Chain complexes for Alperin s weight conjecture and Dade s ordinary conjecture in the abelian defect group case

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1 Chain complexes for Alperin s weight conjecture and Dade s ordinary conjecture in the abelian defect group case Robert Boltje Department of Mathematics University of California Santa Cruz, CA U.S.A. boltje@math.ucsc.edu revised January 25, 2005 Abstract We give an axiomatic concept for the construction of contractible chain complexes whose existence implies Dade s ordinary conjecture and Alperin s weight conjecture and we show that in the abelian defect group case such chain complexes exist if a weaker form of Broué s conjecture holds. Introduction In [Bo01b] we showed that Alperin s weight conjecture [Al87] is equivalent to the existence of contractible chain complexes whose degree n terms have as rank the n-th summand of the alternating sum formulation of Alperin s weight conjecture. Moreover, we gave numerical evidence that similar contractible chain complexes exist that model Dade s ordinary conjecture. In this article we suggest a way to build such chain complexes. The construction we suggest is of a simplicial nature, which is not surprising given the fact that the above conjectures involve alternating sums indexed by G-conjugacy classes of a G-simplicial complex consisting of non-trivial p-subgroups. In fact, there are several choices for the type of chains that lead to equivalent conjectures. Also for the summands themselves there are different equivalent choices possible. One can count the number of indecomposable linear source modules or trivial source modules (cf. [Bo01a]), or the number of irreducible characters MR Subject Classification 20C20, 20C15 Research supported by the NSF, DMS

2 or Brauer characters (cf. [KR89]) attached to a given block of positive defect. Even finer, one can count those irreducible characters that have a given defect in the case of Dade s ordinary conjecture (cf. [Da92]). Our suggested construction takes all these different formulations into account. More precisely we postulate axioms that yield a contractible chain complex involving the representation ring of linear source modules which induces via natural maps other contractible chain complexes that involve the trivial source ring, character ring, and Brauer character ring. These axioms are quite numerous and might look confusing on a first glance. But we show that they are satisfied for blocks with abelian defect groups provided that a splendid equivalence (a notion introduced by Rickard, cf. [Ri96]) exists on the level of representation rings. In particular, our construction works for blocks with cyclic defect groups due to work of Rouquier, cf. [Ro98]. In Section 1 we introduce the simplicial construction and the axioms for our desired chain complexes. The modules of the chain complexes are just representation rings involved in the alternating sum conjectures. The chain maps come from face maps with postulated properties, cf. Definition 1.1 and Hypotheses 1.6. In Section 2 we show that if the face maps are induced by virtual trivial source bimodules with certain relative projectivity properties, then most of the postulated properties follow automatically. Section 3 establishes some results on blocks with abelian defect groups that are used in Section 4 to show that face maps satisfying all required axioms exist in the case that the given block has abelian defect groups, provided that a weaker form of Broué s conjecture holds. Notation Throughout this article G denotes a finite group, p a prime, and (K, O, F ) a p-modular system that is large enough for G. This means that O is a complete discrete valuation ring, K its field of fractions, and F its residue field. It is assumed that F is algebraically closed of characteristic p and that K has characteristic 0 and contains a primitive root of unity of order exp(g), the exponent of G. For the set of blocks of the group ring OG we write Bl(OG). We write H G (resp. H < G) if H is a subgroup (resp. proper subgroup) of G and H G (resp. H G) if H is a normal (resp. proper normal) subgroup of G. For two subgroups H and U of G we write U G H, if U is G-conjugate to a subgroup of H and U = G H if U is G-conjugate to H. For H G and g G we set g H := ghg 1 and H g := g 1 Hg. For g, x G we set g x := gxg 1 and x g := g 1 xg. By P(G) we denote the set of chains σ = (P 0 < < P n ) of non-trivial p-subgroups of G including the empty chain. We consider P(G) as a G-set via conjugation and denote the stabilizer of σ by G σ. The length σ of σ as above is defined to be n. We set := 1 and have G = G. For σ as above and i {0,..., n} we write σ i for the chain arising from σ by omitting P i. Similarly, σ i,j denotes the chain σ with P i and P j omitted. For smoothness of exposition it is sometimes convenient to set P 1 := {1}, the trivial subgroup of G. The chains of length n Z in P(G) are denoted by P n (G). We will also consider various G-subsets of P(G): The set N (G) consisting of all σ as above with P i 2

3 normal in P n for all i {0,..., n}; The set E(G) consisting of all chains of nontrivial elementary abelian subgroups of G; And finally the set U(G) consisting of chains of non-trivial radical p-subgroups P of G, i.e., P = O p (N G (P )). Similar as for P(G), we define X n (G) for any G-subset X (G) of P(G) and n Z. For any ring R we write R mod for the category of finitely generated left R-modules. By OG lin, OG triv, F G triv, we denote the categories of linear source OG-modules, trivial source OG-modules, and trivial source F G-modules. These are modules that are isomorphic to direct summands of monomial modules (resp. permutation modules) in the linear source case (resp. the trivial source case). The representation rings of these categories with respect to direct sums is denoted by L(G), T O (G), and T F (G), respectively. The Grothendieck rings of KG mod and F G mod are denoted by R(G) and R F (G). They are isomorphic to the character ring and Brauer character ring of G. Note that the functor F O induces a bijection on isomorphism classes of indecomposable trivial source OG-modules and F G-modules so that we may identify T O (G) and T F (G) and only write T (G). We write L(G, B), T (G, B),... to denote the subgroups generated by the classes of those modules which lie in the block B Bl(OG). 1 A suggested concept for chain complexes modeling Alperin s weight conjecture and Dade s ordinary conjecture For a block B Bl(OG), a G-subset X (G) of P(G), and n Z we write X n (G, B) for the set of pairs (σ, β), where σ X n (G) and β Bl(OG σ ) with β G = B. Moreover, we write X (G, B) for the disjoint union of the G- sets X n (G, B), n Z. Note that the induced block β G is defined by [KR89, Lemma 3.2]. The sets X n (G, B), n Z, are G-sets under the conjugation action. 1.1 Definition A G-equivariant coefficient system for (G, B) on P(G) and L is a family of group homomorphisms d i (σ,β) : L(G σ, β) L(G σi, β Gσ i ), one for each (σ, β) P(G, B) and each i {0,..., σ }, such that the following axioms hold: (G-equivariance) For each n 0, each (σ, β) P n (G, B), each i {0,..., n}, and each g G, the diagram L(G σ, β) di (σ,β) L(G σi, β Gσ i ) g g L(G ( g σ), g β) di g (σ,β) 3 L(G ( g σ) i, ( g β) G ( g σ) i )

