The extension algebra of some cohomological Mackey functors

Serge Bouc and Radu Stancu

Abstract : Let k be a field of characteristic p. We construct a new inflation functor for cohomological Mackey functors for finite groups over k. Using this inflation functor, we give an explicit presentation of the graded algebra of self extensions of the simple functor S_{1}^{G}, when p is odd and G is an elementary abelian p-group.

AMS Subject Classification : 18A25, 18G10, 18G15, 20J05. Keywords : Cohomological, Mackey functor, extension, simple.

1. Introduction

Let k be a field and G be a finite group. The theory of Mackey functors and cohomological Mackey functors for G over k originates in the work of Green ([4]) and Dress ([3]), at the beginning of the 70’s. It can be viewed as the theory of induction and restriction, when we forget the particular framework of linear representations of G over k. Many important developments have been achieved since, culminating in the comprehensive and seminal paper by Thévenaz-Webb ([5]) in 1995, where the authors introduce the Mackey algebra μ_{k}(G), and show, among many other fundamental results, that the category of Mackey functors for G over k is equivalent to the category of μ_{k}(G)-modules. Similarly, they show that the subcategory M_{k}^{c}(G) of cohomological Mackey functors for G over k is equivalent to the category of coμ_{k}(G)-modules, where coμ_{k}(G) is a specific quotient of μ_{k}(G), called the cohomological Mackey algebra.

The algebras μ_{k}(G) and coμ_{k}(G) share many similarities with the group algebra kG : e.g., they are finite dimensional k-vector spaces, of dimension independent on k, the Maschke theorem holds, there is a good theory of decomposition from characteristic 0 to characteristic p, etc… These resemblances raise some natural questions, whether a given theorem on kG will admit an analogue for μ_{k}(G) or coμ_{k}(G).

This was the main motivation in [2], where the question of complexity of cohomological Mackey functors was solved (in the only non-trivial case where k is a field of positive characteristic p dividing the order of G). It was also shown there how this question can be reduced to the consideration of elementary abelian p-groups E appearing as subquotients of G, and to the knowledge of enough information on the algebra E = Ext^{∗}(S_{1}^{E}, S_{1}^{E}) of self extensions of a particular simple functor S_{1}^{E} for these groups. Along the way, a presentation of this algebra was given when p = 2, together with a formula for the Poincaré series. In the case p > 2, no such presentation was given, and a conjecture was proposed for the Poincaré series of E. This conjecture was only proved in the case p = 3.

This paper settles completely the case p > 2 : a presentation of E is given, and, as a corollary, the forementioned conjecture is proved. The main results are the following, where S_{1,W}^{H} denotes the simple functor for the group H defined as in 2.15. To simplify notation, when W = k is the trivial module, we drop this subscript. We start by the construction of a new inflation functor for cohomological Mackey functors :

1.1. Theorem : Let k be a field, let G be a finite group, let N ⊴ G and let V be a simple k(G/N)-module. Then there exist an exact functor σ_{G/N}^{G} from M_{k}^{c}(G/N) to M_{k}^{c}(G) satisfying

σ_{G/N}^{G}(S_{1,V}^{G/N}) = S_{1,Inf_{G/N}^{G} V}^{G} .

Suppose, moreover, that G = N ⋊ H is the semidirect product of N by a

group H.

1. If V is a kH-module, let Ṽ be the kG-module Inf_{G/N}^{G}Iso_{H}^{G/N}V. Then the restriction of Ṽ to H is isomorphic to V.

2. The composition Res_{H}^{G}σ_{G/N}^{G}Iso_{H}^{G/N} is isomorphic to the identity functor

of M_{k}^{c}(H).

3. Let V and W be simple kH-modules. Then, for any n ∈ ℕ, the restriction from G to H induces a split surjection

r_{H}^{G} : Ext_{M_{k}^{c}(G)}^{n}(S_{1,Ṽ}^{G} , S_{1,W̃}^{G}) → Ext_{M_{k}^{c}(H)}^{n}(S_{1,V}^{H} , S_{1,W}^{H}) .

Let now p be an odd prime, k be a field of characteristic p and G be an elementary abelian p-group of rank r. Then we can give an explicit presentation of Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}), the graded algebra of self extensions of the simple functor S_{1}^{G} = S_{1,k}^{G}. Here is the main result of this paper :

1.2. Theorem : Let A be the graded k-algebra with generators

{τ̂_{φ} | φ ∈ Hom(G, 𝔽_{p}^{+})}, in degree 1, and {γ̂_{X}|X ≤ G, |X| = p}, in degree 2,

subject to the relations

(L1) ∀φ, ψ ∈ Hom(G, 𝔽_{p}^{+}), τ̂_{φ+ψ} = τ̂_{φ} + τ̂_{ψ}.

(L2) If p ≥ 5, then ∀φ ∈ Hom(G, 𝔽_{p}^{+}), τ̂_{φ}^{2} = 0, and [τ̂_{φ}, ∑_{X≰Kerφ} γ̂_{X}] = 0.

If p = 3, then τ̂_{φ}^{2} = − ∑_{X≰Kerφ} γ̂_{X}.

(L3) ∀φ ∈ Hom(G, 𝔽_{p}^{+}), ∀X, |X| = p, X ≤ Kerφ, [τ̂_{φ}, γ_{X}] = 0.

(L4) ∀Q ≤ G, |Q| = p^{2}, ∀X < Q, |X| = p, [γ̂_{X}, ∑_{Y <Q} γ̂_{Y}] = 0.

Then there is an isomorphism of graded algebras from A to Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}).

In view of (L1), this set of generators is redundant : we choose a direct sum decomposition G = Y_{1} ⊕ Y_{2} ⊕ ⋯ ⊕ Y_{r−1} ⊕ Y_{r}, and for 1 ≤ i ≤ r, we choose a group homomorphism φ_{i} : G → 𝔽_{p} with kernel ⨁_{{1≤j≤r \atop j≠i}} Y_{j}.

1.3. Theorem : Let A be the graded k-algebra with generators

{τ̂_{i}|1 ≤ i ≤ r} in degree 1, and {γ̂_{X}|X ≤ G, |X| = p} in degree 2,

subject to the relations

(R1) τ̂_{i}τ̂_{i} = 0, if p ≥ 5, or

τ̂_{i}τ̂_{i} = − ∑_{X≰Ker φ_{i}} γ̂_{X} if p = 3.

(R2) τ̂_{i}τ̂_{j} + τ̂_{j}τ̂_{i} = 0 for 1 ≤ j ≤ i ≤ r, if p ≥ 5, or

τ̂_{i}τ̂_{j} + τ̂_{j}τ̂_{i} = ∑_{X≰Ker(φ_{i}+φ_{j})} γ̂_{X} − ∑_{X≰Ker φ_{i}} γ̂_{X} − ∑_{X≰Ker φ_{j}} γ̂_{X} if p = 3.

(R3) [τ̂_{i}, ∑_{X≰Ker φ_{i}} γ̂_{X}] = 0;

(R4) [φ_{j}(x)τ̂_{i} − φ_{i}(x)τ̂_{j}, γ̂_{⟨x⟩}] = 0, for 1 ≤ i < j ≤ 1, x ∈ G;

(R5) [γ̂_{X}, ∑_{Y <Q} γ̂_{Y}] = 0, for all X < Q ≤ G, |X| = p, |Q| = p^{2}.

Then there is an isomorphism of graded algebras from A to Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}).

As a consequence of the above theorem, we get the following result, which was proved for p = 3 in ([2], Theorem 14.2) and conjectured for p ≥ 5:

1.4. Proposition : Let k be a field of odd characteristic p, and G ≅ (C_{p})^{r}. 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(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^{r−1}−1)t^{2})} .

The paper is organized as follows. In Section 2 we construct a new inflation functor for cohomological Mackey functors for a finite group G over a field k. In the case where the group is a semi-direct product, more properties of this functor are given. For completeness, Section 3 recalls a series of results on extensions of cohomological Mackey functors when G is an elementary abelian p-group G and k is a field of characteristic p. These results, taken from [2], are crucial in the second part of the paper. The use of the inflation functor constructed in Section 2 makes the proof of the exactness of the sequence in Corollary 3.10 straightforward. Section 4 deals with finding relations in the graded algebra E = Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}). Section 5 gives a recursive direct decomposition of this algebra. The decomposition is obtained through an involved induction on the rank of G, constructed on the technical support of Lemma 5.1. Based on this direct decomposition and the relations stated in Section 4, Section 6 builds a presentation of E. Some arithmetics of extensions in abelian categories, needed in the paper, are presented in the Appendix.

