Blockwise relations between triples, and derived equivalences for wreath products
aa r X i v : . [ m a t h . R T ] A ug Blockwise relations between triples, and derivedequivalences for wreath products
Andrei Marcus and Virgilius-Aurelian Minut , ˘a Babes , -Bolyai University of Cluj-Napoca, RomaniaFaculty of Mathematics and Computer ScienceDepartment of Mathematicsemail : [email protected] email : [email protected] Abstract.
Motivated by the reduction techniques involving character triples for the local-global conjectures, we show that a blockwise relation between module triples is a consequenceof a derived equivalence with additional properties. Moreover, we show that this relation iscompatible with wreath products.
MSC 2010.
Key words.
Group algebra, block, Brauer map, graded algebra, wreath product, derived equiv-alence
One of the main approaches to the local-global conjectures in the modular representationtheory of finite groups is to show that they are a consequence of some inductive conditionson simple groups. Recent results (see Britta Sp¨ath’s surveys [17] and [18]) use the language ofcharacter triples and of the various relations between them (denoted by ≥ c and ≥ b ), in order toobtain these reduction theorems.This paper comes, first, as a followup to the study done in [13], in which we have given aversion for module triples of the relation ≥ c , and we have proved in [13, Theorem 6.7] thatit is a consequence of a group graded Rickard equivalence with additional properties. Here, inDefinition 5.1 we also provide a module triple version of the relation ≥ b (see [18, Definition 4.2]),and we prove in Proposition 5.6 that this too is a consequence of a special type of group gradedderived equivalences which is compatible in a certain sense with the Brauer map.Our second objective is to build group graded derived and Rickard equivalences for wreathproducts. Some technical details are already developed in [14] for Morita equivalences. Suchconstructions are again motivated by the reduction methods, which require the compatibilityof the relations between character triples and the wreath product constructions (see Sp¨ath [16,Theorem 5.2] and [18, Theorem 2.21]). Theorem 3.7 below improves [11, Theorem 5.2.12] inseveral ways, by taking into account all the additional structure that we deal with. As alreadynoted by Zimmermann [19], a certain “ p ′ -condition” on the order of the grading groups, whichappears in [11, Theorem 5.2.12], is actually not needed in the case of derived equivalences, butis needed in the case of Rickard equivalences. Finally, Theorem 5.8 and Corollary 5.9 are themain results of this paper, and establish the compatibility of the relation ≥ b between moduletriples with wreath products of derived equivalences.The material is organized as follows: In Section 2 we introduce the general notations and werecall from [13] our basic definitions of a G -graded algebra over a G -graded G -acted algebra C , of a G -graded bimodule over C , and of a G -graded Morita equivalence over C , where G is afinite group. We also recall from Harris [4, 5] some needed facts on the behavior of the Jacobsonradical, of centralizers, and of the Brauer map with respect to tensor products. For generalconcepts and results we refer to [9] and [11].In Section 3 we recall from [14] the construction of wreath product for group graded algebrasand bimodules over C , and we extend them to chain complexes of G -graded bimodules over C . Marcus and V. A. Minut , ˘a Our main result of this section, Theorem 3.7, says that the wreath product between a chaincomplex of G -graded bimodules over C and the symmetric group of order n , S n , is a complexof G ≀ S n -graded bimodules over C ⊗ n , and moreover, if the given complex induces a G -gradedderived (respectively Rickard) equivalence over C , then its wreath product with S n (respectivelya p ′ -subgroup of S n ) will induce a group graded derived (respectively Rickard) equivalence over C ⊗ n . Our group graded algebras here are block extensions, but it is clear that most of thestatements are true for more general group graded algebras.In Section 4 we recall from [13] the definitions of a module triple and that of the relation ≥ c between module triples. In Proposition 4.3 we prove that the relation ≥ c is compatible withwreath products of G -graded derived equivalences over C .In Section 5 we introduce the relation ≥ b between module triples as a refinement of the relation ≥ c , by using the Harris-Kn¨orr correspondence (see Definition 5.1). Note that our definition doesnot fully cover [18, Definition 4.2], because there block induction in a more general situationis considered. We also introduce in Definition 5.5 a notion of a derived equivalence compatiblewith the Brauer map. This a a weaker condition that that of a splendid or basic equivalence,and is inspired by the results of [12], which connect basic Morita equivalences with the mainresult of Dade [3]. We prove in Proposition 5.6 that the relation ≥ b between module triples is aconsequence of a certain group graded derived equivalence compatible with the Brauer map. InTheorem 5.8 and Corollary 5.9 we prove that these equivalences, and the relations ≥ b betweenmodule triples induced by them are compatible with wreath products. All rings in this paper are associative with identity = and all modules are left (unlessotherwise specified) unital and finitely generated. Throughout this article n will represent anarbitrary nonzero natural number.We consider a finite group G , a p -modular system ( K , O , k ) , where O is a complete discretevaluation ring, K is the field of fractions of O and k = O /J ( O ) is its residue field. We assumethat k is algebraically closed, and that K contains all the | G | -th roots of unity. Let N be a normal subgroup of G , and denote ¯ G := G/N . Note that most results in thispaper will use “ ¯ G -gradings”, although this is not everywhere needed. The reason is given by thefact that our main applications concern the ¯ G -graded algebra A = b O G , where b is a ¯ G -invariantblock of O N .We recall from [13] the following definitions: Definition 2.3. An O -algebra C is a ¯ G -graded ¯ G -acted algebra if(1) C is ¯ G -graded, and we write C = L ¯ g ∈ ¯ G C ¯ g ;(2) ¯ G acts on C (on the left);(3) for all ¯ g, ¯ h ∈ ¯ G and for all c ∈ C ¯ h we have c ¯ g ∈ C ¯ h ¯ g . Definition 2.4.
