Complexity and cohomology of cohomological Mackey functors

Serge Bouc

Abstract : Let k be a field of characteristic p > 0. Call a finite group G a poco group over k if any finitely generated cohomological Mackey functor for G over k has polynomial growth. The main result of this paper is that G is a poco group over k if and only if the Sylow p-subgroups of G are cyclic, when p > 2, or have sectional rank at most 2, when p = 2.

A major step in the proof is the case where G is an elementary abelian p-group. In particular, when p = 2, all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra of self extensions of the simple functor S_{1}^{G}, by explicit generators and relations.

AMS Subject Classification : 16P90, 18G10, 18G15, 20J05. Keywords : Cohomological, Mackey functor, complexity, growth.

1. Introduction

This paper addresses the question of the cohomology and rate of growth of cohomological Mackey functors for a finite group G over a field k. The three main results are the following :

1.1. Theorem : Let G be a finite group and k be a field of positive characteristic p. Then the following conditions are equivalent :

1. Every finitely generated cohomological Mackey functor for G over k has a projective resolution with polynomial growth.

2. Let S be a Sylow p-subgroup of G. Then S is cyclic if p > 2, or S has sectional 2-rank at most 2, if p = 2.

The key argument in the proof of this theorem is a reduction to the case where G is an elementary abelian p-group. The case of elementary abelian 2-groups can be described quite completely :

1.2. Theorem : Let G = (C_{2})^{m} be an elementary abelian 2-group of rank m, and k be a field of characteristic 2. Let M_{k}^{c}(G) denote the category of cohomological Mackey functors for G over k. Let S_{1}^{G} denote the simple cohomological Mackey functor defined by

∀Q ≤ G, S_{1}^{G}(Q) = \{ \begin{matrix}k & if Q = 1 \\ \{0\} & otherwise \end{matrix} .

Then the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) is finitely generated by elements of degree 2, and its Poincaré series

P (t) = ∑_{j∈ℕ}dim_{k} Ext_{M_{k}^{c}(G)}^{j}(S_{1}^{G}, S_{1}^{G}) t^{j}

is equal to

P (t) = \frac{1}{(1 − t^{2})(1 − 3t^{2})(1 − 7t^{2}) … (1 − (2^{m−1} − 1)t^{2})} .

1.3. Theorem : Let G = (C_{2})^{m} be an elementary abelian 2-group of rank m, and k be a field of characteristic 2. Then the algebra E = Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) admits the following presentation :

• The generators γ_{x}, of degree 2, are indexed by the elements x of G−{0}.

• The relations are the following :

1. Whenever H is a subgroup of index 2 of G,

∑_{x∉H} γ_{x} = 0 .

2. For any distinct elements x and y of G − {0},

[γ_{x} + γ_{y}, γ_{x+y}] = 0 ,

where [a, b] = ab + ba denotes the commutator of two elements a

and b in E.

1.4. This paper is divided in two parts : the first one focuses on complexity, and the second one on cohomology. Section 2 of Part I quickly recalls the definitions and basic results on the rate of growth of a module for a finite dimensional algebra over a field. In Section 3, the categories of cohomological Mackey functors are introduced, from different points of view, which are equivalent thanks to Yoshida’s theorem. Next some functors associated to bisets between categories of cohomological Mackey functors are defined, with nice adjunction properties. In Section 4, it is shown how to reduce the question of complexity of cohomological Mackey functors for a finite group G over a field of characteristic p to the same question for a Sylow p-subgroup of G. Section 5 exposes a sketch of the proof of Theorem 1.1. In Sections 6, 7, and 8, some simple cohomological functors and extensions between them are discussed. In Section 9, the case of cyclic p-groups is recalled from Samy Modeliar’s thesis ([7]). In Section 10, the case of elementary abelian p-groups is settled, and Section 11 handles the case of 2-groups with sectional 2 rank (at most) equal to 2.

The first section of Part II states further results on extensions of simple cohomological functors for elementary abelian p-groups. This leads to the proof of Theorem 1.3, in Section 13. In Section 14, a similar partial result is stated, for p = 3, which is conjectured for any odd prime p. Finally, Section 15 exposes some results on extensions of simple functors for an arbitrary finite p-group G, which show in particular how to reduce the computation of these extensions to the computation of self extensions for simple functors indexed by the trivial subgroup of some subquotients of G.

Acknowledgments : I wish to thank the MSRI, where this work was completed during my stay there for the program on Representation Theory of Finite Groups and Related Topics, in spring 2008. I also thank Dave Benson for stimulating conversations about all this.

I - Complexity

2. Polynomial growth

2.1. Definition : Let A be a finite dimensional (unital) algebra over a field k. A finitely generated A-module M is said to have polynomial growth if there exists a resolution

P_{∗} : ⋯ ⟶ P_{n} ⟶ P_{n−1} ⋯ ⟶ P_{0} ⟶ M ⟶ 0

of M by projective A-modules, and constants c, d and e such that

∀n ∈ ℕ, dim_{k} P_{n} ≤ c n^{d} + e .

The module M is said to have exponential growth if for any projective resolution P_{∗} of M, there are constants c, d, and e, with c > 0 and d > 1, such

{that \atop ∀n ∈ ℕ, dim_{k} P_{n} ≥ c d^{n} + e ,}

The module M is said to have intermediate growth if M has neither polynomial nor exponential growth.

2.2. Remark : If S is a generating set of M as an A-module, then there is a surjective map of A-modules A^{|S|} ⟶ M. In particular, the projective cover of M has dimension at most d dim_{k} M, where d = dim_{k} A. By induction, this shows that there exists a projective resolution as above such that

dim_{k} P_{n} ≤ (d − 1)^{n−1}d dim_{k} M .

In particular, this dimension is always bounded by some exponential function of n.

2.3. Lemma : Let A be a finite dimensional algebra over a field k, and M be a finitely generated A-module.

{1. If \atop ⋯ ⟶ P_{n} ⟶ P_{n−1} ⋯ ⟶ P_{0} ⟶ M ⟶ 0}

is a minimal projective resolution of M, then

P_{n} ≅ ⊕_{S∈Irr(A)} P_{S}^{dim_{k} Ext_{A}^{n}(M,S)/ dim_{k} End_{A}(S)} ,

where Irr(A) is a set or representatives of isomorphism classes of simple A-modules, and P_{S} denotes a projective cover of S.

2. In particular M has polynomial growth if and only if for any simple A-module S, there exists constants c, d and e such that

∀n ∈ ℕ, dim_{k} Ext_{A}^{n}(M, S) ≤ c n^{d} + e .

Proof : Assertion 1 follows by décalage from the case n = 0, and from the fact that the largest semisimple quotient of M is isomorphic to

⊕_{S∈Irr(A)} S^{dim_{k} Ext_{A}^{0}(M,S)/ dim_{k} End_{A}(S)} .

Assertion 2 is a straightforward consequence of Assertion 1, since there are finitely many simple A-modules, up to isomorphism.

2.4. Remark : In particular, if M has polynomial growth, then any direct summand of M has polynomial growth.

Conversely, the class of modules with polynomial growth is closed under extensions. More precisely :

2.5. Lemma : Let A be a finite dimensional algebra over a field k.

1. If L ⟶^{f} M ⟶^{g} N is an exact sequence of finite dimensional k-vector

spaces, then

dim_{k} M ≤ dim_{k} L + dim_{k} N ,

with equality if and only if f is injective and g is surjective.

{2. Let \atop 0 ⟶ L ⟶ M ⟶ N ⟶ 0}

be a short exact sequence of finitely generated A-modules. If two of the modules L, M, and N have polynomial growth, so does the third.

Proof : For Assertion 1, there is a short exact sequence

0 ⟶ Im f ⟶ M ⟶ Im g ⟶ 0 ,

hence dim_{k} M = dim_{k} Im f +dim_{k} Im g ≤ dim_{k} L+dim_{k} N. Equality holds if and only if dim_{k} Im f = dim_{k} L and dim_{k} Im g = dim_{k} N, i.e. if f is injective and g is surjective.

For Assertion 2, let S be a finitely generated A-module. Consider the long exact sequence of Ext-groups

⋯ → Ext_{A}^{n}(N, S) → Ext_{A}^{n}(M, S) → Ext_{A}^{n}(L, S) → Ext_{A}^{n+1}(N, S) → ⋯

It follows from Assertion 1 that

dim_{k} Ext_{A}^{n}(M, S) ≤ dim_{k} Ext_{A}^{n}(N, S) + dim_{k} Ext_{A}^{n}(L, S) ,

hence if L and N have polynomial growth, so does M. Similarly,

dim_{k} Ext_{A}^{n}(N, S) ≤ dim_{k} Ext_{A}^{n}(M, S) + dim_{k} Ext_{A}^{n−1}(L, S) ,

It follows that N has polynomial growth if L ands M have. Finally,

dim_{k} Ext_{A}^{n}(L, S) ≤ dim_{k} Ext_{A}^{n}(M, S) + dim_{k} Ext_{A}^{n+1}(N, S) ,

hence L has polynomial growth if M and N do.

2.6. Corollary : Let A be a finite dimensional algebra over a field k. The following conditions are equivalent :

1. Every finitely generated A-module has polynomial growth.

2. Every simple A-module has polynomial growth.

Proof : Obviously Condition 1 implies Condition 2. The converse follows by induction on the length of a finitely generated A-module.

3. Cohomological Mackey functors

3.1. Definition. Let R denote an arbitrary commutative unital ring. Recall that A Mackey functor M for G over R consists of the assignment H ↦ M(H) of an R-module to each subgroup H of G, together with maps of R-modules

t_{H}^{K} : M(H) ⟶ M(K) ,

called transfer maps, and

r_{H}^{K} : M(K) ⟶ M(H) ,

called restriction maps, whenever H ≤ K ≤ G, and maps of R modules

c_{x,H} : M(H) ⟶ M(^{x}H) ,

for each x ∈ G, subject to a list of compatibility conditions, in particular the Mackey formula (cf. [9] for details). A Mackey functor M is called cohomological if the additional conditions

t_{H}^{K}r_{H}^{K} = |K : H|Id_{M(H)}

are fulfilled, for any H ≤ K ≤ G.

There is an obvious notion of morphism of (cohomological) Mackey functors, and this yields the category M_{R}^{c}(G) of cohomological Mackey functors for G over R.

3.2. Example (fixed points functors) : In particular, when V is an RGmodule, then the fixed point functor F P_{V} is the Mackey functor defined by F P_{V} (H) = V^{H}, i.e. the set of elements of V which are invariant by H. When H ≤ K ≤ H, the transfer map V^{H} ⟶ V^{K} is the relative trace map Tr_{H}^{K}, defined by

Tr_{H}^{K}(v) = ∑_{x∈[K/H]} yv ,

for any v ∈ V^{H}, where [K/H] is a transversal of H in K. The restriction map V^{K} ⟶ V^{H} is the inclusion map, and for x ∈ G, the conjugation map V^{H} ⟶ V^{^{x}H} is the map v ↦ xv. The functor F P_{V} is obviously cohomological.

The correspondence V ↦ F P_{V} is a functor from RG-Mod to M_{R}^{c}(G). This functor is fully faithful : if V and W are RG-modules, then any morphism φ : F P_{V} → F P_{W} in M_{k}^{c}(G) is equal to F P_{f}, where f is the morphism of RG-modules from V = F P_{V} (1) to W = F P_{W} (1) obtained by evaluating φ at the trivial subgroup of G.

3.3. The cohomological Mackey algebra. It was shown by Thévenaz and Webb (cf. [9]) that M_{R}^{c}(G) is equivalent to the category of modules over the cohomological Mackey algebra coμ_{R}(G) for G over R. More precisely, if M is a cohomological Mackey functor for G over R, then the corresponding coμ_{R}(G)-module is equal to ⊕_{H≤G} M(H).

The algebra coμ_{R}(G) is a finitely generated free R-module, so in particular when R is a field, it is a finite dimensional R-algebra. In this case, there is a natural notion of projective cover of a finitely generated coμ_{R}(G)-module, hence a natural notion of minimal projective resolution of a cohomological Mackey functor.

3.4. Yoshida’s theorem. Let perm_{R}(G) denote the full subcategory of the category RG-mod of finitely generated RG-modules, consisting of permutation RG-modules, i.e. modules admitting a (globally) G-invariant R-basis. It is an R-linear category, which is naturally equivalent to the opposite category : indeed, the dual V^{∗} = Hom_{R}(V, R) of a finitely generated permutation RG-module is again a finitely generated permutation RG-module, and the correspondence δ : V ↦ V^{∗} is an equivalence of categories from perm_{R}(G) to perm_{R}(G)^{op}, which is its own inverse (up to a slight abuse of notation).

When M is a cohomological Mackey functor for G over R, then the correspondence

V ↦ Hom_{M_{R}^{c}(G)}(F P_{V} , M)

is an R-linear contravariant functor M̃ from perm_{R}(G) to R-mod. This yields in turn an R-linear functor M ↦ M̃ from M_{R}^{c}(G) to Fun_{R}(G), where Fun_{R}(G) denotes the category of R-linear contravariant functors from perm_{R}(G) to R-mod.

Conversely, if F is such a functor, and H is a subgroup of G, one can define F̂(H) = F (R(G/H)), where R(G/H) is the free R-module with basis the G-set G/H of H-cosets in G. If H ≤ K ≤ G, then the projection map p_{H}^{K} : G/K ⟶ G/H gives a map of RG-modules Rp_{H}^{K} : R(G/K) ⟶ R(G/H), and taking image by F gives a transfer map t_{H}^{K} = F (p_{H}^{K}) : F̂(H) ⟶ F̂(K).

Similarly, applying first the equivalence δ gives a map r_{H}^{K} = F (δ(Rp_{H}^{K})) : F̂(K) ⟶ F̂(H). Finally, if x ∈ G, then the map g^{x}H ↦ gxH induce a map R(G/^{x}H) ⟶ R(G/H), whose image by F yields a conjugation map c_{x,H} : F̂(H) ⟶ F̂(^{x}H). With these definitions F̂ becomes a cohomological Mackey functor for G over R, and the correspondence F ↦ F̂ is a functor from Fun_{R}(G) to M_{R}^{c}(G). The following theorem is essential :

3.5. Theorem : [Yoshida [10]] The functors M ↦ M̃ and F ↦ F̂ are mutual inverse equivalences of categories between M_{R}^{c}(G) and Fun_{R}(G).

3.6. Remark : One checks easily that if V is an RG-module, then the functor F P_{V} is mapped to the functor Hom_{RG}(−, V) by this equivalence. For this reason, this functor will also be denoted by F P_{V}. More generally, Yoshida’s equivalence allows for an identification of M_{R}^{c}(G) with Fun_{R}(G), that will be used freely throughout the rest of this paper.

3.7. Remark : In particular, if V is a permutation RG-module, then the Yoneda functor F P_{V} = Hom_{perm_{R}(G)}(−, V) is a projective object in Fun_{R}(G). More precisely, if M is a cohomological Mackey functor for G over R, and H is a subgroup of G, there is an isomorphism of R-modules

(3.8)  Hom_{M_{R}^{c}(G)}(F P_{R(G/H)}, M) ≅ M(H) .

It follows more generally that if V is a direct summand of a permutation RGmodule, then the functor F P_{V} is a projective object in M_{R}^{c}(G). Thévenaz and Webb (cf. [9]) have shown conversely that any projective object in M_{R}^{c}(G) is isomorphic to F P_{V}, where V is a direct summand of a permutation RGmodule.

3.9. Remark : It also follows that the category Fun_{R}(G) is equivalent to the category of modules over the Hecke algebra

Y_{R}(G) = End_{RG}( ⊕_{H≤G} R(G/H)) .

One can show easily that this algebra is actually isomorphic to the cohomological Mackey algebra coμ_{R}(G) (cf. [4]). The Y_{R}(G)-module corresponding to the object F of Fun_{R}(G) (resp. to the cohomological Mackey functor M for G over R) under these equivalences, is equal to ⊕_{H≤G} F (R(G/H)) (resp.

to ⊕_{H≤G} M(H)). In particular F is finitely generated (resp. M is finitely gen-

erated) if and only if F (W) is a finitely generated R-module, for any finitely generated permutation RG-module W (resp. M(H) is a finitely generated R-module, for any H ≤ G).

3.10. The dual of a Mackey functor. If M is a Mackey functor for G over R, then the dual Mackey functor M^{∗} is defined by

∀H ≤ G, M^{∗}(G) = Hom_{R}(M(G), R) .

The transfer, restriction, and conjugation maps for M^{∗} are defined by

r_{H}^{K} = ^{τ}(t_{H}^{K}) , t_{H}^{K} = ^{τ}(r_{H}^{K}) , c_{x,H} = ^{τ}(c_{x^{−1},^{x}H}) ,

for any H ≤ K ≤ G and any x ∈ G, where the exponent τ denotes transposed maps.

If M is cohomological, then M^{∗} is also cohomological. Through the equivalence given by Yoshida’s Theorem 3.5, this duality maps the functor F of Fun_{R}(G) to the functor F^{∗} defined as the composition

perm_{R}(G) ⟶^{∗} perm_{R}(G)^{op} ⟶^{F} R-Mod ⟶^{∗} R-Mod^{op} .

3.11. Remark : The correspondence M ↦ M^{∗} is a functor from M_{R}^{c}(G) to the opposite category. The canonical morphism from M to its bidual (M^{∗})^{∗} is functorial in M, and it is an isomorphism when R is a field k and M is finitely generated. In other words, the correspondence M ↦ M^{∗} induces an equivalence from the category of finitely generated cohomological Mackey functors for G over k to the opposite category. Thus, for any finitely generated cohomological Mackey functors M and N for G over k, there is a natural isomorphism

Hom_{M_{k}^{c}(G)}(M, N) ≅ Hom_{M_{k}^{c}(G)}(N^{∗}, M^{∗}) .

The functor M is a finitely generated projective object in M_{k}^{c}(G) if and only if M^{∗} is a finitely generated injective object in M_{k}^{c}(G), and the previous isomorphism extends to natural isomorphisms

Ext_{M_{k}^{c}(G)}^{n}(M, N) ≅ Ext_{M_{k}^{c}(G)}^{n}(N^{∗}, M^{∗}) ,

for any n ∈ ℕ.

3.12. Construction of functors. Let G and H be finite groups. If U is a finite (H, G)-biset consider the R-linear functor

t_{U} : V ↦ RU ⊗_{RG} V

from RG-mod to RH-mod maps permutation RG-modules to permutation RH-modules. By composition, this induces a functor

L_{U} : F ↦ F ∘ t_{U}

from Fun_{R}(H) to Fun_{R}(G). Since the right adjoint to the functor t_{U} is the functor

h_{U} : W ↦ Hom_{RH}(RU, W) ,

it follows from standard results of category theory that the functor

R_{U} : F ↦ F ∘ h_{U}

from Fun_{R}(G) to Fun_{R}(H), is right adjoint to L_{U}.

3.13. Remark : The functors L_{U} and R_{U} are a generalization of functors considered by Tambara (see [8] Section 4).

3.14. Proposition :

1. Let G and H be finite groups, and U be a finite (H, G)-biset. Then the functors R_{U} and L_{U} are exact.

2. Let G and H be finite groups. If U and U′ are isomorphic finite (H, G)- bisets, then there are isomorphisms of functors L_{U} ≅ L_{U′} and R_{U} ≅ R_{U′}.

3. Let G, H, and K be finite groups. Let U be a finite (H, G)-biset, and V be a finite (K, H)-biset. Then there are isomorphisms of functors

R_{V} ∘ R_{U} ≅ R_{V ×_{H}U} , L_{U} ∘ L_{V} ≅ L_{V ×_{H}U} .

4. Let G be a finite group, and let Id_{G} denote the identity biset for G, i.e. the set G for its (G, G)-biset structure given by multiplication. Then the functor L_{Id_{G}} and R_{Id_{G}} are isomorphic to the identity functor.

5. Let G and H be finite groups. If U and U′ are finite (H, G)-bisets, there are isomorphisms of functors

L_{U⊔U′} ≅ L_{U} ⊕ L_{U′} , R_{U⊔U′} ≅ R_{U} ⊕ R_{U′} .

6. Let G and H be finite groups, and U be a finite (H, G)-biset. Then for any object F of Fun_{R}(G), there is an isomorphism

R_{U^{op}}(F)^{∗} ≅ L_{U}(F^{∗})

in Fun_{R}(H), which is functorial in F.

Proof : Assertion 1 is obvious, since the functors R_{U} and L_{U} are obtained by pre-composition with some functor (in other words, they are both restriction functors along a suitable functor).

Assertion 2 is a consequence of the isomorphisms of functors t_{U} ≅ t_{U′}, and h_{U} ≅ h_{U′}, which both follow from the isomorphism RU ≅ RU′ of (RH, RG)- bimodules.

For Assertion 3, the associativity of tensor product gives an isomorphism of functor

t_{V} ∘ t_{U} ≅ t_{V ×_{H}U} ,

which by adjunction, gives the isomorphism of functors

h_{U} ∘ h_{V} ≅ h_{V ×_{H}U} .

The isomorphisms of Assertion 3 follow by composition.

Assertion 4 follows from the fact that the functors t_{Id_{G}} and t_{Id_{G}} are both isomorphic to the identity functor. Similarly, Assertion 5 follows from similar additivity properties of the functors t_{U} and h_{U} with respect to U.

For Assertion 6, let W be an object of perm_{R}(G). Then

R_{U^{op}}(F)^{∗}(W) = Hom_{R}(R_{U^{op}}(F)(W^{∗}), R)
= Hom_{R} (F (Hom_{RG}(RU^{op}, Hom_{R}(W, R))), R)
≅ Hom_{R}(F (Hom_{R}(RU ⊗_{RG} W, R)), R)
= L_{U}(F^{∗})(W) ,

and these isomorphisms are functorial with respect to W.

3.15. Proposition : Let G and H be finite groups, and U be a finite

(H, G)-biset.

1. If V is an RH-module, then

L_{U}(F P_{V}) = F P_{Hom_{RH}(RU,V)} .

In particular, the functor L_{U} maps projective objects to projective objects.

2. If G acts freely on U, then L_{U} ≅ R_{U^{op}}, where U^{op} denotes the opposite

biset of U.

Proof : For Assertion 1, there is an isomorphism of functors

Hom_{RG}(−, Hom_{RH}(RU, V)) ≅ Hom_{RH}(RU ⊗_{RG} −, V) .

In other words F P_{V} ∘ t_{U} ≅ F P_{Hom_{RH}(RU,V)}.

The last part of Assertion 1 follows from the facts that if V is a permutation RH-module, then Hom_{RH}(RU, V) = h_{U}(V) is a permutation RGmodule. An alternative proof consists in observing that since L_{U} is left adjoint to an exact functor, it maps projective objects to projective objects.

For Assertion 2, for any RG-module W, there is an isomorphism of RHmodules

RU ⊗_{RG} W ⟶ Hom_{RG}(RU^{op}, W) ,

defined by sending u ⊗ w, for u ∈ U and w ∈ W, to the map sending v ∈ U^{op} (recall that U = U^{op} as a set) to gw, if there exists an element g ∈ G such that v = ug^{−1} (and in this case there is a unique such g, since G acts freely on U), and to 0 otherwise. This isomorphism is obviously functorial in W, and this completes the proof.