2. Yet another inflation functor

2.1. Let k be a commutative ring with identity element, and G be a finite group. Let M_{k}^{c}(G) denote the category of cohomological Mackey functors for G over k. By Yoshida’s Theorem, the category M_{k}^{c}(G) is equivalent to the category Fun_{k}(G) of k-linear contravariant functors from the category perm_{k}(G) of finitely generated permutation kG-modules, to the category k-Mod of kmodules. 2.2. When G and H are finite groups, any k-linear functor from perm_{k}(G) to perm_{k}(H) gives by precomposition a functor from Fun_{k}(H) to Fun_{k}(G). Two special cases of this situation have been considered in [2] (Section 3.12) : if U is a finite (H, G)-biset, the functor

t_{U} : V ↦ kU ⊗_{kG} V

from perm_{k}(G) to perm_{k}(H) yields a functor

L_{U} : F ↦ F ∘ t_{U}

from Fun_{k}(H) to Fun_{k}(G). Similarly, the functor

h_{U} : W ↦ Hom_{kH}(kU, W)

from perm_{k}(H) to perm_{k}(G) yields a functor

R_{U} : F ↦ F ∘ h_{U}

from Fun_{k}(G) to Fun_{k}(H), which is right adjoint to L_{U}. 2.3. Several special cases of this special case occur when H = G/N, where N is a normal subgroup of G. First, if U = G/N, viewed as a (G, H)-biset in the obvious way, then the functor L_{U} : Fun_{k}(G/N) → Fun_{k}(G) is denoted by ı_{G/N}^{G}, and R_{U} : Fun_{k}(G) → Fun_{k}(G/N) is denoted by ρ_{G/N}^{G}.

Reversing the actions, and viewing V = G/N as an (H, G)-biset, the functor R_{V} : Fun_{k}(G/N) → Fun_{k}(G) is denoted by ȷ_{G/N}^{G}, and L_{V} : Fun_{k}(G) → Fun_{k}(G/N) again by ρ_{G/N}^{G} (there is no notational conflict here, as L_{V} ≅ R_{U} by [2] Proposition 3.15).

The functors ı_{G/N}^{G} and ȷ_{G/N}^{G} are two inflation-like functors for cohomological Mackey functors. They are not isomorphic in general, and also both different from the usual inflation^{1} Inf_{G/N}^{G} for Mackey functors (as defined in [5] Section 2). Similarly, the functor ρ_{G/N}^{G} is a deflation functor. 2.4. When C is a k-linear category, recall ([1] Exemple 8.7.8 p. 97) that the karoubian envelope C^{+} of C is the category defined as follows : the object of C^{+} are the pairs (X, e), where X is an object of C and e is an idempotent in End_{C}(X). If (X, e) and (Y, f) are objects of C^{+}, then by definition

1Note that this inflation functor Inf_{G/N}^{G} does not preserve cohomological Mackey functors over k unless N is a p-group

Hom_{C^{+}}((X, e), (Y, f)) = fHom_{C}(X, Y)e ,

and the composition of morphism in C^{+} is induced by the composition of morphism in C.

The category C^{+} is a k-linear category. Moreover, the correspondence

\{ \begin{matrix}X ∈ C & ↦ & (X, Id_{X}) ∈ C^{+} \\ f ∈ Hom_{C}(X, Y) & ↦ & f ∈ Hom_{C^{+}}((X, Id_{X}), (Y, Id_{Y})) \end{matrix}

is a fully faithful k-linear functor i from C to C^{+}.

By composition, this functor induces a functor I : M ↦ M ∘ i from the category F_{k}^{+} of k-linear functors from C^{+} to k-Mod, to the category F_{k} of k-linear functors from C to k-Mod, which is easily seen to be an equivalence of categories : the functor J : F_{k} → F_{k}^{+} defined by

J(L)(X, e)) = L(e)(L(X))

is a quasi-inverse to I. 2.5. Applying this to the category C = perm_{k}(G), we observe that the category C^{+} is equivalent to the full subcategory perm_{k}^{+}(G) of kG-mod consisting of direct summands of finitely generated permutation kG-modules : this equivalence is induced by the functor (X, e) ↦ e(X). It follows that the category M_{k}^{c}(G) is equivalent to the category Fun_{k}^{+}(G) of contravariant k-linear functors from perm_{k}^{+}(G) to k-Mod. 2.6. It follows that when G and H are finite groups, any k-linear functor from perm_{k}^{+}(G) to perm_{k}^{+}(H) induces by composition a functor from Fun_{k}^{+}(H) to Fun_{k}^{+}(G), and by the above remarks, this yields a corresponding functor M_{k}^{c}(H) → M_{k}^{c}(G). In particular, when X is a direct summand of a finitely generated permutation (kH, kG)-bimodule, the functor t_{X} = X ⊗_{kG} − yields a functor L_{X} : M_{k}^{c}(H) → M_{k}^{c}(G), and the functor h_{X} = Hom_{kH}(X, −) yields a functor R_{X} : M_{k}^{c}(G) → M_{k}^{c}(H). 2.7. We will consider here another special case of this kind of construction : let N be a normal subgroup of G, and let H = G/N. Also denote by U the set G/N, viewed as a (G, H)-biset. As for any kG-module W, there is an isomorphism of kH-modules

Hom_{kG}(kU, W) ≅ W^{N} ,

the functor h_{U} : perm_{k}^{+}(G) → perm_{k}^{+}(H) is the fixed points functor W ↦ W^{N}.

Now W^{N} ⊇ tr_{1}^{N}(W), and this inclusion is functorial with respect to W : the functor W ↦ tr_{1}^{N}(W) is a functor from RG-Mod to RH-Mod, which is a subfunctor of the functor W ↦ W^{N}.

A natural question is then to know whether it can happen that the functor W ↦ tr_{1}^{N}(W) preserves direct summands of permutation modules. This is equivalent to requiring that the following condition holds :

2.8. Condition : For any finite G-set X, the module tr_{1}^{N}(kX) isomorphic to a direct summand of a permutation kH-module.

2.9. Lemma : Let G be a finite group, let N be a normal subgroup of G, let H = G/N, and let X be a finite G-set. Then there is an isomorphism of kH-modules

tr_{1}^{N}(kX) ≅ ⨁_{x∈G\X} Ind_{NG_{x}/N}^{G/N}(|N_{x}|k) ,

where G_{x} is the stabilizer of x in G, and N_{x} = N ∩ G_{x}.

Proof : The module (kX)^{N} has a k-basis consisting of the elements

b_{x} = ∑_{n∈N/N_{x}} nx ,

for x ∈ N\X. This basis is G/N-invariant, and the stabilizer of b_{x} in G/N is equal to NG_{x}/N. Thus

(kX)^{N} ≅ ⨁_{x∈G\X} Ind_{NG_{x}/N}^{G/N}k .

The submodule tr_{1}^{N}(kX) is generated by the elements tr_{1}^{N}(x) = |N_{x}|b_{x}, for x ∈ N\X, and these elements are permuted by G/N. The lemma follows.

In particular, as a k-module,

tr_{1}^{N}(kX) ≅ ⨁_{x∈N\X}(|N_{x}|k) .

Thus if tr_{1}^{N}(kX) is isomorphic to a direct summand of a permutation kHmodule, since any permutation kH-module is free as a k-module, it follows that |N_{x}|k is a projective k-module. Hence, if Condition 2.8 holds, then the following condition also holds :

2.10. Condition : For any subgroup M of N, the module |M|k is a projective k-module.

Conversely, if Condition 2.10 holds, and if X is a finite G-set, then for any x ∈ X, the k-module |N_{x}|k is projective, hence it is isomorphic to a direct summand of k (since the map λ ↦ |N_{x}|λ is a surjective morphism of kmodules from k to |N_{x}|k). It follows that |N_{x}|k is also isomorphic to a direct summand of k as k(NG_{x}/N)-module, since the action of NG_{x}/N on k and |N_{x}|k is trivial. Thus Ind_{NG_{x}/N}^{G/N}(|N_{x}|k) is isomorphic to a direct summand

of Ind_{NG_{x}/N}^{G/N}(k) as a k(G/N)-module. Since this is a permutation module, it follows that tr_{1}^{N}(kX) is isomorphic to a direct summand of a permutation kH-module. This shows that Conditions 2.8 and 2.10 are equivalent. 2.11. Now saying that for some m ∈ ℕ, the k-module mk is projective, is equivalent to saying that there exists an idempotent e_{m} of k such that e_{m}k is equal to the annihilator Ann_{k}(m) of m in k. Moreover Condition 2.10 obviously implies the following :

2.12. Condition : For any prime factor p of |N|, the k-module pk is projective. Equivalently, there exists an idempotent e_{p} of k such that

Ann_{k}(p) = e_{p}k.

Conversely, if Condition 2.12 holds, let m be any integer dividing |N|, and denote by e_{m} the sum ∑_{p|m} e_{p} of the idempotents e_{p} corresponding to the

distinct prime factors of m. Then e_{m} is an idempotent : indeed, if p and q are distinct prime numbers, then e_{p}e_{q} = 0, since it is both a p-torsion and a q-torsion element. Note that, more generally, if m and n are integers such that m|n, then e_{m}e_{n} = e_{m}.