Let C be a ¯ G -graded ¯ G -acted algebra. We say the A is a ¯ G -graded O -algebraover C if there is a ¯ G -graded ¯ G -acted algebra homomorphism ζ : C → C A ( B ) , where B := A and C A ( B ) is the centralizer of B in A , i.e. for any ¯ h ∈ ¯ G and c ∈ C ¯ h , we have ζ ( c ) ∈ C A ( B ) ¯ h , and for every ¯ g ∈ ¯ G , ζ ( c ¯ g ) = ζ ( c ) ¯ g . Definition 2.5.
Let A and A ′ be two ¯ G -graded crossed products over a ¯ G -graded ¯ G -actedalgebra C , with structure maps ζ and ζ ′ , respectively.a) We say that ˜ M is a ¯ G -graded ( A, A ′ ) -bimodule over C if:(1) ˜ M is an ( A, A ′ ) -bimodule;(2) ˜ M has a decomposition ˜ M = L ¯ g ∈ ¯ G ˜ M ¯ g such that A ¯ g ˜ M ¯ x A ′ ¯ h ⊆ ˜ M ¯ g ¯ x ¯ h , for all ¯ g, ¯ x, ¯ h ∈ ¯ G ;(3) ˜ m ¯ g c = c ¯ g ˜ m ¯ g , for all c ∈ C , ˜ m ¯ g ∈ ˜ M ¯ g , ¯ g ∈ ¯ G , where c ˜ m = ζ ( c ) ˜ m and ˜ mc = ˜ mζ ′ ( c ) , forall c ∈ C , ˜ m ∈ ˜ M . erived equivalences for wreath products b) ¯ G -graded ( A, A ′ ) -bimodules over C form a category, where the morphisms between ¯ G -graded ( A, A ′ ) -bimodules over C are just homomorphisms between ¯ G -graded ( A, A ′ ) -bimodules. Definition 2.6.
Let A and A ′ be two ¯ G -graded crossed products over a ¯ G -graded ¯ G -acted alge-bra C , and let ˜ M be a ¯ G -graded ( A, A ′ ) -bimodule over C . Clearly, the A -dual ˜ M ∗ = Hom A ( ˜ M, A ) of ˜ M is a ¯ G -graded ( A ′ , A ) -bimodule over C . We say that ˜ M induces a ¯ G -graded Moritaequivalence over C between A and A ′ , if ˜ M ⊗ A ′ ˜ M ∗ ≃ A as ¯ G -graded ( A, A ) -bimodules and˜ M ∗ ⊗ A ˜ M ≃ A ′ as ¯ G -graded ( A ′ , A ′ ) -bimodules.We will need certain properties of tensor products of algebras endowed with group actions orgroup gradings. We rely on the results of Harris [4, 5], which extend the results of K¨ulshammer[8], and of Alghamdi and Khammash [1].Let A and A ′ be two ¯ G -graded crossed products, hence A ⊗ A ′ is a ¯ G × ¯ G -graded crossedproduct with 1-component B ⊗ B ′ . We assume from now on, that A and A ′ are free and finitelygenerated as O -modules. We start with the graded Jacobson radical. By [11, Proposition 1.5.11] and [4, Section 2]we have: J gr ( A ⊗ A ′ ) = J ( B ⊗ B ′ )( A ⊗ A ′ )= ( J ( B ) ⊗ B ′ + B ⊗ J ( B ′ ))( A ⊗ A ′ )= J ( B ) A ⊗ A ′ + A ⊗ J ( B ′ ) A ′ = J gr ( A ) ⊗ A ′ + A ⊗ J gr ( A ′ ) . From this equality it easily follows that A ⊗ A ′ /J gr ( A ⊗ A ′ ) ≃ A/J gr ( A ) ⊗ A ′ /J gr ( A ′ ) . Moreover, these results imply the following: J gr ( A ⊗ n ) = A ⊗ n J ( B ⊗ n ) and A ⊗ n /J gr ( A ⊗ n ) ≃ ( A/J gr ( A )) ⊗ n , where A ⊗ n := A ⊗ . . . ⊗ A ( n times) and B ⊗ n is its identity component. Because under our assumptions the Hom functors behave well with respect to tensor prod-ucts, we have the isomorphisms C A ( B ) ⊗ C A ′ ( B ′ ) ≃ End A ⊗ B op ( A ) op ⊗ End A ′ ⊗ B ′ op ( A ′ ) op ≃ End ( A ⊗ A ′ ) ⊗ ( B ⊗ B ′ ) op ( A ⊗ A ′ ) op ≃ C A ⊗ A ′ ( B ⊗ B ′ ) of ¯ G × ¯ G ′ -graded ¯ G × ¯ G ′ -acted algebras. Finally, for the Brauer construction, if A is a G -acted O -algebra and Q is a p -subgroup of G , by [5, Section 1], we have the commutative diagram ( A Q ) ⊗ n / / Br ⊗ nQ (cid:15) (cid:15) ( A ⊗ n ) Q n Br Qn (cid:15) (cid:15) A ( Q ) ⊗ n / / A ⊗ n ( Q n ) of N G ( Q ) n -acted algebras, where the horizontal maps are isomorphisms. Consider the notations from Section 2.