4 is commutative, where for any H G we denote by g : L(H) L( g H) the map induced by restriction along the group isomorphism g H H given by x x g. (Compatibility) For each n 1, each (σ, β) P n (G, B), and each pair (i, j) N 2 0 with 0 i < j n, the diagram L(G σ, β) di (σ,β) L(G σi, β Gσ i ) d j (σ,β) d j 1 (σ i,β Gσ i ) d i L(G σj, β Gσ (σ j ) j,β Gσ j ) L(G σi,j, β Gσ i,j ) commutes. 1.2 Remark (a) Similarly, one defines G-equivariant coefficient systems for (G, B) on X and M, where X {N, E, U} and M {L, T, R, R F }. More generally, we may choose X (G) to be any G-subset of P(G) which is stable under deleting subgroups from a chain. This is the reason that we do not consider the set R(G) of radical chains, cf. [Da92]. (b) If we work with the set of chains N (G), then with σ = (P 0 < < P n ) N n (G), i {0,..., n}, Q := P i, and H := G σi the map d i (σ,β) becomes a map d H,Q,β : L(N H (Q), β) L(H, β H ), (1.2.a) with H G, Q H a p-subgroup and β Bl(ON H (Q)). The compatibility condition in Definition 1.1 then translates into the commutativity of the diagram L(N H (Q 1 ), β N H(Q 1) ) d NH (Q 1),Q 2,β L(N H (Q 1 Q 2 ), β) L(H, β H ) d NH (Q 2),Q 1,β L(N H (Q 2 ), β N H(Q 2) ) d H,Q1,β N H (Q 1 ) d H,Q2,β N H (Q 2 ) (1.2.b) for non-trivial p-subgroups Q 1 Q 2 in H. This suggests the speculative question: Given a finite group H, a p-subgroup Q H, and b Bl(ON H (Q)), is there a natural map as in (1.2.a) such that Diagram (1.2.b) commutes (by naturality)? 4

5 1.3 Examples Let M {L, T, R, R F } and X, Y {P, N, E, U}. (a) If (d i (σ,β)) is a G-equivariant coefficient system for (G, B) on X (G) and M, and if X (G) Y(G), then (d i (σ,β)) can be extended to a coefficient system on Y(G) as follows. There exists a G-equivariant involution φ on Z(G) := Y(G) X (G) (in particular, preserving stabilizers) with the property that φ either inserts or deletes a subgroup into or from a chain, cf. for example [Bo01b]. This induces a decomposition Z n (G) = I n D n into G-subsets such that the restrictions φ: I n D n+1 and φ: D n I n 1 are G-equivariant bijections. For (σ, β) Z n (G) and i {0,..., n} we then set { d i (σ,β) := id M(Gσ,β), if σ D n and φ(σ) = σ i, (1.3.a) 0, otherwise. It is straight forward to check that this extension defines a G-equivariant coefficient system for (G, B) on Y(G). (b) If O p (G) 1, then there exists again a G-equivariant involution φ: X (G) X (G), that deletes or inserts subgroups, and a decomposition X n (G) = I n D n into G-subsets such that the restrictions φ: I n D n+1 and φ: D n I n 1 are bijective, cf. [Bo01b, Proposition 4.3]. Similar to part (a), we define d i (σ,β) : M(G σ, β) M(G σi, β Gσ i ) by (1.3.a) and obtain a G-equivariant coefficient system for (G, B) on X (G) and M. 1.4 Chain complexes associated to G-equivariant coefficient systems. To a G- equivariant coefficient system (d i (σ,β)) for (G, B) on X {P, N, E, U} and L we associate a chain complex L X (G, B) of ZG-modules in the usual way. We set, for n Z, L X n (G, B) := L(G σ, β) (σ,β) X n(g,b) and define d n : L X n (G, B) L X n n 1(G σ, B) as i=0 ( 1)i d i n with d i n : L X n (G, B) L X n 1(G, B) defined on the (σ, β)-component with image in the (σ i, β Gσ i )-component as d i (σ,β) : L(G σ, β) L(G σi, β Gσ i ). The compatibility property in Definition 1.1 ensures that d n d n+1 = 0 for all n Z, and the G-equivariance ensures that d n is ZG-linear if we view L n (G, B) as ZG-module via the maps g : L(G σ, β) L(G ( g σ), g β) introduced there. Note that if g G σ, then g acts as the identity on L(G σ, β) so that L X n (G, B) is a permutation ZG-module. It follows that the chain complex of abelian groups, L X (G, B) G, defined by applying to L X (G, B) the functor of taking cofixed points, or equivalently, Z ZG, consists of free Z-modules, the n-th entry having rank lin(g σ, β) = lin(g σ, B σ ) (σ,β) X n(g,b)/g σ X n(g)/g where lin(g, B) denotes the rank of L(G, B) and B σ the sum of blocks β Bl(OG σ ) with β G = B. Thus, if L X (G, B) G is exact, then the alternating sum 5