Since moreover

me_{m} = ∑_{p|m} \frac{m}{p}pe_{p} = 0 ,

it follows that e_{m}k ⊆ Ann_{k}(m).

Now write m as a product p_{1}p_{2} ⋯ p_{l} of (possibly equal) prime numbers. We prove by induction on l that Ann_{k}(m) = e_{m}k.

If l = 0, then m = 1, and e_{1} = 0 generates Ann_{k}(1) = {0}. If l > 0, let p = p_{1}, and n = m/p. If x ∈ Ann_{k}(m), then pnx = 0, so there exists y ∈ k such that nx = e_{p}y. Thus e_{p}nx = nx, i.e. n(x − e_{p}x) = 0. By induction hypothesis, there exists z ∈ k such that x = e_{p}x + e_{n}z. Thus

e_{m}x = e_{m}e_{p}x + e_{m}e_{n}z = e_{p}x + e_{n}z = x ,

since e_{m}e_{r} = e_{r} if r|m. It follows that x ∈ e_{m}k, as was to be shown. This shows that Conditions 2.8, 2.10, and 2.12 are equivalent.

2.13. Remark : These conditions are fulfilled in particular if the ring k is hereditary, since any ideal of k is projective in this case.

2.14. Notation : When the normal subgroup N of G fulfills Condition 2.12, the functor W ↦ tr_{1}^{N}W from perm_{k}^{+}(G) to perm_{k}^{+}(G/N) induces an exact functor denoted by σ_{G/N}^{G} from M_{k}^{c}(G/N) to M_{k}^{c}(G).

The exactness of σ_{G/N}^{G} follows from the fact that this functor is obtained by pre-composition with some functor. 2.15. From now on we will assume that k is a field of characteristic p ≥ 0. Remark 2.13 shows that Condition 2.12 is fulfilled, for any normal subgroup N of G.

Recall that the simple cohomological Mackey functors for G over k are indexed by pairs (Q, V), where Q is a p-subgroup of G (which in the case p = 0 should be understood as Q = 1), and V is a simple k\overline{N}_{G}(Q)-module, where, as usual, \overline{N}_{G}(Q) = N_{G}(Q)/Q. As an object of Fun_{k}^{+}(G), the functor S_{Q,V}^{G} indexed by the pair (Q, V) can be described as follows : if W is a direct summand of a permutation kG-module, then

S_{Q,V}^{G} (W) = tr_{1}^{\overline{N}_{G}(Q)}Hom_{k}(W [Q], V) ,

where W [Q] is the Brauer quotient of W at Q. The functorial structure with respect to W is the obvious one. In other words S_{Q,V}^{G} (W) is the set of k\overline{N}_{G}(Q)-homomorphisms from W [Q] to V which factor through a projective k\overline{N}_{G}(Q)-module.

2.16. Proposition : Let k be a field. If N ⊴ G, and if V is a simple k(G/N)-module, then

σ_{G/N}^{G}(S_{1,V}^{G/N}) = S_{1,Inf_{G/N}^{G} V}^{G} .

Proof : The value of the functor S_{1,V}^{G/N} on a direct summand of a permutation k(G/N)-module W is equal to

S_{1,V}^{G/N}(W) = tr_{1}^{G/N}Hom_{k}(W, V) .

So for U ∈ perm_{k}^{+}(G)

σ_{G/N}^{G}(S_{1,V}^{G/N})(U) = S_{1,V}^{G/N}(tr_{1}^{N}(U))
= tr_{1}^{G/N}Hom_{k}(tr_{1}^{N}(U), V ) .

On the other hand

S_{1,Inf_{G/N}^{G} V}^{G} (U) = tr_{1}^{G}Hom_{k}(U, Inf_{G/N}^{G}V) .

Let i : tr_{1}^{N}U ↪ U denote the inclusion map, and let s : U → tr_{1}^{N}U be a k-linear map such that s ∘ i = Id. If ψ ∈ Hom_{k}(tr_{1}^{N}U, V), then ψ = ψ̃ ∘ i, where ψ̃ = ψ ∘ s ∈ Hom_{k}(U, Inf_{G/N}^{G}V). Setting θ = tr_{1}^{G/N}ψ, for any u ∈ U

θ ∘ tr_{1}^{N}(u) = ∑_{g∈G/N} ∑_{n∈N} gψ̃(g^{−1}nu)
= ∑_{g∈G/N} ∑_{n∈N} gn^{−1}ψ̃(g^{−1}nu)
= ∑_{g∈G} gψ(g^{−1}u)
= tr_{1}^{G}(ψ̃)(u) ,

thus θ ∘ tr_{1}^{N} = tr_{1}^{G}(ψ̃). This shows that the map

I : Hom_{k}(tr_{1}^{N}U, V) → Hom_{k}(U, Inf_{G/N}^{G}V)

sending θ to θ ∘ tr_{1}^{N} maps tr_{1}^{G/N}Hom_{k}(tr_{1}^{N}U, V) into tr_{1}^{G}Hom_{k}(U, Inf_{G/N}^{G}V). The map I is obviously injective, as tr_{1}^{N} : U → tr_{1}^{N}U is surjective.

Conversely, let φ ∈ Hom_{k}(U, Inf_{G/N}^{G}V), and set Ξ = tr_{1}^{G}φ. Then for u ∈ U

Ξ(u) = ∑_{g∈G} gφ(g^{−1}u)
= ∑_{g∈G/N} ∑_{n∈N} gnφ(n^{−1}g^{−1}u)
= ∑_{g∈G/N} ∑_{n∈N} gφ(g^{−1}n^{−1}u)
(since n^{−1}g^{−1}w = g^{−1}·gn^{−1}g^{−1} · u)
= (tr_{1}^{G/N}(φ ∘ i))(tr_{1}^{N}u) ,

so Ξ = I(tr_{1}^{G/N}(φ ∘ i)), showing that I induces an isomorphism

tr_{1}^{G/N}Hom_{k}(tr_{1}^{N}U, V) → tr_{1}^{G}Hom_{k}(U, Inf_{G/N}^{G}V) ,

which is obviously functorial in U. This completes the proof.

2.17. Proposition : Let G = N ⋊ H be the semidirect product of N by a

group H.

1. If V is a kH-module, let Ṽ be the kG-module Inf_{G/N}^{G}Iso_{H}^{G/N}V. Then the restriction of Ṽ to H is isomorphic to V.

2. The composition Res_{H}^{G}σ_{G/N}^{G}Iso_{H}^{G/N} is isomorphic to the identity functor

of M_{k}^{c}(H).

3. Let V and W be simple kH-modules. Then, for any n ∈ ℕ, the restriction from G to H induces a split surjection

r_{H}^{G} : Ext_{M_{k}^{c}(G)}^{n}(S_{1,Ṽ}^{G} , S_{1,W̃}^{G}) → Ext_{M_{k}^{c}(H)}^{n}(S_{1,V}^{H} , S_{1,W}^{H}) .

Proof : Assertion 1 is obvious. For Assertion 2, let W be a direct summand of a permutation kH-module, and let W′ be the kH-module defined by

W′ = Iso_{G/N}^{H}tr_{1}^{N}Ind_{H}^{G}W .

The set N is a set of representatives of G/H, so

Ind_{H}^{G}W ≅ ⨁_{n∈N} n ⊗ W .

Moreover, for any n ∈ N and w ∈ W,

tr_{1}^{N}(n ⊗ w) = ∑_{x∈N} xn ⊗ w = ∑_{x∈N} x ⊗ w .

This shows that the map θ : w ∈ W ↦ ∑_{x∈N} x ⊗ w ∈ W′ is a k-linear

isomorphism from W to W′. Moreover, for any h ∈ H

h(∑_{x∈N} x ⊗ w) = ∑_{x∈N} hx ⊗ w
= ∑_{x∈N} hxh^{−1} · h ⊗ w = ∑_{x∈N} xh ⊗ w
= ∑_{x∈N} x ⊗ hw .

Thus θ is actually an isomorphism of kH-modules W → W′, which is obviously functorial with respect to W. So the functor F = Iso_{G/N}^{H}tr_{1}^{N}Ind_{H}^{G} is isomorphic to the identity functor of perm_{k}^{+}(H). Assertion 2 follows, since the functor Res_{H}^{G}σ_{G/N}^{G}Iso_{H}^{G/N} is the endofunctor of M_{k}^{c}(H) obtained by composition with F.

Assertion 1 implies that Res_{H}^{G}S_{1,Ṽ}^{G} ≅ S_{1,V}^{H}, so the restriction functor Res_{H}^{G} induces a k-linear map

r_{H}^{G} : Ext_{M_{k}^{c}(G)}^{n}(S_{1,Ṽ}^{G} , S_{1,W̃}^{G}) → Ext_{M_{k}^{c}(H)}^{n}(S_{1,V}^{H} , S_{1,W}^{H}) .