3.16. Proposition : Let G and H be finite groups, and U be a finite (H, G)- biset. Then for any n ∈ ℕ, the adjunction of the pair (L_{U}, R_{U}) induces an isomorphism of bifunctors

Ext_{Fun_{R}(G)}^{n}(L_{U}(−), −) ≅ Ext_{Fun_{R}(H)}^{n}(−, R_{U}(−)) .

Proof : This follows from the fact that L_{U} and R_{U} are both exact functors.

3.17. Remark : In terms of Yoneda extensions, this isomorphism can be viewed as follows : the adjunction of the pair (L_{U}, R_{U}) is equivalent to the existence of natural transformation of functors

η : Id ⟶ R_{U} ∘ L_{U} and ε : L_{U} ∘ R_{U} ⟶ Id ,

called respectively the unit and counit of the adjunction, with the property that for each object M in Fun_{R}(G) and each object N in Fun_{R}(H),

ε_{L_{U} (N)} ∘ L_{U}(η_{N}) = Id_{L_{U} (N)} and R_{U}(ε_{M}) ∘ η_{R_{U} (M)} = Id_{R_{U} (M)} .

The bijection Hom_{Fun_{R}(G)}(L_{U}(N), M) ⟶ Hom_{Fun_{R}(H)}(N, R_{U}(M)) is given by α ↦ R_{U}(α) ∘ η_{N}. The inverse bijection is given by β ↦ ε_{M} ∘ L(β).

In other words, these bijections consist in taking images by one of the functors, and then compose on the suitable side with unit or counit.

Now interpreting the extension group Ext_{Fun_{R}(G)}^{n}(L_{U}(N), M) a the set of equivalence classes of exact sequences in Fun_{R}(G) of the form

\xymatrix{0 \ar[r] & M \ar[r] & X_{n−1} \ar[r] & ⋯ \ar[r] & X_{1} \ar[r] & X_{0} \ar[r] & L_{U}(N) \ar[r] & 0 ,}

the procedure is the same : first apply the functor R_{U}(−), to get an exact sequence

\xymatrix{0 \ar[r] & R_{U}(M) \ar[r] & R_{U}(X_{n−1}) \ar[r] & ⋯ \ar[r] & R_{U}(X_{1}) \ar[r] & R_{U}(X_{0}) \ar[r] & R_{U} ∘ L_{U}(N) \ar[r] & 0 ,}

and then compose with the map η_{N}, i.e. complete the cartesian square at the right of the following diagram

[figure: 0; R_{U}(M); R_{U}(X_{n−1}); ⋯; R_{U}(X_{1}); R_{U}(X_{0}); R_{U} ∘ L_{U}(N); 0; 0; R_{U}(M); R_{U}(X_{n−1}); ⋯; R_{U}(X_{1}); Y_{0}; N; η_{N}; 0 .]

The inverse bijection is obtained similarly by first applying the functor L_{U}, and composing with the map ε_{M}, i.e. completing a cocartesian square at the left of the resulting diagram.

3.18. Remark : It follows easily that the isomorphisms of functors

α_{n} : Ext_{Fun_{R}(G)}^{n}(L_{U}(−), −) ⟶^{≅} Ext_{Fun_{R}(H)}^{n}(−, R_{U}(−)) .

of Proposition 3.16, are compatible with the Yoneda product, in the following sense : if P is an object of Fun_{R}(H), and if M and N are objects of Fun_{R}(G), if e ∈ Ext_{Fun_{R}(G)}^{n}(L_{U}(P), N) and f ∈ Ext_{Fun_{R}(G)}^{m}(N, M), then

α_{m+n}(f ∘ e) = R_{U}(f) ∘ α_{n}(e) ∈ Ext_{Fun_{R}(H)}^{m+n}(P, R_{U}(M)) .

3.19. Example (induction and restriction) : Let G be a subgroup of H. Set U = H, viewed as an (H, G)-biset by left and right multiplication. Then the functor V ↦ RU ⊗_{RG} V from RG-mod to RH-mod is isomorphic to the induction functor V ↦ Ind_{G}^{H}V. It follows that the functor L_{U} is isomorphic to the restriction functor Res_{G}^{H} : M_{R}^{c}(H) ⟶ M_{R}^{c}(G) in this case.

In the same situation, the functor W ↦ Hom_{RH}(RU, W) from RH-mod to RG-mod is isomorphic to the restriction functor Res_{G}^{H}. It follows that the functor R_{U} is isomorphic to the induction functor Ind_{G}^{H} : M_{R}^{c}(G) ⟶ M_{R}^{c}(H).

3.20. Example (restriction and induction) : Suppose now that H is a subgroup of G, and consider U = G as an (H, G)-biset by left and right multiplication. Then the functor V ↦ RU ⊗_{RG} V from RG-mod to RH-mod is isomorphic to the restriction functor V ↦ Res_{H}^{G}V. It follows that the functor L_{U} is isomorphic to the induction functor Ind_{H}^{G} : M_{R}^{c}(H) ⟶ M_{R}^{c}(G) in this case.

In the same situation, the functor W ↦ Hom_{RH}(RU, W) from RH-mod to RG-mod is isomorphic to the induction functor Res_{G}^{H}. It follows that the functor R_{U} is isomorphic to the induction functor Ind_{G}^{H} : M_{R}^{c}(G) ⟶ M_{R}^{c}(H).

This yields another proof of the well known fact that the induction and restriction functors between categories of cohomological Mackey functors are left and right adjoint to each other (cf. [9] for details).

3.21. Example (the functor ρ_{G/N}^{G}) : Let H be a finite group, and N be a normal subgroup of H, and set G = H/N. Also set U = G, viewed as an (H, G)-biset in the obvious way. In this case, the functor t_{U} is the inflation functor from RG-mod to RH-mod. Using the equivalences of categories of Theorem 3.5, the functor L_{U} gives a functor denoted by ρ_{H/N}^{H}, from M_{R}^{c}(H) to M_{R}^{c}(H/N). One checks easily that if M is a cohomological Mackey functor for H over R, then, denoting by x ↦ \overline{x} the projection map H ⟶ H/N

(ρ_{H/N}^{H}(M))(\overline{K}) = M(K) .

Similarly, the transfer, restriction, and conjugation maps for the functor ρ_{H/N}^{H}(M) are obtained by just “removing the bars”, i.e.

t\overline{L} ; \overline{K} = t_{K}^{L} , r\overline{L} ; \overline{K} = r_{K}^{L} , c_{\overline{x},\overline{K}} = c_{x,K} .

The right adjoint to ρ_{H/N}^{H} is the functor R_{U}, hereafter denoted by ȷ_{G/N}^{G}.

Moreover Assertion 3 of Proposition 3.15 shows that the functor ρ_{G/N}^{G} is also equal to the functor R_{U^{op}}. In particular, the left adjoint to ρ_{G/N}^{G} is the functor L_{U^{op}}, hereafter denoted by ı_{G/N}^{G}.

3.22. Example (the functor ı_{G/N}^{G}) : The left adjoint to the functor ρ_{G/N}^{G} will be denoted by ı_{G/N}^{G}. It is obtained as follows : if M is a cohomological Mackey functor for G/N, and K is a subgroup of G, then

(ı_{G/N}^{G}(M))(K) = M(KN/N) .

If K ≤ L ≤ G, then the transfer, restriction, and conjugation maps for the functor ı_{G/N}^{G}(M) are given by

t_{K}^{L} = |L ∩ N : K ∩ N|t_{KN/N}^{LN/N} , r_{K}^{L} = r_{KN/N}^{LN/N} , c_{x,K} = c_{xN,KN/N} .

3.23. Example (the functor ȷ_{G/N}^{G}) : The right adjoint to the functor ρ_{G/N}^{G} will be denoted by ȷ_{G/N}^{G}. It is obtained as follows : if M is a cohomological Mackey functor for G/N, and K is a subgroup of G, then

(ȷ_{G/N}^{G}(M))(K) = M(KN/N) .

If K ≤ L ≤ G, then the transfer, restriction, and conjugation maps for the functor ȷ_{G/N}^{G}(M) are given by

t_{K}^{L} = t_{KN/N}^{LN/N} , r_{K}^{L} = |L ∩ N : K ∩ N|r_{KN/N}^{LN/N} , c_{x,K} = c_{xN,KN/N} .

3.24. Remark : The functors ı_{G/N}^{G} and ȷ_{G/N}^{G} should not be confused with the inflation functor Inf_{G/N}^{G} : recall (cf. [9]) that if N is a normal subgroup of a finite group G, and M is a Mackey functor for G/N over R, then the functor Inf_{G/N}^{G}M is the Mackey functor for G over R defined by

(Inf_{G/N}^{G}M)(H) = \{ \begin{matrix}M(H/N) & if H ≥ N \\ \{0\} & otherwise \end{matrix} .

The transfer, restriction, and conjugation maps are the obvious ones. In general, this inflation procedure does not preserve cohomological Mackey functors : indeed, if M is cohomological, and if K < H are subgroups of G such that H ≥ N but K ≱ N, then the composition

t_{K}^{H}r_{K}^{H} : (Inf_{G/N}^{G}M)(H) ⟶ (Inf_{G/N}^{G}M)(H)

is equal to 0, so it is not equal in general to the multiplication by |H : K|.

However, if N is a p-group, for some prime number p, then p divides |KN : K| = |N : K ∩ N|, hence p divides |H : K|. If moreover R has characteristic p, then |H : K| = 0 in R in this situation, and one checks then that Inf_{G/N}^{G}M is cohomological. In this case, the inflation functor M_{R}^{c}(G/N) ⟶ M_{R}^{c}(G) corresponds via Yoshida’s equivalence of categories to the functor Fun_{R}(G/N) ⟶ Fun_{R}(G) obtained by composition with the Brauer quotient functor V ↦ V [N] from perm_{R}(G) to perm_{R}(G/N).

3.25. Example : group isomorphism : Let f : G → H be a group isomorphism. Then the set U = H has a natural structure of (H, G)-biset, for which h ∈ H acts by left multiplication by h, and g ∈ G acts by right multiplication by f(g). It is clear in this case that the functor R_{U} is the transport by isomorphism via f. This functor will be denoted by Iso(f). The functor L_{U} is isomorphic to Iso(f^{−1}), by Assertion 3 of Proposition 3.15.

Recall that a section (B, A) of G is a pair of subgroups of G such that A ⊴ B.

3.26. Lemma : Let G be a finite group, and (B, A) be a section of G. Then there is an isomorphism

Ind_{B}^{G} ı_{B/A}^{B} ≅ L_{A\G}

of functors from M_{R}^{c}(B/A) to M_{R}^{c}(G), where A\G is endowed with its natural (B/A, G)-biset structure. Similarly, there is an isomorphism

ρ_{B/A}^{B}Res_{B}^{G} ≅ L_{G/A}

of functors from M_{R}^{c}(G) to M_{R}^{c}(B/A), where G/A is endowed with its natural (G, B/A)-biset structure.

Proof : The functor Ind_{B}^{G} ı_{B/A}^{B} is equal to the composition of Ind_{B}^{G} = L_{V}, where V is the set G for its natural (B, G)-biset structure, and ı_{B/A}^{B} = L_{U}, where U is the set B/A, for its natural (B/A, B)-biset structure. It follows from Proposition 3.14 that Ind_{B}^{G}ı_{B/A}^{B} ≅ L_{W}, where W = (B/A)×_{B}G is clearly isomorphic to the (B/A, G)-biset A\G.

By adjunction, it follows that ρ_{B/A}^{B}Res_{B}^{G} ≅ R_{A\G}. But R_{A\G} ≅ L_{G/A} by Assertion 3 of Proposition 3.15, since B/A acts freely on G/A.

3.27. Proposition : Let G be a finite group, and (B, A), (D, C) be two sections of G. Then there is an isomorphism

ρ_{D/C}^{D} Res_{D}^{G} Ind_{B}^{G} ı_{B/A}^{B} ≅ ⊕_{g∈[D\G/B]} Ind\overline{D} ; \overline{D}_{g} ı_{\overline{D}_{g}/\overline{C}_{g}}^{\overline{D}_{g}} Iso(f_{g}) ρ_{\overline{B}_{g}/\overline{A}_{g}}^{\overline{B}_{g}} Res\overline{B} ; \overline{B}_{g}

of functors from M_{k}^{c}(B/A) to M_{k}^{c}(D/C), where

\overline{D} = D/C, \overline{D}_{g} = (D ∩^{g}B)C/C, \overline{C}_{g} = (D ∩^{g}A)C/C ,

\overline{B} = B/A, \overline{B}_{g} = (D^{g} ∩ B)A/A, \overline{A}_{g} = (C^{g} ∩ B)A/A ,

and where f_{g} : \overline{B}_{g}/\overline{A}_{g} ⟶ \overline{D}_{g}/\overline{C}_{g} is the group isomorphism sending x\overline{A}_{g} to

gx\overline{C}_{g}, for x ∈ D^{g} ∩ B.

Proof : (see Proposition7.1 of [5] for details) By Lemma 3.26 and Proposition 3.14, there are isomorphisms of functors

ρ_{D/C}^{D}Res_{D}^{G}Ind_{B}^{G}ı_{B/A}^{B} ≅ L_{G/C} ∘ L_{A\G}
≅ L_{A\G×_{G}G/C} ≅ L_{A\G/C} .

Now the (B/A, D/C)-biset A\G/C splits as a disjoint union of transitive ones

A\G/C ≅ ⨆_{g∈[D\G/B]} A\Bg^{−1}D/C .

Moreover for each g ∈ [D\G/B], with the above notation, there is an isomorphism of (\overline{B}, \overline{D})-bisets

A\Bg^{−1}D/C ≅ (\overline{B}/\overline{A}_{g}) ×_{\overline{B}_{g}/\overline{A}_{g}} Iso(f_{g}) ×_{\overline{D}_{g}/\overline{C}_{g}} (\overline{C}_{g}\\overline{D}) .

By Lemma 3.26 and Proposition 3.14 again, there is an isomorphism of functors

L_{A\Bg^{−1}D/B} ≅ Ind\overline{D} ; \overline{D}_{g} ∘ ı_{\overline{D}_{g}/\overline{C}_{g}}^{\overline{D}_{g}} ∘ Iso(f_{g}) ∘ ρ_{\overline{B}_{g}/\overline{A}_{g}}^{\overline{B}_{g}} ∘ Res\overline{B} ; \overline{B}_{g} ,

as was to be shown.

4. Reduction to p-groups

4.1. Lemma : Let k be a field, let G and H be finite groups, and U be a finite (H, G)-biset.

1. If F is an object of Fun_{k}(H), then

dim_{k} L_{U}(F) ≤ C_{U} dim_{k} F ,

where C_{U} = ∑_{K≤G} |H\U/K|.

2. If F is an object of Fun_{k}(H), and if F has polynomial growth, then so

does L_{U}(F).

Proof : Let F be an object of Fun_{k}(H). The coμ_{k}(G)-module corresponding to L_{U}(F) is equal to ⊕_{K≤G} L_{U}(F)(k(G/K)), by Remark 3.9. Thus

dim_{k} L_{U}(F) = ∑_{K≤G} dim_{k} F (kU ⊗_{kG} k(G/K))
= ∑_{K≤G} dim_{k} F (k(U/K))
= ∑_{K≤G} ∑_{u∈[H\U/K]} dim_{k} F (k(H/H_{uK})) ,

where H_{uK} = {h ∈ H | ∃g ∈ K, hu = ug}. Now for each K ≤ G and each u ∈ [H\U/K]

dim_{k} F (k(H/H_{uK})) ≤ dim_{k} F = ∑_{L≤H} dim_{k} F (k(H/L)) ,

thus

dim_{k} L_{U}(F) ≤ ( ∑_{K≤G} ∑_{u∈[H\U/K]} 1) dim_{k} F ,

showing Assertion 1.

Now if

⋯ P_{n} ⟶ P_{n−1} ⟶ ⋯ ⟶ P_{0} ⟶ F ⟶ 0

is a projective resolution with polynomial growth, there are constants c, d, and e, such that dim_{k} P_{n} ≤ cn^{d} + e, for all n ∈ ℕ. Now the complex

⋯ L_{U}(P_{n}) ⟶ L_{U}(P_{n−1}) ⟶ ⋯ ⟶ L_{U}(P_{0}) ⟶ L_{U}(F) ⟶ 0

is a projective resolution L_{U}(F), since L_{U} is exact and maps projective objects to projective objects. Moreover

dim_{k} L_{U}(P_{n}) ≤ C_{U} dim_{k} P_{n} ≤ C_{U}cn^{d} + C_{U}e ,

so L_{U}(F) has polynomial growth.

4.2. Proposition : Let G be a finite group, and (B, A) be a section of G. Let k be a field, and N be a cohomological Mackey functor for B/A over k. Then the following conditions are equivalent :

1. The functor N has polynomial growth.

2. The functor Ind_{B}^{G} ı_{B/A}^{B}(N) has polynomial growth.

Proof : The functor Ind_{B}^{G} ı_{B/A}^{B} is isomorphic to L_{W}, where W is the (B/A, G) biset A\G, by Lemma 3.26, so Condition 1 implies Condition 2, by Lemma 4.1.

Conversely, if L_{W} (N) has polynomial growth, then so does L_{W^{op}}∘L_{W} (N) ≅ L_{W ×_{G}W^{op}}(N), by Lemma 4.1. But W ×_{G} W^{op} ≅ A\G/A, as a (B/A, B/A)- biset. In particular, it is the disjoint union of the identity biset Id_{B/A} = B/A and some other (B/A, B/A)-biset. By Assertions 4 and 5 of Proposition 3.14, it follows that the identity functor is a direct summand of L_{W ×_{G}W^{op}}. In particular N is isomorphic to a direct summand of L_{W ×_{G}W^{op}}(N), and N has polynomial growth, by Remark 2.4. Thus Condition 2 implies Condition 1.

4.3. Lemma : Let G be a finite group and M be a cohomological Mackey functor for G over R. Then for any subgroup H of G the composition

M ⟶ Ind_{H}^{G}Res_{H}^{G}M ⟶ M

of the unit and counit morphisms of the adjoint pairs of functors (Res_{H}^{G}, Ind_{H}^{G}) and (Ind_{H}^{G}, Res_{H}^{G}) is equal to the multiplication by |G : H|.

Proof : This is because the same is true for the categories of RG-modules and RH-modules, and the induction and restriction functors between them.

4.4. Proposition : Let k be a field of positive characteristic p. Let G be a finite group, and S be a subgroup of G, containing a Sylow p-subgroup of G. Then for any cohomological Mackey functor M for G over k, the following conditions are equivalent :

1. The functor M has polynomial growth.

2. The functor Res_{S}^{G}M has polynomial growth.

Proof : Condition 1 implies Condition 2, by Lemma 4.1. Conversely, if Res_{S}^{G}M has polynomial growth, so does Ind_{S}^{G}Res_{S}^{G}M, by Lemma 4.1 again. Now Lemma 4.3 shows that M is a direct summand of Ind_{S}^{G}Res_{S}^{G}M, since |G : S| is non zero in k. Hence M has polynomial growth, by Remark 2.4.

4.5. Definition : Let k be a field. A finite group G is called a poco group over k if every finitely generated cohomological Mackey functor for G over k has polynomial growth.

4.6. Corollary : Let k be a field of characteristic p. Let G be a finite group, and S be a Sylow p-subgroup of G. Then G is a poco group over k if and only if S is a poco group over k.

Proof : Indeed, if G is a poco group, and N is a cohomological Mackey functor for S over k, then Ind_{S}^{G}N has polynomial growth, and N has polynomial growth, by Proposition 4.2, applied to the section (S, 1) of G. So S is a poco group.

Conversely, if S is a poco group, and M is a cohomological Mackey functor for G over k, the Res_{S}^{G}M has polynomial growth, so M has polynomial growth, by Proposition 4.4.

4.7. Proposition : Let k be a field, and G be a finite group. The following conditions are equivalent :

1. The group G is a poco group over k.

2. For any finitely generated kG-module V, the functor F P_{V} has a polynomial growth.

Proof : Obviously Condition 1 implies Condition 2. Conversely, observe that since any morphism from F P_{L} to F P_{N}, where L and N are kG-modules, is determined by a morphism of kG-modules from L to N, for any cohomological Mackey functor M for G over k, there is a short exact sequence

0 ⟶ F P_{V} ⟶ F P_{L} ⟶ F P_{N} ⟶ M ⟶ 0 ,

where L and N are permutation kG-modules, and V is the kernel of a morphism of kG-modules from L to N. Now if F P_{V} has polynomial growth, so does M.

5. Proof of Theorem 1.1

Corollary 4.6 shows that G is a poco group if and only if S is a poco group, where S is a Sylow p-subgroup of G. Now by Proposition 4.2, if (B, A) is a section of S, the factor group B/A is also a poco group.

It will be shown in Sections 9 and 10 that an elementary abelian p-group is a poco group if and only if it has rank at most 1, when p > 2, or at most 2, when p = 2. It will follow that if G is a poco group, then S has sectional rank at most 1 if p > 2, which implies that S is cyclic, or at most 2 if p = 2. In other words, Condition 1 of Theorem 1.1 implies Condition 2.

Conversely, it will be shown in Section 9 that cyclic p-groups are poco groups. So Condition 2 of Theorem 1.1 implies Condition 1, when p > 2. The corresponding assertion for p = 2 will be stated in Section 11, completing the proof of Theorem 1.1.

6. Simple cohomological functors for p-groups

Recall (see [9] for details) that if G is a p-group, the simple cohomological Mackey functors for G over k, up to isomorphism, are in one to one correspondence with the subgroups of G, up to conjugation. The simple functor S_{Q}^{G} (also denoted by S_{Q} if G is clear from context) corresponding to the subgroup Q is defined by

∀T ≤ G, S_{Q}^{G}(T) = \{ \begin{matrix}k & if T =_{G} Q \\ \{0\} & otherwise. \end{matrix}

The projective cover of the functor S_{Q}^{G} is the fixed point functor F P_{kG/Q}. Moreover End_{M_{k}^{c}(G)}(S_{Q}^{G}) ≅ k, and S_{Q}^{G} is self dual (i.e. (S_{Q}^{G})^{∗} ≅ S_{Q}^{G}).

6.1. Lemma : Let k be a field of characteristic p, and G be a p-group. There is a short exact sequence in M_{k}^{c}(G)

0 ⟶ F P_{Ω_{G}} ⟶ F P_{kG} ⟶ S_{1}^{G} ⟶ 0 ,

where Ω_{G} is the kernel of the augmentation map ε : kG → k.

Proof : Consider the exact sequence of kG-modules

0 ⟶ Ω_{G} ⟶ kG ⟶^{ε} k ⟶ 0 .

Since fixed point functors are left exact, the inclusion Ω_{G} ⊆ kG yields an inclusion F P_{Ω_{G}} ⊆ F P_{kG}. Let S denote the quotient functor. In particular S(1) ≅ k. And if Q is a non-trivial subgroup of G, then (kG)^{Q} = Tr_{1}^{Q}(kG) ⊆ Ω_{G}. Thus (Ω_{G})^{Q} = (kG)^{Q}, and S(Q) = {0}. Hence S ≅ S_{1}^{G}.