6 of the above numbers vanishes, which is one of the reformulations of Alperin s weight conjecture, cf. [Bo01a]. Similarly we define a chain complex M X (G, B) if (d i (σ,β) ) is a G-equivariant coefficient system on X (G) and M {T, R, R F }. For a p-subgroup P of G we set L X,P n (G, B) := (σ,β) X n(g,b) L P (G σ, β) with L P (G σ, β) being the span of the classes of those indecomposable linear source OG σ -modules in β whose vertices are G-conjugate to a subgroup of P. If d i (σ,β) (LP (G σ, β)) L P (G σi, β Gσ i ) for all (σ, β) X (G, B), i {0,..., σ }, and all p-subgroups P of G, then we say that (d i (σ,β)) respects the vertex filtration. In this case, the chain complex L X (G, B) has a filtration by ZG-subcomplexes L X,P (G, B) with L X,Q (G, B) L X,P (G, B) if Q G P. Similar definitions apply to T instead of L. Now assume that (d i (σ,β)) is a G-equivariant coefficient system on X and R. For d N 0 we define R d (G, B) as the span of all irreducible characters of G in B whose p-defect is less than or equal to d. Moreover, for n Z, we set R X,d n (G, B) := (σ,β) X n(g,b) R d (G σ, β). If d i (σ,β) (Rd (G σ, β)) R d (G σi, β Gσ i ) for all (σ, β) X (G, B), i {0,..., σ }, and all d N 0, then we say that (d i (σ,β)) respects the defect filtration. In this case, the chain complex R X (G, B) has a filtration by ZG-subcomplexes R X,d (G, B), d N 0, with R X,d (G, B) R X,d+1 (G, B) for d N 0. If, moreover, B has positive defect and each of the chain complexes R d (G, B) is exact, then obviously Dade s ordinary conjecture holds for G and B. Note that in all cases, Mn X (G, B) = 0 for n 2 and M 1(G, X B) = M 1(G, X B) G = M(G, B), if X (G) contains the empty chain. If X (G) = P(G) or if there is no risk of confusion we will omit the superscript X in M X (G, B). 1.5 For every block B Bl(OG) one has a diagram T (G, B) ν T R F (G, B) ι π ν R L(G, B) ν L R(G, B) with ι, ν T, ν L the maps coming from the inclusion functor OG triv OG lin, the functor F O : OG triv F G mod, and the functor K O : OG lin KG mod. Note that ν T and ν L in the above diagram are surjective by Brauer s induction 6

7 theorems for R(G) and for R F (B). Moreover, ν R denotes the decomposition map and π is induced by the functor F O : OG lin F G triv. Obviously, one has the following relations: π ι = id T (G,B), ν R ν L ι = ν T, ν T π = ν R ν L. (1.5.a) We suggest to construct for given B Bl(OG) with positive defect a G- equivariant coefficient system (d i (σ,β)) for (G, B) on P (or any other type of chains, cf. Proposition 1.8(a)) and L satisfying the following hypotheses: 1.6 Hypotheses (i) d i (σ,β) (T (G σ, β)) T (G σi, β Gσ i ) for all (σ, β) P(G, B) and i {0,..., σ }. Moreover, the diagram L(G σ, β) di (σ,β) L(G σi, β Gσ i ) π π T (G σ, β) di (σ,β) T (G σi, β Gσ i ) commutes for all (σ, β) P(G, B) and i {0,..., σ }. (ii) d i (σ,β) : T (G σ, β) T (G σi, β Gσ i ) induces a map d i (σ,β) : R F (G σ, β) R F (G σi, β Gσ i ) such that the diagram T (G σ, β) di (σ,β) T (G σi, β Gσ i ) ν T ν T R F (G σ, β) di (σ,β) R F (G σi, β Gσ i ) commutes for all (σ, β) P(G, B) and i {0,..., σ }. (iii) d i (σ,β) : L(G σ, β) L(G σi, β Gσ i ) induces a map d i (σ,β) : R(G σ, β) R(G σi, β Gσ i ) such that the diagram L(G σ, β) di (σ,β) L(G σi, β Gσ i ) ν L ν L R(G σ, β) di (σ,β) 7 R(G σi, β Gσ i )