The wreath product ¯ G ≀ S n is the semidirect product ¯ G n ⋊ S n , where the symmetric group S n acts on ¯ G n (on the left) by permuting the components: ( g , . . . , g n ) σ := ( g σ − ( ) , . . . , g σ − ( n ) ) , . Marcus and V. A. Minut , ˘a for all g , . . . , g n ∈ ¯ G and σ ∈ S n . More exactly, the elements of ¯ G ≀ S n are of the form (( g , . . . , g n ) , σ ) , and the multiplication is: (( g , . . . , g n ) , σ )(( h , . . . , h n ) , τ ) := (( g , . . . , g n ) · ( h , . . . , h n ) σ , στ ) , for all g , . . . , g n , h , . . . , h n ∈ ¯ G and σ, τ ∈ S n .Similarly, if A is an O -algebra, the wreath product A ≀ S n is the skew group algebra A ≀ S n := A ⊗ n ⊗ O S n between A ⊗ n and S n , with multiplication (( a ⊗ . . . ⊗ a n ) ⊗ σ ) (( b ⊗ . . . ⊗ b n ) ⊗ τ ) = ( a b σ − ( ) ⊗ . . . ⊗ a n b σ − ( n ) ) ⊗ στ, for all ( a ⊗ . . . ⊗ a n ) ⊗ σ , ( b ⊗ . . . ⊗ b n ) ⊗ τ ∈ A ≀ S n .We recall from [14], Lemma 4.3 under the following form: Lemma 3.2.
Let A be a ¯ G -graded crossed product over the ¯ G -graded ¯ G -acted algebra C . Thefollowing statements hold: C ⊗ n is a ¯ G ≀ S n -acted ¯ G n -graded algebra, where ( c ⊗ . . . ⊗ c n ) (( g ,...,g n ) ,σ ) := c σ − ( ) g ⊗ . . . ⊗ c σ − ( n ) g n . A ≀ S n is a ¯ G ≀ S n -graded crossed product over C ⊗ n , with (( g , . . . , g n ) , σ ) -component ( A ≀ S n ) (( g ,...,g n ) ,σ ) := (( A g ⊗ . . . ⊗ A g n ) ⊗ O σ ) , for each (( g , . . . , g n ) , σ ) ∈ ¯ G ≀ S n , and with structural ¯ G ≀ S n -graded ¯ G ≀ S n -acted algebrahomomorphism ζ wr : C ⊗ n → C A ≀ S n ( B ⊗ n ) given by the composition ζ ⊗ n : C ⊗ n → C A ( B ) ⊗ n ⊆ C A ≀ S n ( B ⊗ n ) . Let A and A ′ be two ¯ G -graded crossed products over the ¯ G -graded ¯ G -acted algebra C , withidentity components B and B ′ respectively.If ˜ M is an ( A, A ′ ) -bimodule, the action of S n on ˜ M ⊗ n is defined by ( ˜ m ⊗ . . . ⊗ ˜ m n ) σ := ˜ m σ − ( ) ⊗ . . . ⊗ ˜ m σ − ( n ) , for all ˜ m , . . . , ˜ m n ∈ ˜ M and σ ∈ S n . As an O -module, the wreath product ˜ M ≀ S n is˜ M ≀ S n := ˜ M ⊗ n ⊗ O S n . Note that regarding A ≀ S n and A ′ ≀ S n as S n -graded algebras, we may consider the diagonalsubalgebra: ∆ S n := ∆ S n ( A ≀ S n ⊗ ( A ′ ≀ S n ) op ) := ( A ≀ S n ⊗ ( A ′ ≀ S n ) op ) δ ( S n ) , where δ ( S n ) := (cid:8) ( σ, σ − ) | σ ∈ S n (cid:9) . It is easy to see that: ∆ S n ( A ≀ S n ⊗ ( A ′ ≀ S n ) op ) ≃ ( A ⊗ A ′ op ) ≀ S n , as ¯ G ≀ S n -graded algebras, and thus we have that˜ M ≀ S n ≃ ( A ≀ S n ⊗ ( A ′ ≀ S n ) op ) ⊗ ∆ Sn ˜ M ⊗ n , as ¯ G ≀ S n -graded ( A ≀ S n , A ′ ≀ S n ) -bimodules.We recall [14, Theorem 5.3], which extends [11, Theorem 5.1.21] to the case of group gradedMorita equivalences over a group graded group acted algebra: Theorem 3.4.
