Cohomology of groups, abelian Sylow subgroups and splendid equivalences

Alexander Zimmermann

Abstract Let G be a finite group and let R be a complete discrete valuation domain of characteristic 0 with residue field k of characteristic p and let S be R or k. The cohomology rings H^{∗}(K, S) for subgroups K of G together with restriction to subgroups of G, transfer from subgroups of G and conjugation by elements of G gives H^{∗}(−, S) the structure of a Mackey functor. Moreover, the group HSplen_{S}(K) of splendid auto-equivalences of the bounded derived category of finitely generated SG-modules fixing the trivial module acts S-linearly on H^{∗}(K, S). In this note we study the compatibility of these structures and get some consequences when G has an abelian Sylow p subgroup. In particular we see that in case G has an abelian Sylow p subgroup, then HSplen_{R}(G) acts by automorphisms of the Sylow subgroup on the cohomology.

2000 AMS Subject Classification : 20C05, 20J06, 16G30, 18E30, 14L30

Let G be a finite group and let R be a commutative ring, considered as a trivial RG-module. The bounded derived category D^{b}(RG) of finitely generated RG-modules is used in modular representation theory of finite groups to provide a geometric framework for the classical conjectures like Dade’s conjecture or Alperin’s conjecture [3, 4]. These two are consequences of Broué’s conjecture [1, 4] which states that in case R = k is a field of characteristic p and G is a finite group with abelian Sylow p subgroup P, the derived categories of the principal block B_{0}(kG) of kG and the principal block B_{0}(kN_{G}(P)) of kN_{G}(P) are equivalent. Besides the above conjectures of Alperin and Dade a positive answer to Broué’s conjecture implies for example that the K-theory, the cyclic and the Hochschild (co-)homology of the principal blocks of kG and of kN_{G}(P) coincide. For an account of other consequences and most known results see [4].

If there is an equivalence between two derived categories, an immediate question is, how many equivalences there are. This way, one is lead to the definition of the group T rP ic_{R}(B_{0}(RG)) of autoequivalences of the derived category D^{b}(B_{0}(RG)) (see [8]). This group comes into the play from a very different approach as well. The Mirror symmetry conjecture of Kontsevitch imply that symplectic automorphisms of a symplectic manifold with vanishing first Chern class induce auto-equivalences of the derived category of sheaves of the mirror Calabi-Yau manifold. From there as well one is lead to the group of auto-equivalences of the derived category (see [9]).

Studying a group is done most naturally by studying its modules. So, one should look for a natural module on which these groups act on. From many points of view the derived category is the right object to consider homology. Since the derived categories, we are interested in, are derived categories of group rings, in the context of auto-equivalences of group rings the natural module we asked for is the cohomology of groups.

In the present note we construct a module structure on H^{∗}(G, R) coming from interpreting this object in the derived category. Let A be an R-algebra. Bernhard Keller proved [2] that D^{b}(B_{0}(RG)) ≃ D^{b}(A) as triangulated categories if and only if there is an X ∈ D^{b}(A⊗_{R}B_{0}(RG)^{op}) so that X⊗_{B_{0}(RG)}^{𝕃}− is an equivalence. Such an X is called a two-sided tilting complex. In [11] it is proved that if R is hereditary, then A is R-projective again and by [2] the inverse equivalence is again a derived tensor product by a complex of bimodules. For a complex X note by [X] its isomorphism class in the derived category. Set [8]

T rP ic_{R}(B_{0}(RG)) := {[X]| X ∈ D^{b}(B_{0}(RG) ⊗ B_{0}(RG)^{op}) is a 2-sided tilting complex}

and HD_{R}(G) := {[X] ∈ T rP ic_{R}(B_{0}(RG))| X ⊗_{B_{0}(RG)} R ≃ R} . It is shown (cf. [12]) that the group cohomology H^{∗}(G, R) is an R HD_{R}(G)-module by composing the following morphisms: Take X with [X] ∈ HD_{R}(G). Then X acts on H^{∗}(G, R) by the composition F_{X} of the following maps

\begin{matrix}H^{n}(G, R) & & H^{n}(G, R) \\ ∥ & & ∥ \\ Hom_{D^{b}(RG)}(R, R[n]) & ⟶ Hom_{D^{b}(RG)}(X ⊗_{RG} R, X ⊗_{RG} R[n]) ≃ & Hom_{D^{b}(RG)}(R, R[n]) \end{matrix} .