Conversely, the functor σ_{G/N}^{G}Iso_{H}^{G/N} is an exact functor from M_{k}^{c}(H) to M_{k}^{c}(G), which sends S_{1,V}^{H} to S_{1,Ṽ}^{G}, by Proposition 2.16. This yields a k-linear map

s_{H}^{G} : Ext_{M_{k}^{c}(H)}^{n}(S_{1,V}^{H} , S_{1,W}^{H}) → Ext_{M_{k}^{c}(G)}^{n}(S_{1,Ṽ}^{G} , S_{1,W̃}^{G}) .

Since by Assertion 2, the functor Res_{H}^{G}σ_{G/N}^{G}Iso_{H}^{G/N} is isomorphic to the identity functor of M_{k}^{c}(H), the composition r_{H}^{G} ∘ s_{H}^{G} is an isomorphism, so r_{H}^{G} is split surjective.

3. Extensions of cohomological Mackey functors

3.1. Hypothesis : From now on, we assume that k is a field of positive characteristic p and that G is a finite p-group.

In this section we recall some notation and results from [2] about extensions of cohomological Mackey functors for elementary abelian p-groups. When H is a subgroup of G, we denote by K_{G}(H) the set of complements of H in G, i.e. the set of subgroups T of G such that H ⊕ T = G. 3.2. ([2] Corollary 8.3) When X is a subgroup of order p of G, there exist a

unique cohomological Mackey functor \binom{S_{X}^{G}}{S_{1}^{G}}, up to isomorphism, which fits

into a non split exact sequence in M_{k}^{c}(G) of the form

(3.3)  0 → S_{1}^{G} → \binom{S_{X}^{G}}{S_{1}^{G}} → S_{X}^{G} → 0 .

There is also a unique cohomological Mackey functor \binom{S_{1}^{G}}{S_{X}^{G}}, up to isomor-

phism, which fits into a non split exact sequence in M_{k}^{c}(G) of the form

(3.4)  0 → S_{X}^{G} → \binom{S_{1}^{G}}{S_{X}^{G}} → S_{1}^{G} → 0 ,

and \binom{S_{1}^{G}}{S_{X}^{G}} is isomorphic to the dual \binom{S_{X}^{G}}{S_{1}^{G}}^{∗} of \binom{S_{X}^{G}}{S_{1}^{G}}.

We denote by γ_{X}^{G} ∈ Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}) the class of the exact sequence

Γ_{X}^{G} : 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 two previous short exact sequences ([2] Notation 7.4).

3.5. ([2] Section 14) When p > 2, and φ ∈ Hom(G, k^{+}), let U_{φ}^{G} denote the kG-module k ⊕ k, on which G acts by

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

Let T_{φ}^{G} denote the unique Mackey functor for G over k such that T_{φ}^{G}(H) is zero if H is a non trivial subgroup of G, and T_{φ}^{G}(1) ≅ U_{φ}^{G}. The functor T_{φ}^{G} is cohomological, and fits in an exact sequence

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

in M_{k}^{c}(G), which is non split if φ ≠ 0. We denote by τ_{φ}^{G} the class of this extension in Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}). When φ ∈ Hom(G, 𝔽_{p}^{+}), we denote by the same symbol the composition of φ with the inclusion 𝔽_{p}^{+} ↪ k^{+}, and by τ_{φ} the corresponding element of Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}). 3.6. The following conjecture was proposed in [2], and proved there for p = 3 ([2] Theorem 14.2) :

3.7. Conjecture : Let k be a field of odd characteristic p, and G ≅ (C_{p})^{r}. 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(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^{r−1}−1)t^{2})} .

3.8. Proposition : [[2] Proposition 8.7 and Proposition 10.1] Let k be a field of characteristic p > 0, let G be an elementary abelian p-group, and let H be a subgroup of index p in G. Set I = Ind_{H}^{G}S_{1}^{H}.

1. 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}, and I/R ≅ S ≅ S_{1}^{G}. Moreover

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.

2. Let Y ∈ K_{G}(H). The functor R has a subfunctor J isomorphic to ι_{G/Y}^{G} (S_{1}^{G/Y}), and there is an isomorphism

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

3.9. Corollary : [[2] Corollary 10.3] 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) − {Y }.

Moreover, it is easy to check that the map E_{G}(n) → E_{H}(n) in this corollary is induced by the restriction functor Res_{H}^{G}, since Res_{H}^{G}S_{1}^{G} ≅ S_{1}^{H}. As k is a field by Hypothesis 3.1, the conclusion of Proposition 2.17 holds and this map is (split) surjective. Thus :

3.10. Corollary : With the same notation, for any n ∈ ℕ, there is a short exact sequence of extension groups

0 ⟶ L(n − 1) ⊕ ⊕_{X∈X} E_{G}(n − 2) → E_{G}(n) → E_{H}(n) → 0 .

4. Mackey functors concentrated at 1 and relations in E

We first consider some generalizations of the functor L of Proposition 3.8 :

4.1. Definition : Let k be a field of characteristic p > 0, and G be a finite p-group.

• A Mackey functor M for G over k is said to be concentrated at 1 if M(H) = {0} for any non trivial subgroup H of G.

• A kG-module V is said to have zero traces if

tr_{1}^{X}(V) = {0}

for any non trivial subgroup X of G.

4.2. Remark : 0) A Mackey functor concentrated at 1 is cohomological.

1) A (finitely generated) Mackey functor for G over k is concentrated at 1 if and only if all its composition factors are isomorphic to S_{1}^{G}.

2) By transitivity of traces, a kG-module V has zero traces if and only if tr_{1}^{X}(V) = {0} for any subgroup X of order p of G.

4.3. Proposition : Let k be a field of characteristic p, and G be a finite p-group.

1. Let M be a Mackey functor for G over k. If M is concentrated at 1, then M is cohomological, and the kG-module V = M(1) has zero traces.

2. If V is a kG-module having zero traces, then there is a unique Mackey functor V̂ for G over k such that V̂ is concentrated at 1 and V̂(1) ≅ V as kG-modules.

3. If M is an object of M_{k}^{c}(G), let M^{0} denote the kG-submodule

M^{0} = ⋂_{1<X≤G} Ker t_{1}^{X}

of M(1). Then M^{0} has zero traces, and M̂^{0} is the largest subfunctor of M concentrated at 1.

4. The correspondences M ↦ M(1) and V ↦ V̂ are mutual inverse equivalences of categories between the full subcategory of M_{k}^{c}(G) whose objects are concentrated at 1, and the full subcategory of kG-Mod whose objects are modules with zero traces.

Proof : For Assertion 1, observe that for any subgroup X of G, and any v ∈ V = M(1)

r_{1}^{X}t_{1}^{X}v = ∑_{x∈X} x · v = tr_{1}^{X}(v) .

This has to be zero if X is non-trivial, since M(X) = {0} by assumption. Moreover M is obviously cohomological.

For Assertion 2, it is straightforward to check that if V has zero traces, then the assignments M(1) = V and M(H) = {0} for 1 < H ≤ G, define a Mackey functor M for G over k. This proves the existence part of Assertion 2. Uniqueness is straightforward.

Assertion 3 is also straightforward, and Assertion 4 follows easily.

4.4. Example : Let H be a subgroup of index p of G, and let W = Ind_{H}^{G}k. Let ε : W → k denote the augmentation map, and set V = Ker ε. Then the kG-module V has zero traces : indeed, if X is a subgroup of order p of G, then either X ≤ H, and then tr_{1}^{X}(W) = {0}, since X acts trivially on W. Or X ≰ H, and then Res_{X}^{G}W ≅ Ind_{1}^{X}k ≅ kX, so there is an exact sequence

0 → Res_{X}^{G}V → kX → k → 0 ,

showing that tr_{1}^{X}(V) = {0} also in this case.

The module Ind_{H}^{G}k is inflated from the free module of rank 1 for the cyclic group G/H. Hence, it is uniserial, and all its subquotients are indecomposable, and characterized by their dimension, up to isomorphism. We denote by U_{a} the subquotient of dimension a, for a ∈ {1, … p}. If a ≤ p − 1, the module U_{a} is isomorphic to a submodule of V, hence it has zero traces, and we denote by T_{a} the functor Û_{a}.