8 commutes, for all (σ, β) P(G, B) and i {0,..., σ }. (iv) The G-equivariant coefficient system (d i (σ,β)) on L and the induced coefficient system (d i (σ,β)) on T respect the vertex filtration. (v) The induced G-equivariant coefficient system (d i (σ,β)) on R respects the defect filtration. (vi) There exists a chain map h : L (G, B) G L (G, B) G of degree 1 with h d + d h = id L (G,B) G such that h induces chain maps h : M (G, B) G M (G, B) G for M = T, R, R F with commuting diagrams similar to those in (i) (iii). In particular, one also has h d + d h = id M (G,B) G. Moreover, h on L (G, B) G and T (G, B) G respects the vertex filtration and h on R (G, B) G respects the defect filtration. 1.7 Remark Let B Bl(OG), X {P, N, E, U}, and let (d i (σ,β)) be a G- equivariant coefficient system for (G, B) on X (G) and L satisfying (i)-(iii) in Hypotheses 1.6. Then also the diagram R(G σ, β) di (σ,β) R(G σi, β Gσ i ) ν R ν R R F (G σ, β) di (σ,β) R F (G σi, β Gσ i ) commutes for all (σ, β) X (G, B) and i {0,..., σ }. In fact, since ν L is surjective it suffices to show that d i (σ,β) ν R ν L = ν R d i (σ,β) ν L. But d i (σ,β) ν R ν L = d i (σ,β) ν T π = ν T d i (σ,β) π = ν T π d i (σ,β) Thus, one obtains a diagram = ν R ν L d i (σ,β) = ν R d i (σ,β) ν L. T (G, B) (ν T ) (R F ) (G, B) ι π (ν R ) (1.7.a) L (G, B) (ν L ) R (G, B) in the category of chain complexes in ZG mod satisfying the relations in (1.5.a), and also an induced diagram 8

9 T (G, B) G (ν T ) (R F ) (G, B) G ι π (ν R ) L (G, B) G (ν L ) R (G, B) G of chain complexes of abelian groups still satisfying theses relations. Note that, since the functor ( ) G = Z ZG is right exact, the maps π, (ν L ), (ν R ), and (ν T ) in the last diagram are still surjective, and since ι (T (G, B)) has a ZGcomplement in L (G, B), the map ι in the last diagram is still injective. 1.8 Proposition Let B Bl(OG) have positive defect and let X, Y {P, N, E, U} with X (G) Y(G). (a) If (d i (σ,β)) is a G-equivariant coefficient system for (G, B) on X (G) and L such that Hypotheses 1.6(i) (vi) are satisfied then the extended G-equivariant coefficient system on Y(G) and L introduced in Examples 1.3(a) also satisfies Hypotheses 1.6(i) (vi). (b) Assume that O p (G) 1. Then the G-equivariant coefficient system for (G, B) on L and X introduced in 1.3(b) satisfies Hypotheses 1.6(i) (vi). Proof In both cases we have a G-subset Z(G) P(G) and a G-equivariant involution φ: Z(G) Z(G) as in Examples 1.3 and we use the notation from there. For (σ, β) Z n (G, B) and i {0,..., σ } we defined d i (σ,β) by (1.3.a). It is obvious that the identity map and the trivial map satisfy Hypotheses 1.6(i) (v). In order to establish (vi) we define (in part (b)) or extend (in part (a)) for n Z and (σ, β) Z n (G, B) the map h n on L(G σ, β) by { L n (G σ, β) ( 1)i id L n+1 (G φ(σ), β), if σ I n and σ = φ(σ) i, 0, otherwise. Note that G φ(σ) = G σ. Obviously, h n : L n (G, B) L n+1 (G, B) is ZG-linear and satisfies the Hypotheses 1.6(i) (v). Thus, it suffices to show that h n 1 d n + d n+1 h n = id on L(G σ, β) for (σ, β) Z n (G, B). If σ I n and σ = φ(σ) i, then d n = 0 and d n+1 h n = ( 1) i ( 1) i id = id on L(G σ, β). And if σ D n and φ(σ) = σ i, then h n = 0 and h n 1 d n = ( 1) i ( 1) i id = id on L(G σ, β) as well. 2 Tensoring with trivial source bimodules Throughout this section let k denote a commutative ring. For a homomorphism ϕ: G k into the unit group of k, we write k ϕ for the kg-module which has as 9

10 underlying k-module the ring k and on which G acts via ϕ. We assume further that H is a finite group. For U H, a homomorphism ϕ: U k, and h H, we define h ϕ: h U k by h ϕ(x) := ϕ(x h ). We will frequently view a left khmodule M as right kh-module by setting mh := h 1 m for h H and m M. Similarly, we will view a left k[g H]-module also as (kg, kh)-bimodule and vice versa. 2.1 Lemma Let U, V H, let ϕ: U k and ψ : V k be homomorphisms, and set M := Ind H U (k ϕ ) and N := Ind H V (k ψ ). If we view M as right kh-module and if [V \H/U] denotes a fixed set of representatives for V \H/U, then one has a k-module isomorphism f : M kh N k/ (ψ hϕ)(v) 1 v V h U k h [V \H/U] that sends (h 1 ku 1) kh (h 2 kv 1) to the class of ϕ(u)ψ(v 1 ) in the h- component, where h, u, and v are determined by h 1 2 h 1 = vhu with h [V \H/U], u U, and v V. Its inverse map sends the class of an element α k in the h-component to (h ku α) kh (1 kv 1). Proof It is straight forward to see that the map f is well-defined. Moreover, if α = ψ(v) h ϕ(v 1 ) with h H and v V h U, then (h ku α) kh (1 kv 1) = (h ku ψ(v)) kh (1 kv 1) (h ku h ϕ(v 1 )) kh (1 kv 1) = (h ku 1) kh (1 kv ψ(v)) (h ku ϕ(h 1 v 1 h)) kh (1 kv 1) = (h ku 1) kh (v kv 1) (hh 1 v 1 h ku 1) kh (1 kv 1) = (h ku 1) kh (v kv 1) (v 1 h ku 1) kh (1 kv 1) = 0. Therefore, also the inverse map is well-defined. Finally, it is clear that the two maps are inverses of each other. From now on we assume that the p-modular system (K, O, F ) is also large enough for H, i.e., that O contains a root of unity of order exp(h). 2.2 Lemma Let M = Ind G H U (k ϕ ) k[g H] mod and N = Ind H V (k ψ ) kh mod for subgroups U G H and V H and homomorphisms ϕ: U k and ψ : V k. Set G U := {g G (g, 1) U}, G U := {g G (g, h) U for some h H}, and use similar definitions for H U and H U. Moreover let ϕ GU H U = ϕ 1 ϕ 2 and let [G/G U ] and [V \H/H U ] be fixed sets of representatives of G/G U and V \H/H U. If we view M as (kg, kh)-bimodule then the following hold: (a) G acts from the left on the set [G/G U ] [V \H/U] as follows: For x [G/G U ], y [V \H/H U ], and g G one determines x [G/G U ] and g G U such that gx = x g and chooses h H and v V such that (g, h ) U and 10