In [12] it is shown that F_{X} does not depend on the isomorphisms chosen.

1 Splendid Equivalences

Let now R be a complete discrete valuation ring with field of fractions K of characteristic 0 and residue field k of characteristic p. Let P be a p-Sylow subgroup of G and let Δ : G ⟶ G × G be the codiagonal Δ(g) = (g, g^{−1}).

(1.1) A two-sided tilting complex X with isomorphism class in T rP ic_{R}(RG) is called splendid [5] provided all homogeneous components of X are ΔP -projective p-permutation modules, projective as B_{0}(RG)-modules from the left and from the right, and provided Hom_{B_{0}(RG)^{op}}(X, X) ≃ B_{0}(RG) ≃ Hom_{B_{0}(RG)}(X, X) in the homotopy category of complexes of B_{0}(RG)-bimodules. Let SplenP ic_{R}(G) be the group (!) of homotopy equivalence classes (X) of splendid tilting complexes of B_{0}(RG)⊗_{R} B_{0}(RG)^{op}-modules. Then, there is a natural group homomorphism φ : SplenP ic_{R}(G) ⟶ T rP ic_{R}(B_{0}(RG) by taking isomorphism classes of the objects in the derived category instead of in the homotopy category. Let HSplen_{R}(RG) := φ^{−1}(HD_{R}(G)) ∩ SplenP ic_{R}(RG). We use similar notations for k as base ring.

(1.2) A second major ingredient in what follows is the Brauer construction. For any p-subgroup Q of G and any kG-module M set M(Q) := M^{Q}/ ∑_{R<Q,R≠Q} T r_{R}^{Q}M^{R} where M^{Q} denotes Q-fixed points and T r is the transfer map [10]. The mapping M ↦ M(Q) is functorial.

We shall use the Brauer functor in the following way: Results of Rickard [5] imply that −(ΔQ) : SplenP ic_{k}(G) ⟶ SplenP ic_{k}(C_{G}(Q)) is a homomorphism of groups.

We come to the main theorem.

Theorem 1 Let G be a finite group, let k be a field of characteristic p and let Q be a p-subgroup of G. For any (X) ∈ HSplen_{R}(G) with (X(ΔQ)) ∈ HSplen_{R}(C_{G}(Q)) we have

F_{X(ΔQ)} ∘ res_{C_{G}(Q)}^{G} = res_{C_{G}(Q)}^{G} ∘ F_{X} and F_{X} ∘ tr_{C_{G}(Q)}^{G} = tr_{C_{G}(Q)}^{G} ∘ F_{X(ΔQ)}.

The proof of the theorem is based essentially on the isomorphism X(ΔQ) ⊗_{kC_{G}(Q)} k ≃ X^{G}(Q) for elements (X) ∈ SplenP ic_{k}(G). In order to show that the action of X commutes with the transfer one uses in addition an abstract variant of Frobenius reciprocity. Showing that the action of X commutes with restriction is more straightforward.

We shall be concerned with functorial properties of the above constructions.