Let ˜ M be a ¯ G -graded ( A, A ′ ) -bimodule over C , with identity component M . Then,the following statements hold: erived equivalences for wreath products (1) ˜ M ≀ S n is a ¯ G ≀ S n -graded ( A ≀ S n , A ′ ≀ S n ) -bimodule over C ⊗ n , with scalar multiplication (( a ⊗ . . . ⊗ a n ) ⊗ σ ) (( ˜ m ⊗ . . . ⊗ ˜ m n ) ⊗ τ ) (( a ′ ⊗ . . . ⊗ a ′ n ) ⊗ π )= ( a ⊗ . . . ⊗ a n ) · σ ( ˜ m ⊗ . . . ⊗ ˜ m n ) · στ ( a ′ ⊗ . . . ⊗ a ′ n ) ⊗ στπ, and with (( g , . . . , g n ) , σ ) -component ( ˜ M ≀ S n ) (( g ,...,g n ) ,σ ) = ( ˜ M g ⊗ . . . ⊗ ˜ M g n ) ⊗ O σ. (2) There are isomorphisms of ¯ G ≀ S n -graded ( A ≀ S n , A ′ ≀ S n ) -bimodules over C ⊗ n : f : ( A ≀ S n ) ⊗ B ⊗ n M ⊗ n → ˜ M ≀ S n , (( a ⊗ . . . ⊗ a n ) ⊗ σ ) ⊗ ( m ⊗ . . . ⊗ m n ) → (( a ⊗ . . . ⊗ a n ) · σ ( m ⊗ . . . ⊗ m n )) ⊗ σ, and g : M ⊗ n ⊗ B ′⊗ n ( A ′ ≀ S n ) → ˜ M ≀ S n , ( m ⊗ . . . ⊗ m n ) ⊗ (( a ′ ⊗ . . . ⊗ a ′ n ) ⊗ σ ) → (( m ⊗ . . . ⊗ m n ) · ( a ′ ⊗ . . . ⊗ a ′ n )) ⊗ σ. (3) If ˜ M induces a ¯ G -graded Morita equivalence over C between A and A ′ , then ˜ M ≀ S n induces a ¯ G ≀ S n -graded Morita equivalence over C ⊗ n between A ≀ S n and A ′ ≀ S n . Now, if ˜ X is a chain complex of ¯ G -graded ( A, A ′ ) -bimodules over C which induces a ¯ G -graded derived or Rickard equivalence between A and A ′ , we want to extend the results of [11,Section 5.1.C], to obtain a ¯ G ≀ S n -graded derived or Rickard equivalence over C ⊗ n between A ≀ S n and A ′ ≀ S n . In the case of Rickard equivalences, some additional condition will be needed.Note that by a derived equivalence we mean an equivalence between the bounded derivedcategories D b ( A ) and D b ( A ′ ) induced by a two-sided tilting complex as in [7, Section 6.2],while by a Rickard equivalence, we mean an equivalence between the bounded chain homotopycategories H b ( A ) and H b ( A ′ ) induced by a split endomorphism tilting complex, as presented byRickard in [7, Section 9.2.2]; in this case it is essential that A and A ′ are symmetric algebras. Recall (see, for instance [2, Section 4.1]) that S n acts on ˜ X ⊗ n := ˜ X ⊗ . . . ⊗ ˜ X ( n times).By [11, Lemma 5.2.11], this action can be defined as follows: Denote C = { ± } , and observethat S n acts on the abelian group Fun ( C n2 , C ) of functions from C n2 to C ; for i ∈ Z denote also ^ i = (− ) i . Then there is a -cocycle ǫ ∈ Z ( S n , Fun ( C n2 , C )) such that σ ( x i ⊗ · · · ⊗ x i n ) = ǫ σ (^ i , . . . , ^ i n ) x i σ − ( ) ⊗ · · · ⊗ x i σ − ( n ) , where x i j belongs to the j -th factor of ˜ X ⊗ n , and has degree i j ∈ Z . In our situation, ˜ X ⊗ n is acomplex of ¯ G n -graded ( A ⊗ n , A ′⊗ n ) -bimodules over C ⊗ n , and even more, a complex of ¯ G n -graded ( A ⊗ A ′ op ) ≀ S n -modules.We may therefore consider the wreath product˜ X ≀ S n = ˜ X ⊗ n ⊗ O S n . Theorem 3.7.