11 y := vyh 1 [V \H/H U ]. Then g (x, y) = (x, y ). With this action one has a kg-module isomorphism M kh N = (k/ (ψ yϕ ) 2 )(v) 1 v V y H U k, (x,y) Ind G stab G (x,y) where (x, y) runs through a set of representatives for the G-orbits of [G/G U ] [V \H/H U ]. Moreover the stabilizer stab G (x, y) of (x, y) under the above action consists of all elements g G such that there exists v V with (g x, v y ) U, and such an element g acts on the factor module of k by multiplication with ϕ(g x, v y )ψ(v). (b) Assume that k is a field or that k = O and H U is a p -group. Then one has a kg-module isomorphism M kh N = Ind G stab G (x,y)(k µx,y ) (x,y) ([G/G U ] [V \H/H U ])/G y (ϕ 2 )=ψ 1 y on V (H U ) where µ x,y : stab G (x, y) k is defined by µ x,y (g) := ϕ(g x, v y )ψ(v) if v V is chosen such that (g x, v y ) U. Proof (a) We have Res G H ( H Ind G H U (k ϕ ) ) = Ind H ( (x,y) Res U ( (x,y) k H (x,y) U H (x,y) ϕ ) ), U (x,y) H\(G H)/U where we abbreviated the subgroup 1 H of G H by H. Since H\(G H)/U = (G H)/HU and HU = G U H, the last expression is equal to x G/G U Ind H H (x,1) U ( Res (x,1) U H (x,1) U ( (x,1) k ϕ ) ). It is easy to see that H (x,1) U = H U and Res (x,1) ( U (x,1) ) H U k ϕ = kϕ2. Thus, Res G H H (M) = Now Lemma 2.1 applies and yields M kh N = (x,y) [G/G U ] [V \H/H U ] x G/G U Ind H H U (ϕ 2 ). k/ (ψ y(ϕ 2 ))(v) 1 v V y (H U ) k as k-modules. Retracing the G-action on M through all these isomorphisms yields that the element 1 in the (x, y)-component is mapped under g G to the element ϕ(g, h )ψ(v) in the (x, y )-component, with x, y, g, h, v as in the statement of the lemma. Thus, the summands in the last direct sum are permuted by G and the remaining assertions are verified in a straight forward way. 11

12 (b) If k is a field this follows immediately from part (a). If k = O it follows from part (a) and the fact that a non-trivial root of unity of p -order in O is not congruent to 1 modulo the radical of O. This shows that ψ(v) y (ϕ 2 )(v) 1 is either zero or a unit in O if v is a p -element. 2.3 Proposition Assume that H and G have a common p-subgroup D and let M be a trivial source O[G H]-module that is δ(d)-projective, where δ(d) := {(x, x) x D}. Moreover, assume that M if viewed as an (OG, OH)-bimodule is a left B-module and a right b-module for sums B and b of blocks of OG and OH. (a) If N OH triv (resp. N OH lin), then M OH N OG triv (resp. M OH N OG lin). (b) Let f : L(H, b) L(G, B) be the map induced by M OH according to part (a). Then there exist unique maps f : T (H, b) T (G, B), f : R F (H, b) R F (G, B), and f : R(H, b) R(G, B) such that the diagrams T (H, b) f ι L(H, b) f T (G, B) L(H, b) f ι π L(G, B) T (H, b) f L(G, B) π T (G, B) L(H, b) f L(G, B) T (H, b) f T (G, B) ν L ν L ν T ν T R(H, b) f R(G, B) R F (H, b) f R F (G, B) commute. (c) For every subgroup P D one has f(l P (H, b)) L P (G, B) and f(t P (H, b)) T P (G, B). Proof Part (a) follows immediately from Lemma 2.2(b), since M is a direct summand of a direct sum of modules of the form Ind G H U (O ϕ ) with ϕ = 1 and U δ(d). This implies that H U = 1 and that M OH N is a direct summand of a permutation (resp. monomial) OG-module. Part (b) follows from Lemma 2.2(b) and the fact that for an (OG, OH)- bimodule M and an OH-module N, both free as O-modules, one has OGmodule isomorphisms F O (M OH N) = (F O M) F H (F O N) and K O (M OH N) = (K O M) KH (K O N). Part (c) also follows from Lemma 2.2(b), since in the Lemma s notation, V P implies stab G (x, y) xy 1 P. In fact, if g stab G (x, y), then (g x, v y ) U δ(d) for some v V. This implies g x = v y D P y y 1 P. 12