(1.3) Let 𝔓_{(P,G)} be a set of p-subgroups of G, partially ordered by inclusion and let Sub_{G} be the set of subgroups of G partially ordered by inclusion. Then, ℭ_{(P,G)} := {C_{G}(Q)| Q ∈ 𝔓_{(P,G)}} is a partially ordered set as well, and C_{G}(−) is an inclusion reversing mapping 𝔓_{(P,G)} ⟶ Sub_{G} with image ℭ_{(P,G)}. A partially ordered set (S, ≤) may be seen as category with objects being the elements of the set and the set of morphisms from one element x to another y of S is a singleton if x ≤ y and empty otherwise. A group sheaf is then a contravariant functor of (S, ≤) to the category of groups. Suppose that each element of 𝔓_{p,G} is abelian. Then the functor given by SplenP ic_{R}(−) on objects and the Brauer functor on morphisms is a group sheaf over the partially ordered set ℭ_{(P,G)}. In fact, Q_{1} ⊆ Q_{2} ⇒ C_{G}(Q_{1}) ⊇ C_{G}(Q_{2}) and this inclusion is mapped to the homomorphism SplenP ic_{R}(C_{G}(Q_{1})) ⟶ SplenP ic_{R}(C_{G}(Q_{2})) given by (X) ↦ (X(ΔQ_{2})).

(1.4) The cohomology of groups H^{∗}(−, R) together with restriction res on morphisms is a contravariant functor Sub_{G} ⟶ R − mod and H^{∗}(−, R) together with transfer tr is a covariant functor Sub_{G} ⟶ R − mod. Moreover, for any g ∈ G and any K < G there is an R-linear mapping c_{g} : H^{∗}(K, R) ⟶ H^{∗}( ^{g}K, R). This mapping is a natural transformation for (H^{∗}(−, R), res) as well as for (H^{∗}(−, R), tr). Each c_{g} acts as the identity on H^{∗}(K, R) if g ∈ K and c_{g}c_{h} = c_{gh}. Furthermore, one has the Mackey formula res_{L}^{H}tr_{K}^{H} = ∑_{x∈L\H/K} tr_{L∩ ^{x}K}^{L}c_{x}res_{^{x}L∩K}^{K} for subgroups L, K ≤ H ≤ G.

All these properties may be subsumed by saying that (H^{∗}(−, R), res, tr, c) is a Mackey functor (see [10, section 53]). A natural transformation η : H^{∗}(−, R) ⟶ H^{∗}(−, R) with respect to both res and tr is called a morphism of Mackey functors.

(1.5) Let S be a partially ordered set and F be a group sheaf on S. Let M be a sheaf of R-modules on S. Then, we say that F acts on M if there is a natural transformation F × M ⟶ M satisfying the usual properties of a group action locally for any fixed x ∈ S. This means that we demand for this action in addition to the usual axioms of a group action to be natural with respect to the order relation: More precisely, if x < y, then the natural diagram

F (x) × M(x) ⟶ M(x)
↑  ↑
F (y) × M(y) ⟶ M(y)

is commutative. We say that F acts on a covariant functor N : S ⟶ R − mod if F acts in the above sense on the sheaf N : S ⟶ (R − mod)^{op}. We define an action of a group sheaf F on a Mackey functor (M, res, tr, c) as an action of F on (M, res) and on (M, tr). Remark that for technical reasons we consider (M, tr) : X ⟶ (R−mod)^{op} being valued in the opposite category. Note that a particular case is the constant sheaf where F (x) equals a fixed group Γ for any x ∈ S.

Define for any partially ordered set of abelian p-subgroups 𝔓_{p,G} of the finite group G the following:

HSplen_{k,𝔓_{p,G}}(C_{G}(Q)) := {(X) ∈ HSplen_{k}(C_{G}(Q)) | Q ≤ Q′ ∈ 𝔓_{p,G} ⇒ (X(ΔQ′)) ∈ HSplen_{k}(C_{G}(Q′))}

It is clear that G = C_{G}(1). With this definition we have the

Theorem 2 Let 𝔓_{p,G} be a partially ordered set of p-subgroups of the finite group G and let ℭ_{p,G} := {C_{G}(Q)| Q ∈ 𝔓_{p,G}}.

If ℭ_{p,G} is closed under intersection and conjugation, then the constant sheaf HSplen_{k,𝔓_{p,G}}(G) acts as morphisms of Mackey functors on H^{∗}(−, k).