Let ˜ X be a complex of ¯ G -graded ( A, A ′ ) -bimodules over C , with identity compo-nent X . Then, the following statements hold:
1) ˜ X ≀ S n is a complex of ¯ G ≀ S n -graded ( A ≀ S n , A ′ ≀ S n ) -bimodules over C ⊗ n , isomorphic to ( A ≀ S n ) ⊗ B ⊗ n X ⊗ n and to X ⊗ n ⊗ B ′⊗ n ( A ′ ≀ S n ) . If ˜ X induces a ¯ G -graded derived equivalence between A and A ′ , then ˜ X ≀ S n induces a ¯ G ≀ S n -graded derived equivalence over C ⊗ n between A ≀ S n and A ′ ≀ S n . If ˜ X induces a ¯ G -graded Rickard equivalence between A and A ′ , and if Σ is a p ′ -subgroupof S n , then ˜ X ≀ Σ induces a ¯ G ≀ Σ -graded Rickard equivalence over C ⊗ n between A ≀ Σ and A ′ ≀ Σ . Proof:
1) We use the fact that the constructions presented in 3.3, Theorem 3.4 and 3.6 arefunctorial. More precisely, by using [11, Lemma 1.6.3] and [13, Proposition 2.11] we deducethat ( A ≀ S n ⊗ ( A ′ ≀ S n ) op ) ⊗ ∆ Sn − , A ≀ S n ⊗ B ⊗ n − and − ⊗ B ′⊗ n A ′ ≀ S n are naturally isomorphicequivalences of categories, from the category of complexes of ( A ⊗ A ′ op ) ≀ S n -modules over C ⊗ n to the category of complexes of ¯ G ≀ S n -graded ( A ≀ S n , A ′ ≀ S n ) -bimodules over C ⊗ n . . Marcus and V. A. Minut , ˘a
2) Let ˜ Y = R Hom A ( ˜ X, A ) be the A -dual of ˜ X , hence ˜ Y is a complex of ¯ G -graded ( A ′ , A ) -bimodules over C , by [13, Proposition 2.12.(2)]. By assumption, the canonical map˜ X L ⊗ A ′ ˜ Y → A of complexes of ¯ G -graded ( A, A ) -bimodules is an isomorphism in the derived category D b ( A ⊗ A op ) , that is, it induces an isomorphism between the homology groups of the above complexes.Consequently, we get a map f : ˜ X ⊗ n L ⊗ A ′⊗ n ˜ Y ⊗ n → A ⊗ n of complexes of ¯ G n -graded ( A ⊗ n , A ⊗ n ) -bimodules over C ⊗ n , which is a quasi-isomorphism. Now,by 3.6, ˜ X ⊗ n extends to a complex of ∆ S n ≃ ( A ⊗ A ′ op ) ≀ S n -modules. It follows, as in the proofof [11, Theorem 5.2.5], that ˜ Y ⊗ n extends to a complex of ( A ′ ⊗ A op ) ≀ S n -modules, and that thecanonical map f is ( A ⊗ A op ) ≀ S n -linear. Observe that f still induces an isomorphism betweenhomology groups, hence f is an isomorphism in the derived category D b (( A ⊗ A op ) ≀ S n ) , and italso preserves ¯ G n -gradings. By [13, Proposition 2.11] we deduce that f induces an isomorphism f wr : ˜ X ≀ S n ⊗ ˜ Y ≀ S n → A ≀ S n , in the bounded derived category of ¯ G ≀ S n -graded ( A ≀ S n , A ≀ S n ) -bimodules. This argument showsthat ˜ X ≀ S n induces a ¯ G ≀ S n -graded derived equivalence over C ⊗ n between A ≀ S n and A ′ ≀ S n .3) Let ˜ Y be the O -dual of ˜ X , hence ˜ Y is a complex of ¯ G -graded ( A ′ , A ) -bimodules over C . Byassumptions, there are ¯ G -grade preserving canonical isomorphisms: h : ˜ X ⊗ A ′ ˜ Y → A and h ′ : A → ˜ X ⊗ A ′ ˜ Y, in the homotopy category H b ( A ⊗ A op ) , inverse of each other. Then, h ⊗ n : ˜ X ⊗ n ⊗ A ′⊗ n ˜ Y ⊗ n → A ⊗ n and h ′⊗ n : A ⊗ n → ˜ X ⊗ n ⊗ A ′⊗ n ˜ Y ⊗ n , are isomorphisms in the homotopy category of ¯ G n -graded ( A ⊗ n , A ⊗ n ) -bimodules. As above, itfollows by 3.6 that h ⊗ n and h ′⊗ n are in fact ( A ⊗ A op ) ≀ S n -linear. Since Σ is a p ′ -subgroup of S n , the final part of the proof of [10, Theorem 3.4.(b)] shows that we get the isomorphisms: h ≀ Σ : ˜ X ≀ Σ ⊗ A ′ ≀ Σ ˜ Y ≀ Σ → A ≀ Σ and h ′ ≀ Σ : A ≀ Σ → ˜ X ≀ Σ ⊗ A ′ ≀ Σ ˜ Y ≀ Σ, in the homotopy category of ¯ G ≀ Σ -graded ( A ≀ Σ, A ≀ Σ ) -bimodules. By symmetry, the statementis proved. (cid:4) Additionally to the assumptions from 2.1, we will consider G ′ to be a subgroup of G suchthat G = G ′ N and C G ( N ) ⊆ G ′ , and let N ′ = G ′ ∩ N , hence ¯ G = G/N ≃ G ′ /N ′ .Let b ∈ Z ( O N ) and b ′ ∈ Z ( O N ′ ) be two ¯ G -invariant block idempotents. We denote A := b O G, A ′ := b ′ O G ′ , B := b O N, B ′ := b ′ O N ′ , hence A and A ′ are ¯ G -graded crossed products, with 1-components B and B ′ respectively. Wealso have that A and A ′ are ¯ G -graded algebras over a ¯ G -graded ¯ G -acted O -algebra C = O C G ( N ) ,with structural maps ζ : C → C A ( B ) and ζ ′ : C → C A ′ ( B ′ ) , as in Definition 2.4, given by inclusion.We denote K B = K ⊗ O B = ( ⊗ b ) K N, K B ′ = K ⊗ O B ′ = ( ⊗ b ′ ) K N ′ . Let V be a G -invariant simple K B -module and V ′ be a G ′ -invariant simple K B ′ -module. Inthis situation, we say that ( A, B, V ) is a module triple, and we will consider its endomorphismalgebra E ( V ) := End K A ( K A ⊗ K B V ) op . We recall from [13], the relation ≥ c between module triples. erived equivalences for wreath products Definition 4.2.