6.2. Theorem : [Samy Modeliar [7]] Let k be a field of characteristic p, and G be a finite p-group. Then

Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}) ≅ Hom_{ℤ}(G/(Φ(G)I(G)), k^{+}) ,

where k^{+} is the additive group of k, where Φ(G) is the Frattini subgroup of G, and I(G) is the subgroup of G generated by elements of order 2 (so I(G) = 1

if p > 2).

Proof : By Lemma 6.1, there is an exact sequence

0 ⟶ k → k ⟶ Hom_{M_{k}^{c}(G)}(F P_{Ω_{G}}, S_{1}^{G}) ⟶ Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}) ⟶ 0 ,

hence Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}) ≅ Hom_{M_{k}^{c}(G)}(F P_{Ω_{G}}, S_{1}^{G}).

Now a morphism φ : F P_{Ω_{G}} → S_{1}^{G} is entirely determined by its evaluation at the trivial group, which is a morphism of kG-modules from Ω_{G} to k. Conversely, a morphism of kG-modules f : Ω_{G} → k is the evaluation at 1 of a morphism of Mackey functors from F P_{Ω_{G}} to S_{1}^{G} if and only if it maps (Ω_{G})^{Q} = r_{1}^{Q}F P_{Ω_{G}}(Q) to r_{1}^{Q}S_{1}^{Q}(Q) = {0}, for any non-trivial subgroup Q of G, or equivalently, for any subgroup Q of order p of G.

In other words f is a G-invariant linear form on the space

Ω_{G}/ ∑_{{Q≤G \atop |Q|=p}} Tr_{1}^{Q}kG ,

i.e. a linear form on Ω_{G} such that

(6.3)  f(h(g − 1) − (g − 1)) = 0, ∀g, h ∈ G ,

and

(6.4)  f(1 + x + ⋯ + x^{p−1}) = 0, ∀x ∈ G, |x| = p .

Now Ω_{G} is generated as a k-vector space by the elements d_{g} = g − 1, for g ∈ G, and the only relations between these generators are d_{1} = 0. Hence a linear form f on Ω_{G} is determined by the values u(g) = f(g − 1), subject to u(1) = 0. Since d_{hg} = (hd_{g} − d_{g}) + d_{h} + d_{g}, for any g, h ∈ G, Equation 6.3 is equivalent to

u(gh) = u(g) + u(h), ∀g, h ∈ G .

In other words u is a group homomorphism from G to k^{+} (note that this implies u(1) = 0, as required). Equivalently u factors through a group homomorphism G/Φ(G) → k^{+}.

Now if x is an element of order p of G

1 + x + ⋯ + x^{p−1} = d_{1} + d_{x} + d_{x^{2}} + ⋯ + d_{x^{p−1}} ,

so Equation 2 is equivalent to

u(1) + u(x) + u(x^{2}) + ⋯ + u(x^{p−1}) = 0, ∀x ∈ G, |x| = p .

If u is a group homomorphism from G to k^{+}, this is equivalent to

(0 + 1 + 2 + ⋯ + (p − 1))u(x) = \binom{p}{2}u(x) = 0, ∀x ∈ G, |x| = p .

Now if p is odd, the integer \binom{p}{2} is a multiple of p, so this condition is satisfied whenever u is a group homomorphism to k^{+}. And if p = 2, this condition is equivalent to

u(x) = 0, ∀x ∈ G, |x| = 2 .

This shows finally that Hom_{M_{k}^{c}(G)}(F P_{Ω_{G}}, S_{1}^{G}) is isomorphic to the group of homomorphisms from G/(Φ(G)I(G)) to k^{+}, and this completes the proof.

6.5. Lemma : Let G be a finite p-group, and Q be a subgroup of G. Set B = N_{G}(Q), and denote by A = Φ(Q) the Frattini subgroup of Q. Then

S_{Q}^{G} ≅ Ind_{B}^{G} ı_{B/A}^{B}(S_{Q/A}^{B/A}) ≅ L_{A\G}(S_{Q/A}^{B/A}) ,

where A\G is viewed as a (B/A, G)-biset.

Proof : The second isomorphism follows from Lemma 3.26. As for the first one, recall (see [9]) that S_{Q}^{G} = Ind_{B}^{G}S_{Q}^{B}, so it suffices to show that S_{Q}^{G} = ı_{G/A}^{G}(S_{Q/A}^{G/A}), when Q ⊴ G. In this case, if H ≤ G

ı_{G/A}^{G}(S_{Q/A}^{G/A})(H) = S_{Q/A}^{G/A}(HA/A) .

This is equal to k if HA = Q, and to zero otherwise. But HA = Q if and only if H = Q, and this completes the proof.

6.6. Corollary : With the same notation,

S_{Q}^{G} ≅ Ind_{B}^{G} ȷ_{B/A}^{B}(S_{Q/A}^{B/A}) ≅ R_{G/A}(S_{Q/A}^{B/A}) .

Proof : This follows from Lemma 6.5, using Proposition 3.14, and the fact that S_{Q}^{G} is self dual.

6.7. Lemma : Let G be a p-group, let Q be a subgroup of G, and (B, A) be a section of G. Then

ρ_{B/A}^{B}Res_{B}^{G}(S_{Q}^{G}) ≅ ⊕_{Q′} S_{Q′/A}^{B/A} ,

where Q′ runs through the set of G-conjugates of Q which contain A and are contained in B, modulo B-conjugation.

Proof : Let H/A be a subgroup of B/A. Then

(ρ_{B/A}^{B}Res_{B}^{G}(S_{Q}^{G}))(H/A) = (Res_{B}^{G}(S_{Q}^{G}))(H) = S_{Q}^{G}(H) ,

and this is equal to k if H is conjugate to Q in G, and to {0} otherwise.

7. Some cohomological functors for p-groups

7.1. Notation : Let G be a finite group. A subset S of the set of subgroups of G will be called convex if

∀H ≤ K ≤ L ≤ G, H, L ∈ S ⇒ K ∈ S .

The set S will be called G-stable if it is invariant by G-conjugation.

The following is an extension of Assertion (ii) of Corollary 15.3 of [9] :

7.2. Proposition : Let k be a field of characteristic p, and G be a finite group. The correspondence

M ↦ Supp_{p}(M) = {H ≤ G | H is a p-group and M(H) ≠ {0}}

induces a one to one correspondence between the set of isomorphism classes of subquotients of the functor F P_{k} and the set of G-stable convex subsets of the set of p-subgroups of G. The inverse bijection maps the G-stable convex subset S of p-subgroups of G to the class of the functor k_{S} defined by

∀H ≤ G, k_{S}(H) = \{ \begin{matrix}k & if H_{p} ∈ S \\ \{0\} & otherwise , \end{matrix}

where H_{p} denotes a Sylow p-subgroup of H.

When H ≤ K ≤ G, the restriction map r_{H}^{K} is equal to 0, except if H_{p} ∈ S and K_{p} ∈ S, in which case r_{H}^{K} = 1. The transfer map t_{H}^{K} is multiplication by |K : H| for all H ≤ K ≤ G. The conjugation map c_{x,H} is always the identity

map k_{S}(H) → k_{S}(^{x}H).

Proof : By Corollary 15.3 of [9], the correspondences

F ↦ Supp_{p}(F) = {Q | Q is a p-subgroup of G and F (Q) ≠ {0}}

and

T ↦ k_{T} = <F P_{k}(Q)>_{Q∈T}

are mutual inverse bijections between the set of subfunctors of F P_{k} and the set of G-stable subsets of the set s_{p}(G) of p-subgroups of G, which are closed under taking subgroups.

By Proposition 2.4 of [9], the value of k_{T} at some subgroup H of G is equal to

k_{T} = ∑_{{Q∈T \atop Q≤H}} |H : Q|k .

This is equal to k if H_{p} ∈ T , and to zero otherwise. The restriction maps between non zero values of k_{T} are equal to the identity map of k. The transfer map t_{H}^{K} is multiplication by |K : H|, and the conjugation maps are identity maps (possibly zero).

It follows that any subquotient M of F P_{k} is equal to k_{T} /k_{T′}, where T and T′ are G-stable subsets of s_{p}(G), which are closed under taking subgroups, and such that T′ ⊆ T . With the notation of Proposition 7.2, this means that M ≅ k_{S}, where S = T − T′ is a G-stable convex subset of s_{p}(G), equal to Supp_{p}(M).

Conversely, let S be a G-stable convex subset of s_{p}(G). Set

T = {Q ∈ s_{p}(G) | ∃S ∈ S, Q ≤ S}
T′ = T − {Q ∈ s_{p}(G) | ∃S ∈ S, S ≤ Q} .

Then T and T′ are G-stable, and closed under taking subgroups. Moreover

T − T′ = {Q ∈ s_{p}(G) | ∃S, S′ ∈ S, S′ ≤ Q ≤ S} .

Thus T − T′ = S, and k_{S} = k_{T} /k_{T′} is a subquotient of F P_{k}.

These correspondences M ↦ Supp_{p}(M) and S ↦ k_{S} are clearly mutual inverse bijections between the set of isomorphism classes of subquotients of F P_{k} and the set of G-stable convex subsets of s_{p}(G).

7.3. Corollary : Let G be a finite p-group, and Q ≤ R be subgroups of G. Then there exists a unique object Σ_{Q,R}^{G} of M_{k}^{c}(G) such that

∀H ≤ G, Σ_{Q,R}^{G}(H) = \{ \begin{matrix}k & if Q ≤_{G} H ≤_{G} R \\ \{0\} & otherwise . \end{matrix}

and such that

∀H < K ≤ G, t_{H}^{K} = 0, r_{H}^{K} = \{ \begin{matrix}1 & if Q ≤_{G} H and K ≤_{G} R \\ 0 & otherwise . \end{matrix}

Moreover the socle of Σ_{Q,R}^{G} is isomorphic to S_{Q}^{G}, and the head of Σ_{Q,R}^{G} is isomorphic to S_{R}^{G}.

Proof : Let S = {S ≤ G | Q ≤_{G} S ≤_{G} R}. Then S is a G-stable convex set of subgroups of G, and the corresponding functor k_{S} fulfills the required conditions, so the functor Σ_{Q,R}^{G} exists. The uniqueness follows from the fact that the values of Σ_{Q,R}^{G} are given, as well as the transfer and restriction maps. The non zero conjugation maps c_{x,Q} are determined by elements of H^{1}(G, Ind_{Q}^{G}k^{×}) ≅ Hom_{ℤ}(Q, k^{×}), which is equal to zero since G is a p-group. So the conjugation maps are all identity maps (possibly zero).

Now a morphism φ from Σ_{Q,R}^{G} to some functor N is entirely determined by its evaluation φ_{R} : Σ_{Q,R}^{G}(R) = k → N(R) : indeed, if S ≤ G and Σ_{Q,R}^{G}(S) ≠ {0}, then S^{g} ≤ R for some g ∈ G, and then the map ψ = c_{g,S^{g}}r_{S^{g}}^{Q} is an isomorphism from Σ_{Q,R}^{G}(R) to Σ_{Q,R}^{G}(S), such that φ_{Q} = c_{g,S^{g}}r_{S^{g}}^{Q}φ_{R}ψ^{−1}. Hence the only simple quotient of Σ_{Q,R}^{G} is S_{R}^{G}, with multiplicity one.

By a similar argument, a morphism from N to Σ_{Q,R}^{G} is determined by its value at Q, so the only simple subfunctor of Σ_{Q,R}^{G} is S_{Q}^{G}, with multiplicity one.

7.4. Notation : Let G be a finite p-group. If Q < R are subgroups of G

such that |R : Q| = p, the functor Σ_{Q,R}^{G} will be denoted by \binom{S_{R}^{G}}{S_{Q}^{G}}. The dual

functor (Σ_{Q,R}^{G})^{∗} will be denoted by \binom{S_{Q}^{G}}{S_{R}^{G}}. With this notation, there are non

split exact sequences in M_{k}^{c}(G)

(7.5)  D_{Q,R} : 0 ⟶ S_{Q}^{G} ⟶ \binom{S_{R}^{G}}{S_{Q}^{G}} ⟶ S_{R}^{G} ⟶ 0 .

(7.6) D_{Q,R}^{∗} : 0 ⟶ S_{R}^{G} ⟶ \binom{S_{Q}^{G}}{S_{R}^{G}} ⟶ S_{Q}^{G} ⟶ 0 .

In particular, if X is a subgroup of order p of G, set D_{X} = D_{1,X} and D_{X}^{∗} = D_{1,X}^{∗}, and denote by γ_{X}^{G} (or γ_{X} if G is clear from the context) the element of Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}) represented by the exact sequence

Γ_{X} : 0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ \binom{S_{1}^{G}}{S_{X}^{G}} ⟶ S_{1}^{G} ⟶ 0 ,

obtained by splicing the sequences D_{X} and D_{X}^{∗}.

8. Extensions of simple functors for p-groups

8.1. Lemma : Let G be a p-group, and Z be a central subgroup of order p of G. Then there are isomorphisms

ı_{G/Z}^{G}(S_{1}^{G/Z}) ≅ \binom{S_{Z}^{G}}{S_{1}^{G}} , ȷ_{G/Z}^{G}(S_{1}^{G/Z}) ≅ \binom{S_{1}^{G}}{S_{Z}^{G}}

in M_{k}^{c}(G). In particular, there are non-split exact sequences in M_{k}^{c}(G)

D_{Z} : 0 ⟶ S_{1}^{G} ⟶ ı_{G/Z}^{G}(S_{1}^{G/Z}) ⟶ S_{Z}^{G} ⟶ 0 .

D_{Z}^{∗} : 0 ⟶ S_{Z}^{G} ⟶ ȷ_{G/Z}^{G}(S_{1}^{G/Z}) ⟶ S_{1}^{G} ⟶ 0 .

Proof : Let H be a subgroup of G. Then ı_{G/Z}^{G}(S_{1}^{G/Z})(H) = S_{1}^{G/Z}(HZ/Z) is equal to k if HZ = Z, i.e. if H ≤ Z, and to zero otherwise. The conjugation maps are all identity maps (possibly zero), the restriction map r_{1}^{Z} is the

identity map of k, and the transfer map t_{1}^{Z} is zero. Thus ı_{G/Z}^{G}(S_{1}^{G/Z}) ≅ \binom{S_{Z}^{G}}{S_{1}^{G}}.

The other isomorphism follows by duality, and the two exact sequences are special cases of Sequences 7.5 and 7.6.

8.2. Corollary : For any n ∈ ℕ, composition by D_{Z} induces a group isomorphism

Ext_{M_{k}^{c}(G)}^{n−1}(S_{1}^{G}, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(G)}^{n}(S_{Z}^{G}, S_{1}^{G})

Proof : Indeed, for n ∈ ℕ,

Ext_{M_{k}^{c}(G)}^{n}(\binom{S_{Z}^{G}}{S_{1}^{G}}, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(G/Z)}^{n}(S_{1}^{G/Z}, ρ_{G/Z}^{G}(S_{1}^{G})) = \{0\}

by Proposition 3.16 and Lemma 6.7.

8.3. Corollary : Let k be a field of characteristic p, and G be a p-group. If X is a subgroup of order p of G, then

Ext_{M_{k}^{c}(G)}^{1}(S_{X}^{G}, S_{1}^{G}) ≅ k ≅ Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{X}^{G}) .

Proof : Indeed N_{G}(X) is equal to the centralizer C of X, since |X| = p, and S_{X}^{G} = Ind_{C}^{G}S_{X}^{C}. Thus for any n ∈ ℕ

Ext_{M_{k}^{c}(G)}^{n}(S_{X}^{G}, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(C)}^{n}(S_{X}^{C}, Res_{C}^{G}S_{1}^{G})
≅ Ext_{M_{k}^{c}(C)}^{n}(S_{X}^{C}, S_{1}^{C})
≅ Ext_{M_{k}^{c}(C)}^{n−1}(S_{1}^{C}, S_{1}^{C})

by Corollary 8.2, since X is a central subgroup of order p of C. Corollary 8.3 follows, taking n = 1.

The isomorphism k ≅ Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{X}^{G}) follows from Remark 3.11, since the simple functors are self dual.

8.4. Notation : If N is a normal subgroup of a group G, contained in the subgroup H of G, denote by K_{H}(N) the set of complements of N in H, i.e. the set of subgroups X of G such that NX = H and N ∩ X = 1.

Denote by [N_{G}(H)\K_{H}(N)] a set of representatives of N_{G}(H)-conjugacy classes of subgroups in K_{H}(N).

The following proposition is a generalization of Lemma 8.1 :

8.5. Proposition : Let G be a p-group, let Q be a subgroup of G, and Z be a central subgroup of order p of G contained in Q. Then there are non-split exact sequences in M_{k}^{c}(G)

0 ⟶ ⊕_{X∈[N_{G}(Q)\K_{Q}(Z)]} S_{X}^{G} ⟶ ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}) ⟶ S_{Q}^{G} ⟶ 0 ,

0 ⟶ S_{Q}^{G} ⟶ ȷ_{G/Z}^{G}(S_{Q/Z}^{G/Z}) ⟶ ⊕_{X∈[N_{G}(Q)\K_{Q}(Z)]} S_{X}^{G} ⟶ 0 .

Proof : Since the simple functors are self dual, the second sequence is obtained from the first by applying duality, by definition of the functor ȷ_{G/Z}^{G} and Assertion 6 of Proposition 3.14. So it suffices to prove the existence of the first one.

Denote by I the functor ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}), and by R its radical. If H is a

subgroup of G, then I(H) = S_{Q/Z}^{G/Z}(HZ/Z) is equal to k if HZ =_{G} Q, and to zero otherwise. Moreover HZ = Q if and only if H = Q, or H ∈ K_{Q}(Z), since Z has order p. Equation 3.8 shows that the composition factors of I are the functors S_{Q}^{G} and the functors S_{X}^{G}, for X ∈ [N_{G}(Q)\K_{Q}(Z)], each with multiplicity 1.

Moreover, if Y is any subgroup of G, consider

H_{Y} = Hom_{M_{k}^{c}(G)}(I, S_{Y}^{G}) ≅ Hom_{M_{k}^{c}(G/Z)}(S_{Q/Z}^{G/Z}, ρ_{G/Z}^{G}(S_{Y}^{G})) .

Now ρ_{G/Z}^{G}(S_{Y}^{G}) = {0} if Z ≰ Y, and ρ_{G/Z}^{G}(S_{Y}^{G}) = S_{Y/Z}^{G/Z}, by Lemma 6.7. Thus H_{Y} = {0} unless Y =_{G} Q, and H_{Y} ≅ k in this case.

This means that I/R is simple, isomorphic to S_{Q}^{G}. Moreover, if H is a subgroup of G, then R(H) = {0} except if HZ =_{G} Q and H ∩ Z = 1, i.e. if H is conjugate to some element of K_{Q}(Z) in G, and R(H) ≅ k in this case. Hence R is isomorphic to the direct sum of the simple functors S_{H}^{G}, where H runs in a set of representatives of G-conjugacy classes of such subgroups, i.e. equivalently H ∈ [N_{G}(Q)\K_{Q}(Z)].

8.6. Theorem : Let k be a field of characteristic p > 0, let G be a finite p-group. Let Q and R be subgroups of G, and let Z be a subgroup of order p of Q ∩ R ∩ Z(G). Set \overline{G} = G/Z, \overline{Q} = Q/Z, \overline{R} = R/Z, and

K = [N_{G}(Q)\K_{Q}(Z)].

Then, for any n ∈ ℕ, there are short exact sequences of vector spaces

0 → ⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}) → Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) →^{π_{n}} Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) → 0 ,

where the map π_{n} is induced by ρ_{G/Z}^{G}.

Proof : Applying Hom_{M_{k}^{c}(G)}(−, S_{R}^{G}) to the first exact sequence of Proposition 8.5 gives a long exact sequence

[figure: ⋯; ⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}); Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}); π_{n}; Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}); ⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{n}(S_{X}^{G}, S_{R}^{G}); Ext_{M_{k}^{c}(G)}^{n+1}(S_{Q}^{G}, S_{R}^{G}); π_{n+1}; ⋯]

where the image of an extension u ∈ Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) under the map π_{n} is obtained by first taking the image under the map

φ : ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}) → S_{Q}^{G} ∈ Ext_{M_{k}^{c}(G)}^{0}(ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}), S_{Q}^{G}) ,

and then using the adjunction isomorphisms

Ext_{M_{k}^{c}(G)}^{n}(ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}), S_{R}^{G}) ≅ Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, ρ_{G/Z}^{G}(S_{R}^{G})) ≅ Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) .

In other words, with the notation of Remark 3.18

π_{n}(u) = α_{n}(u ∘ φ) = ρ_{G/Z}^{G}(u) ∘ α_{0}(φ) = ρ_{G/Z}^{G}(u) ,

since the map α_{0}(φ) : S_{Q/Z}^{G/Z} → ρ_{G/Z}^{G}(S_{Q}^{G}) ≅ S_{Q/Z}^{G/Z} obtained from φ by adjunction is the identity map. This shows that π_{n} is induced by ρ_{G/Z}^{G}.

Now the inflation functor Inf_{G/Z}^{G} : Fun_{k}(G/Z) → Fun_{k}(G) is an exact functor such that ρ_{G/Z}^{G} ∘ Inf_{G/Z}^{G} is isomorphic to the identity functor. Since S_{Q}^{Z} ≅ Inf_{G/Z}^{G}(S_{Q/Z}^{G/Z}) and S_{R}^{Z} ≅ Inf_{G/Z}^{G}(S_{R/Z}^{G/Z}), this inflation functor induces a map

σ_{n} : Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) → Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G})

such that π_{n} ∘ σ_{n} = Id. In particular π_{n} is surjective, so the long exact sequence above splits as a series of short exact sequences.

8.7. Proposition : Let k be a field of characteristic p > 0, let G be a finite p-group, and H be a subgroup of index p in G. Set I = Ind_{H}^{G}S_{1}^{H}, and let R and S denote respectively the radical and the socle of I as an object of M_{k}^{c}(G). Then I ⊃ R ⊇ S ⊃ {0}. Moreover I/R ≅ S ≅ S_{1}^{G}, and

R/S ≅ L ⊕ ⊕_{X∈[G\K_{G}(H)]} S_{X}^{G} ,

where L is a functor all of whose composition factors are isomorphic to S_{1}^{G}, with multiplicity p − 2.

Proof : Let Q be a subgroup of G. Then by Proposition 3.16

Hom_{M_{k}^{c}(G)}(I, S_{Q}^{G}) ≅ Hom_{M_{k}^{c}(H)}(S_{1}^{H}, Res_{H}^{G}S_{Q}^{G}) .

Moreover, by Lemma 6.7, the functor Res_{H}^{G}S_{Q}^{G} is isomorphic to the direct sum of functors S_{Q′}^{H}, where Q′ runs through a set of H-conjugacy classes of G-conjugates of Q which are contained in H.

Thus Hom_{M_{k}^{c}(G)}(I, S_{Q}^{G}) = {0} if Q ≠ 1, and Hom_{M_{k}^{c}(G)}(I, S_{1}^{G}) ≅ k. This shows that I/R ≅ S_{1}^{G}.

A similar argument shows that Hom_{M_{k}^{c}(G)}(S_{Q}^{G}, I) is equal to {0} if Q ≠ 1, and isomorphic to k if Q = 1. Thus S ≅ S_{1}^{G}.