Suppose that each element of 𝔓_{p,G} is abelian. Let Res: ℭ_{p,G} ⟶ k − 𝔐od and Trans: ℭ_{p,G} ⟶ (k − 𝔐od)^{op} be the two functors which are identical on objects: H^{∗}(−, k) : ℭ_{p,G} ⟶ k − 𝔐od while on morphisms Res(C_{G}(Q_{1}) ≤ C_{G}(Q_{2})) := res_{C_{G}(Q_{1})}^{C_{G}(Q_{2})} and Trans(C_{G}(Q_{1}) ≤ C_{G}(Q_{2})) := tr_{C_{G}(Q_{1})}^{C_{G}(Q_{2})} Then, HSplen_{k,𝔓_{p,G}}(−) : ℭ_{p,G} ⟶ 𝔊roup acts by natural transformations on Res and on Trans.

Observe that the second statement makes sense since we assume that the groups in 𝔓_{p,G} are abelian. This implies that for Q_{2} > Q_{1} one has Q_{2} ≤ C_{G}(Q_{1}) for all Q_{1}, Q_{2} ∈ 𝔓_{p,G}.

If Q_{2} is not abelian it is false in general that Q_{1} < Q_{2} imply that Q_{2} centralize Q_{1} and the Brauer functor −(ΔQ_{2}) is not defined on SplenP ic_{R}(C_{G}(Q_{1})).

Note however that we did not have to assume that the Sylow p subgroup of G is abelian.

2 Lifting to characteristic 0 and Abelian Sylow subgroups

In this section we shall give some consequences of Theorem 2 in case G is a finite group with abelian Sylow p subgroup P. We keep the hypotheses on R and k at the beginning of section 1.

(2.1) It is easy to see that the group homomorphism − ⊗_{R} k : T rP ic_{R}(B_{0}(RG)) ⟶ T rP ic_{k}(B_{0}(kG)) gives rise to a commutative diagram

T rP ic_{R}(RG) ⟶ T rP ic_{k}(kG)
↑  ↑
SplenP ic_{R}(RG) ˜⟶ SplenP ic_{k}(kG)
∪  ∪
HSplen_{R}(G) ⟶ HSplen_{k}(G)

where the middle horizontal morphism is an isomorphism as a consequence of a result of Rickard [5, Theorem 5.2]. This implies that the lower mapping is injective.

(2.2) The Künneth sequence

0 ⟶ H^{i}(X ⊗_{RG} R) ⊗_{R} k ⟶ H^{i}(X ⊗_{RG} k) ⟶ T or_{1}^{R}(k, H^{i+1}(X ⊗_{RG} R)) ⟶ 0 implies that X ⊗_{RG} R ≃ M ∈ RG − mod with M ⊗_{R} k ≃ k. Since X is a two-sided tilting complex, this is true for K ⊗_{R} X as well and K ⊗_{R} X ⊗_{RG} R ≃ X ⊗_{RG} K ≠ 0. Hence, M is an RG-lattice and since M ⊗_{R} k ≃ k, we conclude that the module M is R-free of rank 1. There are only a finite number of such RG-modules since their G-structure is entirely fixed by their rational (one-dimensional) character. Let M_{1}, M_{2}, …, M_{n} be representatives of the isomorphism classes of such modules. Then, the mapping R ⟶ End_{R}(M_{i}) induced by scalar multiplication is an isomorphism of RG-modules. Hence, the induced map H^{m}(G, R) ⟶ H^{m}(G, End_{R}(M_{i})) ≃ Ext_{RG}^{m}(M_{i}, M_{i}) is an isomorphism. Set Γ := {(X) ∈ SplenP ic_{R}(RG)| (X ⊗_{R} k) ∈ HSplen_{R}(G)} . Then,

− ⊗_{R} k : ⨁_{i=1}^{n} Ext_{RG}^{m}(M_{i}, M_{i}) ⟶ ⨁_{i=1}^{n} H^{m}(G, k)

is a homomorphism of RΓ-modules. Any (X) ∈ Γ acts on the right hand side as (X ⊗_{R} k) and therefore diagonally. It is clear that (X) acts on the left hand side as monomial matrices with the same induced permutation in each degree. For m = 0 each of the summands on the right hand side is a copy of k and each of the summands on the left hand side is a copy of R. One concludes that (X) acts diagonally on the left hand side as well. We proved the following

Lemma 2.1 − ⊗_{R} k : HSplen_{R}(G) ≃ HSplen_{k}(G) .