Let ( A, B, V ) and ( A ′ , B ′ , V ′ ) be two module triples. We write ( A, B, V ) ≥ c ( A ′ , B ′ , V ′ ) if there exists a ¯ G -graded algebra isomorphism E ( V ) = End K A ( K A ⊗ K B V ) op → E ( V ′ ) = End K A ′ ( K A ′ ⊗ K B ′ V ′ ) op such that the diagram E ( V ) ∼ / / E ( V ′ ) KC O O id KC KC , O O of ¯ G -graded K -algebras is commutative, where KC = K C G ( N ) is regarded as a ¯ G -graded ¯ G -acted K -algebra, with 1-component K Z ( N ) .The next result is motivated by [18, Theorem 2.21]. Proposition 4.3.
Consider the module triples ( A, B, V ) and ( A ′ , B ′ , V ′ ) . If A and A ′ are ¯ G -graded derived equivalent over C such that V corresponds to V ′ , then ( A ≀ S n , B ⊗ n , V ⊗ n ) ≥ c ( A ′ ≀ S n , B ′⊗ n , V ′⊗ n ) . Proof:
Observe that ( A ≀ S n , B ⊗ n , V ⊗ n ) and ( A ′ ≀ S n , B ′⊗ n , V ′⊗ n ) are module triples. Indeed, N n is a normal subgroup of G ≀ S n , G ′ ≀ S n is a subgroup of G ≀ S n , N ′ n = ( G ′ ≀ S n ) ∩ ( N n ) , and G ≀ S n = ( G ′ ≀ S n )( N n ) . Moreover, it is easy to see that G ≀ S n /N n ≃ G ′ ≀ S n /N ′ n ≃ ¯ G ≀ S n . It is also clear that b ⊗ n and b ′⊗ n are ¯ G ≀ S n -invariant block idempotents in Z ( O N n ) and Z ( O N ′ n ) respectively, and moreover, A ≀ S n ≃ b ⊗ n O ( G ≀ S n ) and A ′ ≀ S n ≃ b ′⊗ n O ( G ′ ≀ S n ) . By Lemma3.2 we have that A ≀ S n and A ′ ≀ S n are strongly ¯ G ≀ S n -graded algebras over C ⊗ n with identitycomponents B ⊗ n = b ⊗ n O N n and B ′⊗ n = b ′⊗ n O N ′ n respectively. Finally, it is straightforwardthat V ⊗ n is a G ≀ S n -invariant simple K B ⊗ n -module, that V ′⊗ n is a G ′ ≀ S n -invariant simple K B ′⊗ n -module, and that V ′⊗ n corresponds to V ⊗ n .Now, by Theorem 3.7, A ≀ S n and A ′ ≀ S n are ¯ G ≀ S n -graded derived equivalent over C ⊗ n , hencethe conclusion follows by [13, Theorem 6.7]. (cid:4) We keep the notations of the preceding section.Sp¨ath also considered in [16], [17] and [18] the relation ≥ b between character triples. Thisrelation is a refinement of ≥ c , and involves block induction, see [18, Definition 4.2]. We showin this section that certain group graded derived equivalences compatible with the Brauer mapimply the relation ≥ b between the corresponding triples, and are also compatible with wreathproducts.We are going to use the Brauer map and basic equivalences between blocks, introduced byL. Puig in [15]. Then [18, Remark 4.3 (c)] leads us to the following setting. Definition 5.1.
