) ⊗ F T f F ˜ T ∼ = F ˜ Se ⊗ F S ( Y V ⊗ F F Q ⊗ F U ◦ ) ⊗ F T f F ˜ T . R be a Sylow p -subgroup of ˜ T containing Q . Then ˜ T = RM , and by (7), R is alsoas Sylow p -subgroup of ˜ S so that ˜ S = RN . For every r ∈ R , one has an isomorphism F ˜ Se ⊗ F S ( Y V ⊗ F F Q ⊗ F U ◦ ) ⊗ F T f F ˜ T ∼ = F ˜ S r e ⊗ F S ( r Y V ⊗ F F Q ⊗ F r ( U ◦ )) ⊗ F T r f F ˜ T of chain complexes of ( F ˜ Se ′ , F ˜ T f ′ ) -bimodules. Proof
For any M ∈ F S mod , consider the map F ˜ Se ⊗ F S ( M ⊗ F F Q ⊗ F U ◦ ) ⊗ F T f F ˜ T → F ˜ S r e ⊗ F S ( r M ⊗ F F Q ⊗ F r ( U ◦ )) ⊗ F T r f F ˜ T , mapping a ⊗ ( y ⊗ q ⊗ u ) ⊗ b to ar − ⊗ ( y ⊗ rqr − ⊗ u ) ⊗ rb . It is straightforward to checkthat it is well-defined, an isomorphism of ( F ˜ Se ′ , F ˜ T f ′ )-bimodules, and functorial in M .Thus, it yields the desired isomorphism of chain complexes.For easy reference we formulate the following hypotheses. (i) Every subgroup of Q is normal in R .(ii) Res SQ ( V ) has an F p -form W ∈ F p Q mod and W has an endosplit p -permutationresolution X W , see also (8). (a) By the classification of capped endopermutation modules, Hypothesis(ii) is satisfied if p is odd. If p = 2, it is satisfied if Q does not have a subsection isomorphicto the quaternion group of order 8. See [T07] for more details.(b) If R is abelian, both hypotheses are satisfied. Suppose that the hypotheses in 5.10(i) and (ii) hold. For any ˜ s ∈ ˜ S , one has an isomorphism ˜ s (Ind SQ X W ) ∼ = Ind SQ X W of complexes of F p S -modules. Inparticular, for any ˜ s ∈ ˜ S , one has ˜ s (Ind SQ Res SQ V ) ∼ = Ind SQ Res SQ V as F S -modules.
Proof
Recall that the complex Ind SQ X W consists of direct sums of permutation F p S -modules of the form F p [ S/Q ] where Q Q . Let ˜ s ∈ ˜ S = N R and write ˜ s = nr for some n ∈ N and r ∈ R . By Hypothesis 5.10 we have r Q = Q and hence ˜ s Q = n Q . This implies that the permutation modules F p [ S/Q ] and ˜ s ( F p [ S/Q ]) ∼ = F p [ S/ n Q ]are isomorphic. It is straightforward to show that this isomorphism commutes with thedifferentials of the complexes and hence we have ˜ s (Ind SQ X W ) ∼ = Ind SQ X W . For the lastassertion note that this also implies that ˜ s ( F ⊗ F p Ind SQ X W ) ∼ = F ⊗ F p Ind SQ X W . Since themodule Ind SQ Res SQ V is the homology of the complex F ⊗ F p Ind SQ X W the result follows. Suppose that the hypotheses in 5.10(i) and (ii) hold. Then one has stab Γ ( Z ) = stab Γ ( ω ) = p ( X ) = stab Γ ( e ′ ) = stab Γ ( f ′ ) = p ( Y ) . roof Note that p ( X ) = stab Γ ( e ′ ) = stab Γ ( f ′ ) = p ( Y ) hold by (6).Since the complex Z induces a splendid Rickard equivalence between F ˜ Se ′ and F ˜ T f ′ ,the inclusion stab Γ ( Z ) stab Γ ( e ′ ) is immediate. Thus, stab Γ ( Z ) p ( X ). Conversely,if σ ∈ p ( X ), then σ e = ˜ s e for some ˜ s ∈ ˜ S . Write ˜ s = rn for some r ∈ R and n ∈ N and note that we have σ e = r e . This implies that σ V ∼ = r V as F S -modules. ByProposition 5.12, we have σ ( F ⊗ F p Ind SQ X W ) ∼ = F ⊗ F p Ind SQ X W ∼ = r ( F ⊗ F p Ind SQ X W ) ascomplexes of F S -modules and σ (Ind SQ Res SQ V ) ∼ = Ind SQ Res SQ V ∼ = r (Ind SQ Res SQ V ) as F S -modules. Therefore Lemma 3.3 implies that σ Y V ∼ = r Y V as complexes of F S -modules, as Y V was chosen to be contractible-free, see 5.7. Since the idempotents e of F S and f of F T are Brauer correspondents, also r e and r f are Brauer correspondents. Since the Galoisaction commutes with the Brauer correspondence, σ e = r e implies σ f = r f . Therefore wehave σ U ∼ = r U as F T -modules. By Lemma 5.9, we obtain σ Z ∼ = σ (cid:16) F ˜ Se ⊗ F S ( Y V ⊗ F F Q ⊗ F U ◦ ) ⊗ F T f F ˜ T (cid:17) ∼ = F ˜ S σ e ⊗ F S ( σ Y V ⊗ F F Q ⊗ F σ U ◦ ) ⊗ F T σ f F ˜ T ∼ = F ˜ S r e ⊗ F S ( r Y V ⊗ F F Q ⊗ F r U ◦ ) ⊗ F T r f F ˜ T ∼ = F ˜ Se ⊗ F S ( Y V ⊗ F F Q ⊗ F U ◦ ) ⊗ F T f F ˜ T ∼ = Z .