Let X be a subgroup of G. Then

I(X) ≅ ⊕_{x∈G/HX} S_{1}^{H}(H ∩^{x}X) .

This is isomorphic to k^{p} if X = 1, to k if X is non trivial and H ∩ X = 1 (which implies that HX = G), and to {0} otherwise. In particular I ≠ S (since I(1) = k^{p} ≠ k = S(1)), so S ⊆ R.

Moreover, the short exact sequence

0 ⟶ R ⟶ I → S_{1}^{G} ⟶ 0

shows that Ext_{M_{k}^{c}(G)}^{n}(S_{X}^{G}, R) ≅ Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{1}^{G})) for any n ∈ ℕ, since Ext_{M_{k}^{c}(G)}^{n}(S_{X}^{G}, I) ≅ Ext_{M_{k}^{c}(H)}^{n}(Res_{H}^{G}S_{X}^{G}, S_{1}^{H}), and since Res_{H}^{G}S_{X}^{G} = {0}. In particular Ext_{M_{k}^{c}(G)}^{1}(S_{X}^{G}, R) ≅ Hom_{M_{k}^{c}(G)}(S_{X}^{G}, S_{1}^{G}) = {0}.

Now the short exact sequence

0 ⟶ S_{1}^{G} ⟶ R → R/S ⟶ 0

yields a long exact sequence of Ext_{M_{k}^{c}(G)}(S_{X}^{G}, −) groups starting with

0 ⟶ 0 ⟶ 0 ⟶ Hom(S_{X}, R/S) → Ext^{1}(S_{X}, S_{1}) → Ext^{1}(S_{X}, R) = {0} .

It follows that Hom(S_{X}, R/S) ≅ k, by Corollary 8.3. Moreover I is self dual, since S_{1}^{H} is self dual and Ind_{H}^{G} commutes with duality, by Proposition 3.14. Thus

Hom(R/S, S_{X}) ≅ Hom(S_{X}, R/S) ≅ k ,

by Remark 3.11, since S_{X} is also self dual.

In particular, there exists an injection i : S_{X} ⟶ R/S and a surjection s : R/S ⟶ S_{X}. If s ∘ i = 0, then (R/S)(X) has dimension at least equal to 2. But (R/S)(X) ≅ I(X) ≅ k, so s ∘ i ≠ 0. Hence s ∘ i is invertible, and S_{X} is a direct summand of R/S.

It follows that ⊕_{X∈[G\K_{G}(H)]} S_{X}^{G} is also a direct summand of R/S, for it is a

direct sum of non isomorphic simple summands of R/S. Hence there exists a subfunctor L of R/S such that

R/S ≅ L ⊕ ⊕_{X∈[G\K_{G}(H)]} S_{X}^{G} .

Now if Q is a subgroup of G, then (R/S)(Q) is equal to {0}, unless Q = 1 or Q is a complement of H in G, in which case (R/S)(Q) ≅ k. It follows that L(Q) is equal to {0}, except if Q = 1 (and L(1) has dimension p − 2 over k). This completes the proof.

9. The case of cyclic p-groups

9.1. It has been shown by M. Samy Modeliar ([7]) that cyclic p-groups are poco groups over a field k of characteristic p. This result relies on the construction of periodic projective resolutions for the fixed points functors, which can be seen has follows : if G is cyclic of order p^{m}, then the group algebra kG is isomorphic to a truncated polynomial algebra A = k[X]/(X^{p^{m}}), via the map sending X ∈ A to g − 1 ∈ kG, where g is some chosen generator of G.

The indecomposable A-modules are the modules E_{d} = k[X]/(X^{d}), for 1 ≤ d ≤ p^{m}, so E_{d}, where the chosen generator g of G acts by multiplication by 1 + X, is the unique indecomposable kG-module of dimension d, up to isomorphism. The indecomposable permutation kG-modules have dimension equal to a power of p, so they are the modules E_{p^{j}}, for 0 ≤ j ≤ m.

This means that the functors F P_{E_{p^{j}}} are the projective indecomposable objects, in M_{k}^{c}(G), and they are their own projective resolution.

Now if 1 ≤ d ≤ p^{m} and d is not a power of p, there exists a unique integer h in {0, ⋯, m − 1} such that p^{h} < d < p^{h+1}. Let

e_{d} : E_{p^{h}} ⊕ E_{p^{h+1}} ⟶ E_{d}

denote the morphism of A-modules induced by the map

(Q, R) ∈ k[X] × k[X] ↦ X^{d−p^{h}}Q + R ∈ E_{d} .

This map is well defined, and surjective, and it is a morphism of kG-modules.

The unique subgroup G_{l} of G of index p^{l}, where 0 ≤ l ≤ m, is the subgroup generated by g^{p^{l}}. If W is a kG-module, then the subspace W^{G_{l}} of W on which G_{l} acts trivially is equal to

W^{G_{l}} = {w ∈ W | g^{p^{l}}w = w} = {w ∈ W | (g − 1)^{p^{l}}w = 0} .

So viewing W as an A-module, the space W^{G_{l}} is equal to the kernel of X^{p^{l}} on W. In particular, for any j ∈ {1, ⋯, m}, the space (E_{j})^{G_{l}} is equal to k[X]X^{r}/(X^{j}), where r = max(j − p^{l}, 0). This shows that

e_{d}((E_{p^{h}} ⊕ E_{p^{h+1}})^{G_{l}}) = (k[X]X^{d−p^{h}+max(0,p^{h}−p^{l})} + k[X]X^{max(0,p^{h+1}−p^{l})})/(X^{d}) .

If l ≤ h, this is equal to k[X]X^{d−p^{l}}/(X^{d}), and if l > h, this is equal to k[X]/(X^{d}). In other words

e_{d}((E_{p^{h}} ⊕ E_{p^{h+1}})^{G_{l}}) = k[X]X^{max(0,d−p^{l})}/(X^{d}) = (E_{d})^{G_{l}} .

It follows that the restriction of the map e_{d} to the spaces of fixed points by any subgroup of G is surjective : in the terminology of Samy Modeliar ([7]), the map e_{d} is supersurjective. Equivalently, it induces a surjection

F P_{e_{d}} : F P_{E_{p^{h}}} ⊕ F P_{E_{p^{h+1}}} ⟶ F P_{E_{d}}

in M_{k}^{c}(G).

Now the kernel of e_{d} consists of the images in E_{p^{h}} ⊕ E_{p^{h+1}} of the pairs (Q, R) ∈ k[X] × k[X] such that X^{d−p^{h}}Q + R is a multiple of X^{d}, i.e. of the pairs (−T, X^{d−p^{h}}T), for T ∈ k[X]. Hence the map

T ∈ E_{p^{h}+p^{h+1}−d} ↦ (−T, X^{d−p^{h}}T) ∈ Ker e_{d}

is well defined (note that p^{h} < p^{h}+p^{h+1}−d < p^{h+1}), and it is an isomorphism of A-modules. This yields a short exact sequence

0 ⟶ E_{p^{h}+p^{h+1}−d} ⟶ E_{p^{h}} ⊕ E_{p^{h+1}} ⟶^{e_{d}} E_{d} ⟶ 0

of kG-modules, leading to a short exact sequence

0 ⟶ F P_{E_{p^{h}+p^{h+1}−d}} ⟶ F P_{E_{p^{h}}} ⊕ F P_{E_{p^{h+1}}} ⟶^{F P_{e_{d}}} F P_{E_{d}} ⟶ 0

in M_{k}^{c}(G). This shows in particular that F P_{E_{d}} has a projective resolution in M_{k}^{c}(G), which is periodic of period 2, of the form

⋯ ⟶^{F P_{α_{d}}} F P_{E_{p^{h}}⊕E_{p^{h+1}}} ⟶^{F P_{β_{d}}} F P_{E_{p^{h}}⊕E_{p^{h+1}}} ⟶^{F P_{α_{d}}} F P_{E_{p^{h}}⊕E_{p^{h+1}}} ⟶^{F P_{e_{d}}} F P_{E_{d}} ⟶ 0

where α_{d} and β_{d} are the endomorphisms of E_{p^{h}} ⊕ E_{p^{h+1}} given by

α_{d}(Q, R) = (−X^{p^{h+1}−d}Q − R, X^{p^{h+1}−p^{h}}Q + X^{d−p^{h}}R)
β_{d}(Q, R) = (−X^{d−p^{h}}Q − R, X^{p^{h+1}−p^{h}}Q + X^{p^{h+1}−d}R) .

Hence all the functors F P_{V}, where V is an indecomposable kG-module, have a periodic resolution. In particular G is a poco group, by Proposition 4.7. 9.2. If p^{m} ≥ 3, this applies to the case d = p^{m} − 1 (which is not a power of p), and h = m − 1, consequently. In this case E_{d} ≅ Ω_{G}, and E_{p^{h+1}} ≅ kG. Splicing the above resolution with the short exact sequence

0 ⟶ F P_{E_{p^{m}−1}} ⟶ F P_{E_{p^{m}}} → S_{1}^{G} ⟶ 0

of Lemma 6.1 gives a projective resolution of S_{1}^{G} of the form

⋯ ⟶^{F P_{β}} F P_{E_{p^{m−1}}⊕E_{p^{m}}} ⟶^{F P_{α}} F P_{E_{p^{m−1}}⊕E_{p^{m}}} ⟶^{F P_{γ}} F P_{E_{p^{m}}} ⟶ S_{1}^{G} ⟶ 0 ,

where α and β are the above maps α_{d} and β_{d}, for d = p^{m}−1 (and h = m−1), and γ : E_{p^{m−1}} ⊕ E_{p^{m}} → E_{p^{m}} is the map defined by

γ(Q, R) = X^{p^{m}−p^{m−1}}Q + XR .

Now F P_{E_{p^{m−1}}⊕E_{p^{m}}} ≅ F P_{E_{p^{m−1}}} ⊕ F P_{E_{p^{m}}}. Moreover, by Equation 3.8,

Hom_{M_{k}^{c}(G)}(F P_{E_{p^{m−1}}}, S_{1}^{G}) ≅ S_{1}^{G}(G_{p^{m−1}}) = {0} ,

and

Hom_{M_{k}^{c}(G)}(F P_{E_{p^{m}}}, S_{1}^{G}) ≅ S_{1}^{G}(G_{p^{m}}) = S_{1}^{G}(1) = k .

More precisely, a morphism F P_{E_{p^{m}}} ⟶ S_{1}^{G} is determined by its evaluation at the trivial subgroup, which is a scalar multiple of the augmentation morphism ε.

It follows easily that the groups Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}) are the cohomology groups of the complex whose terms are all isomorphic to k, with zero differentials. Thus Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}) ≅ k, for n ∈ ℕ.

9.3. Proposition : Let k be a field of characteristic p, and G be a cyclic p-group. Then :

∀n ∈ ℕ − \{0\}, Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}) ≅ \{ \begin{matrix}k & if |G| ≥ 3 \\ \{0\} & if |G| ≤ 2 \end{matrix} .

Proof : The case |G| ≥ 3 follows from the above discussion. The case |G| ≤ 2 follows essentially from the fact that in this case, any kG-module is a permutation module : if G is trivial, there is nothing to prove. And if G has order 2, then S_{1}^{G} has a projective resolution

0 ⟶ F P_{k} ⟶ F P_{kG} ⟶ S_{1}^{G} ⟶ 0 ,

and moreover Hom_{M_{k}^{c}(G)}(F P_{k}, S_{1}^{G}) = {0}, by Equation 3.8.

10. The case of elementary abelian p-groups

10.1. Proposition : Let k be a field of characteristic p, and G be an elementary abelian p-group. Let H be a subgroup of index p in G, and T ∈ K_{G}(H). Then the functor I = Ind_{H}^{G}S_{1}^{H} of M_{k}^{c}(G) has a subfunctor J isomorphic to ι_{G/T}^{G} (S_{1}^{G/T}). Moreover J is contained in the radical R of I, and there is an isomorphism

R/J ≅ L ⊕ ⊕_{X∈K_{G}(H)−{T }} S_{X}^{G} ,

where L is a functor all of whose composition factors are isomorphic to S_{1}^{G}, with multiplicity p − 2.

Proof : Let S be the socle of I. By Proposition 8.7, there is a filtration I ⊃ R ⊇ S ⊃ {0}, and I/R ≅ S ≅ S_{1}^{G}. Moreover

(10.2)  R/S ≅ L ⊕ ⊕_{X∈K_{G}(H)} S_{X}^{G} ,

where L is a functor all of whose composition factors are isomorphic to S_{1}^{G}, with multiplicity p − 2.

By Lemma 8.2, the functor J = ı_{G/T}^{G} (S_{1}^{G/T}) has simple socle, equal to its radical, and isomorphic to S_{1}^{G}, and simple top, isomorphic to S_{T}^{G}. Moreover

Hom_{M_{k}^{c}(G)}(J, I) ≅ Hom_{M_{k}^{c}(H)}(Res_{H}^{G}J, S_{1}^{H}) ≅ k ,

since Res_{H}^{G}J ≅ S_{1}^{H}, for the only non zero evaluation of this functor is at the trivial group, where it is equal to k.

Hence there is a non zero morphism f : J → I. If f is not injective, then its image is isomorphic to S_{T}^{G}, since this is the only proper non zero quotient of J. But S_{T}^{G} is not isomorphic to a subfunctor of I, since the socle of I is simple and isomorphic to S_{1}^{G}.

It follows that f is injective, and one can identify J with a subfunctor of I, containing S, and J/S ≅ S_{T}^{G}. Moreover J is a proper subfunctor of I, since S_{1}^{G} is not a quotient of J. Hence J ⊆ R, and

R/J ≅ L ⊕ ⊕_{X∈K_{G}(H)−{T }} S_{X}^{G} ,

by (10.2).

10.3. Corollary : With the same notation, there is a long exact sequence of extension groups

[figure: ⋯; L(n − 1) ⊕ ⊕_{X∈X} E_{G}(n − 2); E_{G}(n); E_{H}(n); L(n) ⊕ ⊕_{X∈X} E_{G}(n − 1); E_{G}(n + 1); E_{H}(n + 1) ⋯,]

where E_{G}(n) = Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}), E_{H}(n) = Ext_{M_{k}^{c}(H)}^{n}(S_{1}^{H}, S_{1}^{H}), L(n) =

Ext_{M_{k}^{c}(G)}^{n}(L, S_{1}^{G}), and X = K_{G}(H) − {T }.

Proof : There is a short exact sequence

0 ⟶ J ⟶ I ⟶ I/J ⟶ 0

in M_{k}^{c}(G). Since Ext_{M_{k}^{c}(G)}^{n}(J, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(G/T)}^{n}(S_{1}^{G/T} , ρ_{G/T}^{G} (S_{1}^{G})) = {0} for any n ∈ ℕ, by Proposition 3.16 and Lemma 6.7, and since Res_{H}^{G}S_{1}^{G} = S_{1}^{H}, it follows that

Ext_{M_{k}^{c}(G)}^{n}(I/J, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(G)}^{n}(I, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(H)}^{n}(S_{1}^{H}, S_{1}^{H}) = E_{H}(n) . Now by Proposition 10.1, there is a short exact sequence

0 ⟶ R/J ⟶ I/J ⟶ S_{1}^{G} ⟶ 0 ,

and R/J ≅ L ⊕ ⊕_{X∈K_{G}(H)−{T }} S_{X}^{G}. Applying the functor Hom_{M_{k}^{c}(G)}(−, S_{1}^{G})

gives the long exact sequence of extension groups in M_{k}^{c}(G)

[figure: 0; Hom(S_{1}^{G}, S_{1}^{G}); Hom(I/J, S_{1}^{G}); Hom(R/J, S_{1}^{G}); Ext^{1}(S_{1}^{G}, S_{1}^{G}); Ext^{1}(I/J, S_{1}^{G}); Ext^{1}(R/J, S_{1}^{G}); ⋯; ⋯; ⋯; Ext^{n}(S_{1}^{G}, S_{1}^{G}); Ext^{n}(I/J, S_{1}^{G}); Ext^{n}(R/J, S_{1}^{G}); Ext^{n+1}(S_{1}^{G}, S_{1}^{G}); Ext^{n+1}(I/J, S_{1}^{G}); Ext^{n+1}(R/J, S_{1}^{G}) ⋯]

Now

Ext^{n}(R/J, S_{1}^{G}) ≅ Ext^{n}(L, S_{1}^{G}) ⊕ ⊕_{X∈K_{G}(H)−{T }} Ext^{n}(S_{X}^{G}, S_{1}^{G})
≅ Ext^{n}(L, S_{1}^{G}) ⊕ ⊕_{X∈K_{G}(H)−{T }} Ext^{n−1}(S_{1}^{G}, S_{1}^{G})
= L(n) ⊕ ⊕_{X∈X} E_{G}(n − 1) ,

by Corollary 8.2. This completes the proof.

10.4. The case p > 2.

10.5. Theorem : Let k be a field of odd characteristic p. Let G be an elementary abelian p-group of rank 2. Then the simple functor S_{1}^{G} of M_{k}^{c}(G) has exponential growth.

Proof : Let H be a subgroup of index p in G, and consider the long exact sequence of Corollary 10.3. The set K_{G}(H) consists of p subgroups of order p in H, so X has cardinality p − 1. Moreover

E_{H}(n) = Ext_{M_{k}^{c}(H)}^{n}(S_{1}^{H}, S_{1}^{H}) ≅ k ,

by Proposition 9.3, since H is cyclic of order p ≥ 3.

This yields a long exact sequence of the form

⋯→k →L(n)⊕E_{G}(n−1)^{p−1} →E_{G}(n+1)→k →L(n+1)⊕E_{G}(n)^{p−1} →· · · .

In particular, by Lemma 2.5, this shows that

dim_{k} (L(n) ⊕ E_{G}(n − 1)^{p−1}) ≤ 1 + dim_{k} E_{G}(n + 1) ,

thus dim_{k} E_{G}(n + 1) ≥ (p − 1) dim_{k} E_{G}(n − 1) − 1, for any n ≥ 1. Hence, by induction on n,

dim_{k} E_{G}(2n) − \frac{1}{p − 2} ≥ (p − 1)^{n}( dim_{k} E_{G}(0) − \frac{1}{p − 2}) = \frac{(p − 1)^{n}(p − 3)}{p − 2} ,

dim_{k} E_{G}(2n+1)−\frac{1}{p − 2} ≥ (p−1)^{n}( dim_{k} E_{G}(1)−\frac{1}{p − 2}) = \frac{(p − 1)^{n}(2p − 5)}{p − 2} ,

since dim_{k} E_{G}(1) = 2 by Lemma 6.2. Thus S_{1}^{G} has exponential growth if p > 3.

Now if p = 3, the functor L is simple, and isomorphic to S_{1}^{G}, by Proposition 10.1. It follows that in this case

dim_{k} E_{G}(n) + (p − 1) dim_{k} E_{G}(n − 1) − 1 ≤ dim_{k} E_{G}(n + 1) .

In particular dim_{k} E_{G}(2) ≥ 2+2−1 = 3, since dim_{k} E_{G}(1) = 2 by Lemma 6.2. Since dim_{k} E_{G}(n + 1) ≥ 2 dim_{k} E_{G}(n − 1) − 1, it follows that

dim_{k} E_{G}(2n) − 1 ≥ 2^{n−1}( dim_{k} E_{G}(2) − 1) ≥ 2^{n} ,
dim_{k} E_{G}(2n + 1) − 1 ≥ 2^{n}( dim_{k} E_{G}(1) − 1) ≥ 2^{n} ,

hence S_{1}^{G} has exponential growth in this case also.

10.6. The case p = 2.

10.7. Theorem : Let k be a field of characteristic 2, and G be an elementary abelian 2-group. Then

∀j ∈ ℕ, Ext_{M_{k}^{c}(G)}^{2j+1}(S_{1}^{G}, S_{1}^{G}) = {0} .

Proof : By induction on the rank m of G, starting with m = 0, i.e. G = 1, where the result is trivial. Suppose m > 0, and that the result holds for all elementary abelian 2-groups of rank smaller than m. Choose a subgroup H of index p in G, and a complement T ∈ K_{G}(H), and consider the long exact sequence of Corollary 10.3.

In this case by Proposition 10.1, since p − 2 = 0, the functor L is equal to zero. Thus L(n) = {0} for any n ∈ ℕ. Moreover Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}) = {0}, by Theorem 6.2, since G is generated by involutions. This starts an induction argument on j : suppose that Ext_{M_{k}^{c}(G)}^{2r+1}(S_{1}^{G}, S_{1}^{G}) = {0}, for 0 ≤ r < j. The exact sequence of Corollary 10.3 becomes

(10.8)  [figure: ⋯; ⊕_{X∈X} E_{G}(n − 2); E_{G}(n); E_{H}(n); ⊕_{X∈X} E_{G}(n − 1); E_{G}(n + 1); E_{H}(n + 1) ⋯,]

where E_{G}(n) = Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}) and E_{H}(n) = Ext_{M_{k}^{c}(H)}^{n}(S_{1}^{H}, S_{1}^{H}), and X = K_{G}(H) − {T }.

Set n = 2j + 1 in this sequence. By induction hypothesis on m

E_{H}(n) = E_{H}(2j + 1) = Ext_{M_{k}^{c}(H)}^{2j+1}(S_{1}^{H}, S_{1}^{H}) ≅ {0} ,

since H is elementary abelian of rank m − 1. Also

E_{G}(n − 2) = E_{G}(2j − 1) = Ext_{M_{k}^{c}(G)}^{2j−1}(S_{1}^{G}, S_{1}^{G}) = {0} ,

by induction hypothesis on j.

It follows that E_{G}(n) = Ext^{2j+1}(S_{1}^{G}, S_{1}^{G}) = {0}, and this completes the inductive step on j, hence the inductive step on m.

10.9. Theorem : Let k be a field of characteristic 2. Let G be an elementary abelian 2-group, and H be a subgroup of index 2 of G. Choose

T ∈ K_{G}(H), and set S = K_{G}(H) − {T }.

Then for any j ∈ ℕ, there is a short exact sequence of extension groups

0 → ⊕_{X∈S} Ext^{2j−2}(S_{1}^{G}, S_{1}^{G}) →^{γ} Ext^{2j}(S_{1}^{G}, S_{1}^{G}) →^{r_{H}^{G}} Ext^{2j}(S_{1}^{H}, S_{1}^{H}) ⟶ 0

where r_{H}^{G} is induced by restriction to H, and γ is the direct sum of the maps induced by the Yoneda product with the element γ_{X} of Ext^{2}(S_{1}^{G}, S_{1}^{G}), for

X ∈ S.

Proof : This follows from the exact sequence 10.8, in which all the terms E_{G}(2j − 1), E_{G}(2j + 1), and E_{H}(2j + 1) are equal to zero. The exact sequence 10.8 splits as a series of short exact sequences

0 → ⊕_{X∈S} Ext^{2j−2}(S_{1}^{G}, S_{1}^{G}) →^{γ} Ext^{2j}(S_{1}^{G}, S_{1}^{G}) →^{r} Ext^{2j}(S_{1}^{H}, S_{1}^{H}) ⟶ 0 ,

for each j ∈ ℕ. It remains to show that the morphisms γ and r are as stated in Theorem 10.9.