Let φ ∈ Hom(G, k^{+}) with kernel H. Then the functor T_{φ}^{G} introduced in Paragraph 3.5 is isomorphic to T_{2}. Similarly, the functor L of Proposition 3.8 is isomorphic to T_{p−2}, since L(1) ≅ U_{p−2}. We also denote by M the functor T_{p−1} = V̂. The two short exact sequences

(4.5)  0 → k → U_{a+1} → U_{a} → 0 , and 0 → U_{a} → U_{a+1} → k → 0

of kG-modules yield corresponding exact sequences

(4.6) 0 → S_{1}^{G} → T_{a+1} → T_{a} → 0 , and 0 → T_{a} → T_{a+1} → S_{1}^{G} → 0

of cohomological Mackey functors for G over k. In particular,

(4.7)  0 → k → V → U → 0 , and 0 → U → V → k → 0

yield

(4.8)  0 → S_{1}^{G} → M → L → 0 , and 0 → L → M → S_{1}^{G} → 0 .

We will denote by τ_{φ,L} the element of Ext_{M_{k}^{c}(G)}^{1}(L, S_{1}^{G}), respectively τ_{L,φ} the element of Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, L) corresponding to this last two short exact sequences.

Assertion 2 of Proposition 3.8 can now be rephrased as follows : there exists a subfunctor Σ of R, containing S, such that

R = M + Σ, S = M ∩ Σ, Σ/S ≅ ⊕_{X∈K_{G}(H)} S_{X}^{G} .

Setting K = L + Σ, this gives the following diagram of subfunctors of I :

[figure: I; R; K; M; Σ; L; S; {0}.]

We give now a series of relations between elements in Ext_{M_{k}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}) and Ext_{M_{k}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}). For simplicity, we will drop the exponent G, and write τ_{φ}, γ_{X}, S_{1}, T_{φ}, U_{φ} instead of τ_{φ}^{G}, γ_{X}^{G}, S_{1}^{G}, T_{φ}^{G}, U_{φ}^{G} respectively. We start with a linearity relation between the τ’s

4.9. Lemma : Let φ and ψ be two morphisms in Hom(G, k^{+}). Then

τ_{φ} + τ_{ψ} = τ_{φ+ψ} in Ext_{M_{k}^{c}(G)}^{1}(S_{1}, S_{1}).

Proof : Let 0 → S_{1} → T_{φ} → S_{1} → 0 be the representative for τ_{φ} and 0 → S_{1} → T_{ψ} → S_{1} → 0 be the representative for τ_{ψ} described in Paragraph 3.5. Given that both τ_{φ} and τ_{ψ} are concentrated at the trivial subgroup, this will also be the case for τ_{φ} + τ_{ψ}.

We construct a representative 0 → S_{1} → T → S_{1} → 0 for the sum as in Lemma A.2 working with the G-modules U_{φ} and U_{ψ} that give the functors T_{φ}, respectively T_{ψ}. Then T (1) is given by the following sequence of pushout and pullback

\xymatrix{k ⊕ k \ar[r]^{i⊕j} \ar[d]_{π_{1}+π_{2}} & U_{φ} ⊕ U_{ψ} \ar[d] \\ k \ar[r] & U′} , \xymatrix{U′ \ar[r]^{{ \atop s⊕t}} & k ⊕ k \\ T (1) \ar[r] \ar[u] & k \ar[u]_{Δ}} .

The same computation as in Lemma A.2 gives

T (1) ≃ {(a_{1}, a_{2}, b_{1}, b_{2}, c) ∈ k^{5}|a_{2} = b_{2}}/{(0, d_{1}, 0, d_{2}, −d_{1} − d_{2})|d_{1}, d_{2} ∈ k} .

This module has dimension 2, the action of G on the class of (a_{1}, a_{2}, b_{1}, b_{2}, c) is induced by the action on U_{φ} for the first two terms, the action on U_{ψ} for the next two terms and is trivial on the last. More precisely,

g[(a_{1}, a_{2}, b_{1}, b_{2}, c)] = [(a_{1} + φ(g)a_{2}, a_{2}, b_{1} + ψ(g)b_{2}, b_{2}, c)] .

Thus T (1) is a G-module with k-basis {[(0, 1, 0, 1, 0)], [(0, 0, 0, 0, 1)]} and action given by

g[(0, 0, 0, 0, 1)] = [(0, 0, 0, 0, 1)]

and

g[(0, 1, 0, 1, 0)]=[(φ(g)1, 1, ψ(g), 1, 0)]=[(0, 1, 0, 1, 0)]+[(0, 0, 0, 0, φ(g)+ψ(g))] .

This means that T (1) ≅ U_{φ+ψ} and that T ≅ T_{φ+ψ}.

The following lemmas give relations involving (τ_{φ})^{2}. We obtain different relations in the cases p = 3 and p ≥ 5. This is tidily related to the existence of the functor T_{3} only when p ≥ 5. This functor is constructed in Example 4.4 as an application of Proposition 4.3 . The main ingredient used in both lemmas is Lemma 12.2 of [2], that we use to detect zero elements in Ext_{M_{k}^{c}(G)}^{2}(S_{1}, S_{1}).

4.10. Lemma : Let k be a field of characteristic p ≥ 5 and let φ ∈ Hom(G, k^{+}). Then (τ_{φ})^{2} = 0 in Ext_{M_{k}^{c}(G)}^{2}(S_{1}, S_{1}).

Proof : Recall from Example 4.4 that T_{φ} = \binom{S_{1}}{S_{1}} ≅ T_{2}, when H = Ker φ.

Since p ≥ 5, we have p − 1 ≥ 3, so the module U_{3} defined in Example 4.4 has zero traces. This module has a filtration

{0} ⊂ k ⊂ U_{2} ⊂ U_{3} ,

such that U_{3}/k ≅ U_{2} and U_{3}/U_{2} ≅ k. It follows that T_{3} has a filtration

{0} ⊂ S_{1} ⊂ T_{2} ⊂ T_{3}

such that T_{3}/S_{1} ≅ T_{2} and T_{3}/T_{2} ≅ S_{1}. Applying Lemma 12.2 of [2] to this filtration shows that the sequence

0 → S_{1} → \binom{S_{1}}{S_{1}} → \binom{S_{1}}{S_{1}} → S_{1} → 0

represents the zero element of Ext_{M_{k}^{c}(G)}^{2}(S_{1}, S_{1}). But this represents precisely (τ_{φ})^{2}.

In the case p = 3, using the notation of Example 4.4, we have M = T_{2} and L = S_{1}. Hence, the construction in the proof of previous lemma cannot be used. We use instead the decomposition of the functor I from Proposition 3.8.

4.11. Hypothesis : We assume from now on that G ≅ (C_{p})^{r} is an elementary abelian p-group of rank r.

4.12. Lemma : Let k be a field of characteristic p = 3. Let G be an elementary abelian p-group, and φ ∈ Hom(G, 𝔽_{p}^{+}). Then

(τ_{φ})^{2} = − ∑_{X≰Ker φ} γ̂_{X}

in Ext_{M_{k}^{c}(G)}^{2}(S_{1}, S_{1}).

Proof : If φ = 0, there is nothing to prove, since τ_{φ} = 0 and the summation in the right hand side is empty. So we assume φ ≠ 0, and set H = Kerφ. With the notation of Proposition 3.8, let I ⊃ R ⊃ S_{1} be a filtration of the functor I = Ind_{H}^{G}S_{1}^{H}. Then we have I/R ≃ S_{1} and, using that R/S_{1} ≅

S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}, we obtain I/S_{1} ≃ \binom{S_{1}}{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}}. Moreover, we have

that R ≅ \binom{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}}{S_{1}}. Lemma 12.2 of [2] applied to this filtration

gives that the exact sequence

0 → S_{1} → \binom{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}}{S_{1}} → \binom{S_{1}}{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}} → S_{1} → 0

represents the zero element of Ext_{M_{k}^{c}(G)}^{2}(S_{1}, S_{1}). The result follows as this sequence also represents (τ_{φ})^{2} + ∑_{X∈K_{G}(H)} γ̂_{X}.

We prove now two commutation relations satisfied by the elements τ_{φ} and γ_{X} in Ext_{M_{k}^{c}(G)}^{3}(S_{1}, S_{1}) :

4.13. Lemma : Let k be a field of characteristic p ≥ 3 and G be an elementary abelian p-group. Then

γ_{X}τ_{φ} = τ_{φ}γ_{X}

for all φ : G → k^{+} and X ≤ Ker φ, |X| = p.

Proof : We fix φ : G → k^{+} and X ≤ Ker φ, |X| = p. Let M be the functor defined by the following data. First set M(1) = U_{φ}. Recall that this means M(1) ≅ k ⊕ k as k-vector spaces and the action by g ∈ G is given by g(x, y) = (x + φ(g)y, y). Then set M(X) = k ⊕ k = {(s, t)|s, t ∈ k} with trivial G-action and M(H) = 0 for all subgroups H of G different from 1 and X. Also set t_{1}^{X}(1, 0) = (0, 0), t_{1}^{X}(0, 1) = (1, 0), r_{1}^{X}(1, 0) = (0, 0) and r_{1}^{X}(0, 1) = (1, 0). It is easy to check that M is a cohomological Mackey functor. The fact that X ≤ Ker φ gives that the action of u ∈ X is trivial

on U_{φ} and, thus ∑_{u∈X} u · (x, y) = p · (x, y) = 0 = r_{1}^{X}t_{1}^{X}(x, y).