We assume that the block b has defect group Q , G ′ = N G ( Q ) , N ′ = N N ( Q ) ,and b ′ is the Brauer correspondent of b . Let ( A, B, V ) and ( A ′ , B ′ , V ′ ) be two module triples.We write ( A, B, V ) ≥ b ( A ′ , B ′ , V ′ ) if the following conditions are satisfied:(1) ( A, B, V ) ≥ c ( A ′ , B ′ , V ′ ) ;(2) For any subgroup N ≤ J ≤ G , if the simple O J -module W covering V corresponds (viacondition (1)) to the simple O J ′ -module W ′ covering V ′ (where J ′ = G ′ ∩ J ), then theblock β of O J to which W belongs is the Harris-Kn¨orr correspondent of the block β ′ of O J ′ to which W ′ belongs. . Marcus and V. A. Minut , ˘a Recall that the Harris-Kn¨orr correspondence [6] is a bijection between the blocks of A withdefect group D (where Q ≤ D ) and the blocks of A ′ with defect group D . This bijection ininduced by the Brauer map Br Q : A Q → A ( Q ) (see [12, Lemma 3.4] for an alternative proof). Denote ¯ C = ¯ C A ( B ) = C A ( B ) /J gr ( C A ( B )) . we know from [3, 2.9] that ¯ C is a ¯ G [ b ] -graded crossed product, where¯ G [ b ] = { ¯ g ∈ ¯ G | A ¯ g ≃ B as ( B, B ) -bimodules } = { ¯ g ∈ ¯ G | A ¯ g A ¯ g − = B } . Denote also ¯ C ′ = ¯ C A ′ ( B ′ ) = C A ′ ( B ′ ) /J gr ( C A ′ ( B ′ )) .The main result of Dade [3] says that the Brauer map Br Q induces an isomorphism ¯ C ≃ ¯ C ′ of ¯ G [ b ] -graded ¯ G -acted algebras. Moreover, by [12, Theorem 3.7], this isomorphism induces thesame Harris-Kn¨orr correspondence between the blocks of A and the blocks of A ′ . Recall also from [11, Corollary 5.2.6] that a ¯ G -graded derived equivalence between A and A ′ induces yet another isomorphism ¯ C ≃ ¯ C ′ of ¯ G [ b ] -graded ¯ G -acted algebras. Definition 5.5.
We say that a ¯ G -graded derived equivalence between A and A ′ is compatiblewith the Brauer map if the induced isomorphism ¯ C ≃ ¯ C ′ of ¯ G [ b ] -graded ¯ G -algebras from 5.4coincides with the isomorphism induced by the Brauer map Br Q from 5.3. Proposition 5.6.
Assume that the complex ˜ X induces a ¯ G -graded derived equivalence between A and A ′ compatible with the Brauer map Br Q , such that the simple K B -module V correspondsto the simple K B ′ -module V ′ . Then ( A, B, V ) ≥ b ( A ′ , B ′ , V ′ ) . Proof:
The proof of [13, Theorem 6.7] works for derived equivalences as well, and shows that ( A, B, V ) ≥ c ( A ′ , B ′ , V ′ ) . Condition (2) of Definition 5.1 follows from the fact (see [11, Corollary5.2.6]) that the truncated complex ˜ X ¯ J = ⊕ ¯ g ∈ ¯ J ˜ X ¯ g induces a ¯ J -graded derived equivalence between A ¯ J and A ′ ¯ J ′ , which is still compatible with the Brauer map Br Q . (cid:4) Remark 5.7.
By [12, Corollary 4.4], a ¯ G -graded basic Morita equivalence between A and A ′ iscompatible with the Brauer map Br Q in the sense of the above definition.Note also that a direct product of Dade Q -algebras is also a Dade Q -algebra. It follows easilythat we get a ¯ G n -graded basic Morita equivalence between A ⊗ n and A ′⊗ n .This in turn, induces a Morita equivalence between A ≀ S n ≃ b ⊗ n O ( G ≀ S n ) and A ′ ≀ S n ≃ b ′⊗ n O ( G ′ ≀ S n ) , by Theorem 3.4 or [11, Theorem 5.1.21]. However, this equivalence need not bebasic (see also [19, Remark 3.4]), so we cannot apply the results of [12] to deduce its compatibilitywith the Brauer map Br Q n .Nevertheless, we still have the following result. Theorem 5.8.