First, with the notation of Proposition 10.1, the morphism

t : Ext^{2j−1}(R/J, S_{1}^{G}) ⟶ Ext^{2j}(S_{1}^{G}, S_{1}^{G})

is the transition morphism associated to the short exact sequence

T : 0 ⟶ R/J ⟶ I/J ⟶ S_{1}^{G} ⟶ 0 ,

so it it given by Yoneda product with T.

Via the isomorphism R/J ≅ ⊕_{X∈S} S_{X}^{G}, the morphism t can be viewed as

the direct sum of morphisms t_{X} : Ext^{2j−1}(S_{X}^{G}, S_{1}^{G}) → Ext^{2j}(S_{1}^{G}, S_{1}^{G}), where t_{X} is given by composition with the sequence T_{X} obtained from T by the projection π_{X} to the summand S_{X}^{G}, i.e. by completing the diagram

[figure: T : 0; ⊕_{X∈S} S_{X}^{G}; π_{X}; I/J; f; S_{1}^{G}; 0; T_{X} : 0; S_{X}^{G}; Y; S_{1}^{G}; 0 .]

Now f is surjective, since π_{X} and Id_{S_{1}^{G}} are. Since S_{X}^{G} is not a quotient of I/J, it follows that Y is a non-split extension of S_{1}^{G} by S_{X}^{G}. Thus Y ≅ ȷ_{G/X}^{G}(S_{1}^{G/X}), by Lemma 8.3. More precisely T_{X} is isomorphic to the sequence D_{X}^{∗} of Lemma 8.1.

Moreover, by Corollary 8.2, the isomorphism

Ext^{2j−2}(S_{1}^{G}, S_{1}^{G}) ≅ Ext^{2j−1}(S_{X}^{G}, S_{1}^{G})

is given by Yoneda composition with D_{X}. Hence the component index by X ∈ S of the morphism

γ : ⊕_{X∈S} Ext^{2j−2}(S_{1}^{G}, S_{1}^{G}) ⟶ Ext^{2j}(S_{1}^{G}, S_{1}^{G})

is given by composition with Γ_{X} = D_{X} ∘D_{X}^{∗}, i.e. by Yoneda product with γ_{X}, as claimed.

Now the morphism r_{H}^{G} : Ext_{M_{k}^{c}(G)}^{2j}(S_{1}^{G}, S_{1}^{G}) → Ext_{M_{k}^{c}(H)}^{2j}(S_{1}^{H}, S_{1}^{H}) is composed of two steps : first, taking the image by the projection map q : I → S_{1}^{G}, i.e. taking the Yoneda product with q ∈ Ext_{M^{c}(G)}^{0}(Ind_{H}^{G}S_{1}^{H}, S_{1}^{G}), and then using the adjunction (Ind_{H}^{G}, Res_{H}^{G}), which gives a map

Ext_{M_{k}^{c}(G)}^{2j}(I, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(H)}^{2j}(S_{1}^{H}, Res_{H}^{G}S_{1}^{G}) ≅ Ext_{M_{k}^{c}(H)}^{2j}(S_{1}^{H}, S_{1}^{H}) .

In other words, with the notation of Remark 3.18, for u ∈ Ext_{M_{k}^{c}(G)}^{2j}(S_{1}^{G}, S_{1}^{G})

r_{H}^{G}(u) = α_{2j}(u ∘ q) = Res_{H}^{G}(u) ∘ α_{0}(q) = Res_{H}^{G}(u) ,

since the map α_{0}(q) : S_{1}^{H} → Res_{H}^{G}S_{1}^{G} ≅ S_{1}^{H} obtained by adjunction is the identity map. Hence r_{H}^{G} is induced by Res_{H}^{G}, and this completes the proof.

The following corollary is Theorem 1.2 of Section 1 :

10.10. Corollary : Let G be an elementary abelian 2-group of rank m. Then the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) is finitely generated by the elements γ_{X}, where X is a subgroup of order 2 of G. Its Poincaré series

P (t) = ∑_{j∈ℕ}dim_{k} Ext_{M_{k}^{c}(G)}^{j}(S_{1}^{G}, S_{1}^{G}) t^{j}

is equal to

P (t) = \frac{1}{(1 − t^{2})(1 − 3t^{2})(1 − 7t^{2}) … (1 − (2^{m−1} − 1)t^{2})} .

Proof : By induction on m, the case m = 0 being trivial. In Theorem 10.9, one can assume that the algebra Ext^{∗}(S_{1}^{H}, S_{1}^{H}) is generated by the elements γ_{X}^{H}, where X is a subgroup of order 2 of H. This means that for n ∈ ℕ, any element in Ext^{2n}(S_{1}^{H}, S_{1}^{H}) is a k-linear combination of products of the form

γ_{X_{1}}^{H}γ_{X_{2}}^{H} ⋯ γ_{X_{n}}^{H} ,

where X_{1},…, X_{n} are subgroups of order 2 of H.

Denote by Γ the k-linear subspace of Ext^{2n}(S_{1}^{G}, S_{1}^{G}) generated by the similar products γ_{X_{1}}^{G}γ_{X_{2}}^{G} ⋯ γ_{X_{n}}^{G}.

Now for X ≤ H, by Proposition 3.14, there are isomorphisms of functors

Res_{H}^{G} ∘ ı_{G/X}^{G} ≅ L_{G} ∘ L_{G/X} ≅ L_{G×_{G}(G/X)} ,

where G is viewed as an (H, G)-biset and G/X as a (G, G/X)-biset. Now G ×_{G} (G/X) ≅ (H/X) ×_{H/X} (G/X) as (H, G/X)-bisets, thus

Res_{H}^{G} ∘ ı_{G/X}^{G} ≅ ı_{H/X}^{H} ∘ Res_{H/X}^{G/X} .

It follows that Res_{H}^{G} \binom{S_{X}^{G}}{S_{1}^{G}} ≅ \binom{S_{X}^{H}}{S_{1}^{H}}. Similarly Res_{H}^{G} \binom{S_{1}^{G}}{S_{X}^{G}} ≅ \binom{S_{1}^{H}}{S_{X}^{H}}. Thus

r_{H}^{G}γ_{X}^{G} = γ_{X}^{H}, because Res_{H}^{G} respects the Yoneda composition.

It follows that

γ_{X_{1}}^{H}γ_{X_{2}}^{H} ⋯ γ_{X_{n}}^{H} = r_{H}^{G}(γ_{X_{1}}^{G}γ_{X_{2}}^{G} ⋯ γ_{X_{n}}^{G}) ,

thus Ext^{2n}(S_{1}^{G}, S_{1}^{G}) = Γ + Ker r_{H}^{G}. But

Ker r_{H}^{G} = γ(Ext^{2n−2}(S_{1}^{G}, S_{1}^{G})) = ∑_{X∈S} Ext^{2n−2}(S_{1}^{G}, S_{1}^{G}) ∘ γ_{X}^{G} ,

and by induction on n, this shows that Γ = Ext^{2n}(S_{1}^{G}, S_{1}^{G}).

Denote by P_{m}(t) the Poincaré series of the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}), where G is an elementary abelian 2-group of rank m. The short exact sequence of Theorem 10.9 gives the relation

P_{m}(t) = |S|t^{2}P_{m}(t) + P_{m−1}(t) ,

and moreover |S| = 2^{m−1} − 1. Thus

P_{m}(t) = P_{m−1}(t) · \frac{1}{1 − (2^{m−1} − 1)t^{2}} ,

and the claimed formula follows by induction on m.

10.11. Theorem : Let k be a field of characteristic 2, and G be an elementary abelian 2-group of rank m.

1. [Samy Modeliar [7]] If m ≤ 2, the group G is a poco group over k.

2. If m ≥ 3, the group G is not a poco group over k. More precisely, the simple functor S_{1}^{G} of M_{k}^{c}(G) has exponential growth.

For Assertion 2, observe that

\frac{1}{(1 − t^{2})(1 − 3t^{2}) … (1 − (2^{m−1} − 1)t^{2})} = ∏_{j=1}^{m−1} (∑_{n_{j}=0}^{∞}(2^{j} − 1)^{n_{j}}t^{2n_{j}} ) ,

Thus

dim_{k} Ext^{2n}(S_{1}^{G}, S_{1}^{G}) = ∑_{n_{1}+···+n_{m−1}=n} ∏_{j=1}^{m−1}(2^{j} − 1)^{n_{j}}
≥ (2^{m−1} − 1)^{n} .

By Corollary 8.2, since moreover Ext^{2n+1}(S_{Z}^{G}, S_{1}^{G}) ≅ Ext^{2n+1}(S_{1}^{G}, S_{Z}^{G}) as simple functors are self dual, it follows that

dim_{k} Ext^{2n+1}(S_{1}^{G}, S_{Z}^{G}) ≥ (2^{m−1} − 1)^{n} ,

thus S_{1}^{G} has exponential growth, by Lemma 2.3, if 2^{m−1} − 1 > 1, i.e. if m ≥ 3.

Assertion 1 is trivial if m = 0. It is straightforward if m = 1 : since every kG-module for a group G of order 2 is a permutation module, it follows that all the fixed points functors are projective in this case, so M_{k}^{c}(G) has finite projective dimension in this case.

For m = 2, M. Samy Modeliar has described explicit eventually periodic resolutions of the fixed points functors F P_{V}, for all indecomposable kGmodules V. An alternative proof can be sketched as follows : the Poincaré series of Ext^{∗}(S_{1}^{G}, S_{1}^{G}) is equal to \frac{1}{1−t^{2}} in this case. It follows that Ext^{n}(S_{1}^{G}, S_{1}^{G}) is equal to {0} if n is odd, and one dimensional if n is even. If Q is a non trivial subgroup of G, then there is an exact sequence

(10.12)  0 ⟶ ⊕_{X∈K_{Q}(Z)} S_{X}^{G} ⟶ ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}) ⟶ S_{Q}^{G} ⟶ 0 ,

where Z is a subgroup of order 2 of Q. Moreover ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}) has finite projective dimension, since G/Z has order 2, and since ı_{G/Z}^{G} is exact and preserves projectives. It follows easily by induction on |Q| that there exists a constant c_{Q} such that dim_{k} Ext^{j}(S_{Q}^{G}, S_{1}^{G}) ≤ c_{Q} for any j ∈ ℕ.

Thus dim_{k} Ext^{j}(S_{1}^{G}, S_{R}^{G}) ≤ c_{R} for any j ∈ ℕ, and any R ≤ G. Using again the exact sequence 10.12, one can show by induction on |Q| that for any Q, R ≤ G, there is a constant c_{Q,R} such that dim_{k} Ext^{j}(S_{Q}^{G}, S_{R}^{G}) ≤ c_{Q,R} for any j ∈ ℕ. By Lemma 2.3 and Corollary 2.6, the group G is a poco group.

11. The case of 2-groups of sectional rank at most 2

11.1. Proposition : Let k be a field of characteristic 2, let G be a 2group, and let H be a subgroup of index 2 of G. If the functor S_{1}^{H} (over k) has polynomial growth, and if the functor S_{X}^{C_{G}(X)} has polynomial growth, for any X ∈ K_{G}(H), then the functor S_{1}^{G} has polynomial growth.

Proof : Consider the functor I = Ind_{H}^{G}S_{1}^{H}. By Proposition 8.7, there is a filtration

I ⊃ R ⊇ S ⊃ {0} ,

where R is the radical and S is the socle of I, such that I/R ≅ S ≅ S_{1}^{G}, and R/S ≅ ⊕_{X∈[G\K_{G}(H)]} S_{X}^{G}. This gives two short exact sequences in M_{k}^{c}(G)

(11.2)  0 ⟶ S_{1}^{G} ⟶ R ⟶ ⊕_{X∈[G\K_{G}(H)]} S_{X}^{G} ⟶ 0
(11.3)  0 ⟶ R ⟶ I ⟶ S_{1}^{G} ⟶ 0 .

Let M be a finitely generated cohomological Mackey functor for G over k. Applying the functor Hom_{M_{k}^{c}(G)}(−, M) to the sequence (11.2) gives the following long exact sequence of Ext groups in M_{k}^{c}(G)

⋯ ⟶ ⊕_{X∈[G\K_{G}(H)]} Ext^{n}(S_{X}^{G}, M) ⟶ Ext^{n}(R, M) ⟶ Ext^{n}(S_{1}^{G}, M) ⟶ ⋯

By Lemma 2.5, this gives

r_{n} ≤ e_{n} + ∑_{X∈[G\K_{G}(H)]} e_{X,n} ,

where r_{n} = dim_{k} Ext^{n}(R, M), e_{X,n} = dim_{k} Ext^{n}(S_{X}^{G}, M), for X ∈ K_{G}(H), and e_{n} = dim_{k} Ext^{n}(S_{1}^{G}, M).

Now applying the functor Hom_{M_{k}^{c}(G)}(−, M) to the sequence (11.3) gives the following long exact sequence of Ext groups in M_{k}^{c}(G)

⋯ ⟶ Ext^{n}(R, M) ⟶ Ext^{n+1}(S_{1}^{G}, M) ⟶ Ext^{n+1}(I, M) ⟶ ⋯

By Lemma 2.5, this gives

e_{n+1} ≤ r_{n} + i_{n+1} ,

where i_{n} = dim_{k} Ext^{n}(I, M). Thus

(11.4)  e_{n+1} ≤ e_{n} + ( ∑_{X∈[G\K_{G}(H)]} e_{X,n}) + i_{n+1} .

Now S_{X}^{G} = Ind_{C_{G}(X)}^{G}S_{X}^{C_{G}(X)}, for any X ∈ K_{G}(H), thus

e_{X,n} = dim_{k} Ext_{M_{k}^{c}(C_{G}(X))}^{n}(S_{X}^{C_{G}(X)}, Res_{C_{G}(X)}^{G}M) .

Hence if S_{X}^{C_{G}(X)} has polynomial growth, there are constants c_{X}, d_{X}, and e_{X}, such that

∀n ∈ ℕ, e_{X,n} ≤ c_{X}n^{d_{X}} + e_{X} .

Thus

∀n ∈ ℕ, ∑_{X∈[G\K_{G}(H)]} e_{X,n} ≤ Cn^{D} + E ,

where C = ∑_{X∈[G\K_{G}(H)]} c_{X}, D = max_{X∈[G\K_{G}(H)]} d_{X}, and E = ∑_{X∈[G\K_{G}(H)]} e_{X}.

Similarly,

i_{n} = dim_{k} Ext_{M_{k}^{c}(H)}^{n}(S_{1}^{H}, Res_{H}^{G}M) ,

so if S_{1}^{H} has polynomial growth, there are constants c, d, and e such that

∀n ∈ ℕ, i_{n} ≤ cn^{d} + e .

Thus

∀n ∈ ℕ, i_{n+1} ≤ c|(n + 1)^{d} − 1| + c + e ≤ c(2n)^{d} + c + e = c2^{d}n^{d} + c + e .

Inequality 11.4 now gives

∀n ∈ ℕ, e_{n+1} ≤ e_{n} + γn^{δ} + ϵ

where γ = c2^{d} + C, δ = max(D, d), and ϵ = E = c = e. By induction, it follows that

∀n ∈ ℕ, e_{n} ≤ γ ∑_{j=1}^{n} j^{δ} + nϵ + e_{0} ≤ γn^{δ+1} + nϵ + e_{0} ≤ (γ + ϵ)n^{δ+1} + e_{0} ,

so S_{1}^{G} has polynomial growth.

11.5. Theorem : Let k be a field of characteristic 2, and G be a 2-group of sectional 2-rank at most 2. Then G is a poco group over k.

Proof : By induction on the order of G, one may assume that for any section (B, A) ≠ (G, 1), the group B/A is a poco group. By Corollary 2.6, proving that G is a poco group is equivalent to proving that for any subgroup Q of G, the simple functor S_{Q}^{G} has polynomial growth. First consider the case where Q = 1.

Let H be a subgroup of index 2 of G. Then H is a poco group by induction hypothesis. Similarly, if X ∈ K_{G}(H) and C_{G}(X) ≠ G, then C_{G}(X) is a poco group. Thus, if H has no central complement in G, by Proposition 11.1, the functor S_{1}^{G} has polynomial growth. This holds in particular if H has no complement at all in G, i.e. if H contains all involutions of G. Thus, if G is not generated by involutions, then S_{1}^{G} has polynomial growth. So one can assume that G is generated by involutions.

Now if H has some central complement X in G, then G ≅ H ×X. Since G has sectional 2-rank at most 2, the group H has sectional 2-rank at most 1, hence it is cyclic. Thus G is abelian, and generated by involutions, hence elementary abelian, and of sectional 2-rank at most 2. So the group G is elementary abelian of rank at most 2, and the results of Section 10 show that S_{1}^{G} has polynomial growth.

In any case S_{1}^{G} has polynomial growth. By induction on the order of Q, it follows that S_{Q}^{G} has polynomial growth, for any subgroup Q of G : indeed, setting B = N_{G}(Q) and A = Φ(Q), Lemma 6.5 shows that

S_{Q}^{G} ≅ L_{U}(S_{Q/A}^{B/A}) ,

where U is the set A\G, for its natural structure of (B/A, G)-biset structure. If (B, A) ≠ (G, 1), then S_{Q/A}^{B/A} has polynomial growth, hence S_{Q}^{G} has polynomial growth, by Lemma 4.1.

So it remains to consider the case where (B, A) = (Q, 1), i.e. the case where Q is a non trivial normal elementary abelian subgroup of G. Let Z be a subgroup of order 2 of Q ∩ Z(G).

By Proposition 8.5, there is a short exact sequence

0 ⟶ ⊕_{X∈[G\K_{Q}(Z)]} S_{X}^{G} ⟶ ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}) ⟶ S_{Q}^{G} ⟶ 0 .

By the induction hypothesis, all the functors S_{X}^{G}, for X ∈ K_{Q}(Z), have polynomial growth. Moreover, the functor ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}) has polynomial growth, by Lemma 4.1. Hence S_{Q}^{G} has polynomial growth, by Lemma 2.5, and this completes the proof.

11.6. Remark : The 2-groups of sectional rank at most 1 are the cyclic 2groups. In a recent preprint, Andersen, Oliver and Ventura have shown that the 2-groups of sectional rank 2 are the metacyclic 2-groups (see Lemma 10.2 of [1]). For 2-groups of order at least 32, this also follows from a theorem of Blackburn (see [2], or Satz 11.13 in [6]).

II - Cohomology

12. Extension of simple functors for elementary abelian p-groups

In this section k is a field of characteristic p, the group G ≅ (C_{p})^{m} is an elementary abelian p-group of rank m, in additive notation, and all the cohomological Mackey functors have values in k-vector spaces.

Recall from Notation 7.4 that γ_{X} ∈ Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}) is the element represented by the 2-fold extension

0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ \binom{S_{1}^{G}}{S_{X}^{G}} ⟶ S_{1}^{G} ⟶ 0 .

12.1. Notation : If G = (C_{p})^{m} and x ∈ G − {0}, set γ_{x} = γ_{<x>}^{G}. Thus

γ_{x} ∈ Ext^{2}(S_{1}^{G}, S_{1}^{G}).

12.2. Lemma : Let A be a ring. If L ⊇ M ⊇ N ⊇ {0} is a filtration of an A-module L, then the exact sequence

0 ⟶ N ⟶ M ⟶ L/N ⟶ L/M ⟶ 0

obtained by splicing the short exact sequences

\xymatrix{0 \ar[r] & N \ar[r] & M \ar[rr] \ar[dr] &  & L/N \ar[r] & L/M \ar[r] & 0 \\  &  &  & M/N \ar@{-->}[dr] \ar@{-->}[ur] \\  &  & 0 \ar@{-->}[ur] &  & 0}

represents the zero class of Ext_{A}^{2}(L/M, N).

Proof : There is a commutative diagram

[figure: 0; N; M; i; L/N; p; (Id,p); L/M; 0; 0; N; L; (q,0); L/N ⊕ L/M; _{\binom{0}{Id}}; L/M; 0]

where i is the inclusion map, and p, q are projection maps. Moreover, the bottom sequence represents zero in Ext_{A}^{2}(L/M, N), since the map \binom{0}{Id} is split surjective.

12.3. Lemma : Let G be an elementary abelian p-group. If X and Y are distinct subgroups of order p of G, set Q = XY. Then the sequences

0 ⟶ S_{X}^{G} ⟶ \binom{S_{1}^{G}}{S_{X}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{1}^{G}} ⟶ S_{Y}^{G} ⟶ 0

{and \atop 0 ⟶ S_{X}^{G} ⟶ \binom{S_{Q}^{G}}{S_{X}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{Q}^{G}} ⟶ S_{Y}^{G} ⟶ 0}

represent opposite elements of Ext_{M_{k}^{c}(G)}^{2}(S_{Y}^{G}, S_{X}^{G}).

Proof : Indeed, the sum of the corresponding elements in Ext_{M_{k}^{c}(G)}^{2}(S_{Y}^{G}, S_{X}^{G}) is represented by the sequence

(12.4)  0 ⟶ S_{X}^{G} ⟶ \binom{S_{1}^{G} S_{Q}^{G}}{S_{X}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{1}^{G} S_{Q}^{G}} ⟶ S_{Y}^{G} ⟶ 0 ,

where the functor L_{X} = \binom{S_{1}^{G} S_{Q}^{G}}{S_{X}^{G}} is the Mackey functor for G over k whose values at 1, X and Q are equal to k, and other values are zero. The conjugation maps for this functor are all identity maps (possibly zero), and the possibly non zero transfer and restriction maps are given in the following diagram

\xymatrix{L_{X}(Q) = k \ar[d]_{r_{X}^{Q}=1} \\ L_{X}(X) = k \ar[u]_{t_{X}^{Q}=0} \ar[d]_{r_{1}^{X}=0} \\ L_{X}(1) = k \ar[u]_{t_{1}^{X}=1}}

It follows in particular that S_{X}^{G} is isomorphic to a subfunctor of L_{X}, and this yields an exact sequence

(12.5)  0 ⟶ S_{X}^{G} ⟶ \binom{S_{1}^{G} S_{Q}^{G}}{S_{X}^{G}} ⟶ S_{1}^{G} ⊕ S_{Q}^{G} ⟶ 0

in M_{k}^{c}(G).

The functor \binom{S_{Y}^{G}}{S_{1}^{G} S_{Q}^{G}} is isomorphic to the dual L_{Y}^{∗} of L_{Y} = \binom{S_{1}^{G} S_{Q}^{G}}{S_{Y}^{G}} : its non zero values are at 1, Y and Q, and they are equal to k. The conjugation maps for L_{Y}^{∗} are all identity maps (possibly zero), and the possibly non zero transfer and restriction maps are given in the following diagram

\xymatrix{L_{Y}^{∗} (Q) = k \ar[d]_{r_{Y}^{Q}=0} \\ L_{Y}^{∗} (Y) = k \ar[u]_{t_{Y}^{Q}=1} \ar[d]_{r_{1}^{Y} =1} \\ L_{Y}^{∗} (1) = k \ar[u]_{t_{1}^{Y} =0}}

There is an exact sequence

(12.6)  0 ⟶ S_{1}^{G} ⊕ S_{Q}^{G} ⟶ \binom{S_{Y}^{G}}{S_{1}^{G} S_{Q}^{G}} ⟶ S_{Y}^{G} ⟶ 0

in M_{k}^{c}(G), and the exact sequence 12.4 is obtained by splicing the exact sequences 12.5 and 12.6.