13 2.4 Remark The previous proposition suggests that in order to construct a G-equivariant coefficient system (d i (σ,β)) for (G, B) on P(G) and L it is helpful to define d i (σ,β) : L(G σ, β) L(G σi, β Gσ i ) as being induced by tensoring with virtual (OG σi, OG σ )-bimodules that are δ(d)-projective for a Sylow p-subgroup D of G σ. But there are still three main problems left, namely to find these virtual bimodules in a way that the compatibility axiom in Definition 1.1 is satisfied, that the defect filtration is respected, and that the associated G-cofixed point complex becomes contractible. As far as the defect filtration is concerned there might be an internal structural property of the virtual bimodule that implies the desired behaviour. For the other two problems the only hope we have is that there is a natural construction of the virtual bimodules that will imply the rest. 3 Some generalities on blocks with abelian defect groups We will have to make use the following well-known lemma. See for instance Theorem V.5.21 in [NT89]. 3.1 Lemma Let B Bl(OG) have abelian defect groups, let Q G be a p- subgroup, and let b Bl(ON G (Q)) satisfy b G = B. Then every defect group of b is also a defect group of B. 3.2 Corollary Let B Bl(OG) have abelian defect groups and let (σ, β) P(G, B). Then every defect group of β is also a defect group of B. Proof We proceed by induction on σ. If σ = 1, there is nothing to show, since β = B. If σ = 0, this is the statement of Lemma 3.1. Now we assume that σ = (P 0 < < P n ) with n 1 and set σ := (P 1 < < P n ). Then P 0 < P 1 G σ and N G σ (P 0 ) = G σ G σ. Let β := β G σ, then β G = (β G σ ) G = β G = B. By Lemma 3.1, every defect group of β is a defect group of β, and by induction, every defect group of β is a defect group of B. In the proof of the following proposition we will frequently use the following fact, cf. [Al86, Theorem 15.1, Lemma 15.3]: If N is a normal subgroup of G and for each defect group D of each block b Bl(ON) one has C G (D) N, then the map Bl(OG) Bl(ON)/G defined by the notion of covering is a bijection with inverse induced by b b G. In particular, for b Bl(ON) and B Bl(OG) one has: B covers b if and only if b G = B. 3.3 Proposition Let B Bl(OG), let D be a defect group of B, and let b Bl(OH) be the Brauer correspondent of B, where H := N G (D). (a) If (σ, β) P n (G, B) and σ = (P 0 < < P n ), then P n G D. 13

14 (b) Assume that D is abelian and let n 1. Then the map γ : P n (H, b) P n (G, B), (τ, β) (τ, β Gτ ), is injective, H-equivariant, and induces a bijection γ : P n (H, b)/h P n (G, B)/G. Proof (a) We proceed by induction on n. If n = 1, by convention we have P 1 = 1 and 1 D. If n = 0, then G σ = N G (P 0 ). Let Q be a defect group of β. Then P 0 Q, since P 0 G σ, and Q G D, since β G = B. Thus, P 0 G D. Now assume that n 1 and set σ := (P 1 < < P n ) so that we have P 0 G σ G σ G and G σ = N G σ (P 0 ). Thus, β := β G σ Bl(OG σ ) is defined and β G = (β G σ ) G = β G = B so that ( σ, β) P n 1 (G, B). By induction one has P n G D. (b) Let (τ, β) P(H, b). Since H τ = N Gτ (D), the block β Gτ is defined, and (β Gτ ) G = β G = (β H ) G = b G = B so that (τ, β Gτ ) P(G, B) and γ is well-defined. The map γ is obviously H-equivariant. Moreover it is injective by Corollary 3.2 and by Brauer s first main theorem. First we show that γ is surjective. Let (σ, β) P n (G, B) with σ = (P 0 < < P n ). By Part (a) and Corollary 3.2 we may assume that P n D and that D is a defect group of β. Since D G σ and H σ = N Gσ (D), the Brauer correspondent β Bl(OH σ ) of β has the property ( β H ) G = β G = ( β Gσ ) G = β G = B. This implies that β H = b by Brauer s first main theorem. In fact, β H also has D as a defect group, since defect groups can only become bigger under block induction, and since D is a defect group of B. Thus, (σ, β) = γ(σ, β). Next we show that γ is injective by induction on n. If n = 1 then this holds, since P 1 (G, B) = {(, B)} and P 1 (H, b) = {(, b)}. Now we assume that n 0. Let (τ, β), (τ, β ) P n (H, b) and let g G be such that (τ, (β ) G τ ) = g (τ, β Gτ ). We want to show that there exists h H such that (τ, β ) = h (τ, β). We write τ = (P 0 < < P n ) and τ = (P 0 <... < P n). Note that P n, P n D by part (a). Let (D, β D ) and (D, β D ) be subpairs such that β covers β D Bl(OC G (D)) and β covers β D Bl(OC G(D)). Then β D and β D have defect group D and β Hτ D = β and (β D )H τ = β so that βd H = b = (β D )H. Thus, b covers β D and β D. This implies that β D = h β D for some h H. Replacing (τ, β) with h (τ, β) we may assume from now on that β D = β D. For the reader s convenience we arrange the relevant subgroups and blocks in Figure 1. All occurring blocks are defined as Brauer induced blocks of β D. Note that all these blocks have D as defect group. Dotted lines indicate that the blocks are in correspondence via Brauer s first main theorem. Since g (τ, β) = (τ, β ), we have g (P n, β n ) = (P n, g βn ) and ( g βn ) G τ = ( g βn ) ggτ g 1 = g β = ( β n ) G τ so that g βn = g ( β n) for some g G τ, since both g βn and β n are blocks of OC G (P n) covered by β. Thus, the B-subpairs (P n, β n ) and (P n, β n) satisfy g 1 g (Pn, β n ) = (P n, β n) and are both contained in the Sylow B-subpair (D, β D ). By [AB79, Proposition 4.21], there exist y N G (D, β D ) H and z C G (P n ) 14