The functor M has a subfunctor isomorphic to \binom{S_{X}}{S_{1}} and the quotient

by this subfunctor is isomorphic to \binom{S_{1}}{S_{X}}. Define the Mackey functor M′

by M′(1) = U_{φ}, M′(X) = k, r_{1}^{X}(1) = (1, 0) and t_{1}^{X}(x, y) = 0. It is straight forward that M′ is a cohomological Mackey functor, that T_{φ} is a subfunctor

of M′ and that M′ = (\begin{matrix}( & {S_{X} \atop S_{1}} & ) \\ S_{1} \end{matrix}). Also, define the Mackey functor M′′ by

M′′(1) = U_{φ}, M′′(X) = k, r_{1}^{X}(1) = (0, 0) and t_{1}^{X}(x, y) = y. Again, it is straight forward that M′′ is a cohomological Mackey functor, that T_{φ} is a

factor of M′′ when factoring out the socle S_{X} and that M′′ = (\begin{matrix}S_{1} \\ ( & {S_{1} \atop S_{X}} & ) \end{matrix}).

Moreover, we get an exact sequence

0 → S_{1} → M′ → M → M′′ → S_{1} → 0 .

It is now straightforward to show that this sequence is Yoneda equivalent to

0 → S_{1} → \binom{S_{X}}{S_{1}} → \binom{S_{1}}{S_{X}} → \binom{S_{1}}{S_{1}} → S_{1} → 0

via the canonical projections M′ → \binom{S_{X}}{S_{1}}, M → \binom{S_{1}}{S_{X}} and, respectively,

M′′ → \binom{S_{1}}{S_{1}} and also equivalent to

0 → S_{1} → \binom{S_{1}}{S_{1}} → \binom{S_{X}}{S_{1}} → \binom{S_{1}}{S_{X}} → S_{1} → 0

via the canonical injections \binom{S_{1}}{S_{1}} → M′, \binom{S_{X}}{S_{1}} → M and, respectively,

\binom{S_{1}}{S_{X}} → M′′. Given that the last two exact sequences represent γ_{X}τ_{φ},

respectively, τ_{φ}γ_{X}, the result follows.

4.14. Lemma : Let k be a field of characteristic p ≥ 3 and G be an elementary abelian p-group. Then

( ∑_{X≰Ker φ}γ_{X})τ_{φ} = τ_{φ}( ∑_{X≰Ker φ}γ_{X}) .

Proof : Again if φ = 0, there is nothing to prove, so we assume φ ≠ 0, and set H = Ker φ. Let T = T_{φ}^{G} be the functor described in Paragraph 3.5 and let L, respectively M, be the functors described in Example 4.4.

• First step.

(∗) τ_{L,φ}τ_{φ} + ∑_{X∈K_{G}(H)} γ_{L,X} = 0

where the notation is as follows : the functor L was defined in Proposition 3.8, and the element τ_{L,φ} in Example 4.4.

Let i : S_{1} → L be the inclusion map corresponding to the isomorphism from S_{1} to the socle of L. Taking the image under i of Extension 3.3 yields the following diagram

[figure: 0; S_{1}; i; \binom{S_{X}}{S_{1}}; S_{X}; 0; 0; L; _{\binom{S_{X}}{L}}; S_{X}; 0]

whose bottom line can be spliced with Extension 3.4 to give the exact sequence

0 → L → \binom{S_{X}}{L} → \binom{S_{1}}{S_{X}} → S_{1} → 0 ,

which defines the element γ_{L,X} ∈ Ext_{M_{k}^{c}(G)}^{2}(S_{1}, L).

We apply Lemma 12.2 in [2] to the sequence of functors

{0} ⊂ L ⊂ R ⊂ I ,

where R and I are defined in Proposition 3.8. The quotient R/L is isomorphic

to S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}, so R can be represented by \binom{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}}{L}.

The quotient I/R is isomorphic to S_{1}. The quotient I/L is isomorphic

to \binom{S_{1}}{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}}. Thus one gets a four terms exact sequence

0 → L → \binom{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}}{L} → \binom{S_{1}}{S_{1} ⊕ ⊕_{X∈K_{G}(H)} S_{X}} → S_{1} → 0

representing a zero extension in Ext^{2}(S_{1}, L), which is in fact (*).

• Second step.

(∗∗) τ_{φ,L}γ_{L,X} = τ_{φ}γ_{X}

where τ_{φ,L} = (0 → S_{1} → M → L → 0)

The left hand side of (**) is represented by

0 → S_{1} → M → \binom{S_{X}}{L} → \binom{S_{1}}{S_{X}} → S_{1} → 0 ,

and the right hand side is represented by

0 → S_{1} → T → \binom{S_{X}}{S_{1}} → \binom{S_{1}}{S_{X}} → S_{1} → 0 .

One constructs an equivalence of these extensions by taking inclusions

maps T ↪ M, respectively \binom{S_{X}}{S_{1}} ↪ \binom{S_{X}}{L}, and the identity elsewhere.

The equality in (**) follows.

• Third step. Recall that we have

(*) τ_{L,φ}τ_{φ} + ∑_{X∈K_{G}(H)} γ_{L,X} = 0

(**) τ_{φ,L}γ_{L,X} = τ_{φ}γ_{X}

(***) τ_{φ,L}τ_{L,φ} + ∑_{X∈K_{G}(H)} γ_{X} = 0

Left-multiplying (*) by τ_{φ,L} and right-multiplying (***) by τ_{φ} we get:

( ∑_{X∈K_{G}(H)} γ_{X})τ_{φ} = τ_{φ,L}( ∑_{X∈K_{G}(H)} γ_{L,X})

Then by (**) we replace the right term to get

( ∑_{X∈K_{G}(H)} γ_{X})τ_{φ} = τ_{φ}( ∑_{X∈K_{G}(H)} γ_{X})

hence ( ∑_{X∈K_{G}(H)} γ_{X}) and τ_{φ} commute. If G is cyclic, the sum has only one

term γ_{X} commuting with (the unique) τ_{φ}.

5. Inductive construction of a basis of E

Recall that k be a field of characteristic p ≥ 3 and G is an elementary abelian p-group of rank r. Our aim is to construct a basis of the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}). We do this by induction. The technical structure of the induction is build on the following general result on commutative diagrams :

5.1. Lemma : Suppose we have the following diagram of finite dimensional k-vector spaces, where the four exterior triangles and the triangle (E_{0}, F_{2}, F_{1}) are commutative.

[figure: E_{1}; F_{2}; F_{1}; E_{1}; E_{2}; F_{1}; F_{0}; E_{0}]

Suppose that, in the above diagram, the two maps E_{0} → F_{1} are the same and that all the sequences

\xymatrix{E_{r} \ar[r] & F_{i} \ar[r]^{α} & F_{j} \ar[r] & E_{s} ,}

are exact, where α is any of the horizontal or vertical maps. Suppose moreover that

(H1) Ker(E_{0} → F_{0}) = Ker(E_{0} → F_{1}),

(H2) Ker(E_{1} → F_{1}) = Ker(E_{1} → F_{2}),

(H3) E_{0} → F_{0} → E_{2} is exact,

(H4) Im(E_{1} → F_{1}) = Im(F_{0} → F_{1}).

Then we have

(C1) Im(F_{1} → E_{1}) = Im(F_{2} → E_{1})

(C2) Im(F_{0} → E_{2}) = Im(F_{1} → E_{2})

(C3) E_{0} → F_{1} → E_{2} is exact (C4) Im(E_{1} → F_{2}) = Im(F_{1} → F_{2}) and F_{0} ≅ F_{1} ≅ F_{2} (C5) E_{1} → F_{2} → E_{1} is exact.

Proof : Choose a direct decomposition F_{0} = U ⊕ V such that

E_{0} ≅ Ker(E_{0} → F_{0}) ⊕ U .

Given (H1) and the fact that the triangle (E_{0}, F_{1}, F_{0}) is commutative, the horizontal map in this triangle induces an isomorphism between Im(E_{0} → F_{1}) and Im(E_{0} → F_{0}). Similarly, (H2) and the fact that the triangle (E_{1}, F_{1}, F_{2}) is commutative, implies that the vertical map in this triangle induces an isomorphism between Im(E_{1} → F_{1}) and Im(E_{1} → F_{2}).

Moreover, one has Im(E_{0} → F_{0}) ≅ U and, using (H4) and the exactness of the sequence E_{0} → F_{0} → F_{1}, that Im(E_{1} → F_{1}) ≅ V.