Let M_{X,Y} denote the functor defined for H ≤ G by M_{X,Y} (H) = k if H ∈ {1, X, Y, Q}, and by M_{X,Y} (H) = {0} otherwise. The conjugation maps for M_{X,Y} are all identity maps (possibly zero), and the possibly non zero transfer and restriction maps for M_{X,Y} are given in the following diagram

(12.7)  \xymatrix{ & M_{X,Y} (Q) = k \ar[dl]_{r_{X}^{Q}=1} \ar[dr]_{r_{Y}^{Q}=0} \\ M_{X,Y} (X) = k \ar[ur]_{t_{X}^{Q}=0} \ar[dr]_{r_{1}^{X}=0} &  & M(Y) = k \ar[ul]_{t_{Y}^{Q}=1} \ar[dl]_{r_{1}^{Y} =1} \\  & M_{X,Y} (1) = k \ar[ul]_{t_{1}^{X}=1} \ar[ur]_{t_{1}^{Y} =0}}

One checks easily that M_{X,Y} is a cohomological Mackey functor for G over k, that L_{X} is a subfunctor of M_{X,Y} (represented by the left half of diagram 12.7), and that L_{Y}^{∗} is a quotient functor of M_{X,Y} (represented by the right half of 12.7). Hence there is a filtration

M_{X,Y} ⊃ L_{X} ⊃ S_{X}^{G} ⊃ {0} ,

such that M_{X,Y} /L_{X} ≅ S_{Y}^{G} and M_{X,Y} /S_{X}^{G} ≅ L_{Y}^{∗}. Now Lemma 12.2 shows that the exact sequence 12.4 represents zero in Ext_{M_{k}^{c}(G)}^{2}(S_{Y}^{G}, S_{X}^{G}), and this completes the proof.

12.8. Lemma : Let G be an elementary abelian p-group, and let Q < R be subgroups of G such that |R : Q| = p^{2}. Let M denote the functor Σ_{Q,R}^{G} of Corollary 7.3. Then there is a filtration M ⊃ J ⊃ S ⊃ {0}, where J is the radical of M and S is its socle. Moreover

M/J ≅ S_{R}^{G} , S ≅ S_{Q}^{G} , J/M ≅ ⊕_{Q<X<R} S_{X}^{G} .

Proof : Recall that Σ_{Q,R}^{G} is the subquotient of F P_{k} whose value at H ≤ G is equal to k if Q ≤ H ≤ R, and to zero otherwise. By Corollary 7.3, the head of Σ_{Q,R}^{G} is simple, isomorphic to S_{R}^{G}, and its socle is simple, isomorphic to S_{Q}^{G}. In particular S ≤ J. Moreover, if H ≤ G, then (J/S)(H) is equal to zero, except if Q < H < R, and in this case (J/S)(H) = k. It follows that

J/M ≅ ⊕_{Q<X<R} S_{X}^{G} ,

as was to be shown.

12.9. Proposition : Let G be an elementary abelian p-group, and Q be a subgroup of order p^{2} of G. Then for any subgroup Y of order p of Q

[ ∑_{1<X<Q} γ_{X}, γ_{Y}] = 0 ,

in Ext_{M_{k}^{c}(G)}^{4}(S_{1}^{G}, S_{1}^{G}), where [a, b] = ab − ba, for a, b ∈ Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}).

Proof : Recall that if X is a subgroup of order p of G, the element γ_{X} of Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}) is represented by the sequence

0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ \binom{S_{1}^{G}}{S_{X}^{G}} ⟶ S_{1}^{G} ⟶ 0 .

It follows that if Y is another subgroup of order p of G, the Yoneda product γ_{X}γ_{Y} is represented by the sequence

0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ \binom{S_{1}^{G}}{S_{X}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{1}^{G}} ⟶ \binom{S_{1}^{G}}{S_{Y}^{G}} ⟶ S_{1}^{G} ⟶ 0 .

This sequence is obtained by splicing the following three exacts sequences :

0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ S_{X}^{G} ⟶ 0

(12.10)  0 ⟶ S_{X}^{G} ⟶ \binom{S_{1}^{G}}{S_{X}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{1}^{G}} ⟶ S_{Y}^{G} ⟶ 0

0 ⟶ S_{Y}^{G} ⟶ \binom{S_{Y}^{G}}{S_{1}^{G}} ⟶ S_{1}^{G} ⟶ 0 .

By Lemma 12.3, if XY = Q, i.e. if X and Y are distinct subgroups of order p of Q, the sequence 12.10 and the sequence

(12.11)  0 ⟶ S_{X}^{G} ⟶ \binom{S_{Q}^{G}}{S_{X}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{Q}^{G}} ⟶ S_{Y}^{G} ⟶ 0

represent opposite elements of Ext_{M_{k}^{c}(G)}^{2}(S_{Y}^{G}, S_{X}^{G}). It follows that for a given subgroup Y of order p of Q

( ∑_{{1<X<Q \atop X≠Y}} γ_{X})γ_{Y} = −( ∑_{{1<X<Q \atop X≠Y}} f_{X,Y}^{Q})γ_{Y} ,

where f_{X,Y}^{Q} is the element represented by the sequence

0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ \binom{S_{Q}^{G}}{S_{X}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{Q}^{G}} ⟶ \binom{S_{1}^{G}}{S_{Y}^{G}} ⟶ S_{1}^{G} ⟶ 0 .

This in turn is the splice of the following two exact sequences

U_{X}^{Q} : 0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ \binom{S_{Q}^{G}}{S_{X}^{G}} ⟶ S_{Q}^{G} ⟶ 0
(U_{Y}^{Q})^{∗} : 0 ⟶ S_{Q}^{G} ⟶ \binom{S_{Y}^{G}}{S_{Q}^{G}} ⟶ \binom{S_{1}^{G}}{S_{Y}^{G}} ⟶ S_{1}^{G} ⟶ 0 .

The sum of the elements u_{X}^{Q} of Ext_{M_{k}^{c}(G)}^{2}(S_{Q}^{G}, S_{1}^{G}) represented by the sequences U_{X}^{Q}, for 1 < X < Q, is represented by the sequence

0 ⟶ S_{1}^{G} ⟶ \binom{Σ_{Q}^{G}}{S_{1}^{G}} ⟶ \binom{S_{1}^{G}}{Σ_{Q}^{G}} ⟶ S_{Q}^{G} ⟶ 0 ,

where \binom{Σ_{Q}^{G}}{S_{1}^{G}} has simple socle isomorphic to S_{1}^{G}, and head isomorphic to Σ_{Q}^{G} = ⊕_{1<X<Q} S_{X}^{G}, and \binom{S_{1}^{G}}{Σ_{Q}^{G}} is isomorphic to the dual of \binom{Σ_{Q}^{G}}{S_{1}^{G}}. By Lemma 12.2 and Lemma 12.8, it follows that

∑_{1<X<Q} u_{X}^{Q} = 0 .

Hence ∑_{{1<X<Q \atop X≠Y}} u_{X}^{Q} = −u_{Y}^{Q}, so

( ∑_{{1<X<Q \atop X≠Y}} γ_{X})γ_{Y} = u_{Y}^{Q}γ_{Y} .

This is represented by the sequence

0 ⟶ S_{1}^{G} ⟶ \binom{S_{Y}^{G}}{S_{1}^{G}} ⟶ \binom{S_{Q}^{G}}{S_{Y}^{G}} ⟶ \binom{S_{Y}^{G}}{S_{Q}^{G}} ⟶ \binom{S_{1}^{G}}{S_{Y}^{G}} ⟶ S_{1}^{G} ⟶ 0 ,

which is obviously self dual. It follows that

( ∑_{{1<X<Q \atop X≠Y}} γ_{X})γ_{Y} = (( ∑_{{1<X<Q \atop X≠Y}} γ_{X})γ_{Y} )^{∗} = γ_{Y} ( ∑_{{1<X<Q \atop X≠Y}} γ_{X}) .

Adding γ_{Y}^{2} to both side, this shows that γ_{Y} and ∑_{1<X<Q} γ_{X} commute, which completes the proof.

13. The algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) for G ≅ (C_{2})^{m}

In this section k is a field of characteristic 2, the group G ≅ (C_{2})^{m} is an elementary abelian 2-group of rank m, in additive notation, and all the cohomological Mackey functors have values in k-vector spaces.

By Corollary 10.10 and Notation 12.1, the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) is generated by the elements γ_{x} of Ext^{2}(S_{1}^{G}, S_{1}^{G}), for x ∈ G − {0}.

13.1. Proposition : Let H be a subgroup of index 2 of G. Then

∑_{x∉H} γ_{x} = 0 in Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}) .

Proof : By Proposition 10.1, there is a filtration I ⊃ R ⊃ S ⊃ {0}, where I = Ind_{H}^{G}S_{1}^{H}, such that S ≅ I/R ≅ S_{1}^{G}, and i

R/S ≅ ⊕_{X∈K} S_{X}^{G} ,

where K = K_{G}(H). By Lemma 12.2, the sequence

0 ⟶ S_{1}^{G} ⟶ R ⟶ I/S ⟶ S_{1}^{G} ⟶ 0

represents 0 in Ext^{2}(S_{1}^{G}, S_{1}^{G}).

For each X ∈ K, the functors ρ_{G/X}^{G}(I), ρ_{G/X}^{G}(R), and ρ_{G/X}^{G}(I/S) are all isomorphic to S_{1}^{G/X}, since their non zero evaluation is at the trivial subgroup, where it is isomorphic to k. By adjunction, this gives morphisms

p_{X} : \binom{S_{X}^{G}}{S_{1}^{G}} → R and q_{X} : I/S → \binom{S_{1}^{G}}{S_{X}^{G}} .

This gives a commutative diagram with exact lines

[figure: 0; ⊕_{X∈K} S_{1}^{G}; Σ; ⊕_{X∈K} \binom{S_{X}^{G}}{S_{1}^{G}}; p; ⊕_{X∈K} \binom{S_{1}^{G}}{S_{X}^{G}}; ⊕_{X∈K} S_{1}^{G}; 0; 0; S_{1}^{G}; R; ⊕_{X∈K}_{\binom{S_{1}^{G}}{S_{X}^{G}}}; ⊕_{X∈K} S_{1}^{G}; 0; 0; S_{1}^{G}; R; I/S; q; S_{1}^{G}; Δ; 0 ,]

where p is the sum of the maps p_{X} and q is the sum of the maps q_{X}, for X ∈ K, where Σ is the summation map, and Δ the diagonal inclusion. The top line of this diagram is the direct sum of the sequences Γ_{X}, for X ∈ K (see Notation 7.4).

One checks easily that the top left square in this diagram is cocartesian, and that the bottom right square is cartesian. It follows that the bottom line represents the sum in Ext^{2}(S_{1}^{G}, S_{1}^{G}) of the 2-fold extensions in the top line, i.e. the sum ∑_{X∈K} γ_{X}.

Hence this sum is equal to 0, i.e. equivalently ∑_{x∉H} γ_{x} = 0.

13.2. Proof of Theorem 1.3. Let G ≅ (C_{2})^{m} be an elementary abelian 2-group of rank m, in additive notation, and k be a field of characteristic 2. Denote by E the graded algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}).

• By Corollary 10.10 and Notation 12.1, the algebra E is generated by the elements γ_{x}, for x ∈ G − {0}, where γ_{x} has degree 2.

Let x and y be distinct elements in G − {0}. Then x and y generate a subgroup Q of order 4 of G, and the non zero elements of Q are x, y, and x + y. By Proposition 12.9, the commutator

[γ_{x} + γ_{y} + γ_{x+y}, γ_{x+y}]

in E is equal to 0. Equivalently

(13.3)  [γ_{x} + γ_{y}, γ_{x+y}] = 0 .

Now if H is a subgroup of index 2 of G, Proposition 13.1 shows that

(13.4)  ∑_{x∉H} γ_{x} = 0

in E.

• Conversely, let Ẽ denote the graded associative k-algebra with generators γ̃_{x} in degree 2, for x ∈ G − {0}, subject to the relations

\{ \begin{matrix}∀H ≤ G, |G : H| = 2, ∑_{x∉H} γ̃_{x} = 0 , \\ ∀x, y ∈ G − \{0\}, x ≠ y, [γ̃_{x} + γ̃_{y}, γ̃_{x+y}] = 0 . \end{matrix}

Then there is a unique surjective homomorphism s : Ẽ → E of graded kalgebras such that s(γ̃_{x}) = γ_{x}, for all x ∈ G − {0}. Thus showing that s is an isomorphism is equivalent to showing that for any integer n ∈ ℕ, the restriction s_{n} of s to the subspace Ẽ_{n} of elements of degree n in Ẽ is an isomorphism onto the corresponding subspace E_{n} of E. Since s_{n} is surjective, this amounts to showing that dim_{k} Ẽ_{n} ≤ dim_{k} E_{n}.

13.5. Lemma : Let B be an 𝔽_{2}-basis of G. Then the set

{γ̃_{x} | x ∈ G − (B ⊔ {0})}

is a k-basis of Ẽ_{2}, and s_{2} is an isomorphism.

Proof : By Corollary 10.10

dim_{k} E_{2} = (2^{1} − 1) + (2^{2} − 1) + ⋯ + (2^{m−1} − 1) = 2^{m} − m − 1 .

Thus dim_{k} Ẽ_{2} ≥ 2^{m} − m − 1, since s_{2} is surjective. As 2^{m} − m − 1 is precisely equal to the cardinality of G − (B ⊔ {0}), it is enough to show that the subspace Ẽ′_{2} of Ẽ_{2} generated by the elements γ̃_{x}, for x ∈ G − (B ⊔ {0}), is equal to Ẽ_{2}.

Let b ∈ B, and denote by H the subgroup of G generated by B − {b}. Then |G : H| = 2, so

∑_{x∉H} γ̃_{x} = 0 .

Since G−H = H +b, the only element of G−H which is also in B is b itself. This gives

γ̃_{b} = ∑_{y∈H−{0}} γ̃_{b+y} ,

so γ̃_{b} ∈ Ẽ′_{2}. Since this holds for any b ∈ B, and since Ẽ_{2} is the set of k-linear combinations of elements γ̃_{x}, for x ∈ G − {0}, it follows that Ẽ′_{2} = Ẽ_{2}, as was to be shown.

13.6. Notation : Fix a linear ordering b_{1} < b_{2} < ⋯ < b_{m} on B. If x ∈ G − {0}, let d_{B}(x) denote the least integer i ∈ {1, …, m} such that

x ∈ <b_{1}, …, b_{i}>.

Then d_{B}(x) = i if and only if x = y + b_{i}, for some y ∈ <b_{1}, …, b_{i−1}>.

13.7. Definition : Let n ∈ ℕ. If M = γ̃_{x_{1}} γ̃_{x_{2}} … γ̃_{x_{n}} ∈ Ẽ_{2n}, where

x_{i} ∈ G − {0} for 1 ≤ i ≤ n, set w(M) = ∑_{i=1}^{n} d_{B}(x_{i}). The monomial M

is called special if x_{i} ∈ G − (B ⊔ {0}), for 1 ≤ i ≤ n, and ordered if

d_{B}(x_{i}) ≤ d_{B}(x_{i+1}), for 1 ≤ i < n.

13.8. Lemma : Let x_{1},…,x_{n} be elements of G − (B ⊔ {0}). Then for any i ∈ {1, …, n − 1} such that d_{B}(x_{i}) ≠ d_{B}(x_{i+1}), the sum

γ̃_{x_{1}} … γ̃_{x_{i−1}} γ̃_{x_{i}} γ̃_{x_{i+1}} γ̃_{x_{i+2}} … γ̃_{x_{n}} + γ̃_{x_{1}} … γ̃_{x_{i−1}} γ̃_{x_{i+1}} γ̃_{x_{i}} γ̃_{x_{i+2}} … γ̃_{x_{n}}

is a linear combination of special monomials M′ with w(M′) > w(M).

Proof : Since the assertion is symmetric in x_{i} and x_{i+1}, one can assume that r = d_{B}(x_{i}) > d_{B}(x_{i+1}) = s. In this case x_{i} = u + b_{r}, where u ∈ <b_{1}, …, b_{r−1}> − {0}, and x_{i+1} = v + b_{s}, where v ∈ <b_{1}, …, b_{s−1}> − {0}. Let t = x_{i} + x_{i+1} = (v + b_{s} + u) + b_{r}, so d_{B}(t) = r. Now the relation [γ̃_{x_{i+1}} + γ̃_{t}, γ̃_{x_{i}}] = 0 gives

γ̃_{x_{i}} γ̃_{x_{i+1}} + γ̃_{x_{i+1}} γ̃_{x_{i}} = γ̃_{t}γ̃_{x_{i}} + γ̃_{x_{i}} γ̃_{t} ,

so the sum in the lemma is equal to

(13.9) S = γ̃_{x_{1}} … γ̃_{x_{i−1}} γ̃_{t}γ̃_{x_{i}} γ̃_{x_{i+2}} … γ̃_{x_{n}} + γ̃_{x_{1}} … γ̃_{x_{i−1}} γ̃_{x_{i}} γ̃_{t}γ̃_{x_{i+2}} … γ̃_{x_{n}} .

If t ∉ B, this is a sum of two special monomials M′ with w(M′) > w(M), for d_{B}(t) = d_{B}(x_{i}) > d_{B}(x_{i+1}). And if t ∈ B, then t = b_{r}, so

(13.10)  γ̃_{t} = ∑_{u∈H−{0}} γ̃_{b_{r}+u} ,

where H is the subgroup of index 2 of G generated by B − {b_{r}}. But d_{B}(b_{r} + u) ≥ r, for any u ∈ H. Replacing γ̃_{t} in 13.9 by the right hand side of 13.10 gives an expression of S as a sum of special monomials M′ with w(M′) > w(M).

13.11. Proposition : The set of special ordered monomials is a k-basis of Ẽ, and the map s is an isomorphism.

Proof : This is equivalent to saying that for any n ∈ ℕ, the set of special ordered monomials of degree 2n is a k-basis of Ẽ_{2n}, and that the map s_{2n} is an isomorphism.

The first step consists in showing that the special ordered monomials of degree 2n generate Ẽ_{2n}. In other words, using (13.10), any special monomial

M = γ̃_{x_{1}} γ̃_{x_{2}} … γ̃_{x_{n}}

should be equal to a linear combination of special ordered monomials of degree 2n.

For such an arbitrary monomial w(M) ≤ nm, and this allows for a proof by induction on j = nm − w(M) : if j = 0, then d_{B}(x_{i}) = m, for 1 ≤ i ≤ n, and the monomial M is a special ordered monomial, so there is nothing to prove.

Otherwise, if M is not ordered, there is a least integer i ∈ {1, …, n − 1} such that d_{B}(x_{i}) > d_{B}(x_{i+1}). By Lemma 13.8, the monomial M is equal to the monomial obtained by exchanging γ̃_{x_{i}} and γ̃_{x_{i+1}}, up to a linear combination of monomials M′ with w(M′) > w(M), which are equal to a linear combination of special ordered monomials, by induction hypothesis.

By repeated application of this procedure, the term γ̃_{x_{i+1}} can be moved to the left, until it sits between γ̃_{x_{j}} and γ̃_{x_{j+1}} such that

d_{B}(x_{j}) ≤ d_{B}(x_{i+1}) < d_{B}(x_{j+1}) ,

(possibly j = 0, in which case γ̃_{x_{i+1}} is moved to the first place on the left), and the monomial M is equal to the monomial

γ̃_{x_{1}} γ̃_{x_{2}} … γ̃_{x_{j}} γ̃_{x_{i+1}} γ̃_{x_{j+1}} … γ̃_{x_{i−1}} γ̃_{x_{i}} γ̃_{x_{i+2}} … γ̃_{x_{n}}

up to a linear combination of special ordered monomials. In this monomial, the i + 1 first values

d_{B}(x_{1}), d_{B}(x_{2}), …, d_{B}(x_{j}), d_{B}(x_{i+1}), d_{B}(x_{j+1}) …, d_{B}(x_{i−1}), d_{B}(x_{i})

are linearly ordered. By induction on n − i, this monomial is is equal to a linear combination of special ordered monomials. This shows that the special ordered monomials of degree 2n generate Ẽ_{2n}.

The second step consists in counting the special ordered monomials : such a monomial is a product

\underbrace{γ̃_{x_{1}} … γ̃_{x_{j_{1}}}}_{d_{B}(x_{i})=1} \underbrace{γ̃_{x_{j_{1}+1}} … γ̃_{x_{j_{1}+j_{2}}}}_{d_{B}(x_{i})=2} … \underbrace{γ̃_{x_{j_{1}+···+j_{m−1}+1}} … γ̃_{x_{j_{1}+···+jm}}}_{d_{B}(x_{i})=m}

of j_{1} elements γ_{x} with x ∈ G − (B ⊔ {0}) and d_{B}(x) = 1, followed by j_{2} elements γ_{x} with x ∈ G − (B ⊔ {0}) and d_{B}(x) = 2, and so on, up to a product of j_{m} elements γ_{x} with x ∈ G − (B ⊔ {0}) and d_{B}(x) = m, where j_{1} + j_{2} + ⋯ + j_{m} = n. An element x of G − (B ⊔ {0}) such that d_{B}(x) = i is an element of the form y + b_{i}, where y is a non zero element of <b_{1}, …, b_{i−1}>−{0}. It follows that the number of special ordered monomials of degree 2n is equal to

∑_{j_{1}+j_{2}+···+j_{m}=n} ∏_{i=1}^{m}(2^{i−1} − 1)^{j_{i}} .

The only element x in G − {0} with d_{B}(x) = 1 is b_{1} ∈ B, and this forces j_{1} = 0. So the number of ordered monomials of degree 2n is equal to

∑_{l_{1}+l_{2}+···+l_{m−1}=n} ∏_{i=1}^{m−1}(2^{i} − 1)^{l_{i}} ,

where l_{i} = j_{i+1}, for 1 ≤ i ≤ m−1. But this is precisely equal to the coefficient of the term of degree 2n in the Poincaré series for E, by Corollary 10.10. It follows that dim_{k} Ẽ_{2n} ≤ dim_{k} E_{2n}, hence dim_{k} Ẽ_{2n} = dim_{k} E_{2n}, and the map s_{2n} is an isomorphism. This completes the proof of Corollary 13.11, and also the proof of Theorem 1.3.

14. Partial results for G ≅ (C_{p})^{m}, p > 2

Let G ≅ (C_{p})^{m} be an odd order elementary abelian p-group. The main difference with the case p = 2 is that Theorem 10.7 no longer holds, as can be seen from Theorem 6.2 : if φ : G → k^{+} is a group homomorphism, let E_{φ}^{G} denote the kG-module k ⊕ k, where the G-action is defined by

∀g ∈ G, ∀(x, y) ∈ k^{2}, g(x, y) = (x + yφ(g), y) ,

and let T_{φ}^{G} denote the unique Mackey functor for G over k such that T_{φ}(H) is equal to zero if 1 ≠ H ≤ G, and such that T_{φ}(1) ≅ E_{φ}. Then T_{φ}^{G} is a cohomological Mackey functor, and there is a non split exact sequence

0 ⟶ S_{1}^{G} ⟶ T_{φ}^{G} ⟶ S_{1}^{G} ⟶ 0 ,

whose class is an element τ_{φ}^{G} ∈ Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}).