Assume that the complex ˜ X induces a ¯ G -graded derived equivalence between A and A ′ compatible with the Brauer map Br Q . Then the ¯ G ≀ S n -graded derived equivalence between A ≀ S n and A ′ ≀ S n induced by ˜ X ≀ S n is compatible with the Brauer map Br Q n . Proof:
We have that A ≀ S n ≃ O ( G ≀ S n ) b ⊗ n is a G ≀ S n /N n ≃ ¯ G ≀ S n -graded algebra with -component B ⊗ n . Consider the group ( ¯ G ≀ S n )[ b ⊗ n ] defined in 5.3, hence ¯ C A ≀ S n ( B ⊗ n ) is a ( ¯ G ≀ S n )[ b ⊗ n ] -graded crossed product.By applying the Brauer construction, we get ( O ( G ≀ S n ))( Q n ) ≃ kC G ≀ S n ( Q n ) = kC G n ( Q n ) ≃ ( kC G ( Q )) ⊗ n , and in fact, ( A ≀ S n )( Q n ) ≃ A ( Q ) ⊗ n . This, together with Dade’s result presented in 5.3 showthat ( ¯ G ≀ S n )[ b ⊗ n ] ⊆ ¯ G n , and therefore¯ C A ≀ S n ( B ⊗ n ) ≃ ¯ C A ⊗ n ( B ⊗ n ) . Consequently, it is enough to show that the ¯ G n -graded derived equivalence between A ⊗ n and A ′⊗ n induced by ˜ X ⊗ n is compatible with the Brauer map Br Q n . erived equivalences for wreath products We use that all our constructions behave well with respect to tensor products. First, as in2.8, we have (see also [5, Proposition 1.4.]) C A ⊗ n ( B ⊗ n ) ≃ End A ⊗ n ⊗ B ⊗ n op ( A ⊗ n ) op ≃ ( End A ⊗ B op ( A ) op ) ⊗ n ≃ C A ( B ) ⊗ n . Next, by using 2.7, we have¯ C ⊗ n = ¯ C A ( B ) ⊗ n = ( C A ( B ) /J gr ( C A ( B ) ) ⊗ n ≃ C A ( B ) ⊗ n /J gr ( C A ( B ) ⊗ n ) ≃ C A ⊗ n ( B ⊗ n ) /J gr ( C A ⊗ n ( B ⊗ n )) = ¯ C A ⊗ n ( B ⊗ n ) . Let ϕ ˜ X : ¯ C A ( B ) → ¯ C A ′ ( B ′ ) denote the isomorphism induced by ˜ X , as in [13, Subsection 5.3].Henceforth, we get the following commutative diagram of isomorphisms of ¯ G [ b ] -graded ¯ G -actedalgebras: ¯ C A ( B ) ⊗ n ϕ ⊗ n ˜ X / / ≃ (cid:15) (cid:15) ¯ C A ′ ( B ′ ) ⊗ n ≃ (cid:15) (cid:15) ¯ C A ⊗ n ( B ⊗ n ) ϕ ˜ X ⊗ n / / ¯ C A ′⊗ n ( B ′⊗ n ) . By 2.9, we also have the following commutative diagram of isomorphisms of ¯ G [ b ] -graded ¯ G -actedalgebras: ¯ C A ( B ) ⊗ n Br ⊗ nQ / / ≃ (cid:15) (cid:15) ¯ C A ′ ( B ′ ) ⊗ n ≃ (cid:15) (cid:15) ¯ C A ⊗ n ( B ⊗ n ) Br Qn / / ¯ C A ′⊗ n ( B ′⊗ n ) . By our assumptions, the isomorphism ϕ ˜ X coincides with the isomorphism given by Br Q . Theabove two commutative diagrams imply that the isomorphisms ϕ ˜ X ⊗ n and Br Q n also coincide,hence the equivalence induced by ˜ X ⊗ n is indeed compatible with the Brauer map. (cid:4) From Proposition 4.3 and Theorem 5.8 we immediately deduce:
Corollary 5.9.
Assume that the complex ˜ X induces a ¯ G -graded derived equivalence over C between A and A ′ , and that this equivalence is compatible with the Brauer map Br Q . Assumealso that the simple K B -module V corresponds to the simple K B ′ -module V ′ . Then ( A ≀ S n , B ⊗ n , V ⊗ n ) ≥ b ( A ′ ≀ S n , B ′⊗ n , V ′⊗ n ) . Remark 5.10.
We are interested in the relation ≥ b when induced by derived equivalences.However, it is not difficult to show directly, with the methods already used here, that similarlyto [16, Theorem 5.2], if ( A, B, V ) ≥ b ( A ′ , B ′ , V ′ ) , then ( A ≀ S n , B ⊗ n , V ⊗ n ) ≥ b ( A ′ ≀ S n , B ′⊗ n , V ′⊗ n ) Remark 5.11.
The following situation is considered in [10]. Assume that p ∤ | ¯ G | , b is theprincipal block of O N , and that ˜ X induces a ¯ G -graded derived equivalence between A and A ′ = O N G ( Q ) b ′ , where b ′ is the principal block of O N N ( Q ) (and of O C N ( Q ) ). By [10, Corollary3.9], ˜ X ( Q ) induces a C G ( Q ) /C N ( Q ) -graded derived autoequivalence of kC G ( Q ) b ′ (which actuallylifts to O ). It is not difficult to see that this equivalence extends to an N G ( Q ) /C N ( Q ) -gradedderived autoequivalence of A ′ = O N G ( Q ) b ′ . Moreover, the arguments of [12, Theorem 4.3 andCorollary 4.4] show that the isomorphism ¯ C ≃ ¯ C ′ induced by this equivalence coincides with theisomorphism induced by Br Q .In order to deal with arbitrary blocks, one needs to extend the results of [12] to the case ofbasic Rickard equivalences. We intend to consider this problem in a subsequent paper. . Marcus and V. A. Minut , ˘a References [1] A. M. Alghamdi, A. A. Khammash,
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