Moreover Im(F_{1} → F_{0}) = Ker(F_{0} → E_{2}) = Im(E_{0} → F_{0}) ≅ U. Thus we have F_{1} ≅ Im(E_{1} → F_{1}) ⊕ Im(F_{1} → F_{0}) ≅ V ⊕ U, and it follows that Im(F_{1} → F_{2}) ≅ F_{1}/Im(E_{0} → F_{1}) ≅ V and

Im(F_{1} → E_{1}) ≅ F_{1}/Im(F_{0} → F_{1}) ≅ (U ⊕ V)/V ≅ U .

Since the sequence F_{1} → F_{2} → E_{1} is exact, one gets

F_{2} ≅ V ⊕ Im(F_{2} → E_{1}) .

By the commutativity of the triangle (E_{1}, F_{2}, F_{1}),

Im(F_{2} → E_{1}) ⊆ Im(F_{1} → E_{1}) ≅ U .

Since the sequence E_{1} → F_{2} → F_{1} is exact, we get that

F_{2} ≅ V ⊕ Im(F_{2} → F_{1}) .

By the commutativity of the triangle (E_{0}, F_{2}, F_{1}), we have

Im(F_{2} → F_{1}) ⊇ Im(E_{0} → F_{1}) ≅ U .

Hence F_{2} is contained in and, respectively contains a k-vector space isomorphic to U ⊕ V implying that F_{2} ≅ U ⊕ V and that the previous inclusions are equalities.

In particular Im(F_{2} → E_{1}) ≅ U, hence (C1). Moreover, Im(E_{1} → F_{2}) = Im(F_{1} → F_{2}) ≅ V, and (C4) follows. Also

Im(F_{1} → E_{2}) ≅ F_{1}/Im(F_{2} → F_{1}) ≅ (U ⊕ V)/U ≅ V ,

and, using (H3), we have Im(F_{0} → E_{2}) ≅ F_{0}/Im(E_{0} → F_{0}) ≅ V. The commutativity of the triangle (E_{2}, F_{0}, F_{1}) implies then (C2). In the triangle (E_{0}, F_{1}, F_{2}), we have Im(E_{0} → F_{1}) = Im(F_{2} → F_{1}) and, from the exactness of the sequence F_{2} → F_{1} → E_{2}, we have Im(F_{2} → F_{1}) = Ker(F_{1} → E_{2}). Now (C3) follows. Lastly, Im(E_{1} → F_{2}) = Im(F_{1} → F_{2}) = Ker(F_{2} → E_{1}) and (C5) follows.

5.2. First we fix some notation : we set F_{a}(n) := Ext_{M_{k}^{c}(G)}^{n}(T_{a}, S_{1}^{G}). In particular, E_{G}(n) = F_{1}(n) and L(n) = F_{p−2}(n). Applying Hom_{M_{k}^{c}(G)}(−, S_{1}^{G}) to the short exact sequences of Mackey functors in 4.4, yields two long exact sequences

0 → E_{G}(0) → F_{a+1}(0) → F_{a}(0) → E_{G}(1) → …

and

0 → F_{a}(0) → F_{a+1}(0) → E_{G}(0) → F_{a}(1) → …

In what follows E_{G}(n − 1) → F_{a}(n), E_{G}(n) → F_{a}(n), F_{a}(n) → E_{G}(n), F_{a}(n) → E_{G}(n + 1), F_{a}(n) → F_{a+1}(n) or F_{a+1}(n) → F_{a}(n) are the maps in the long exact sequences above.

5.3. Proposition : Let p > 5. Then, for all n, the sequence

E_{G}(n) → L(n) → E_{G}(n)

given as above is exact and dim L(n) = dim E_{G}(n). Moreover, the sequence

E_{G}(n) → E_{G}(n + 1) → E_{G}(n + 2)

given by multiplication from the right with τ_{φ}, is exact, for any non trivial homomorphism φ : G → k^{+} and we have a direct decomposition of E_{G}(n) inductively given by

E_{G}(n) ≅ E_{H}^{G}(n)  ⊕ E_{H}^{G}(n − 1)τ_{φ}
⊕ E_{H}^{G}(n − 2)s_{φ}  ⊕ E_{H}^{G}(n − 3)τ_{φ}s_{φ}
⊕ E_{H}^{G}(n − 4)s_{φ}^{2}  ⊕ E_{H}^{G}(n − 5)τ_{φ}s_{φ}^{2}
⊕ …
⊕ ⨁_{X∈X} E_{G}(n − 2)γ_{X} ⊕ ⨁_{X∈X} E_{G}(n − 3)γ_{X}τ_{φ}
⊕ ⨁_{X∈X} E_{G}(n − 4)s_{φ}γ_{X} ⊕ ⨁_{X∈X} E_{G}(n − 5)s_{φ}γ_{X}τ_{φ}
⊕ ⨁_{X∈X} E_{G}(n − 6)s_{φ}^{2}γ_{X} ⊕ ⨁_{X∈X} E_{G}(n − 7)s_{φ}^{2}γ_{X}τ_{φ}
⊕ …

where E_{H}^{G}(m) := σ_{G/Y}^{G} Iso_{H}^{G/Y} E_{H}(m) and s_{φ} := ∑_{X∈K_{G}(H)} γ_{X}.

5.4. We do the proof by simultaneous induction on n and |G|. The cases n = 0 or G = 1 are trivial so we have the starting point for the induction.

We suppose the proposition true for all the proper subgroups K of G and all m, and, respectively, for K = G and m ≤ n, and we prove it for G and n + 1. 5.5. By induction, we have that the sequence

E_{K}(m − 1) → E_{K}(m) → E_{K}(m + 1)

given by multiplication from the right with τ_{ψ}, is exact, for K and m as in the previous paragraph, and for any non trivial homomorphism ψ : K → k^{+}. To ease notation, we set E(n) := E_{G}(n), while still writing E_{K}(n) when K is different from G. 5.6. The four exterior triangles and the triangle (E(n−1), F_{d+2}(n), F_{d+1}(n)) in the following diagram

[figure: E(n); F_{d+2}(n); F_{d+1}(n); E(n); E(n + 1); F_{d+1}(n); F_{d}(n); E(n − 1)]

are commutative for all n and 1 ≤ d ≤ p − 4. Setting E_{i} = E(n + i − 1) and F_{i} = F_{d+i}(n) for i ∈ {0, 1, 2}, for d = 1 the corresponding hypothesis (H3) in Lemma 5.1 is the fact that E(n − 1) → E(n) → E(n + 1) is exact and the hypotheses (H1), (H2), (H4) are easy to check. This starts the induction on d. Then, (H1) to (H4) are satisfied for all d ≤ p − 4. Indeed, the conclusions (C3) and (C4) for F_{i} = F_{d+i}(n) in the same Lemma 5.1 are the hypotheses (H3) and (H4) for F_{i} = F_{d+i+1}(n). Also, we have

Ker(E(n) → F_{a}(n+1)) = Ker(E(n) → F_{b}(n+1)), ∀a, b ∈ {1, …, p−3} and

Ker(E(n+1) → F_{a}(n+1)) = Ker(E(n+1) → F_{b}(n+1)), ∀a, b ∈ {2, …, p−2}. Hence (H1)-(H2) for F_{i} = F_{i+d+1}(n) are obtained from (C1) and (C2) for F_{i} = F_{d+i}(n) and the long exact sequences in 5.2:

Im(F_{a}(n) → E(n)) = Ker(E(n) → F_{a−1}(n + 1)), ∀a ∈ {2, …, p − 2} and

Im(F_{a}(n) → E(n + 1)) = Ker(E(n + 1) → F_{a+1}(n + 1)), ∀a ∈ {1, …, p − 3} .

5.7. Thus, by induction on d and using Lemma 5.1, (C4) and (C5) are true for all d ∈ {1, …, p − 4}. In particular, (C5) for d = p − 4, in which case F_{2} = F_{p−2}(n) = L(n), together with the last part of (C4), for any integer d ∈ {1, …, p − 4}, give the first two claims in Proposition 5.3. 5.8. The last ingredient we need to continue the induction on n is that the sequence E(n) → E(n + 1) → E(n + 2) is exact. These maps are given by multiplication from the right by τ_{φ} in Yoneda’s notation. To prove the exactness of the sequence at E(n + 1), we fix a decomposition G = H ⊕ Y, where Y has order p, and we explicitly decompose E(n + 1) with respect to E(m) for m ≤ n and to E_{H}^{G}(m) := σ_{G/Y}^{G} Iso_{H}^{G/Y} E_{H}(m) for m ≤ n + 1. 5.9. Recall that we have the following properties:

• E(n + 1) ≅ L(n)τ_{L,φ} ⊕ ⨁_{X∈X} E(n − 1)γ_{X} ⊕ E_{H}^{G}(n + 1), where we have set

X = K_{G}(H) \ {Y }.

• the multiplication by γ_{X} induces an injective map from E(m) to E(m + 2).