As a consequence we formulate

Theorem 3 Let G be a group with abelian Sylow p subgroup P and let R be a complete discrete valuation domain of characteristic 0 and residue field k of characteristic p. Then, HSplen_{k,{P }}(G) acts on H^{∗}(G, k) by outer automorphisms of P.

Proof. Let S be R or k. Let X be a splendid tilting complex with isomorphism class in HSplen_{S}(G). Then

H^{∗}(G, S) ⟶^{res_{C_{G}(P)}^{G}} H^{∗}(C_{G}(P), S)
↓ F_{X}^{∗}  ↓ F_{X(ΔP)}^{∗}
H^{∗}(G, S) ⟶^{res_{C_{G}(P)}^{G}} H^{∗}(C_{G}(P), S)

is a commutative diagram. The image of res_{C_{G}(P)}^{G} is the set of stable elements in H^{∗}(C_{G}(P), S). By

definition, C_{G}(P) acts trivially on P and so every element of H(P, S) is C_{G}(P)-stable. Hence res_{P}^{C_{G}(P)} is an isomorphism. Since the index [G : C_{G}(P)] is invertible in S, the homomorphism res_{C_{G}(P)}^{G} is injective. Moreover, P is a normal subgroup of C_{G}(P) and the quotient C_{G}(P)/P is a p′-group. So, C_{G}(P) = P × C_{G}(P)/P. It is clear that the principal block of kC_{G}(P) is isomorphic to kP. The isomorphism is induced by the trivial representation of C_{G}(P)/P. By [8], T rP ic_{S}(SP) = P ic_{S}(SP)× C_{∞} where C_{∞} is the group generated by the shift of degree. So, HSplen_{S,{P }}(G) acts on H^{∗}(G, k) as HSplen_{S}(C_{G}(P)) acts on H^{∗}(C_{G}(P), k) and HSplen_{S}(C_{G}(P)) = HSplen_{S}(P) ⊆ Out_{S}(SP) .

But, HSplen_{R}(P) ≃ HSplen_{k}(P) by Lemma 2.1. Roggenkamp and Scott prove (see [6]) that the set of automorphisms of RP which preserve the augmentation equals Inn(RP) · Aut(P). A theorem of Coleman and an improvement due to Leonard Scott (see [7, part I § 2.1, Lemma 2.1]) states that Aut(P) ∩ Inn(RP) = Inn(P). Hence Out_{R}(RP) ∩ HSplen_{R}(RP) ⊆ Out(P). Since P is abelian, Out(P) = Aut(P). This proves the theorem.

Example (2.3) : It is clear that the restriction res_{P}^{G} maps the action of an outer automorphism α of G to the action of the restriction α|_{P} to P, where one may modify if necessary α by an inner automorphism so that α fixes P. From that description it is clear that there are automorphisms of P which do not act as any automorphism of G on the image of the restriction map res_{P}^{G}. One might ask if the action of any automorphism of P on the image of res_{P}^{G} can be realized as the action of an element in the larger group HSplen_{R}(G). This is not the case. Let D_{p} be the dihedral group of order 2p for a prime number p. Then, for R = ℤ̂_{p}, the p-adic integers, the group ring RD_{p} is a Brauer tree algebra with two edges and exceptional vertex in the middle vertex. By the arguments used in [8] and [12], T rP ic_{R}(B_{0}(RD_{p})) is a central extension by an infinite cyclic group of some subgroup of P SL_{2}(ℤ). In fact, it is not difficult to see that T rP ic_{R}(B_{0}(RD_{p})) is generated by the Splendid auto-eqivalences and abelian Sylow subgroups 5