15 G, B H, b N G (P n ), δ n N H (P n ), δ n N G (P n), δ n N G (P n), δ n H τ, β G τ, β G τ, β H τ, β C G (P n ), β n C H (P n ), β n C G (P n), β n C H (P n), β n C G (D), β D C H (D), β D P n D P n Figure 1: Relevant subgroups and blocks such that g 1 g = yz. Note that y 1 g N G (P n ), since g P n = P n and y P n = g 1 gz 1 P n = g 1g P n = P n. Note also that, since β D and β D correspond under Brauer s first main theorem, y stabilizes with β D also β D. If we set (τ, β ) := y 1 (τ, β ) it suffices to show that (τ, β) and (τ, β ) are H-conjugate. Since y P n = P n, the chains τ and τ of length n have the same largest subgroup, namely P n. We consider ˆτ and ˆτ arising from τ and τ by omitting P n and obtain elements (ˆτ, β), (ˆτ, β ) P n 1 (N H (P n ), δ n ) with δ n := β N H(P n) D, since N H (P n )ˆτ = H τ, N H (P n )ˆτ = H τ, and β = y 1 (β ) = y 1 (β H τ D ) = ( y 1β D ) y 1 H τ y = (β D ) H τ. We have y 1 g (ˆτ, β G τ ) = ( y 1gˆτ, y 1 g (β G τ )) = (ˆτ, (β ) G τ ) P n 1 (N G (P n ), δ n ) 15

16 with δ n := δ N G(P n) n, where N G (P n )ˆτ = G τ and N G (P n )ˆτ = G τ, since g (β G τ ) = (β ) G τ and y 1 ((β ) G τ ) = (β ) G τ. By our induction hypothesis for (N G (P n ), δ n ) and (N H (P n ), δ n ) instead of (G, B) and (H, b) there exists h N H (P n ) with h (ˆτ, β) = (ˆτ, β ) and we may conclude the proof with yh (τ, β) = (τ, β ). 3.4 Remark It is easy to see that the result in Proposition 3.3(b) implies the same result also for N (G) and E(G) instead of P(G). However, it is in general not true for U(G) or the radical chains R(G). 4 G-equivariant coefficient systems in the abelian defect group case and Broué s conjecture In this section we will show that if B is a block of OG with non-trivial abelian defect groups then a weak form of Broué s abelian defect group conjecture implies the existence of a G-equivariant coefficient system for (G, B) on P(G) and L that satisfies the Hypotheses 1.6(i) (vi) and thereby provides a contractible chain complex modeling the alternating sum in Dade s ordinary conjecture. 4.1 Definition Let B be a block of OG with defect group D, set H := N G (D), let b Bl(OH) denote the Brauer correspondent of B, and let b Bl(OH) denote the image of b under the antipode OH OH, h h 1. For the purpose of this paper we call an element C T (G H, B O b ) a weak splendid equivalence between B and b if C is a Z-linear combination of classes of indecomposable trivial source O[G H]-modules which are δ(d)-projective and if C Č = [B] and Č C = [b] as elements in T (G G, B O B ) and T (H H, b O b ). Here Č T (H G, b O B ) is the dual element of C, arising from application of the functor Hom O (, O) between the categories of (OG, OH)-bimodules and (OH, OG)-bimodules. Moreover, the multiplication C Č is defined as being induced by taking tensor products over OH after interpreting C as (OG, OH)- bimodule and Č as (OH, OG)-bimodule. If such a C exists, we call B and b weakly splendid equivalent. Recall from [Ri96] that if the two blocks B and b are splendidly equivalent, then the Lefschetz element in T (G H, B O b ) of the chain complex providing the splendid equivalence has the above property. In particular, if D is cyclic, there exists such a weakly splendid equivalence C T (G H, B O b ), by [Ro98]. 4.2 Theorem Let B be a block of OG with a non-trivial abelian defect group D, let H := N G (D), let b Bl(OH) be the Brauer correspondent of B, and let X {P, N, E}. Furthermore, assume that for each (τ, β) X (H, b) the block β of OH τ is weakly splendid equivalent to its Brauer correspondent β Gτ 16

17 Bl(OG τ ). Then there exists a G-equivariant coefficient system (d i (σ,β)) for (G, B) on X (G) and L that satisfies the Hypotheses 1.6(i) (vi). Proof We will use Proposition 3.3(b) to lift the result for (H, b), established in Proposition 1.8(b), to (G, B). We choose a fixed set [X (H, b)/h] of representatives of the H-orbits of X (H, B) and choose for each (τ, β) [X (H, b)/h] a weakly splendid equivalence C (τ,β) T (G τ H σ, β Gτ O β ). Then multiplication with C (τ,β) gives an isomorphism c (τ,β) : L(H τ, β) L(G τ, β Gτ ). Next, for any h H we define c h (τ,β) : L(H ( h τ), h β) L(G ( h τ), ( h β) G ( h τ) ) by the commutative diagram L(H ( h τ), h β) c h (τ,β) L(G ( h τ), ( h β) G ( h τ) ) h h L(H τ, β) c (τ,β) L(G τ, β Gτ ), where the vertical maps are as usually induced by conjugation with h (or rather h 1 ). Note that c h = c h if h (τ, β) = h (τ, β) for some h H, (τ,β) (τ,β) since stab H (τ, β) H τ G τ and H τ (resp. G τ ) acts trivially on L(H τ, β) (resp. L(G τ, β Gτ )). Thus, we have well-defined isomorphisms c (τ,β) for all (τ, β) X (H, b) such that they commute with H-conjugation. Next we define d i (τ,β Gτ ) : L(G τ, β Gτ ) L(G τi, β Gτ i ) for (τ, β) [X (H, b)/h] by the commutative diagram L(G τ, β Gτ ) di (τ,βgτ ) L(G τi, β Gτ i ) c (τ,β) c (τi,β Hτ i ) L(H τ, β) δi (τ,β) L(H τi, β Hτ i ), where (δ(τ,β) i ) is the H-invariant coefficient system for (H, b) on X and L from Example 1.3(b) which we have available, since O p (H) D 1. Note that, by Proposition 3.3(b), we thereby defined the desired G-equivariant coefficient system (d i (σ,β) ) for (G, B) on X (G) and L already for the elements (τ, βgτ ), (τ, β) [X (H, b)/h], which form a set of representatives for X (G, B)/G. Now, 17