If X is a subgroup of order p of G, recall that there is a 2-fold extension γ_{X}^{G} ∈ Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}) represented by the short exact sequence

0 ⟶ S_{1}^{G} ⟶ \binom{S_{X}^{G}}{S_{1}^{G}} ⟶ \binom{S_{1}^{G}}{S_{X}^{G}} ⟶ S_{1}^{G} ⟶ 0 .

In this case, I propose the following conjecture :

14.1. Conjecture : Let k be a field of odd characteristic p, and G ≅ (C_{p})^{m}. Then :

1. The algebra E = Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) is generated by the elements τ_{φ}^{G} in degree 1, for φ ∈ Hom_{ℤ}(G, k^{+}), and by the elements γ_{X}^{G} in degree 2,

for X ≤ G with |X| = p.

2. The Poincaré series for E is equal to

\frac{1}{(1 − t)(1 − t − (p−1)t^{2})(1 − t − (p^{2}−1)t^{2}) … (1 − t − (p^{m−1}−1)t^{2})} .

14.2. Theorem : Conjecture 14.1 is true for p = 3.

Proof : If G = 1, i.e. if m = 0, there is nothing to prove. By induction on m, one can assume that the result holds for any elementary abelian pgroup of rank smaller than m. Let H be a subgroup of index p of G, let T be a complement of H in G, and set X = K_{G}(H) − {T }. By Corollary 10.3, there is a long exact sequence in M_{k}^{c}(G) of the form (14.3)

[figure: ⋯; L(n − 1) ⊕ ⊕_{X∈X} E_{G}(n − 2); E_{G}(n); r_{n}; E_{H}(n); L(n) ⊕ ⊕_{X∈X} E_{G}(n − 1); E_{G}(n + 1); r_{n+1}; E_{H}(n + 1); ⋯,]

where E_{G}(n) = Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}), E_{H}(n) = Ext_{M_{k}^{c}(H)}^{n}(S_{1}^{H}, S_{1}^{H}), L(n) = Ext_{M_{k}^{c}(G)}^{n}(L, S_{1}^{G}), and X = K_{G}(H) − {T }. Recall from Proposition 8.7 that L is a functor all of whose composition factors are isomorphic to S_{1}^{G}, with multiplicity p − 2. Thus if p = 3, the functor L is isomorphic to S_{1}^{G}, and it follows that L(n) ≅ E_{G}(n), for any n ∈ ℕ.

It is easy to see that the map L(n−1) → E_{G}(n) in Sequence 14.3 consists in taking Yoneda product with the sequence

(14.4)  0 ⟶ L ⟶ M ⟶ S_{1}^{G} ⟶ 0

obtained by taking the image of the sequence

0 ⟶ R/J ⟶ I/J ⟶ S_{1}^{G} ⟶ 0

of Proposition 10.1 under the split surjection R/J → L.

On the other hand, the map E_{G}(n − 2) → E_{G}(n) from the component indexed by X ∈ X is equal to the Yoneda product by γ_{X}^{G}, by the argument given in the proof Theorem 10.9.

The map r_{n} in Sequence 14.3 is induced by restriction from G to H, also by the argument used in the proof of Theorem 10.9. In particular, it is compatible with the Yoneda product. By induction hypothesis, any element in E_{H}(n) is a linear combination of Yoneda products of elements τ_{ψ}^{H} and elements γ_{Y}^{H}, where ψ ∈ Hom_{ℤ}(H, k^{+}) and Y ≤ H with |Y | = p.

To show that r_{n} is surjective, it suffices to show that for any such ψ, there exists φ ∈ Hom_{ℤ}(G, k^{+}) such that τ_{ψ}^{H} = r_{1}(τ_{φ}^{G}), and that for any such Y, there exists X ≤ G with |X| = p, such that γ_{Y}^{H} = r_{2}(γ_{X}^{G}). For the latter, the argument of the proof of Corollary 10.10 applies, and one can take X = Y. Now if ψ ∈ Hom_{ℤ}(H, k^{+}), there exists φ ∈ Hom_{ℤ}(G, k^{+}) whose restriction to H is equal to ψ, and it is straightforward to check that the restriction to H of the extension T_{φ}^{G} defining τ_{φ}^{G} is isomorphic to T_{ψ}^{H}. It follows that r_{1}(τ_{φ}^{G}) = τ_{ψ}^{H}, so r_{n} is surjective, for any n ∈ ℕ. Actually this proves more : the submodule E′_{G}(n) of E_{G}(n) generated by products of elements τ_{φ}^{G} and γ_{X}^{G} maps surjectively by r_{n} on E_{H}(n). Thus E′_{G}(n) + Ker r_{n} = E_{G}(n).

Finally, in the case p = 3, the long exact sequence 14.3 splits as a series of short ones

(14.5) 0 ⟶ E_{G}(n − 1) ⊕ ⊕_{X∈X} E_{G}(n − 2) ⟶ E_{G}(n) ⟶^{r_{n}} E_{H}(n) ⟶ 0 .

The image of each component E_{G}(n − 2) in E_{G}(n) is obtained by taking Yoneda product with some γ_{X}^{G}, and the image of E_{G}(n − 1) is obtained by taking the Yoneda product with the sequence 14.4, which for p = 3, is of the form

0 ⟶ S_{1}^{G} ⟶ M ⟶ S_{1}^{G} ⟶ 0 .

It follows easily that it is isomorphic to a sequence T_{φ}, where φ ∈ Hom_{ℤ}(G, k^{+}) has kernel H. So the image of E_{G}(n − 1) in E_{G}(n) is obtained by taking Yoneda product with τ_{φ}^{G}.

An easy induction argument now shows that the kernel of r_{n} is contained in E′_{G}(n), so E′_{G}(n) = E_{G}(n), completing the inductive step for Assertion 1 of Conjecture 14.1.

Assertion 2 now follows from Sequence 14.5 : this sequence shows that if P_{m}(t) denotes the Poincaré series for the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}), where G ≅ (C_{p})^{m}, then

P_{m}(t) = tP_{m}(t) + |X |t^{2}P_{m}(t) + P_{m−1}(t) .

Thus

P_{m}(t) = \frac{P_{m−1}(t)}{1 − t − (p^{m−1} − 1)t^{2}} ,

and Assertion 2 follows easily by induction.

14.6. Remark : The main reason for proposing Conjecture 14.1, apart from Theorem 14.2, is a computer calculation for p = 5 or p = 7, using GAP software (http://www.gap-system.org), showing that the terms of lower degree of the Poincaré series for the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) (up to degree 5 for p = 5 and m = 2) are as predicted by Assertion 2 of Conjecture 14.1.

15. More on extension of simple functors

15.1. Proposition : Let k be a field of characteristic p, and G be a finite p-group. Let Q and R be normal subgroups of G, and set N = [Q, R]. Then for each n ∈ ℕ, the functor ρ_{G/N}^{G} induces an isomorphism

Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) ⟶^{≅} Ext_{M_{k}^{c}(G/N)}^{n}(S_{Q/N}^{G/N}, S_{R/N}^{G/N}) .

Proof : By induction on the order of N : if N is trivial, there is nothing to show. Otherwise, the subgroup N is a non-trivial normal subgroup of G, contained in Q ∩ R, so N contains a central subgroup Z of order p of G. If X is a complement of Z in Q, then for any n ∈ ℕ

Ext_{M_{k}^{c}(G)}^{n}(S_{X}^{G}, S_{R}^{G}) ≅ Ext_{M_{k}^{c}(G)}^{n}(Ind_{N_{G}(X)}^{G}S_{X}^{N_{G}(X)}, S_{R}^{G})
≅ Ext_{M_{k}^{c}(N_{G}(X))}^{n}(S_{X}^{N_{G}(X)}, Res_{N_{G}(X)}^{G}S_{R}^{G}) ,

and Res_{N_{G}(X)}^{G}S_{R}^{G} = {0} if R ≰ N_{G}(X), by Lemma 6.7. But if R ≤ N_{G}(X), then [R, X] ≤ X, hence [R, Q] ≤ X, for Q = X · Z and Z ≤ Z(G). It follows that Z ≤ X, and this contradiction shows that Ext_{M_{k}^{c}(G)}^{n}(S_{X}^{G}, S_{R}^{G}) = {0} for any X ∈ K_{Q}(Z). Then by Theorem 8.6, for any n ∈ ℕ, the map

π_{n} : Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G})⟶Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R})

induced by ρ_{G/Z}^{G} is an isomorphism. Now \overline{Q} and \overline{R} are normal subgroups of \overline{G}, and [\overline{Q}, \overline{R}] = [Q, R]/Z has order smaller that |N|. By induction, the functor ρ_{G/N}^{G/Z} induces an isomorphism

Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) ⟶ Ext_{M_{k}^{c}(G/N)}^{n}(S_{Q/N}^{G/N}, S_{R/N}^{G/N}) ,

and the result follows by composition with π_{n}, since ρ_{G/N}^{G/Z} ∘ ρ_{G/Z}^{G} ≅ ρ_{G/N}^{G}.

The following result shows that for a p-group G and a field k of characteristic p, the computation of extension groups Ext_{M_{k}^{c}(G)}^{j}(S_{Q}^{G}, S_{R}^{G}) comes down to the case where Q and R are elementary abelian normal subgroups of G, which centralize each other. First a notation :

15.2. Notation : Let G be a finite p-group. If Q and R are subgroups

{of G, set \atop S_{R,Q}^{G} = \{g ∈ G | Φ(R) ·^{g}Φ(Q) · [R,^{g}Q] ≤ R ∩^{g}Q\} .}

Denote by \overline{S}_{R,Q}^{G} the set of (N_{G}(R), N_{G}(Q))-double cosets in S_{R,Q}^{G}, and by [\overline{S}_{R,Q}^{G}] a set of representatives of \overline{S}_{R,Q}^{G}.

Let \overline{ℕ} = ℕ ⊔ \{∞\} be the linearly ordered set obtained by adding to ℕ a largest element ∞. Denote by ν_{G}(R, Q) ∈ \overline{ℕ} the element defined by

p^{ν_{G}(R,Q)} = min_{g∈S_{R,Q}^{G}} |R·^{g}Q : R ∩^{g}Q| = min_{g∈[\overline{S}_{R,Q}^{G}]} |R·^{g}Q : R ∩^{g}Q| ,

if S_{R,Q}^{G} ≠ ∅, and by ν_{G}(R, Q) = ∞ otherwise.

15.3. Theorem : Let k be a field of characteristic p, let G be a finite p-group, and let Q and R be subgroups of G. Then for g ∈ S_{R,Q}^{G} :

1. the groups R and ^{g}Q normalize each other.

2. the group Φ(R) ·^{g}Φ(Q) · [R,^{g}Q] is equal to Φ(R·^{g}Q).

3. the groups R̂ = R/Φ(R·^{g}Q) and ^{ ĝ}Q = ^{g}Q/Φ(R·^{g}Q) are elementary abelian normal subgroups of N̂_{G}(R,^{g}Q) = N_{G}(R,^{g}Q)/Φ(R·^{g}Q), which centralize each other.

Moreover for any j ∈ ℕ,

Ext_{M_{k}^{c}(G)}^{j}(S_{Q}, S_{R}) ≅ ⊕_{g∈[\overline{S}_{R,Q}^{G}]} Ext_{M_{k}^{c}(N̂_{G}(R,^{g}Q))}^{j}(SN̂_{G}(R,^{g}Q) ; ^{ĝ}Q, SN̂_{G}(R,^{g}Q) ; R̂) ,

Proof : If g ∈ S_{R,Q}^{G}, then [R,^{g}Q] ≤ R ∩ ^{g}Q, so R and ^{g}Q normalize each other. Let T = Φ(R) ·^{g}Φ(Q) · [R,^{g}Q]. Then the groups R/T and ^{g}Q/T are elementary abelian subgroups of N_{G}(R,^{g}Q)/T, which centralize each other. Thus (R·^{g}Q)/T is elementary abelian, and T ≥ Φ(R·^{g}Q).

Conversely Φ(R) ≤ Φ(R·^{g}Q), since R·^{g}Q is a p-group, and similarly Φ(^{g}Q) = ^{g}Φ(Q) ≤ Φ(R·^{g}Q). Moreover [R,^{g}Q] ≤ [R·^{g}Q, R·^{g}Q] ≤ Φ(R·^{g}Q), so finally T ≤ Φ(R·^{g}Q), hence T = Φ(R·^{g}Q). Now R̂ = R/T and ^{ ĝ}Q = ^{g}Q/T are elementary abelian normal subgroups of N̂_{G}(R,^{g}Q), which centralize each other.

Set A = Φ(Q) and B = N_{G}(Q). Set similarly C = Φ(R) and D = N_{G}(R). Then S_{Q}^{G} ≅ Ind_{B}^{G}ı_{B/A}^{B}(S_{Q/A}^{B/A}) and S_{R}^{G} = Ind_{D}^{G}ȷ_{D/C}^{D}(S_{R/C}^{D/C}), by Lemma 6.5 and Corollary 6.6. Thus, by Proposition 3.16, for any j ∈ ℕ,

Ext_{M_{k}^{c}(G)}^{j}(S_{Q}, S_{R}) ≅ Ext_{M_{k}^{c}(D/C)}^{j}(ρ_{D/C}^{D}Res_{D}^{G}Ind_{B}^{G}ı_{B/A}^{B}(S_{Q/A}^{B/A}), S_{R/C}^{D/C}) .

Now by Proposition 3.27,

ρ_{D/C}^{D}Res_{D}^{G}Ind_{B}^{G}ı_{B/A}^{B}(S_{Q/A}^{B/A})≅ ⊕_{g∈[D\G/B]} Ind\overline{D} ; \overline{D}_{g}ı_{\overline{D}_{g}/\overline{C}_{g}}^{\overline{D}_{g}}Iso(f_{g})ρ_{\overline{B}_{g}/\overline{A}_{g}}^{\overline{B}_{g}}Res\overline{B} ; \overline{B}_{g}(S_{Q/A}^{B/A}) ,

where

\overline{D} = D/C, \overline{D}_{g} = (D ∩^{g}B)C/C, \overline{C}_{g} = (D ∩^{g}A)C/C ,
\overline{B} = B/A, \overline{B}_{g} = (D^{g} ∩ B)A/A, \overline{A}_{g} = (C^{g} ∩ B)A/A ,

and where f_{g} : \overline{B}_{g}/\overline{A}_{g} ⟶ \overline{D}_{g}/\overline{C}_{g} is the group isomorphism sending x\overline{A}_{g} to gx\overline{C}_{g}, for x ∈ D^{g} ∩ B.

By Lemma 6.7, the functor ρ_{\overline{B}_{g}/\overline{A}_{g}}^{\overline{B}_{g}}Res\overline{B} ; \overline{B}_{g}(S_{Q/A}^{B/A}) is isomorphic to the direct

sum of the simple functors S_{X/\overline{A}_{g}}^{\overline{B}_{g}/\overline{A}_{g}} corresponding to subgroup X such that X/A is conjugate to \overline{Q} = Q/A in B/A, up to conjugation by \overline{B}_{g}/\overline{A}_{g}. Since Q ⊴ B, the only possible such subgroup is Q itself, if \overline{A}_{g} ≤ \overline{Q} ≤ \overline{B}_{g}. This gives

Ext_{M_{k}^{c}(G)}^{j}(S_{Q}, S_{R}) ≅ ⨁_{{g∈[D\G/B] \atop \overline{A}_{g}≤\overline{Q}≤\overline{B}_{g}}} Ext_{M_{k}^{c}(\overline{D})}^{j}(Ind\overline{D} ; \overline{D}_{g}ı_{\overline{D}_{g}/\overline{C}_{g}}^{\overline{D}_{g}}Iso(f_{g})(S_{\overline{Q}/\overline{A}_{g}}^{\overline{B}_{g}/\overline{A}_{g}}), S_{R/C}^{D/C}) .

By adjunction, this gives

Ext_{M_{k}^{c}(G)}^{j}(S_{Q}, S_{R})≅ ⨁_{{g∈[D\G/B] \atop \overline{A}_{g}≤\overline{Q}≤\overline{B}_{g}}} Ext_{M_{k}^{c}(\overline{D})}^{j}(Iso(f_{g})(S_{\overline{Q}/\overline{A}_{g}}^{\overline{B}_{g}/\overline{A}_{g}}), ρ_{\overline{D}_{g}/\overline{C}_{g}}^{\overline{D}_{g}}Res\overline{D} ; \overline{D}_{g}(S\overline{D} ; \overline{R})) ,

where \overline{R} = R/C. By the same argument, the functor ρ_{\overline{D}_{g}/\overline{C}_{g}}^{\overline{D}_{g}}Res\overline{D} ; \overline{D}_{g}(S\overline{D} ; \overline{R}) is

equal to zero, unless \overline{C}_{g} ≤ \overline{R} ≤ \overline{D}_{g}, in which case it is isomorphic to S_{\overline{R}/\overline{C}_{g}}^{\overline{D}_{g}/\overline{C}_{g}}. Thus

Ext_{M_{k}^{c}(G)}^{j}(S_{Q}, S_{R})≅ ⨁_{g∈[D\G/B] ; {\overline{A}_{g}≤\overline{Q}≤\overline{B}_{g} \atop \overline{C}_{g}≤\overline{R}≤\overline{D}_{g}}} Ext_{M_{k}^{c}(\overline{D}_{g}/\overline{C}_{g})}^{j}(Iso(f_{g})(S_{\overline{Q}/\overline{A}_{g}}^{\overline{B}_{g}/\overline{A}_{g}}), S_{\overline{R}/\overline{C}_{g}}^{\overline{D}_{g}/\overline{C}_{g}}) .

Now the condition \overline{Q} ≤ \overline{B}_{g} is equivalent to Q ≤ (D^{g} ∩ B)A, i.e. to Q = A(D^{g} ∩ Q), i.e. to D^{g} ∩ Q = Q, since A = Φ(Q). In other words Q ≤ D^{g}. Similarly, the condition \overline{R} ≤ \overline{D}_{g} is equivalent to R ≤ (D ∩ ^{g}B)C, i.e. to R = (R ∩ ^{g}B)C, i.e. to R = R ∩ ^{g}B, since C = Φ(R). In other words R ≤ ^{g}B.

Then the condition \overline{A}_{g} ≤ \overline{Q} is equivalent to C^{g} ∩ B ≤ Q. Since moreover C^{g} ≤ R^{g} ≤ B, it follows that C^{g} ∩ B = C^{g} ≤ Q.

Similarly, the condition \overline{C}_{g} ≤ \overline{R} is equivalent to D ∩ ^{g}A ≤ R. But gA ≤ ^{g}Q ≤ D, thus D ∩^{g}A = ^{g}A ≤ R.

In this situation

\overline{D}_{g} = (D ∩^{g}B)/C, \overline{C}_{g} = ^{g}A·C/C, \overline{B}_{g} = (D^{g} ∩ B)/A, \overline{A}_{g} = C^{g}·A/A ,

and the isomorphism

f_{g} : \overline{B}_{g}/\overline{A}_{g} = (D^{g} ∩ B)/(C^{g}·A) ⟶ (D ∩^{g}B)/(C·^{g}A) = \overline{D}_{g}/\overline{C}_{g}

is induced by conjugation by g. In particular f_{g}(\overline{Q}/\overline{A}_{g}) = ^{g}Q/(C·^{g}A), thus finally

Ext_{M_{k}^{c}(G)}^{j}(S_{Q}, S_{R}) ≅ ⊕_{g∈S} Ext_{M_{k}^{c}((D∩^{g}B)/(C·^{g}A))}^{j}(S_{^{g}Q/(C·^{g}A)}^{(D∩^{g}B)/(C·^{g}A)}, S_{R/(C·^{g}A)}^{(D∩^{g}B)/(C·^{g}A)}) ,

where

S = {g ∈ [D\G/B] | C·^{g}A ≤ R ∩^{g}Q ≤ R·^{g}Q ≤ D ∩^{g}B}
= \{g ∈ [D\G/B] | C·^{g}A · [R,^{g}Q] ≤ R ∩^{g}Q\} = [\overline{S}_{R,Q}^{G}] .

Moreover, by Proposition 15.1, for g ∈ S, the group

Ext_{M_{k}^{c}((D∩^{g}B)/(C·^{g}A))}^{j}(S_{^{g}Q/(C·^{g}A)}^{(D∩^{g}B)/(C·^{g}A)}, S_{R/(C·^{g}A)}^{(D∩^{g}B)/(C·^{g}A)})

is isomorphic to

Ext_{M_{k}^{c}(N̂_{G}(R,^{g}Q))}^{j}(SN̂_{G}(R,^{g}Q) ; ^{ĝ}Q, SN̂_{G}(R,^{g}Q) ; R̂) ,

and this completes the proof.

15.4. Proposition : Let k be a field of characteristic p, let G be a finite p-group, and let Q and R be subgroups of G. Then

n < ν_{G}(R, Q) ⇒ Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) = {0} .

Proof : If ν_{G}(R, Q) = ∞, i.e. if S_{R,Q}^{G} = ∅, then Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) = {0} for any n ∈ ℕ, by Theorem 15.3. So one can assume ν_{G}(R, Q) ∈ ℕ.

The result is trivial if G = 1, so by induction on |G|, one can assume that it holds for any group of order less than |G|. The group G being given with this property, one can proceed by induction on n, to show that Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) = {0} if Q and R are subgroups of G with ν_{G}(R, Q) > n. This starts with the case n = −1, where there is nothing to prove, since ν_{G}(R, Q) ≥ 0 and Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) = {0} for n < 0.

For the inductive step, let n ≥ 0, let Q and R be subgroups of G such that ν_{G}(R, Q) > n, and let g ∈ S_{R,Q}^{G}. Then the index |R·^{g}Q : R ∩ ^{g}Q| is equal to the index |R′·Q′ : R′ ∩ Q′|, where R′ = R/(Φ(R)·^{g}Φ(Q)) and Q′ = ^{g}Q/(Φ(R)·^{g}Φ(Q)). By Theorem 15.3, and by induction hypothesis on G, it suffices to consider the case where Q and R are elementary abelian normal subgroups of G, which centralize each other.

In this case, set j = ν_{G}(R, Q), so p^{j} = |R·Q : R ∩ Q|. If Q ∩ R ≠ 1, let Z be a subgroup of order p of Q ∩ R ∩ Z(G). By Theorem 8.6, for any integer n, there is a short exact sequence

0 → ⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}) → Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) → Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) → 0 ,

where K is a set of representatives of G-conjugacy classes of complements of Z in Q, and \overline{Q} and \overline{R} denote Q/Z and R/Z, respectively.

If X ∈ K, since R ⊴ G, the set N_{G}(X)\G/N_{G}(R) has cardinality 1. Moreover X ∩ R ≠ Q ∩ R, since Z ≤ Q ∩ R and Z ≰ X, so |Q ∩ R : X ∩ R| = p, and

|X·R : X ∩ R| = \frac{|Q||R|}{p|X ∩ R|^{2}} = p^{j+1} .

In other words ν_{G}(R, X) = j + 1 > n − 1, so Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}) = {0} by induction hypothesis on n.