preimage of the level two congruence subgroup Γ(2) of P SL_{2}(ℤ) and the standard element ϕ of order 2. Call s^{2} and t^{2} preimages of the standard generators of Γ(2). The stabilizer of the trivial module equals the group < ϕts(ϕs)^{−3}, t^{2} > remarking that (ϕs)^{3} is shift by 2 degrees. But, as the action of HD_{R}(G) on H^{∗}(G, R) factors via the natural quotient of HD_{R}(G) to the group of autoequivalences of the stable module category (see [12]) and as there are no stable auto-equivalences fixing the trivial module (cf Linckelmann [4, chapter XI]), < ϕts(ϕs)^{−3}, t^{2} > acts trivially on H^{∗}(D_{p}, 𝔽_{p}). Nevertheless, H^{∗}(D_{p}, 𝔽_{p}) = H^{∗}(C_{p}, 𝔽_{p})^{C_{2}} = 𝔽_{p}[X^{2}] where X is a 2-cocycle in H^{∗}(C_{p}, 𝔽_{p}). The action of Aut(C_{p}) = C_{p−1} is multiplication by a multiplicative generator in degree 2, hence by its square in degree 4. As soon as p > 3, there exists α ∈ 𝔽_{p} with α^{2} ≠ 1.

(2.4) Let G be a finite group with abelian Sylow p subgroup P. By the above considerations the action of HSplen_{R,{P,{1}}}(G) on H(G, R) induces a group homomorphism HSplen_{R,{P,{1}}}(G) ⟶ Aut(P).

It might be an interesting question to determine the image of the Brauer functor −(ΔP). For an abelian group P it is clear that Aut(P) acts faithfully on H^{∗}(P, R), but there is a difficult and open question of S. Jackowski if for any p-group P the group Out(P) acts faithfully on H^{∗}(P, R). The answer is negative if R is replaced by a field k of characteristic p; the cyclic group of order p^{2} gives already a counterexample.

References

[1] M. Broué, Isométries Parfaites, Types de Blocs, Catégories Dérivées. Astérisque 181–182 (1990), 61–92.

[2] Bernhard Keller, A remark on tilting theory and DG-algebras, manuscripta mathematica 79 (1993) 247-253.

[3] B. Külshammer, Offene Probleme in der Darstellungstheorie endlicher Gruppen, Jahresbericht der deutschen Mathematiker Vereinigung 94 (1992), 98–104.

[4] Steffen König and Alexander Zimmermann, “Derived equivalences for group rings”, with contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier, Lecture Notes in Mathematics 1685, Springer Verlag, Berlin 1998.

[5] Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331-358.

[6] Klaus Roggenkamp, Subgroup Rigidity of p-adic group rings (Weiss’ argument revisited), J. London Mathematical Society 59 (1992) 534-544.

[7] Klaus Roggenkamp and Martin Taylor, “Group Rings and Class Groups”, Birkhäuser Basel 1992.

[8] Raphaël Rouquier and Alexander Zimmermann, A Picard group for derived module categories, accepted for Proceedings of the London Mathematical Society.

[9] Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, preprint (2000), http://xxx.lanl.gov/abs/math.AG/0001043

[10] J. Thévenaz, “G-algebras and modular representation theory”, Oxford 1995.

[11] Alexander Zimmermann, Tilted symmetric orders are symmetric orders, Archiv der Mathematik 73 (1999) 15-17.

[12] Alexander Zimmermann, Auto-equivalences of derived categories acting on group cohomology, preprint (August 1999).

[13] Alexander Zimmermann, Cohomology of groups and splendid equivalences of derived categories, to appear in Proceedings of the Cambridge Philosophical Society (2001).

Faculté de Mathématiques Université de Picardie 33, rue St Leu 80039 Amiens Cedex; France

electronic mail: Alexander.Zimmermann@u-picardie.fr

Amiens, June 2000

[AMA - Algebra Montpellier Announcements - 01-2000][October, 2000]
Received July 2000, revised September 2000.