18 for any g G, (τ, β) [X (H, b)/h], and i {0,..., τ }, we define d i g (τ,β Gτ ) by the commutative diagram L(G ( g τ),( g β) G ( g τ) ) d i g (τ,β Gτ ) L(G ( g τ) i, ( g β) G ( g τ) i ) g g (4.2.a) L(G τ, β Gτ ) di (τ,βgτ ) L(G τi, β Gτ i ). Again, if g G is such that g (τ, β Gτ ) = g (τ, β Gτ ), then d i g (τ,β Gτ ) = di g (τ,β Gτ ), since stab G (τ, β Gτ ) G τ G τi acts trivially on L(G τ, β Gτ ) and L(G τi, β Gτ i ). With these definitions we obtain the commutative diagram L(G τ, β Gτ ) di (τ,βgτ ) L(G τi, β Gτ i ) c (τ,β) c (τi,β Hτ i ) (4.2.b) L(H τ, β) δi (τ,β) L(H τi, β Hτ i ) for each (τ, β) X (H, b) and i {0,..., τ }. The family of maps (d i (σ,β)), (σ, β) X (G, B), i {0,..., σ }, is a G- equivariant coefficient system for (G, B) on X (G) and L. In fact, it is G- equivariant by definition, cf. Diagram (4.2.a), and using this it suffices to prove the compatibility property from Definition 1.1 only for (τ, β Gτ ) with (τ, β) [X (H, b)/h] and 0 i < j τ. But in this restricted situation, compatibility follows from Diagram (4.2.b) and the compatibility of the maps δ(τ,β) i. By Diagram (4.2.b) and the G-equivariance of d i (σ,β), (σ, β) X (G, B), i {0,..., σ }, it follows from Proposition 2.3 and from [Br90, Théorèm 1.5.(2)] that with the maps δ(τ,β) i also the maps di (σ,β) satisfy Hypotheses 1.6(i) (v). Again by Diagram (4.2.b) and Proposition 3.3(b), the maps c (τ,β), (τ, β) X (H, b), induce a ZH-linear chain map c : L (H, b) L (G, B) between the associated chain complexes. Moreover, c induces an isomorphism of chain complexes c : L (H, b) H L (G, B) G and further isomorphisms T (H, b) H T (G, B) G, R (H, b) H R (G, B) G, and (R F ) (H, b) H (R F ) (G, B) G, 18

19 which commute with the chain maps induced by ι, π, ν T, and ν L (cf. Proposition 2.3(a),(b)). Thus, since the maps δ(τ,β) i satisfy the hypothesis (vi) in 1.6, also the maps d i (σ,β) do, and the proof is complete. 4.3 Remark It was already proved in [U97, Proposition 1.7] that the existence of perfect isometries between B and b, and between associated block on local levels, implies that Dade s ordinary conjecture is true for B. References [Al86] J. L. Alperin: Local representation theory. Cambridge University Press, [Al87] J. L. Alperin: Weights for finite groups. Proc. Sympos. Pure Math. 47 (1987), [AB79] [Bo01a] J. L. Alperin, M. Broué: Local methods in block theory. Ann. of Math. 110 (1979), R. Boltje: Alperin s weight conjecture in terms of linear source modules and trivial source modules. In: Modular Representation Theory of Finite Groups; Proceedings of a Symposium held at the University of Virginia, Charlottesville May 8-15, 1998, de Gruyter 2001, [Bo01b] R. Boltje: Alperin s weight conjecture and chain complexes. J. London Math. Soc. 68 (2003), [Br90] M. Broué: Isométries parfaites, types de blocs, catégories dérivées. Astérisque (1990), [Da92] E. C. Dade: Counting characters in blocks. I. Invent. Math. 109 (1992), [KR89] [NT89] [Ri96] [Ro98] R. Knörr, G. Robinson: Some remarks on a conjecture of Alperin. J. London Math. Soc. (2) 39 (1989), H. Nagao, Y. Tsushima: Representations of finite groups. Academic Press, San Diego J. Rickard: Splendid equivalences: Derived categories and permutation modules. Proc. London Math. Soc. 72 (1996), R. Rouquier: The derived category of blocks with cyclic defect groups. In: Derived equivalences for group rings (eds. S. König, A. Zimmermann), Springer Lecture Notes 1685, (1998),

20 [U97] Y. Usami: Perfect isometries for principal blocks with abellian defect groups and elementary abelian 2-inertial quotients. J. Algebra 196,