On the other hand Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) = 0 since n < j, by induction hypothesis on |G|. It follows that Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) = {0}.

It remains to consider the case where Q∩R = 1. Since ν_{G}(R, Q) > n ≥ 0, it follows that at least one of the groups Q or R is non trivial. By symmetry on Q and R, one can assume that Q ≠ 1, and choose a subgroup Z of order p in Q ∩ Z(G).

Now for any n ∈ ℕ

Ext_{M_{k}^{c}(G)}^{n}(ı_{G/Z}^{G}(S_{Q/Z}^{G/Z}), S_{R}^{Q}) ≅ Ext_{M_{k}^{c}(G/Z)}^{n}(S_{Q/Z}^{G/Z}, ρ_{G/Z}^{G}(S_{R}^{G})) = {0} , by Proposition 6.7, since Z ⊈ R. Applying Hom_{M_{k}^{c}(G)}(−, S_{R}^{G}) to the first exact sequence of Proposition 8.5 gives the isomorphisms

(15.5)  Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) ≅ ⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}) ,

for any n ∈ ℕ, where K is a set of representatives of G-conjugacy classes of complements of Z in Q.

Let X ∈ K, and set l = ν_{G}(R, X). Then

p^{l} = |X·R : X ∩ R| = |X·R| = |X||R| ,

so l = j − 1.

If n < j, then n − 1 < l, hence Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}) = {0} by induction hypothesis on n. It follows that Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) = {0}, as was to be shown.

15.6. Notation : Let Q be an elementary abelian p-group. Denote by St(Q) the only non-zero reduced integral homology group of the poset ]1, Q[ of proper non trivial subgroups of Q, ordered by inclusion.

It is well known (see e.g. [3]) that, if Q has rank r, then St(Q) =

H̃_{r−2}(]1, Q[, ℤ) is isomorphic to ℤ^{p^{\binom{r}{2}}}. The automorphism group A ≅ GL(r, 𝔽_{p}) of Q acts on St(Q), and the module St(Q) is isomorphic to the Steinberg module of A. The restriction of St(Q) to a Sylow p-subgroup P of A is a free ℤP -module of rank 1.

When G is an elementary abelian p-group, the following theorem was proved by Tambara ([8] Theorem 4.1) :

15.7. Theorem : Let k be a field of characteristic p, and G be a finite p-group. Let Q be an elementary abelian normal subgroup of rank q of G, and H be a subgroup of rank h of Q. Then ν_{G}(H, Q) = q − h and :

1. There is a group isomorphism

Ext_{M_{k}^{c}(G)}^{q−h}(S_{Q}^{G}, S_{H}^{G}) ≅ Ext_{M_{k}^{c}(N_{G}(H)/H)}^{q−h}(S_{Q/H}^{N_{G}(H)/H}, S_{H/H}^{N_{G}(H)/H}) .

2. These groups are isomorphic to the space of coinvariants kSt(Q/H)_{N_{G}(H)} of N_{G}(H) on k ⊗_{ℤ} St(Q/H).

3. Let H = H_{0} < H_{1} < … < H_{q−h} = Q be a maximal N_{G}(H)-invariant flag. Then Ext_{M_{k}^{c}(G)}^{q−h}(S_{Q}^{G}, S_{H}^{G}) has a k-basis indexed by the N_{G}(H)- conjugacy classes of flags X_{1} > X_{2} > … > X_{q−h−1} of Q such that X_{i} + H_{i} = Q and X_{i} ∩ H_{i} = H, for 1 ≤ i ≤ q − h − 1. In particular

dim_{k} Ext_{M_{k}^{c}(G)}^{q−h}(S_{Q}^{G}, S_{H}^{G}) = \frac{p^{\binom{q−h}{2}}}{|N_{G}(H) : Z_{G}(H, Q)|} ,

where Z_{G}(H, Q) = {g ∈ G | [g, Q] ≤ H} (in other words Z_{G}(H, Q) is

the preimage in N_{G}(H) of the centralizer of Q/H in N_{G}(H)/H).

Proof : Since H ≤ Q and Φ(Q) = 1, it follows that Φ(H)Φ(Q)[H, Q] = 1 ≤ H ∩ Q, so ν_{G}(H, Q) < ∞. Moreover p^{ν_{G}(H,Q)} = |Q·H : Q ∩ H| = |Q : H|, since Q ⊴ G, so ν_{G}(H, Q) = q − h.

For Assertion 1, observe first that

Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{H}^{G}) ≅ Ext_{M_{k}^{c}(N_{G}(H))}^{n}(S_{Q}^{N_{G}(H)}, S_{H}^{N_{G}(H)}) ,

for any n ∈ ℕ, and ν_{N_{G}(H)}(H, Q) = q − h since Q ⊴ N_{G}(H). By induction on |G|, if N_{G}(H) < G, there is an isomorphism

Ext_{M_{k}^{c}(N_{G}(H))}^{n}(S_{Q}^{N_{G}(H)}, S_{H}^{N_{G}(H)}) ≅ Ext_{M_{k}^{c}(N_{G}(H)/H)}^{n}(S_{Q/H}^{N_{G}(H)/H}, S_{H/H}^{N_{G}(H)/H}) ,

so one can suppose H ⊴ G.

If H = 1, there is nothing to prove. Otherwise, let Z be a subgroup of order p of H ∩ Z(G). By Theorem 8.6, there is an exact sequence

0 → ⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{q−h−1}(S_{X}^{G}, S_{H}^{G}) → Ext_{M_{k}^{c}(G)}^{q−h}(S_{Q}^{G}, S_{H}^{G}) →^{π_{q−h}} Ext_{M_{k}^{c}(\overline{G})}^{q−h}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{H}) → 0 ,

in M_{k}^{c}(G), where K is a set of representatives of G-conjugacy classes of complements of Z in Q, and overlines denote quotients by Z. If X ∈ K, then

p^{ν_{G}(H,X)} = |X·H : X ∩ H| = \frac{|X||H|}{|X ∩ H|^{2}} = \frac{|Q||H|}{p|X ∩ H|^{2}} = p^{q−h−1}|H : X ∩ H|^{2} .

Since Z ≤ H and Z ∩ X = 1, it follows that H ≰ X, thus |H : X ∩ H| = p, and ν_{G}(H, X) = q − h + 1. By Proposition 15.4, the group Ext_{M_{k}^{c}(G)}^{q−h−1}(S_{X}^{G}, S_{R}^{G}) is equal to zero. Hence the map π_{q−h} is an isomorphism. By induction on the the order of G, there is an isomorphism

Ext_{M_{k}^{c}(\overline{G})}^{q−h}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{H}) ≅ Ext_{M_{k}^{c}(G/H)}^{q−h}(S_{Q/H}^{G/H}, S_{H/H}^{G/H}) ,

which completes the proof of Assertion 1.

It follows that for Assertions 2 and 3, it suffices to consider the case H = 1, i.e. h = 0. In this case H_{i} ⊴ G, for 0 ≤ i ≤ q. If q = 0, i.e. if Q = 1, there is nothing to prove, since Ext_{M_{k}^{c}(G)}^{0}(S_{1}^{G}, S_{1}^{G}) ≅ k. This allows to start a proof by induction on q.

If q > 0, by Proposition 8.5, there is an exact sequence

0 ⟶ ⊕_{X_{1}∈K} S_{X_{1}}^{G} ⟶ ι_{G/H_{1}}^{G}(S_{Q/H_{1}}^{G/H_{1}}) ⟶ S_{Q}^{G} ⟶ 0 ,

where K is a set of representatives of G-conjugacy classes of complements of H_{1} in Q. Since ρ_{G/H_{1}}^{G}(S_{1}^{G}) = {0}, this gives an isomorphism

Ext_{M_{k}^{c}(G)}^{q}(S_{Q}^{G}, S_{1}^{G}) ≅ ⊕_{X_{1}∈K} Ext_{M_{k}^{c}(G)}^{q−1}(S_{X_{1}}^{G}, S_{1}^{G}) .

Moreover, for X_{1} ∈ K,

Ext_{M_{k}^{c}(G)}^{q−1}(S_{X_{1}}^{G}, S_{1}^{G}) ≅ Ext_{M_{k}^{c}(N_{G}(X_{1}))}^{q−1}(S_{X_{1}}^{N_{G}(X_{1})}, S_{1}^{N_{G}(X_{1})}) .

The flag 1 < H_{2} ∩ X_{1} < H_{3} ∩ X_{1} < … < H_{q−1} ∩ X_{1} < X_{1} is a maximal N_{G}(X_{1})-invariant flag in X_{1}. By induction hypothesis, the space

Ext_{M_{k}^{c}(N_{G}(X_{1}))}^{q−1}(S_{X_{1}}^{N_{G}(X_{1})}, S_{1}^{N_{G}(X_{1})})

has a k-basis indexed by N_{G}(X_{1})-conjugacy classes of flags

X_{2} > X_{3} > … > X_{q−1}

such that X_{i} ⊕ (H_{i} ∩ X_{1}) = X_{1}, for 2 ≤ i ≤ q − 1. Equivalently X_{2} < X_{1}, and X_{i} ⊕ H_{i} = Q, for 2 ≤ i ≤ q − 1.

Now X_{1} runs through a set of representatives of G-conjugacy classes of complements of H_{1} in Q, and X_{2} > … > X_{q−1} runs through a set of representatives of N_{G}(X_{1})-conjugacy classes of flags such that X_{i} ⊕ H_{i} = Q, for 2 ≤ i ≤ q − 1. Equivalently X_{1} > X_{2} > … > X_{q−1} runs through a set of conjugacy classes of flags of Q such that X_{i} ⊕ H_{i} = Q, for 1 ≤ i ≤ q − 1. This completes the inductive step, and the first part of Assertion 3 follows.

Now the stabilizer C in G of a flag X_{1} > X_{2} > … > X_{q−1} such that X_{i} ⊕ H_{i} = Q, for 1 ≤ i ≤ q − 1 stabilizes two opposite maximal flags of Q. Since G is a p-group, it follows that the image of C in the automorphism group of Q is trivial (since it consists of matrices which are both upper and lower triangular with 1’s on the diagonal). In other words C = C_{G}(Q).

There are p^{\binom{q}{2}} maximal flags opposite to a given maximal flag in Q, and the cardinality of each G-conjugacy class of such flags is |G : C_{G}(Q)|. This completes the proof of Assertion 3.

Assertion 2 follows from the fact that kSt(Q) has a k-basis indexed by maximal flags opposite to a given G-invariant maximal flag of Q, and this basis is permuted by G (see [3] for details). In other words kSt(Q) is a free k(G/C_{G}(Q))-module.

Together with Theorem 15.3, the following theorem allows to compute any extension group between simple functors for a p-group G, from the self extension groups for the simple functor S_{1}^{C}, where C runs through some subquotients of G :

15.8. Theorem : Let k be a field of characteristic p, let G be a finite pgroup, and let Q and R be elementary abelian normal subgroups of G which centralize each other. Let q denote the rank of Q and r the rank of R.

1. For any n ∈ ℕ, the group Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) is isomorphic to

( ⊕_{h∈ℕ ; {H≤Q∩R \atop rank(H)=h}} kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ⊗_{k} kSt(Q/H))_{G} ,

where Z_{H} = Z_{G}(H, Q·R)/H = {g ∈ G | [g, Q·R] ≤ H}/H, and the G-subscript denotes the space of coinvariants.

2. In particular, the dimension of the space Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) is equal to

∑_{h∈ℕ ; {H≤Q∩R, mod.G \atop rank(H)=h}} \frac{p^{\binom{q−h}{2}+\binom{r−h}{2}}}{|N_{G}(H) : Z_{G}(H, Q·R)|} dim_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) .

Proof : For Assertion 1, when Q and R are both trivial, there is nothing to prove. This allows for a proof by induction on |Q||R|. By symmetry on Q and R, one can assume that Q ≠ 1, and choose a subgroup Z of order p of Q ∩ Z(G). There are two cases :

• if Z ≰ R, then ρ_{G/Z}^{G}(S_{R}^{G}) = {0} by Lemma 6.7, and the exact sequence of Proposition 8.5 gives isomorphisms

Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) ≅ ⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}) ,

for any n ∈ ℕ, where K = [G\K_{Q}(Z)] is a set of representatives of G conjugacy classes of complements of Z in Q. Now for each X ∈ K

Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G}) ≅ Ext_{M_{k}^{c}(N_{G}(X))}^{n−1}(S_{X}^{N_{G}(X)}, S_{R}^{N_{G}(X)}) .

Since |X||R| = |Q||R|/p, by induction hypothesis, this group is isomorphic to

( ⊕_{h∈ℕ ; {H≤X∩R \atop rank(H)=h}} kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H}^{X})}^{n−1+2h−x−r}(S_{1}^{Z_{H}^{X}}, S_{1}^{Z_{H}^{X}}) ⊗_{k} kSt(X/H))_{N_{G}(X)} ,

where x = q − 1 is the rank of X, and

Z_{H}^{X} = {g ∈ N_{G}(X) | [g, X·R] ≤ H}/H .

But if g ∈ G, then [g, X·R] ≤ H if and only if [g, Q·R] ≤ H, since Q = X·Z and Z ≤ Z(G). And if [g, Q·R] ≤ H, then

[g, X] ≤ [g, Q] ≤ [g, Q·R] ≤ H ≤ X ,

so g ∈ N_{G}(X). It follows that

Z_{H}^{X} = {g ∈ G | [g, Q·R] ≤ H}/H = Z_{H} .

Finally, the group Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) is isomorphic to

( ⊕_{X∈K_{Q}(Z)} ⊕_{h∈ℕ ; {H≤X∩R \atop rank(H)=h}} kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ⊗_{k} kSt(X/H))_{G} .

Exchanging the order of summation, this gives

( ⊕_{h∈ℕ ; {H≤Q∩R \atop rank(H)=h}}kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ⊗_{k} ( ⊕_{{X∈K_{Q}(Z) \atop X≥H}} kSt(X/H)))_{G} .

Now Z∩H = 1 since Z∩R = 1, hence the conditions X ∈ K_{Q}(Z) and X ≥ H are equivalent to X·(ZH) = Q and X ∩ (ZH) = H, i.e. to H ≤ X ≤ Q and X/H ∈ K_{Q/H}(ZH/H). By classical combinatorial results,

⊕_{X/H∈K_{Q/H}(ZH/H)} kSt(X/H) ≅ St(Q/H) ,

thus Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) is isomorphic to

( ⊕_{h∈ℕ ; {H≤Q∩R \atop rank(H)=h}}kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ⊗_{k} kSt(Q/H))_{G} ,

as was to be shown.

• If Z ≤ R, then by Theorem 8.6, for any n ∈ ℕ, there is an exact sequence

0 → \underbrace{⊕_{X∈K} Ext_{M_{k}^{c}(G)}^{n−1}(S_{X}^{G}, S_{R}^{G})}_{Σ} → Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G})→Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) → 0 ,

in M_{k}^{c}(G), where K = [G\K_{Q}(Z)] as above, and overlines denote quotients by Z. It follows that Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) ≅ Σ ⊕ Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}).

The same argument as above shows that the space Σ is isomorphic to (15.9)

( ⊕_{h∈ℕ ; {H≤Q∩R, H∩Z=1 \atop rank(H)=h}}kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ⊗_{k} kSt(Q/H))_{G} .

On the other hand, by induction hypothesis, the space Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) is isomorphic to

( ⊕_{\overline{h}∈ℕ ; {\overline{H}≤\overline{Q}∩\overline{R} \atop rank(\overline{H})=\overline{h}}} kSt(\overline{R}/\overline{H}) ⊗_{k} Ext_{M_{k}^{c}(Z_{\overline{H}})}^{n+2\overline{h}−\overline{q}−\overline{r}}(S_{1}^{Z_{\overline{H}}}, S_{1}^{Z_{\overline{H}}}) ⊗_{k} kSt(\overline{Q}/\overline{H}))_{\overline{G}} ,

where \overline{q} = q − 1 and \overline{r} = r − 1. Summing over \overline{H} is equivalent to summing over H ≤ Q ∩ R with H ≥ Z. If H has rank h, then \overline{H} = H/Z has rank \overline{h} = h − 1, and n + 2\overline{h} − \overline{q} − \overline{r} = n + 2h − q − r. Moreover \overline{R}/\overline{H} ≅ R/H, and \overline{Q}/\overline{H} ≅ Q/H. Also

Z_{\overline{H}} = \{\overline{g} ∈ \overline{G} | [\overline{g}, \overline{Q}·\overline{R}] ≤ \overline{H}\}/\overline{H} ≅ \{g ∈ G | [g, Q·R] ≤ H\}/H = Z_{H} .

Thus Ext_{M_{k}^{c}(\overline{G})}^{n}(S\overline{G} ; \overline{Q}, S\overline{G} ; \overline{R}) is isomorphic to (15.10)

( ⊕_{h∈ℕ ; {Z≤H≤Q∩R \atop rank(H)=h}}kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ⊗_{k} kSt(Q/H))_{G} .

Now the direct sum of 15.9 and 15.10 is isomorphic to the expression in Theorem 15.8, as was to be shown for Assertion 1.

It follows that the dimension of the space Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) is equal to

∑_{h∈ℕ ; {H≤Q∩R, mod.G \atop rank(H)=h}} dim_{k} (kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ⊗_{k} kSt(G/H))_{N_{G}(H)} .

The image of the group N_{G}(H) in the automorphism group of R/H is isomorphic to N_{G}(H)/Z_{G}(H, R), and kSt(R/H) is a free kN_{G}(H)/Z_{G}(H, R)- module. Similarly the module kSt(Q/H) is a free kN_{G}(H)/Z_{G}(H, Q)-module. It follows that the module V = kSt(R/H)⊗_{k} kSt(Q/H) is a free kN_{G}(H)/Zmodule, where Z = Z_{G}(H, R) ∩ Z_{G}(H, Q) = Z_{G}(H, Q·R).

Moreover since Z_{H} = Z/H, the group Z acts trivially on

W = Ext_{M_{k}^{c}(Z_{H})}^{n+2h−q−r}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ,

so W is a kN_{G}(H)/Z-module. Hence V ⊗ W is a free kN_{G}(H)/Z-module,

and it follows that dim_{k}(V ⊗W)_{N_{G}(H)} = \frac{dim_{k}(V ⊗ W)}{|N_{G}(H)/Z|}, which completes the

proof of Assertion 2.

In the case where G is elementary abelian, the following has been proved by Tambara ([8] Theorem B) :

15.11. Corollary : Let k be a field of characteristic p, let G be a finite p-group, and let Q and R be subgroups of G, such that ν_{G}(R, Q) < ∞. Then Ext_{M_{k}^{c}(G)}^{ν_{G}(R,Q)}(S_{Q}^{G}, S_{R}^{G}) is isomorphic to

⊕_{g∈M_{Q,R}^{G}} (kSt(R/(R ∩^{g}Q)) ⊗_{k} kSt(^{g}Q/(R ∩^{g}Q)))_{N_{G}(R,^{g}Q)} ,

where M_{Q,R}^{G} = \{g ∈ [\overline{S}_{R,Q}^{G}] | |R·^{g}Q : R ∩^{g}Q| = p^{ν_{G}(R,Q)}\}.

In particular ν_{G}(R, Q) = min{n ∈ ℕ | Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) ≠ {0}}.

Proof : If n < ν_{G}(R, Q), then Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) = {0}, by Proposition 12.4.

Suppose that n = ν_{G}(R, Q), and let g ∈ [\overline{S}_{R,Q}^{G}] such that, with the notation of Theorem 15.3,

Ext_{M_{k}^{c}(N̂_{G}(R,^{g}Q))}^{n}(SN̂_{G}(R,^{g}Q) ; ^{ĝ}Q, SN̂_{G}(R,^{g}Q) ; R̂) ≠ 0 .

It follows that p^{n} ≥ p^{ν_{N̂_{G}(R,^{g}Q)}(R̂,^{ĝ}Q)} = |R·^{g}Q : R ∩ ^{g}Q| ≥ p^{ν_{G}(R,Q)}. Hence n = ν_{G}(R, Q) = |R·^{g}Q : R ∩^{g}Q|.

Moreover, in the expression of Ext_{M_{k}^{c}(N̂_{G}(R,^{g}Q))}^{n}(SN̂_{G}(R,^{g}Q) ; ^{ĝ}Q, SN̂_{G}(R,^{g}Q) ; R̂) given

by Theorem 15.8, the non zero terms correspond to subgroups H of R̂ ∩ ^{ ĝ}Q such that n + 2h − q − r ≥ 0, where h, q, and r are the ranks of H, ^{ ĝ}Q and R̂ respectively. Thus

|H|^{2} ≥ |^{ĝ}Q||R̂|/p^{n} = |^{ĝ}Q||R̂||/|R̂·^{ĝ}Q| = |R̂ ∩ ^{ ĝ}Q|^{2} .

It follows that H = R̂∩^{ ĝ}Q, that R̂/H ≅ R/R∩^{g}Q, that ^{ ĝ}Q/H ≅ ^{g}Q/R∩^{g}Q, and that n + 2h − q − r = 0, so Ext_{M_{k}^{c}(Z_{H})}^{0}(S_{1}^{Z_{H}}, S_{1}^{Z_{H}}) ≅ k. This completes the proof.

15.12. Remark : Let G be an abelian p-group, and Q, R be elementary abelian subgroups of G, of rank q and r, respectively. In this case, by Assertion 1 of Theorem 15.3, the group Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}) is isomorphic to

⊕_{h∈ℕ ; {H≤Q∩R \atop rank(H)=h}} kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(G/H)}^{n+2h−q−r}(S_{1}^{G/H}, S_{1}^{G/H}) ⊗_{k} kSt(Q/H) .

Fix H ≤ Q ∩ R, of rank h, and denote by i the inclusion map

kSt(R/H) ⊗_{k} Ext_{M_{k}^{c}(G/H)}^{n+2h−q−r}(S_{1}^{G/H}, S_{1}^{G/H}) ⊗_{k} kSt(Q/H) ↪ Ext_{M_{k}^{c}(G)}^{n}(S_{Q}^{G}, S_{R}^{G}).

It is easy to check that this map can be explicitly described as follows : if y ∈ kSt(R/H) ≅ Ext_{M_{k}^{c}(G)}^{r−h}(S_{H}^{G}, S_{R}^{G}), if e ∈ Ext_{M_{k}^{c}(G/H)}^{n+2h−q−r}(S_{1}^{G/H}, S_{1}^{G/H}), and if

x ∈ kSt(Q/H) ≅ Ext_{M_{k}^{c}(G)}^{q−h}(S_{Q}^{G}, S_{H}^{G}), then i(y ⊗ e ⊗ x) is equal to the Yoneda composition y ∘ Inf_{G/H}^{G}e ∘ x.

In terms of morphisms in the derived category of M_{k}^{c}(G), it means that any morphism from S_{Q}^{G} to some translate of S_{R}^{G} is a linear combination of morphisms which factor through a morphism of the form Inf_{G/H}^{G}e, where H ≤ Q ∩ R, and e is a morphism from S_{1}^{G/H} to some of its translate.

In the case p = 2 and G is elementary abelian, this leads to an explicit description of all the Yoneda products of extensions of simple cohomological Mackey functors.

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Serge Bouc - CNRS - LAMFA Université de Picardie Jules Verne 33, rue St Leu - 80039 - Amiens cedex 1 FRANCE

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