This proves that stab Γ ( Z ) = p ( X ).Since ω is a p -permutation equivalence between F ˜ Se ′ and F ˜ T f ′ , the inclusionstab Γ ( ω ) stab Γ ( e ′ ) = p ( X ) is clear. The inclusion stab Γ ( Z ) stab Γ ( ω ) is immedi-ate, and the proof is complete. Suppose that the hypotheses in 5.10(i) and (ii) hold. (a)
There exists a p -permutation equivalence between F p ˜ S ˜ e and F p ˜ T ˜ f . (b) There exists a p -permutation equivalence between F p G ˜ b and F p H ˜ c . Proof (a) By Lemma 5.13 we have stab Γ ( ω ) = stab Γ ( e ′ ) = stab Γ ( f ′ ). Hence by Lemma4.3 there exists a p -permutation equivalence between F p ˜ S ˜ e and F p ˜ T ˜ f .(b) The p -permutation bimodule F p G ˜ e induces a Morita equivalence, hence a p -permutation equivalence, between F p G ˜ b and F p ˜ S ˜ e . Similarly, the bimodule F p H ˜ f in-duces a p -permutation equivalence between F p H ˜ c and F p ˜ T ˜ f . The result follows now fromPart (a). Suppose that stab Γ ( e ) = stab Γ ( b ) and that Res SQ V has an endosplit p -permutation resolution. (a) There exists a splendid Rickard equivalence between F p S ˜ e and F p T ˜ f . (b) There exists a splendid Rickard equivalence between F p G ˜ b and F p H ˜ c . Proof (a) The equality stab Γ ( e ) = stab Γ ( b ) implies that we havestab Γ ( f ) = stab Γ ( c ) = stab Γ ( b ) = stab Γ ( e ) , ˜ S = S , ˜ T = T , e ′ = e and f ′ = f ,
14y (5) and (6). Let F p [ e ] denote the smallest field containing the coefficients of the idempo-tent e . Then F ′ := F p [ e ] = F p [ f ] ⊆ F . By Corollary 2.5, there exists an absolutely simple F ′ N -module V ′ such that V ∼ = F ⊗ F ′ V ′ , the unique simple module in the block F N e .Since e is S -stable, also V ′ extends to an F ′ S -module that we again denote by V ′ . Then V ∼ = F ⊗ F ′ V ′ also as F Se -modules. Since V ∈ F S mod has an endosplit p -permutation res-olution, also V ′ ∈ F ′ S mod has an endosplit p -permutation resolution X ′ ∈ Ch ( F ′ S mod ),see Remark 3.2(d). Since F ′ is a splitting field of V ′ as F ′ N -module, we may use theresults from Theorem 7.8 and the following Remark (a) in [R96] in order to see thatInd S × Q ∆ Q S ( X ′ ) is a splendid Rickard equivalence between F ′ Se and F ′ Q . Using the unique F ′ -form U ′ ∈ F ′ M mod of U ∈ F M mod and its unique extension to an F ′ T -mdoule, wesimilarly obtain that Ind T × Q ∆ Q T ( U ′ ) induces a splendid Morita equvivalence between F ′ T f and F ′ Q . Thus, the chain complex Z ′ = Ind S × Q ∆ Q S ( X ′ ) ⊗ F Q
Ind Q × T (∆ Q T ) ◦ ( U ′ ) ◦ is a splendid Rickard equivalence between F ′ Se and F ′ T f . The result now follows from[KL18, Theorem 6.5].(b) This follows from Part (a) as in the proof of Corollary 5.14(b).
References [A76]
J.L. Alperin:
The main problem of block theory. In:
Proceedings of the Con-ference on Finite Groups , Academic Press, New York, (1976), 341–356.[BG07]
R. Boltje, A. Glesser:
On p-monomial modules over local domains.
J. GroupTheory (2007), 173–183.[BKY20] R. Boltje, C¸ . Karag¨uzel, D. Yılmaz:
Fusion systems of blocks of finitegroups over arbitrary fields.
Pacific Journal of Mathematics (1) (2020), 29–41.[BP20]
R. Boltje, P. Perepelitsky: p -permutation equialences between blocks ofgroup algebras. arXiv:2007.09253.[BX08] R. Boltje, B. Xu: On p -permutation equivalences: Between Rickard equiv-alences and isotypies. Transactions of the American Mathematical Society (2008), 5067–5087.[Bc10]
S. Bouc:
Biset functors for finite groups. Lecture Notes in Mathematics, 1990.Springer-Verlag, Berlin, 2010.[F82]
W. Feit:
The representation theory of finite groups. North Holland, 1982.[KL18]
R. Kessar, M. Linckelmann:
Descent of equivalences and character bijec-tions. Geometric and topological aspects of the representation theory of finitegroups, 181–212, Springer Proc. Math. Stat., 242, Springer, Cham, 2018.15L09]
M. Linckelmann:
Trivial source bimodule rings for blocks and p-permutationequivalences.
Trans. Amer. Math. Soc. (2009), 1279–1316.[L18a]
M. Linckelmann:
The block theory of finite group algebras. Vol. I. LondonMathematical Society Student Texts, 91. Cambridge University Press, Cam-bridge, 2018.[L18b]
M. Linckelmann:
The block theory of finite group algebras. Vol. II. LondonMathematical Society Student Texts, 92. Cambridge University Press, Cam-bridge, 2018.[M73]
G. Michler:
The blocks of p -nilpotent groups over arbitrary fields. Journal ofAlgebra (1973), 303–315.[R96] J. Rickard:
Splendid equivalences: derived categories and permutation mod-ules.
Proceedings of the London Mathematical Society (1996), 331–358.[T07] J. Th´evenaz:
Endo-permutation modules, a guided tour. In: