SELF-TILTING COMPLEXES YIELD UNSTABLE MODULES

ALEXANDER ZIMMERMANN

Abstract. Let G be a group and let R be a commutative ring. Let T rP ic_{R}(RG) be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of RG-modules. The subgroup HD_{R}(G) of T rP ic_{R}(RG) consisting of self-equivalences fixing the trivial RG-module acts on the cohomology ring H^{∗}(G, R). The action is functorial with respect to R. The self-equivalences which are ’splendid’ in a sense defined by J. Rickard act natural with respect to transfer and restriction to centralizers of p-subgroups in case R is a field of characteristic p. In the present paper we prove that this action of self-equivalences on H^{∗}(G, R) commutes with the action of the Steenrod algebra and study the behaviour of the action of splendid self-equivalences with respect to Lannes’ T -functor.

Introduction

In an earlier joint paper [9] with Raphaël Rouquier I defined the group T rP ic_{R}(A) of standard self-equivalences of a derived module category D^{b}(A) for an R-algebra A which is projective as an R-module. For any A-module M let HD_{M}(A) be the subgroup of T rP ic_{R}(A) which is formed by the self-equivalences mapping M to an isomorphic copy. Then, in an earlier paper [11] I showed that, under some hypothesis on M, the group HD_{M}(A) acts in a natural way on the Ext-algebra Ext_{A}^{∗}(M, M). In case of A being a group algebra RG, with R being a field of characteristic p and G being a finite group, then J. Rickard defines in [8] what is a splendid equivalence by some technical conditions basically by asking that the homogeneous components of a tilting complex are p-permutation modules induced from diagonal p-subgroups, and by some invertibility condition in the homotopy category. These splendid equivalences induce self-equivalences of the derived categories of centralizers of p-subgroups by the Brauer construction. In [12] I showed that then, for M = R being the trivial module, the action of those splendid equivalences commute with transfer and restriction from and to the cohomology rings of centralizers of p-subgroups. In the present paper we enlarge these properties still further. The action of self-equivalences of the bounded derived category D^{b}(RG) on H^{∗}(G, R) commutes with the action of the Steenrod algebra on H^{∗}(G, R) for any prime p.

As consequence of the above statements, any cohomology ring H^{∗}(G, 𝔽_{p}) defines a functor H^{∗}(G, 𝔽_{p}) ⊗_{𝔽_{p} HD_{𝔽p}(B_{0}(𝔽_{p}G))} − from the modules over the group of derived self-equivalences of the principal block of the group ring 𝔽_{p} HD_{𝔽_{p}}(𝔽_{p}G) fixing the trivial module to the category of unstable modules 𝔘_{p} and similarly in the opposite direction. Obviously, we may restrict to splendid self-equivalences.

We shall describe the composition of Lannes’ T -functor with the first functor and the image of free unstable modules by the second. Moreover, by a result of Lannes T_{V} (H^{∗}(G, 𝔽_{p})) decomposes into direct product of cohomology rings as unstable modules. We shall prove that this decomposition is also a decomposition as modules over the action of splendid selfequivalences. This will give evidence that splendid equivalences are the correct objects to study in this context.

The paper is organized as follows. Section 1 recalls the necessary definitions and properties of Even’s norm map as it is used here and the definition of the Steenrod operation. In Section 2 it is shown that the normalized part of the outer automorphism group of the group ring behaves well with respect to the norm map norm_{G}^{G×C_{p}} on the cohomology ring of G. Section 3 proves that splendid self-equivalences of the derived category of a group ring fixing the trivial module act equivariantly with respect to the Steenrod operation on the cohomology ring of the group. Finally, in Section 4 we work on the relation of unstable modules and the action of splendid self-equivalences of the derived category of a group ring.

Date: August 2001, revised version December 2001.

1991 Mathematics Subject Classification. 16E30, 20J06, 55S10, 18E30.

Acknowledgement. I thank Serge Bouc for many fruitful discussions over the last years. In particular he mentioned to me a subtlety concerning the difference between homotopy equivalences and quasi-isomorphisms for tensor product complexes which lead to Remark 3.4. I thank the referee for pointing out an error in the proof of Proposition 2.1 in the original version of the paper.

1. Preliminary definitions and known results

1.1. Equivalences between derived categories of group rings. The notation concerning derived categories and derived equivalences we use here and throughout the whole article is the one in [5]. Let R be a commutative ring and let A and B be R-algebras. A complex T is a tilting complex if it is a bounded complex with finitely generated projective homogeneous components so that Hom_{D^{b}(A)}(T, T [n]) = 0 for n ≠ 0 and so that the smallest triangulated subcategory of D^{b}(A) which is closed under taking direct summands of finite direct sums and which contains T also contains the rank 1 free module A. The fundamental result of Rickard [7] shows that the derived categories D^{b}(A) and D^{b}(B) are equivalent as triangulated categories if and only if there is a tilting complex T in D^{b}(A) so that End_{D^{b}(A)}(T) ≃ B.

If A is flat as R-module, then for any tilting complex T there is a complex X in D^{b}(A ⊗_{R} B^{op}) so that X ⊗_{B}^{𝕃} − is an equivalence which maps B to T. A complex X in D^{b}(A ⊗_{R} B^{op}) so that X ⊗_{B}^{𝕃} − is an equivalence is called a twosided tilting complex. Suppose A = B and A is projective as R-module. Then (cf [9]), the set T rP ic_{R}(A) of isomorphism classes of twosided tilting complexes in D^{b}(A ⊗_{R} A^{op}) is a group with group law − ⊗_{A}^{𝕃} −.

For any A-module M set

HD_{M}(A) := {[X] ∈ T rP ic_{R}(A) | X ⊗_{A}^{𝕃} M ≃ M}.

If any automorphism of M is given by multiplication with a unit in the centre of A, then in [11] it is shown that Ext_{A}^{∗}(M, M) is an R HD_{M}(A)-module by the action

Hom_{D^{b}(A)}(M, M[n]) ⟶ Hom_{D^{b}(A)}(X ⊗_{A}^{𝕃} M, X ⊗_{A}^{𝕃} M[n]) ⟶^{(α_{X}^{−1})^{∗}(α_{X})_{∗}} Hom_{D^{b}(A)}(M, M[n])

where α_{X} : X ⊗_{A}^{𝕃} M ⟶ M is an isomorphism.

In case A = RG is a group ring for a finite group G, a more subtle notion is useful. Let R be a field of characteristic p or a complete discrete valuation ring of characteristic 0 with residue field of characteristic p. Let B be the principal block of RG and let P be a Sylow p subgroup of G. Let Δ be the diagonal embedding of G into G × G. A twosided tilting complex X in D^{b}(B⊗B^{op}) is a splendid complex (cf Rickard [8]) if X is a complex of modules whose homogeneous components are projective as right-RG-modules and projective as left RG-modules, so that Hom_{RG}^{•}(X, X) is homotopy equivalent to B as complex of left and as complex of right modules, and so that each of the homogeneous components of X are Δ(P)- projective p-permutation modules. Let SplenP ic_{R}(G) be the set of homotopy equivalence classes (X) of splendid complexes X whose isomorphism class is in T rP ic_{R}(B). Let

HSplen_{R}(G) := {(X) ∈ SplenP ic_{R}(G) | [X] ∈ HD_{R}(B)}.

Suppose that R is a field of characteristic p. Then, for any p-subgroup Q of G, denote by b the principal block of RC_{G}(Q). The Brauer functor

B ⊗ B^{op} − mod ∋ N ↦ N(ΔQ) := N^{ΔQ}/ ∑_{S<Q; S≠Q} N^{ΔS} ∈ b ⊗ b^{op} − mod

induces a homomorphism

SplenP ic_{R}(G) ⟶ SplenP ic_{R}(C_{G}(Q)).

In [12] it is shown that the action of an X in HSplen_{R}(G) so that X(Δ(Q)) ∈ HSplen_{R}(C_{G}(Q)) on H^{∗}(G, R) commutes with restriction res_{C_{G}(Q)}^{G} and transfer tr_{C_{G}(Q)}^{G} between H^{∗}(G, R) and H^{∗}(C_{G}(Q), R).

1.2. Steenrod operations. Let G be a group and let H be a subgroup of G of finite index. Then, G ≤ H ≀ 𝔖_{|G:H|} via the action of G on the cosets modulo H. Suppose R is a commutative, hereditary coefficient ring.

We recall below the definitions of Even’s norm map and Steenrod operations from Benson’s book [1].

Let P = (P_{i}, d_{i})_{i∈ℕ} be a projective resolution of R as RH-module. Then, ⨂_{i=1}^{|G:H|} P is a (not necessarily projective) resolution of R as R(H ≀ 𝔖_{|G:H|})-module. Let Q = (Q_{i}, ∂_{i})_{i∈ℕ} be a projective resolution of R as R(H ≀ 𝔖_{|G:H|})-module. Then, there is a morphism of R(H ≀ 𝔖_{|G:H|})-module complexes τ : Q ⟶ ⨂_{i=1}^{|G:H|} P.

We denote by R_{s} the sign representation of 𝔖_{|G:H|} and R_{s}^{(r)} the r-fold tensor product of R_{s} with itself.

An element x ∈ H^{n}(H, M) is given as an equivalence class of homomorphisms with representative x : P_{n} ⟶ M and this defines a homomorphism ⨂_{i=1}^{|G:H|} P_{n} ⟶ ⨂_{i=1}^{|G:H|} M of R(H ≀𝔖_{|G:H|})-modules. As ⨂_{i=1}^{|G:H|} P_{n} is a direct factor of the degree |G : H|·n-homogeneous component of ⨂_{i=1}^{|G:H|} P, the cocycle x defines a homomorphism

Q_{n·|G:H|} ⟶ ( ⨂_{i=1}^{|G:H|} P)_{n·|G:H|} ⟶ ⨂_{i=1}^{|G:H|} P_{n} ⟶ ⨂_{i=1}^{|G:H|} M ⊗ R_{s}^{(n)} .

We have to tensor with R_{s}^{(n)} in order to get the mapping well defined with respect to the sign convention for tensor products of complexes. Observe that for even degree cohomology this is tensor product with the identity. The restriction to G ⊆ (H ≀ 𝔖_{|G:H|}) defines an element norm_{H}^{G}(x) in H^{n|G:H|}(G, ⨂_{i=1}^{|G:H|} M ⊗_{R} R_{s}^{(n)}). However, the sign of norm_{H}^{G}(x) depends on the embedding of G into H ≀ 𝔖_{|G:H|}. Since later on the embedding will be fixed once and for all, this ambiguity will not be of importance for us.

The Steenrod operations are defined by the following procedure.

Observing that in case p is odd, any p-cycle is of even parity and so, one can forget the sign representation which had to be introduced earlier in order to get the norm map well defined. We take

norm_{G}^{G×C_{p}} : H^{n}(G, 𝔽_{p}) ⟶ H^{n·p}(G × C_{p}, 𝔽_{p}) ≃ ⊕_{i=0}^{np}H^{np−i}(G, 𝔽_{p}) ⊗_{𝔽_{p}} H^{i}(C_{p}, 𝔽_{p})

and express

norm_{G}^{G×C_{p}}(x) = ⊕_{i=0}^{np}D^{np−i}(x) ⊗ a_{i}

where a_{2i} = a_{1}^{2i} and a_{2i+1} = β(a_{1})a_{2i}. We define

Sq^{i} := D^{n−i} : H^{n}(G, 𝔽_{2}) ⟶ H^{n+i}(G, 𝔽_{2})

if p = 2 and

P^{i} := (−1)^{i+mn(n+1)/2}(m!)^{−n}D^{(p−1)(n−2i)} : H^{n}(G, 𝔽_{p}) ⟶ H^{n+2(p−1)i}(G, 𝔽_{p})

with m = \frac{p−1}{2} for p odd.

We will have to use a more diagrammatic method to express the above in order to be able to prove that the Steenrod operations commute with the action of self-equivalences of the derived category.

As above let P be a projective resolution of R as RG-module and let Q be a projective resolution of R as RG ≀ 𝔖_{p}-module. As usually there is a mapping, unique up to homotopy, Q ⟶^{τ} ⨂_{1≤i≤p} P lifting the identity on R. Then,

Hom_{R(G^{p})}( ⨂_{1≤i≤p} P, ( ⨂_{1≤i≤p} M)[p · n]) ≃ Hom_{R(G^{p})}( ⨂_{1≤i≤p} P, ⨂_{1≤i≤p} M[n])
≃ ⨂_{1≤i≤p} Hom_{RG}(P, M[n])

with obvious isomorphisms. Observe that since the action of 𝔖_{p} does not enter the picture yet, the sign representation does not play a rôle here.

With the diagonal Δ we get

Hom_{RG}(P, M[n]) ⟶^{Δ} ⨂_{1≤i≤p} Hom_{RG}(P, M[n])

and we observe that the image of Δ under the isomorphism

Hom_{R(G^{p})}( ⨂_{1≤i≤p} P, ⨂_{1≤i≤p} M[n]) ≃ ⨂_{1≤i≤p} Hom_{RG}(P, M[n])

is in Hom_{R(G≀𝔖_{p})}(⨂_{1≤i≤p} P, (⨂_{1≤i≤p} M[n]) ⊗_{R} R_{s}^{(n)}). Now, the restriction of the sign representation of 𝔖_{p} to C_{p} always is the trivial representation; for p = 2, since there 1 = −1, and for p an odd prime, since there any p-cycle has even parity. Therefore, Δ factors through res_{G^{p}}^{G≀𝔖_{p}} by a mapping Δ′. So, the norm map norm_{G}^{G×C_{p}} can be seen as left composition of the left vertical homomorphisms in the following commutative diagram.

Hom_{RG}(P, M[n])  ⟶^{Δ} Hom_{R(G^{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} M[n])
↓ Δ′  ∥
Hom_{R(G≀𝔖_{p})}(⨂_{i=1}^{p} P, (⨂_{i=1}^{p} M[n]) ⊗_{R} R_{s}^{(n)}) ⟶^{res_{G^{p}}^{G≀𝔖p}} Hom_{R(G^{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} M[n])
↓
Hom_{R(G≀𝔖_{p})}(Q, (⨂_{i=1}^{p} M[n]) ⊗_{R} R_{s}^{(n)})
↓ res_{G×C_{p}}^{G≀𝔖_{p}}
Hom_{R(G×C_{p})}(Q, ⨂_{i=1}^{p} M[n])

One should observe that Δ is not linear over the base ring unless this base ring R is 𝔽_{p}.

2. Ring automorphisms

The group Out_{R}(RG) = Aut_{R,1}(RG)/Inn(RG) of outer augmented R-algebra automorphisms acts on H^{∗}(G, R) in the obvious way. We shall prove that this action is compatible with Steenrod operations.

Proposition 2.1. Let R = 𝔽_{p}. The action of Out_{𝔽_{p}}(𝔽_{p}G) on H^{∗}(G, 𝔽_{p}) commutes with the action of mod p Steenrod operations on H^{∗}(G, 𝔽_{p}).

To prove this, we shall show that for any automorphism α of 𝔽_{p}G there is an automorphism α̂ of 𝔽_{p}(G ≀ 𝔖_{p}) so that the diagram

α̂Q ⟶^{τ} ⨂_{1}^{p} ^{α}P
ρ ↓  ↓ ⊗σ
Q ⟶^{τ}  ⨂_{1}^{p} P

commutes. Here, the vertical homomorphisms are chain maps which lift the identity on 𝔽_{p} : The complex Q is a projective resolution of 𝔽_{p} and a mapping ρ lifting the identity on 𝔽_{p} exists by the universal property of Q being a projective resolution. P is a projective resolution of 𝔽_{p} as G-module. Then, we use the tensor product of p copies of the lifting σ of the identity on 𝔽_{p} onto P as complex of 𝔽_{p}G-module. This tensor product is easily seen to be 𝔽_{p}(G ≀ 𝔖_{p})-linear.

In order to define α̂ we observe the well known fact that

𝔽_{p}(G ≀ 𝔖_{p}) ≃ 𝔽_{p} \underbrace{(G × G × ⋯ × G)}_{p factors} ⊗_{𝔽_{p}}𝔽_{p}𝔖_{p}

is isomorphic to the skew tensor product of the first factor with the quotient group acted upon : 𝔽_{p}(G ≀ 𝔖_{p}) is the ordinary tensor product as 𝔽_{p}-module, and the multiplication is defined via a torsion by 𝔖_{p}:

(g_{1} ⊗ σ_{1}) · (g_{2} ⊗ σ_{2}) = g_{1}σ_{1}(g_{2}) ⊗ σ_{1}σ_{2} .

Since 𝔽_{p} \underbrace{(G × G × ⋯ × G)}_{p factors} ≃ ⨂_{i=1}^{p} 𝔽_{p}G, we may take α̂ := (⊗_{i=1}^{p}α) ⊗ id_{𝔖_{p}} and we verify

readily that α̂ is a ring automorphism of 𝔽_{p}(G ≀ 𝔖_{p}). Moreover, a projective resolution of ⊗_{i=1}^{p}𝔽_{p} as 𝔽_{p}(G ≀ 𝔖_{p})-module is Q := (⨂_{i=1}^{p} P) ⊗_{𝔽_{p}} R, where P is a projective resolution of 𝔽_{p} as 𝔽_{p}G-module and R is a projective resolution of 𝔽_{p} as an 𝔽_{p}𝔖_{p}-module (cf e.g. [1, Chapter 4.1]). Note that since the Sylow p subgroup of 𝔖_{p} is cyclic, the homogeneous components of R are indecomposable projective 𝔽_{p}𝔖_{p}-modules.

We shall have to compare the diagram

Hom_{𝔽_{p}G}(P, 𝔽_{p}[n])  ⟶^{Δ} Hom_{𝔽_{p}(G^{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} 𝔽_{p}[n])
↓ Δ′  ∥
Hom_{𝔽_{p}(G≀𝔖_{p})}(⨂_{i=1}^{p} P, (⨂_{i=1}^{p} 𝔽_{p}[n]) ⊗_{𝔽_{p}} 𝔽_{p}_{s}^{(n)}) ⟶^{res_{G^{p}}^{G≀𝔖p}} Hom_{𝔽_{p}(G^{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} 𝔽_{p}[n])
↓

Hom_{𝔽_{p}(G≀𝔖_{p})}(Q, (⨂_{i=1}^{p} 𝔽_{p}[n]) ⊗_{𝔽_{p}} 𝔽_{p}_{s}^{(n)})

↓ res_{G×C_{p}}^{G≀𝔖_{p}}
Hom_{𝔽_{p}(G×C_{p})}(Q, ⨂_{i=1}^{p} 𝔽_{p}[n])

with the diagram corresponding to the twisted resolutions

Hom_{𝔽_{p}G}(^{α}P, 𝔽_{p}[n])  ⟶^{Δ} Hom_{𝔽_{p}(G^{p})}(⨂_{i=1}^{p} ^{⊗α}P, ⨂_{i=1}^{p} 𝔽_{p}[n])
↓ Δ′  ∥
Hom_{𝔽_{p}(G≀𝔖_{p})}(α̂⨂_{i=1}^{p} P, (⨂_{i=1}^{p} 𝔽_{p}[n]) ⊗_{𝔽_{p}} 𝔽_{p}_{s}^{(n)}) ⟶^{res_{G^{p}}^{G≀𝔖p}} Hom_{𝔽_{p}(G^{p})}(^{⊗α}⨂_{i=1}^{p} P, ⨂_{i=1}^{p} 𝔽_{p}[n])
↓

Hom_{𝔽_{p}(G≀𝔖_{p})}(α̂Q, (⨂_{i=1}^{p} 𝔽_{p}[n]) ⊗_{𝔽_{p}} 𝔽_{p}_{s}^{(n)})

↓ res_{G×C_{p}}^{G≀𝔖_{p}}
Hom_{𝔽_{p}(G×C_{p})}(α̂Q, ⨂_{i=1}^{p} 𝔽_{p}[n])

and we shall have to define mappings from the morphism groups of the bottom diagram to the corresponding morphism sets of the top diagram in a way such that the squares which are created this way are commutative. Since P as well as^{α}P are projective resolutions of the trivial module, there is a chain homotopy isomorphism σ : P ⟶^{α} P and this isomorphism induces a chain homotopy isomorphism ⊗_{1}^{p}σ : ⨂_{i=1}^{p} P ⟶ ⨂_{i=1}^{p} ^{α}P and a chain homotopy isomorphism ρ : Q ⟶ α̂Q defined by taking the identity on the factors corresponding to R. It is then straightforward to see that all squares which occur are commutative.

In order to compute

norm_{G}^{G×C_{p}} : H^{n}(G, 𝔽_{p}) ⟶ H^{pn}(G × C_{p}, 𝔽_{p})

we have to map a cocycle by res_{G×C_{p}}^{G≀𝔖_{p}} ∘ τ^{∗} ∘ Δ′.

Since res_{G×C_{p}}^{G≀𝔖_{p}} = δ^{∗} for δ being the diagonal embedding of G into G^{p}, and since α̂ ∘ δ = δ ∘ (α × id_{C_{p}}),

norm_{G}^{G×C_{p}} ∘ α^{∗} = res_{G×C_{p}}^{G≀𝔖_{p}} ∘ τ^{∗} ∘ Δ′ ∘ α^{∗}
= res_{G×C_{p}}^{G≀𝔖_{p}} ∘ α̂^{∗} ∘ τ^{∗} ∘ Δ′
= (α × id_{C_{p}})^{∗} ∘ res_{G×C_{p}}^{G≀𝔖_{p}} ∘ τ^{∗} ∘ Δ′
= (α × id_{C_{p}})^{∗} ∘ norm_{G}^{G×C_{p}}

The proof is complete here.

3. Derived self-equivalences

Let R be a field of characteristic p, let B(RG) be the principal block of RG and let X be a twosided tilting complex in D^{b}(B(RG) ⊗_{R} B(RG)^{op}). Starting from X, we shall construct a twosided tilting complex in D^{b}(B(RG ≀ 𝔖_{p}) ⊗_{R} B(RG ≀ 𝔖_{p})^{op}).

It should be noted that Andrei Marcus constructed in [6] a tilting complex in B(RG ≀ 𝔖_{n}) if p does not divide n. However, in order to apply this construction to the Steenrod operation we are obliged to pass to n = p.

We consider the complex

X^{⊗p} := X ⊗_{R} X ⊗_{R} ⋯ ⊗_{R} X

with p factors as object in D^{b}(B(RG^{p}) ⊗_{R} B(RG^{p})^{op}). Of course, this is an invertible complex as for the endomorphism complex we get

Hom_{RG^{p}}(X^{⊗p}, X^{⊗p}) ≃ ⨂_{i=1}^{p} Hom_{RG}(X, X) ≃ ⨂_{i=1}^{p} B(RG) in D^{b}(RG^{p} ⊗_{R} (RG^{p})^{op}).

We consider now the complex of R(G ≀ 𝔖_{p})-right modules

X^{G≀𝔖_{p}} := X^{⊗p} ⊗_{RG^{p}} R(G ≀ 𝔖_{p}).

It is immediate that X^{G≀𝔖_{p}} ≃ X^{⊗p} ⊗_{R} R𝔖_{p} as R(G^{p})-left modules, where the p factors of G × G × . . . · · · × G act on the respective copy of X ⊗ X ⊗ ⋯ ⊗ X respectively and each acts trivially on R𝔖_{p}.

We shall now define a left action of R(G ≀ 𝔖_{p}) on X^{G≀𝔖_{p}} which commutes with the right action of G ≀ 𝔖_{p} (see [1, Chapter 4.1]).

Let σ be an element of 𝔖_{p}. The degree n homogeneous component of X^{⊗p} is

(X^{⊗p})_{n} = ⨁_{k_{1}+···+k_{p}=n} X_{k_{1}} ⊗_{R} X_{k_{2}} ⊗_{R} ⋯ ⊗_{R} X_{k_{p}}.

The differential δ_{n} on the tensor product complex is

∑_{k_{1}+···+k_{p}=n} d⊗id⊗⋯⊗id+(−1)^{k_{1}}id⊗d_{n−k}⊗id⊗⋯⊗id+⋯+(−1)^{k_{1}+···+k_{p−1}}id⊗⋯⊗id⊗d.

Now, σ acts on the tensor product complex permuting the factors

x_{1} ⊗ ⋯ ⊗ x_{p} ∈ X_{k_{1}} ⊗_{R} X_{k_{2}} ⊗_{R} ⋯ ⊗_{R} X_{k_{p}}

accordingly to σ acts on the index set {1, …, p} multiplied by a sign (−1)^{ν} where

ν = ∑_{j<k; σ(j)>σ(k)} deg(x_{j})deg(x_{k}) .

A more standard definition of X^{G≀𝔖_{p}} by tensor induction followed by ordinary induction, can be done using that X itself is a complex of G × G-modules.

Let H be a finite group and let Y be a complex of RH-modules. Then, the tensor induced complex Y^{⊗↑_{H}^{H×Cp}} is isomorphic to \underbrace{Y ⊗_{R} Y ⊗_{R} ⋯ ⊗_{R} Y}_{p factors} as complex of R(H × C_{p})-modules.

But actually, Y^{⊗↑_{H}^{H×Cp}} is a complex of R(H≀𝔖_{p})-modules, as is seen easily from the definition (cf e.g. [1]).

We will not use the two following lemmas, though we cite them since they might become important in later studies.

Lemma 3.1. Lt S be a p-subgroup of H. If Y is a direct sum of relatively S-projective H-modules, then Y^{⊗↑_{H}^{H×Cp}} is relatively S ≀ C_{p}-projective.

Proof. Let U be an RS-modules, so that Y | U ↑_{S}^{H}. Let π : U ↑_{S}^{H}⟶ Y and ι : Y ⟶ U ↑_{S}^{H} with π ∘ ι = id_{Y}. Then, the monomorphism

\underbrace{Y ⊗_{R} Y ⊗_{R} ⋯ ⊗_{R} Y}_{p factors} ⟶^{ι⊗···⊗ι} \underbrace{U ↑_{S}^{H} ⊗_{R}U ↑_{S}^{H} ⊗_{R} ⋯ ⊗_{R} U ↑_{S}^{H}}_{p factors}

is split by π ⊗ π ⊗ ⋯ ⊗ π and both morphisms are clearly morphisms of R(H ≀ 𝔖_{p})-modules. Moreover,

\underbrace{U ↑_{S}^{H} ⊗_{R}U ↑_{S}^{H} ⊗_{R} ⋯ ⊗_{R} U ↑_{S}^{H}}_{p factors} ≃ \underbrace{(U ⊗_{R} U ⊗_{R} ⋯ ⊗_{R} U)}_{p factors} ↑_{S×S×···×S}^{H×H×···×H}

as R(H ×H ×⋯×H)-modules. Nevertheless, the isomorphism consists of re-identifying the order of the tensors, and this isomorphism is clearly H ≀𝔖_{p}-linear. But now, the isomorphism induces an isomorphism

\underbrace{U ↑_{S}^{H} ⊗_{R}U ↑_{S}^{H} ⊗_{R} ⋯ ⊗_{R} U ↑_{S}^{H}}_{p factors} ≃ \underbrace{(U ⊗_{R} U ⊗_{R} ⋯ ⊗_{R} U)}_{p factors} ↑_{S≀𝔖_{p}}^{H≀𝔖_{p}} .

Since Y is a direct factor of this induced module, and since C_{p} is the Sylow p subgroup of 𝔖_{p}, the result follows.

We observe that

X^{G≀𝔖_{p}} := (X^{⊗↑_{G×G}^{(G×G)×Cp}} ) ↑_{(G×G)≀𝔖_{p}}^{(G≀𝔖_{p})×(G≀𝔖_{p})}

Indeed, (X^{⊗↑_{G×G}^{(G×G)×Cp}} ) is a tensor product of p copies of X, the direct product of p copies

of G acting from the left and from the right on the different copies of the p-fold tensor product, and on which C_{p} acts from the left and from the right by permuting the factors of the tensor product. Moreover, (G × G) ≀ 𝔖_{p} embeds into (G ≀ 𝔖_{p}) × (G ≀ 𝔖_{p}) via the mapping (g_{1}, g_{2}, c) ↦ (g_{1}, c) × (g_{2}, c), where g_{1}, g_{2} ∈ G^{p}. It is immediate to see that this construction gives the one defined above in a more elementary manner. However, we did not look at the differentials yet. Here, either we deal with multicomplexes and take the total complex of these multicomplexes, or we have to go back to the elementary definition above.

Lemma 3.2. Let P be a Sylow p subgroup of G. If X is a direct sum of Δ(P)-projective modules, then X^{G≀𝔖_{p}} is a direct sum of Δ(P ≀ C_{p})-projective modules.

Proof. Let P be a p-Sylow subgroup of G. Then, P^{p} is a p-Sylow subgroup of G^{p} and P ≀ C_{p} is a p-Sylow subgroup of G ≀ 𝔖_{p}. Let Δ(P) be the diagonal of P in G × G. Since by

Lemma 3.1 (X^{⊗↑_{G×G}^{(G×G)×Cp}} ) is Δ(P^{p})×C_{p}-projective, X^{G≀𝔖_{p}} is Δ(P ≀C_{p})-projective. Indeed, C_{p} in Δ(P^{p}) ×C_{p} is diagonally embedded in (G ≀ 𝔖_{p}) × (G ≀ 𝔖_{p}) and X^{G≀𝔖_{p}} is induced from

(G×G)^{p}×C_{p}. So, if (X^{⊗↑_{G×G}^{(G×G)×Cp}} ) is Δ(P^{p})×C_{p}-projective, X^{G≀𝔖_{p}} is Δ(P ≀C_{p})-projective.

The result follows from there.

Proposition 3.3. Let R be a field of characteristic p and let G be a finite group with Sylow p subgroup P. Then, for any X with isomorphism class in T rP ic_{R}(B(RG)), the complex X^{G≀𝔖_{p}} is a twosided tilting complex in T rP ic_{R}(B(R(G ≀ 𝔖_{p}))).

If X is splendid, then X^{G≀𝔖_{p}} is a twosided tilting complex with homogeneous components which are Δ(P)-projective p-permutation modules.

Remark 3.4. The complex X^{G≀𝔖_{p}} is in general not splendid even if X is splendid. This surprising fact can be seen if one wants to apply Brauer functors with respect to the Sylow p-subgroup of 𝔖_{p}. The complex X^{G≀𝔖_{p}} will then be transformed into a complex which is not a tilting complex anymore.

Proof of Proposition 3.3. We will show that the above defined complex X^{G≀𝔖_{p}} is a twosided tilting complex in D^{b}(B(R(G ≀ 𝔖_{p})) ⊗_{R} B(R(G ≀ 𝔖_{p}))^{op}).

We get isomorphisms of (the total complexes of) double complexes

Hom_{R(G≀𝔖_{p})}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}}) ≃ Hom_{R(G^{p})}^{•}(X^{⊗p}, X^{G≀𝔖_{p}} ↓_{G^{p}})
≃ ⨁_{σ∈𝔖_{p}} Hom_{R(G^{p})}^{•}(X^{⊗p}, X^{⊗p}) ⊗ σ

≃ ⨁_{σ∈𝔖_{p}} (⨂_{i=1}^{p}Hom_{RG}^{•}(X, X)) ⊗ σ

Moreover, we have to recover the action of R(G ≀ 𝔖_{p}) coming from the left action of R(G ≀ 𝔖_{p}) on the first and of the second variable X^{G≀𝔖_{p}}. The action on the first variable is going to become the right action of R(G≀𝔖_{p}) on Hom_{R(G≀𝔖_{p})}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}}) and the action on the second variable is going to become the left action on this Hom-complex.

The action of the subgroup G^{p} of G ≀ 𝔖_{p} trivially is factorwise on the tensor products, on each of the direct summands. The action of 𝔖_{p} from the right on the complex X^{G≀𝔖_{p}} is given by the induction

X^{⊗p} ↑_{G^{2}≀𝔖_{p}}^{(G≀𝔖_{p})^{2}}≃ X^{⊗p} ⊗_{R} R𝔖_{p}

The action from the right on the Hom^{•}-complex comes from the action on the second variable which is, after having applied Frobenius reciprocity, just the above described. Hence the action of 𝔖_{p} from the right permutes the direct summands of

⨁_{σ∈𝔖_{p}} (⨂_{i=1}^{p}Hom_{RG}^{•}(X, X)) ⊗ σ .

We shall study the left action of G≀𝔖_{p} on the Hom-complex. The action of G^{p} ≤ G≀𝔖_{p} on the left trivially is again the factorwise action on the two factors of the tensor products, in each of the direct summands. Now, the action of σ ∈ 𝔖_{p} from the left permutes the factors in the tensor product on the level of the modules in each degree; and applies a sign according to the above described formula; and henceforth also the factors in the tensor product

⨁_{σ∈𝔖_{p}} (⨂_{i=1}^{p}Hom_{RG}^{•}(X, X)) ⊗ σ .

Now, we may assume that the homogeneous components of X are free as B(RG) ⊗_{R} B(RG)^{op}-modules, except the left most one, which is projective as RG-right and projective as RG-left module. Then, using the Künneth formula,

H^{0}(Hom_{R(G≀𝔖_{p})}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}})) = (⨂_{i=1}^{p} Hom_{K(RG)}(X, X)) ⊗_{R} R𝔖_{p}

and H^{n}(Hom_{RG}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}})) = 0 for n ≠ 0 again by the Künneth formula. The Künneth formula applies without problems since R is a field. Then, at the final step we use that X is a tilting complex from either side. So, Hom_{RG}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}}) is quasiisomorphic to its degree 0 homology as left module and as right module. Therefore, by ordinary Morita theory (see e.g. Curtis-Reiner [2, (3.54)]) Hom_{R(G≀𝔖_{p})}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}}) is an invertible B(R(G ≀ 𝔖_{p}))-bimodule. This implies that X^{G≀𝔖_{p}} is a two-sided tilting complex in T rP ic_{R}(B(R(G ≀ 𝔖_{p}))). As a consequence, since R(G ≀ 𝔖_{p}) is a symmetric algebra,

Hom_{R(G≀𝔖_{p})}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}}) ≃ H^{0}(Hom_{R(G≀𝔖_{p})}^{•}(X^{G≀𝔖_{p}}, X^{G≀𝔖_{p}})) ≃ B(R(G ≀ 𝔖_{p}))

in the derived category of R(G ≀ 𝔖_{p}) ⊗_{R} R(G ≀ 𝔖_{p})^{op}-modules.

By Lemma 3.2, if X is a complex of p-permutation modules over G×G which are induced from the diagonal of a fixed p-Sylow subgroup of G, then X^{G≀𝔖_{p}} also is a complex ppermutation modules over (G ≀ 𝔖_{p}) × (G ≀ 𝔖_{p}), induced from a fixed Sylow p subgroup of G ≀ 𝔖_{p}.

Proposition 3.3 follows.

Remark 3.5. It is straightforward to generalize the statement and the proof of Proposition 3.3 to the case where X is a twosided tilting complex between two blocks B and b of RG and RH for two finite groups G and H and R is a field of finite characteristic. Moreover, it is an easy task to pass to any wreath product by a fixed permutation group Π.

We now study first properties of X^{G≀𝔖_{p}} with respect to the action on Ext-algebras [11].

Lemma 3.6. Let M be an RG-module for a finite group G and let X be a tilting complex with isomorphism class in HD_{M}(G). Let R_{s} be the sign representation of 𝔖_{p}. Then, the isomorphism class of X^{G≀𝔖_{p}} is in HD_{(⊗_{i=1}^{p}M)}(G ≀ 𝔖_{p}) ∩ HD_{((⊗_{i=1}^{p}M)⊗_{R}R_{s})}(G ≀ 𝔖_{p}).

Proof.

X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} (⊗_{i=1}^{p}M) ≃ (X^{⊗p} ⊗_{R} R𝔖_{p}) ⊗_{G^{p}×𝔖_{p}} (⊗_{i=1}^{p}M)
≃ (X^{⊗p}) ⊗_{G^{p}} (⊗_{i=1}^{p}M)
≃ ⨂_{i=1}^{p}(X ⊗_{G} M)
≃ ⊗_{i=1}^{p}M

where the last is an isomorphism in the derived category. Moreover,

X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} ((⊗_{i=1}^{p}M) ⊗_{R} R_{s}) ≃ (X^{⊗p} ⊗_{R} R𝔖_{p}) ⊗_{G^{p}×𝔖_{p}} ((⊗_{i=1}^{p}M) ⊗_{R} R_{s})
≃ ((X^{⊗p}) ⊗_{G^{p}} (⊗_{i=1}^{p}M)) ⊗_{R} (R𝔖_{p} ⊗_{𝔖_{p}} R_{s})
≃ (⨂_{i=1}^{p}(X ⊗_{G} M)) ⊗_{R} R_{s}
≃ (⊗_{i=1}^{p}M) ⊗_{R} R_{s}

is a similar computation. This proves the statement.

We are now able to prove that the mod 2 Steenrod square Sq and the mod p Steenrod operation P commute with the action of HD_{R}(G).

Theorem 1. Let G be a finite group, let R be a field of characteristic p. Then, for any X with isomorphism class in HD_{R}(G), the following squares are commutative. If p = 2, then

H^{n}(G, R) ⟶^{X⊗_{G}−} H^{n}(G, R)
Sq_{n}^{i} ↓  ↓ Sq_{n}^{i}
H^{n+i}(G, R) ⟶^{X⊗_{G}−} H^{n+i}(G, R)

is commutative and if p ≥ 3,

H^{n}(G, R)  ⟶^{X⊗_{G}−}  H^{n}(G, R)
P_{n}^{i} ↓  ↓ P_{n}^{i}
H^{n+2(p−1)i}(G, R) ⟶^{X⊗_{G}−} H^{n+2(p−1)i}(G, R)

is commutative.

Proof. We use the interpretation of the Steenrod operation from section 1.2. We have to apply X ⊗_{G} − or X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} − or (X ⊗_{R} RC_{p}) ⊗_{G×C_{p}} − to the various mappings in the diagram

Hom_{RG}(P, R[n])  ⟶^{Δ}  ⨂_{i=1}^{p} Hom_{RG}(P, R[n])
↓ Δ′  ∥
Hom_{R(G≀𝔖_{p})}(⨂_{i=1}^{p} P, (⨂_{i=1}^{p} R[n]) ⊗_{R} R_{s}^{(n)}) ⟶^{res_{G^{p}}^{G≀𝔖p}} Hom_{R(G^{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} R[n])
↓
Hom_{R(G≀𝔖_{p})}(Q, (⨂_{i=1}^{p} R[n]) ⊗_{R} R_{s}^{(n)})
↓ res_{G×C_{p}}^{G≀C_{p}}
Hom_{R(G×C_{p})}(Q, ⨂_{i=1}^{p} R)

and shall prove that the squares which are induced by this are commutative in each case. The square

Hom_{RG}(P, R[n])  ⟶^{Δ}  ⨂_{i=1}^{p} Hom_{RG}(P, R[n])
↓ X ⊗_{G} −  ↓ ⨂_{i=1}^{p} X ⊗_{G} −
Hom_{RG}(X ⊗_{G} P, X ⊗_{G} R[n]) ⟶^{Δ} ⨂_{i=1}^{p} Hom_{RG}(X ⊗_{G} P, X ⊗_{G} R[n])

is obviously commutative. Moreover, the square

⨂_{i=1}^{p} Hom_{RG}(P, R[n])  ≃ Hom_{RG^{p}}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} R[n])
⨂_{i=1}^{p}(X ⊗_{G} −) ↓  ↓ X^{⊗p} ⊗_{G^{p}} −
⨂_{i=1}^{p} Hom_{RG}(X ⊗_{G} P, X ⊗_{G} R[n]) ≃ Hom_{RG^{p}}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} R[n])

is commutative as well. This proves the announced commutativity of the diagrams with respect to the mappings Δ and Δ′.

The square

Hom_{R(G≀𝔖_{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} R[n] ⊗_{R} R_{s}^{(n)}) ⟶^{res_{G^{p}}^{G≀Cp}} Hom_{R(G^{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} R[n])
↓ X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} −  ↓ X^{⊗p} ⊗_{G^{p}} −

Hom_{R(G≀𝔖_{p})}(X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} ⨂_{i=1}^{p} P,  ⟶^{res_{G^{p}}^{G≀𝔖p}} Hom_{R(G^{p})} (X^{⊗p} ⊗_{G^{p}} ⨂_{i=1}^{p} P ,
X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} ⨂_{i=1}^{p} R[n] ⊗_{R} R_{s}^{(n)})  X^{⊗p} ⊗_{G^{p}} ⨂_{i=1}^{p} R[n])

again is commutative. In fact the complex X^{G≀𝔖_{p}} is an induced complex. And so, X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} − is the same as X^{⊗p} ⊗_{G^{p}} −. This proves that applying the various twosided tilting complexes derived from X to the upper square of the original diagram yields a parallel square with, together with the mappings of the corresponding corners, forms a commutative cube.

Next, the square

Hom_{R(G≀𝔖_{p})}(Q, ⨂_{i=1}^{p} R[n] ⊗_{R} R_{s}^{(n)})  ⟶  Hom_{R(G^{p})}(⨂_{i=1}^{p} P, ⨂_{i=1}^{p} R[n])
↓ X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} −  ↓ X^{⊗p} ⊗_{G^{p}} −
Hom_{R(G≀𝔖_{p})}(X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} Q,  ⟶  Hom_{R(G^{p})} (X^{⊗p} ⊗_{G^{p}} ⨂_{i=1}^{p} P ,
X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} ⨂_{i=1}^{p} R[n] ⊗_{R} R_{s}^{(n)})  X^{⊗p} ⊗_{G^{p}} ⨂_{i=1}^{p} R[n])

is commutative by the same trivial reasons.

The only part which remains to be proven is that the square (*) below

Hom_{R(G≀𝔖_{p})}(Q, ⨂_{i=1}^{p} R[n] ⊗_{R} R_{s}^{(n)}) ⟶^{res_{G×Cp}^{G≀𝔖p}}  Hom_{R(G×𝔖_{p})}(Q, ⨂_{i=1}^{p} R[n])
↓ X^{G≀C_{p}} ⊗_{G≀C_{p}} −  (X ⊗_{R} RC_{p}) ⊗_{G×C_{p}} − ↓

Hom_{R(G≀𝔖_{p})}(X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} Q,  ⟶^{res_{G×Cp}^{G≀𝔖p}} Hom_{R(G×C_{p})} ((X ⊗_{R} RC_{p}) ⊗_{G×C_{p}} Q,
X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} ⨂_{i=1}^{p} R[n] ⊗_{R} R_{s}^{(n)})  (X ⊗_{R} RC_{p}) ⊗_{G×C_{p}} ⨂_{i=1}^{p} R[n])

is commutative.

In order to prove this fact we replace R by a fixed projective resolution. Let P be a projective resolution of R as RG-module and let Q be a minimal projective resolution of R as a R𝔖_{p}-module. Using that R𝔖_{p} has cyclic defect group, we observe that each homogeneous component of Q is indecompoable. A projective resolution P^{G≀𝔖_{p}} of R as R(G ≀ 𝔖_{p})-module is then ⨂_{i=1}^{p} P ⊗_{R} Q and a projective resolution of R_{s}^{(n)} as R(G ≀ 𝔖_{p})- module is ⨂_{i=1}^{p} P ⊗_{R} (Q ⊗_{R} R_{s}^{(n)}). Note Q_{s} := Q ⊗_{R} R_{s}^{(n)}. We shall replace R by this

resolution P^{G≀𝔖_{p}} and R_{s}^{(n)} by the resolution P^{G≀𝔖_{p}} ⊗_{R} R_{s}^{(n)}.

Now,

X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} P^{G≀𝔖_{p}} ≃ X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} (⨂_{i=1}^{p} P ⊗_{R} Q) ≃ (⨂_{i=1}^{p}(X ⊗_{G} P)) ⊗_{R} Q

and so, using Künneth formulas,

Hom_{K(RG≀𝔖_{p})} (X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} P^{G≀𝔖_{p}}, X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} (P^{G≀𝔖_{p}} ⊗_{R} R_{s}^{(n)}) [n])

≃ Hom_{K(RG≀𝔖_{p})} ((⨂_{i=1}^{p}(X ⊗_{G} P)) ⊗_{R} Q, (⨂_{i=1}^{p}(X ⊗_{G} P)) ⊗_{R} Q_{s}[n])

≃ ⨁_{i+j=n} Hom_{K(RG^{p})} ((⨂_{i=1}^{p}(X ⊗_{G} P)) , (⨂_{i=1}^{p}(X ⊗_{G} P)) [i]) ⊗_{R} Hom_{K(R𝔖_{p})}(Q, Q_{s}[j])

≃ ⨁_{i+j=n} ( ⨂ ; ∑_{l=1}^{p} i_{l}=i Hom_{K(RG)} (X ⊗_{G} P, X ⊗_{G} P[i_{l}])) ⊗_{R} Hom_{K(R𝔖_{p})}(Q, Q_{s}[j])

Similarly,

Hom_{K(RG≀𝔖_{p})}(P^{G≀𝔖_{p}}, P^{G≀𝔖_{p}} ⊗_{R} R_{s}^{(n)}[n])

≃ Hom_{K(RG≀𝔖_{p})} ((⨂_{i=1}^{p} P) ⊗_{R} Q, (⨂_{i=1}^{p} P) ⊗_{R} Q_{s}[n])

≃ ⨁_{i+j=n} Hom_{K(RG^{p})} (⨂_{i=1}^{p} P, (⨂_{i=1}^{p} P) [i]) ⊗_{R} Hom_{K(R𝔖_{p})}(Q, Q_{s}[j])

≃ ⨁_{i+j=n} ( ⨂ ; ∑_{l=1}^{p} i_{l}=i Hom_{K(RG)} (P, P[i_{l}])) ⊗_{R} Hom_{K(R𝔖_{p})}(Q, Q_{s}[j])

with the same isomorphisms.

Now, we shall have to study how the restriction res_{G×C_{p}}^{G≀𝔖_{p}} is interpreted in this context.

Actually,

res_{G×C_{p}}^{G≀𝔖_{p}}(Hom_{K(RG≀𝔖_{p})} (P^{G≀𝔖_{p}}, P^{G≀𝔖_{p}} ⊗_{R} R_{s}^{(n)}[n]))

≃ res_{G×C_{p}}^{G≀𝔖_{p}} ( ⨁_{i+j=n} ( ⨂ ; ∑_{l=1}^{p} i_{l}=i Hom_{K(RG)} (P, P[i_{l}])) ⊗_{R} Hom_{K(R𝔖_{p})}(Q, Q_{s}[j]))

≃ ⨁_{i+j=n} Hom_{K(RG)} (P, P[i]) ⊗_{R} Hom_{K(RC_{p})} (Q, Q[j]))

since Hom_{R}^{•} (P, P) ≃ R.

Similarly,

res_{G×C_{p}}^{G≀𝔖_{p}} (Hom_{K(RG≀𝔖_{p})} (X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} P^{G≀𝔖_{p}}, X^{G≀𝔖_{p}} ⊗_{G≀𝔖_{p}} P^{G≀𝔖_{p}} ⊗_{R} R_{s}^{(n)}[n]))

≃ res_{G×C_{p}}^{G≀𝔖_{p}} ( ⨁_{i+j=n} ( ⨂ ; ∑_{l=1}^{p} i_{l}=i Hom_{K(RG)} (X ⊗_{G} P, X ⊗_{G} P[i_{l}])) ⊗_{R} Hom_{K(R𝔖_{p})}(Q, Q_{s}[j]))

≃ ⨁_{i+j=n} Hom_{K(RG)} (X ⊗_{G} P, X ⊗_{G} P[i]) ⊗_{R} Hom_{K(RC_{p})} (Q, Q[j])

by the same isomorphisms. This proves that the square (*) is commutative.

It is now direct to see that the action of X on the Ext-algebra commutes with mod p Steenrod operations. In fact, the left square of the diagram

H^{n}(G, 𝔽_{p}) ⟶^{norm_{G}^{G×Cp}}  H^{pn}(G × C_{p}, 𝔽_{p})  ≃ ⨁_{i=0}^{np} H^{i}(G, 𝔽_{p}) ⊗_{𝔽_{p}} H^{np−i}(C_{p}, 𝔽_{p})
X ⊗ − ↓  (X ⊗ id) ⊗_{G×C_{p}} − ↓  ⨁_{i=0}^{np}(X ⊗ id) ⊗ − ↓

H^{n}(G, 𝔽_{p}) ⟶^{norm_{G}^{G×Cp}}  H^{pn}(G × C_{p}, 𝔽_{p})  ≃ ⨁_{i=0}^{np} H^{i}(G, 𝔽_{p}) ⊗_{𝔽_{p}} H^{np−i}(C_{p}, 𝔽_{p})

is commutative as is shown above. The right square of the diagram is clearly commutative for p = 2, since the cohomology of a cyclic group of order 2 s 𝔽_{2} in each degree. For odd p the right square is commutative up to a scalar, which is easily seen to be the identity.

The proof of the theorem is finished here.

4. Consequences

Let G be a finite group. As in [12] we call HSplen_{𝔽_{p}}(G) be the subgroup of homotopy equivalence classes of splendid tilting complexes in T rP ic_{𝔽_{p}}(𝔽_{p}G) which in addition map the trivial 𝔽_{p}G-module to an (in the derived category of 𝔽_{p}G-modules) isomorphic copy.

By Theorem 1 the action of the mod p Steenrod algebra A_{p} and the action of HD_{𝔽_{p}}(G) on H^{∗}(G, 𝔽_{p}) commute. Hence the cohomology ring H^{∗}(G, 𝔽_{p}) is an A_{p} − HD_{𝔽_{p}}(G)-bimodule.

Let 𝔘_{p} be the category of unstable modules over the mod p Steenrod algebra (cf e.g. Schwartz [10]). Then, for any group G we get a functor D_{p}(G) defined by

H^{∗}(G, 𝔽_{p}) ⊗_{𝔽_{p} HD_{𝔽p}(G)} − : 𝔽_{p} HD_{𝔽_{p}}(G) − mod ⟶ 𝔘_{p}

By the usual adjointness formula

Hom_{𝔽_{p} HD_{𝔽p}(G)} (−, Hom_{𝔘_{p}}(H^{∗}(G, 𝔽_{p}),?)) ≃ Hom_{𝔘_{p}} (H^{∗}(G, 𝔽_{p}) ⊗_{𝔽_{p} HD_{𝔽p}(G)} −,?)

as bifunctors 𝔽_{p} HD_{𝔽_{p}}(G) − mod × 𝔘_{p} ⟶ Ens the functor D_{p}(G) has a right adjoint E_{p}(G) := Hom_{𝔘_{p}}(H^{∗}(G, 𝔽_{p}), −).

Of course there is a dual version of the above defined by E_{p}^{Hom}(G) defined by

− ⊗_{A_{p}} H^{∗}(G, 𝔽_{p}) : 𝔘_{p} ⟶ mod − (𝔽_{p} HD_{𝔽_{p}}(G))

which has a right adjoint D_{p}^{Hom}(G) := Hom_{𝔽_{p} HD_{𝔽p}(G)}(H^{∗}(G, 𝔽_{p}), −).

We get a system of functors

\begin{matrix}D_{p}(G) : & 𝔽_{p} HD_{p}(G) − mod & ⟶ & 𝔘_{p} \\ D_{p}^{Hom}(G) : & 𝔽_{p} HD_{p}(G) − mod & ⟶ & 𝔘_{p} \\ E_{p}(G) : & 𝔽_{p} HD_{p}(G) − mod & ⟵ & 𝔘_{p} \\ E_{p}^{⊗}(G) : & 𝔽_{p} HD_{p}(G) − mod & ⟵ & 𝔘_{p} \end{matrix}

where (D_{p}(G), E_{p}(G)) and (D_{p}^{Hom}(G), E_{p}^{⊗}(G)) are adjoint pairs and where we abbreviate HD_{p}(G) for HD_{𝔽_{p}}(G).

First, not very astonishing, we are able to find the image of the free unstable modules F (n) by E_{p}^{⊗}(G). Recall that F (n) are determined as representing object to the functor M ↦ M^{n} for any n ∈ ℕ :

Hom_{𝔘_{p}}(F (n), −) ≃ (−)^{n} .

Lemma 4.1. We get E_{p}^{⊗}(G)(F (n)) ≃ H^{∗}(G)^{n} as 𝔽_{p} HD_{p}(G)-modules.

Proof. We apply Yoneda’s lemma to the equation

Hom_{𝔽_{p} HD_{p}(G)}(F (n) ⊗_{𝔄_{p}} H^{∗}(G), −) ≃ Hom_{𝔘_{p}}(F (n), Hom_{HD_{p}(G)}(H^{∗}(G), −))
≃ Hom_{𝔽_{p} HD_{p}(G)}(H^{∗}(G), −)^{n}
≃ Hom_{𝔽_{p} HD_{p}(G)}(H^{∗}(G)^{n}, −)

to get the result.

We shall study the compatibility with Lannes’ T -functor (cf e.g. [10]). For this, we observe that by functoriality, if N is an A_{p}-HD_{k}(G) bimodule, then T_{V} (N) is such a bimodule as well.

Proposition 4.2. Let V be an elementary abelian p-group and let M be an HD_{𝔽_{p}}(G)- module. Then,

(T_{V} ∘ (D_{p}(G))) ≃ T_{V} (H^{∗}(G)) ⊗_{𝔽_{p} HD_{𝔽p}(G)} −.

Proof. We use the different adjointness formulas.

Hom_{𝔘_{p}} (T_{V} (D_{p}(G)M), −) ≃ Hom_{𝔘_{p}} (D_{p}(G)(M), H^{∗}(V) ⊗_{𝔽_{p}} −)
≃ Hom_{HD_{p}(G)} (M, Hom_{𝔘_{p}}(H^{∗}(G), H^{∗}(V) ⊗_{𝔽_{p}} −))
≃ Hom_{HD_{p}(G)} (M, Hom_{𝔘_{p}}(T_{V} (H^{∗}(G), −))
≃ Hom_{𝔘_{p}}(T_{V} (H^{∗}(G)) ⊗_{HD_{p}(G)} M, −)

The rest follows by Yoneda’s lemma.

We shall elaborate more on Proposition 4.2. As quoted in Henn, Lannes and Schwartz [4, Theorem 5.2] (see [3] for a proof) Lannes found that

T_{V} (H^{∗}(G, 𝔽_{p})) ≃ ∏_{ρ∈Hom(V,G)/conjugation} H^{∗}(C_{G}(ρ(V)), 𝔽_{p})

Let us recall how the mapping is constructed. For any fixed ρ ∈ Hom(V, G) one gets

V × C_{G}(im ρ) ⟶^{μ_{ϕ}} C_{G}(im ρ) ↪ G
(v, c) ↦ ρ(v)c .

Taking cohomology, this induces a mapping

H^{∗}(G, 𝔽_{p}) ⟶^{res_{C_{G}(im ρ)}^{G}} H^{∗}(C_{G}(im ρ), 𝔽_{p}) ⟶ H^{∗}(V, 𝔽_{p}) ⊗_{𝔽_{p}} H^{∗}(C_{G}(im ρ), 𝔽_{p})

Now, T_{V} is left adjoint to H^{∗}(V, 𝔽_{p}) ⊗_{𝔽_{p}} −, and therefore, this mapping induces a mapping

T_{V} (H^{∗}(G, 𝔽_{p})) ⟶^{ad(μ_{ϕ}^{∗}∘res_{C}^{G})} H^{∗}(C_{G}(im ρ)) .

Taking the product of these mappings gives the required isomorphism.

In [12] we obtained that the Brauer construction with respect to the diagonal of a psubgroup of G in G × G yields a group homomorphism

SplenP ic_{𝔽_{p}}(G) ⟶ SplenP ic_{𝔽_{p}}(C_{G}(Q)) .

In [12] we were not able to prove that under this homomorphism the property to fix the trivial module is preserved. Nevertheless, we conjecture that this is true.

Conjecture 4.3. The mapping SplenP ic_{𝔽_{p}}(G) ⟶ SplenP ic_{𝔽_{p}}(C_{G}(Q)) induces a mapping HSplen_{𝔽_{p}}(G) ⟶ HSplen_{𝔽_{p}}(C_{G}(Q)).

Supposing Conjecture 4.3 is true, we shall show that the decomposition T_{V} (H^{∗}(G, 𝔽_{p})) = ∏_{ρ} H^{∗}(C_{G}(im ρ), 𝔽_{p}) is compatible under the action of HSplen_{𝔽_{p}}(G) from the right.

Actually, the mapping HSplen_{𝔽_{p}}(G) → HSplen_{𝔽_{p}}(C_{G}(Q)) gives then a 𝔽_{p} HSplen_{𝔽_{p}}(G)- module structure on H^{∗}(C_{G}(im ρ), 𝔽_{p}) for any ρ ∈ Hom(V, G).

Proposition 4.4. Suppose that Conjecture 4.3 is true. Then, the mappings above induce a decomposition

T_{V} (H^{∗}(G, 𝔽_{p})) = ∏_{ρ} H^{∗}(C_{G}(im ρ), 𝔽_{p})

as A_{p}-𝔽_{p} HSplen_{𝔽_{p}}(G)-bimodules.

Proof. Let X be a twosided tilting complex in HSplen_{𝔽_{p}}(G). In [12] it is proved that the diagram

H^{∗}(G, 𝔽_{p})  ⟶^{res_{C_{G}(im ρ)}^{G}}  H^{∗}(C_{G}(im ρ), 𝔽_{p})
X ⊗_{G} − ↓  ↓ X(Δ(im ρ)) ⊗_{C_{G}(im ρ)} −
H^{∗}(G, 𝔽_{p})  ⟶^{res_{C_{G}(im ρ)}^{G}}  H^{∗}(C_{G}(im ρ), 𝔽_{p})

is commutative.

Now, V is abelian. So, im ρ ⊆ C_{G}(im ρ) for any ρ ∈ Hom(V, G). Let Q be a free resolution of the trivial 𝔽_{p}V -module and let P be a projective resolution of the trivial 𝔽_{p}C_{G}(im ρ)- module. Then, Q ⊗_{𝔽_{p}} P is a projective resolution of the trivial 𝔽_{p}(V × C_{G}(im ρ))-module. In order to describe the mapping H^{∗}(C_{G}(im ρ), 𝔽_{p}) ⟶ H^{∗}(V, 𝔽_{p}) ⊗_{𝔽_{p}} H^{∗}(C_{G}(im ρ), 𝔽_{p}) we have to determine a mapping τ which makes the diagram of 𝔽_{p}(V × C_{G}(im ρ))-modules

μP  ⟶ 𝔽_{p}
τ ↑  ∥
Q ⊗_{𝔽_{p}} P ⟶ 𝔽_{p}

commutative. Since Q is a free resolution, and since im ρ ⊆ C_{G}(im ρ) the resolution P is a C_{G}(im ρ)-module, and the mapping τ can simply be chosen to be multiplication via ρ. It is obvious that this makes the diagram commutative.

Denote for the moment X′ := X(Δ(C_{G}(im ρ))). Then, choosing these particular resolutions and mappings, it is easily seen that

H^{∗}(C_{G}(im ρ), 𝔽_{p}) ⟶^{μ^{∗}} H^{∗}(C_{G}(im ρ), 𝔽_{p}) ⊗_{𝔽_{p}} H^{∗}(C_{G}(im ρ), 𝔽_{p})
X′ ⊗_{G} − ↓  ↓ (𝔽_{p}V ⊗_{𝔽_{p}} X′) ⊗_{V ×C_{G}(im ρ)} −

H^{∗}(C_{G}(im ρ), 𝔽_{p}) ⟶^{μ^{∗}} H^{∗}(C_{G}(im ρ), 𝔽_{p}) ⊗_{𝔽_{p}} H^{∗}(C_{G}(im ρ), 𝔽_{p})

is commutative.

Now, it is sufficient to consider the adjoint mappings to the above. These yield commutative diagrams

T_{V} (H^{∗}(G, 𝔽_{p})) ⟶ H^{∗}(C_{G}(im ρ), 𝔽_{p})
X ⊗_{G} − ↓  ↓ X(Δ(im ρ)) ⊗_{C_{G}(im ρ)} −
T_{V} (H^{∗}(G, 𝔽_{p})) ⟶ H^{∗}(C_{G}(im ρ), 𝔽_{p})

by the functoriality of the adjointness formulas. This finishes the proof.

Remark 4.5. If Conjecture 4.3 is not true, Proposition 4.4 remains true for the action of ⋂_{ρ∈Hom(V,G)} (−(Δ(C_{G}(im ρ))))^{−1} (HSplen_{𝔽_{p}}(C_{G}(im ρ))) on T_{V} (H^{∗}(G, 𝔽_{p})).

Remark 4.6. Observe that we really needed to have splendid equivalences in the proof of Proposition 4.4. Moreover, Lemma 4.2 does not use any hypothesis on the nature of the equivalences. Since the decomposition of T_{V} (H^{∗}(G, 𝔽_{p})) as unstable module holds in general, one would like to have an interpretation of this decomposition as module over the bigger group of self-equivalences. This seems to include a generalization of [12].

References

[1] David Benson, Representations and cohomology II, Cambridge 1991.

[2] Charles W. Curtis and Irving Reiner, Methods of Representation Theory, Vol I John Wiley and Sons 1981.

[3] Hans-Werner Henn, Cohomology of groups and unstable modules over the Steenrod algebra, Advanced course on classifying spaces and cohomology of groups, Centre de Recerca Matemàtica Bellaterra (Spain). http://www-irma.u-strasbg.fr/ henn/notes.ps

[4] Hans-Werner Henn, Jean Lannes and Lionel Schwartz, Locations of unstable A-modules and equivariant mod p cohomology, Math. Ann. 301 (1995) 23-68.

[5] Steffen König and Alexander Zimmermann, Derived equivalences for group rings, (with contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier) Springer LNM 1685 (1998)

[6] Andrei Marcus, On equivalences between Blocks of Group Algebras: Reduction to the Simple Components, Journal of Algebra 184 (1996) 372-396.

[7] Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. 39 (1989) 436–456.

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

[9] Raphaël Rouquier and Alexander Zimmermann, A Picard group for derived module categories, accepted in Proc. London Math. Soc.

[10] Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, Chicago University Press 1994.

[11] Alexander Zimmermann, Auto-equivalences of derived categories acting on cohomology, to appear in Archiv der Mathematik.

[12] Alexander Zimmermann, Cohomology of groups and splendid equivalences of derived categories, Math. Proc. Cambridge Phil. Soc. 131 (2001) 459-472.

Faculté de Mathématiques et CNRS (LAMFA FRE 2270), Université de Picardie, 33 rue St Leu, 80039 Amiens Cedex, FRANCE

   E-mail address: Alexander.Zimmermann@u-picardie.fr
   URL: http://www.mathinfo.u-picardie.fr/alex/azim.html