• E(n) → L(n) → E(n) is exact and dim L(n) = dim E(n).

• the multiplication by τ_{L,φ} induces an injective map from L(m − 1) to E(m).

5.10. Let’s concentrate on L(n)τ_{L,φ}. It is easy to check that

Im(E(n) → L(n))τ_{L,φ} = E(n)τ_{φ} .

Moreover from the last diagram of the induction on d we have

L(n) = Im(E(n − 1) → L(n)) + Im(E(n) → L(n)) .

In terms of Yoneda’s composition of extensions, this becomes

L(n)τ_{L,φ} = E(n)τ_{φ} + E(n − 1)τ_{φ,L}τ_{L,φ} = E(n)τ_{φ} + E(n − 1)( ∑_{X∈K_{G}(H)} γ_{X}) .

The above sum is not direct but we have a description of the intersection of the two terms. This is given in the following proposition.

5.11. Proposition :

(#) := E(n)τ_{φ} ∩ E(n − 1)s_{φ}γ_{X}
= E(n − 2)τ_{φ}s_{φ}
= E(n − 2)s_{φ}τ_{φ}

Proof : By the induction started in 5.4, the kernel of the multiplication from the left by τ_{φ} from E(n) to E(n+1) is E(n−1)τ_{φ}. Hence dim(E(n−2)s_{φ}τ_{φ}) = dim (E(n − 2)s_{φ}) − dim (E(n − 1)τ_{φ} ∩ E(n − 2)s_{φ})

Also by induction on n one can assume that E(n − 1)τ_{φ} ∩ E(n − 2)s_{φ} = E(n − 3)τ_{φ}s_{φ} and that the multiplication by s_{φ} gives an injective map. Thus

dim (E(n − 2)s_{φ}τ_{φ}) = dim (E(n − 2)s_{φ}) − dim (E(n − 3)τ_{φ}s_{φ})
= dim E(n − 2) − dim (E(n − 3)τ_{φ})
= dim E(n − 2)τ_{φ}

where the last equality is given by the exactness of the sequence

E(n − 3) → E(n − 2) → E(n − 1)

with maps given by multiplication from the right by τ_{φ}. Thus we get that the multiplication from the right by s_{φ} gives a bijection between E(n − 2)τ_{φ} and E(n − 2)τ_{φ}s_{φ}.

Moreover E(n−2)τ_{φ}s_{φ} is contained in the intersection (#) which enables the following computation:

dim (E(n − 2)τ_{φ}s_{φ}) ≤ dim(#)
= dim E(n)τ_{φ} + dim E(n − 1)s_{φ} − dim L(n)τ_{L,φ}
= dim E(n)τ_{φ} + dim E(n − 1)s_{φ} − dim L(n)
= dim E(n − 1)s_{φ} − dim E(n − 1)τ_{φ}
≤ dim E(n − 1) − dim E(n − 1)τ_{φ}
= dim E(n − 2)τ_{φ}

We proved that the first and the last expression in the sequence are equal so all the inequalities are equalities and we get (#) and, moreover, that dim E(n − 1) = dim E(n − 1)s_{φ}.

5.12. Making use of the previous two propositions we get the following decomposition of E(n + 1) as a direct sum :

E(n + 1) ≅ [E(n)τ_{φ} + E(n − 1)( ∑_{X∈K_{G}(H)} γ_{X})] ⊕ ⨁_{X∈X} E(n − 1)γ_{X} ⊕ E_{H}^{G}(n + 1).

5.13. Using this, by induction on n one can get the following direct sum decomposition, which has the advantage that for every term, the multiplications from the right by monomials of type s_{φ}^{α}γ_{X}^{δ}τ_{φ}^{ε}, for α ≥ 0, for δ, ε ∈ {0, 1}, and (α, δ) ≠ (0, 0), induce bijective maps from E(n + 1 − 2α − 2δ − ε) to E(n + 1) :

E(n + 1) ≅ E_{H}^{G}(n + 1)  ⊕ E_{H}^{G}(n)τ_{φ}
⊕ E_{H}^{G}(n − 1)s_{φ}  ⊕ E_{H}^{G}(n − 2)τ_{φ}s_{φ}
⊕ E_{H}^{G}(n − 3)s_{φ}^{2}  ⊕ E_{H}^{G}(n − 4)τ_{φ}s_{φ}^{2}
⊕ …
⊕ ⨁_{X∈X} E(n − 1)γ_{X} ⊕ ⨁_{X∈X} E(n − 2)γ_{X}τ_{φ}
⊕ ⨁_{X∈X} E(n − 3)s_{φ}γ_{X} ⊕ ⨁_{X∈X} E(n − 4)s_{φ}γ_{X}τ_{φ}
⊕ ⨁_{X∈X} E(n − 5)s_{φ}^{2}γ_{X} ⊕ ⨁_{X∈X} E(n − 6)s_{φ}^{2}γ_{X}τ_{φ}
⊕ …

5.14. Now, for every group homomorphism ψ : G → k^{+} with Ker ψ ≠ H = Ker φ, the restriction from E(m) to E_{H}(m) gives an isomorphism between the multiplication by τ_{ψ} on E_{H}^{G}(m) and the multiplication by τ_{Res_{H}^{G}(ψ)}. Hence the short exact sequence E_{H}(m) → E_{H}(m + 1) → E_{H}(m + 2) given by multiplication from the left by τ_{Res_{H}^{G}(ψ)} induces a short exact sequence

E_{H}^{G}(m) → E_{H}^{G}(m + 1) → E_{H}^{G}(m + 2)

given by the multiplication from the left by τ_{ψ}. Summing all up we get an exact sequence E(n) → E(n + 1) → E(n + 2) given by the multiplication from the left by τ_{ψ}. 5.15. The duality sends the multiplication from the left by τ_{ψ} to multiplication from the right by τ_{ψ} and we can exchange the roles of φ and ψ to get that the exact sequence E(n) → E(n + 1) → E(n + 2) given by multiplication from the right by τ_{φ} is exact for any homomorphism φ : G → k^{+}. This finishes the induction step on n and the proof of Proposition 5.3. 5.16. Remark that for p = 3 we have τ_{φ}^{2} = s_{φ} thus, also in this case, τ_{φ} and s_{φ} commute. Now, using in Corollary 3.10 the fact that L(n) = E(n) we get the following decomposition as a direct sum

E(n + 1) ≅ E(n)τ_{φ} ⊕ ⨁_{X∈X} E(n − 1)γ_{X} ⊕ E_{H}^{G}(n + 1) .

Decomposing E(n) in the same way we get

E(n+1)≅E(n−1)s_{φ}⊕⨁_{X∈X}E(n−2)γ_{X}τ_{φ}⊕E_{H}^{G}(n)τ_{φ}⊕⨁_{X∈X}E(n−1)γ_{X}⊕E_{H}^{G}(n+1)

Continuing the replacement, we get the same decomposition as in 5.13. 5.17. Following the proof of Theorem 14.2 in [2] we get now the Poincaré series for Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) for all odd p. Indeed, the only ingredients needed in that proof are the exactness of

0 ⟶ L(n − 1) ⊕ ⊕_{X∈X} E_{G}(n − 2) → E_{G}(n) → E_{H}(n) → 0 .

and that dim E_{G}(n) = dim L(n). These facts are proved in Corollary 3.10, respectively in Proposition 5.3.

6. A presentation of the algebra E

We are now able to give a presentation of the algebra E = Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}). Our aim is to present an algbra A that, a priori, has E as quotient and then to show that the two algebra are isomorphic. Let r be the rank of G and 0 = H_{0} < H_{1} < ⋯ < H_{r−1} = H < H_{r} = G be a maximal flag in G. We choose a direct sum decomposition of G

G = Y_{1} ⊕ Y_{2} ⊕ ⋯ ⊕ Y_{r−1} ⊕ Y_{r} ,

where Y_{i} is a complement of H_{i−1} in H_{i}. Let φ_{i} : G → 𝔽_{p} be a non-trivial morphism with kernel ⨁_{{1≤j≤r \atop j≠i}} Y_{j}. For every i ∈ \{1, …, r\} we define an atom

τ̂_{i} of degree 1, and for every i ∈ {1, …, r} and for every subgroup X ≤ G of order p, such that XH_{i−1} = H_{i}, we say that X has position i, and we define an atom γ̂_{X} of degree 2.

Consider the set A = {τ̂_{i}|1 ≤ i ≤ r} ∪ {γ̂_{X}|X ≤ G, |X| = p}. A word with atoms in A is obtained in the natural way, by concatenation. The degree of a word is the sum of the degrees of its atoms. Denote by pos X the position of X. We say that a word is of type i if the atoms appearing in its decomposition are among τ̂_{i} and γ̂_{X} with pos X = i. The empty word is of type i for all i ∈ {1, …, r}. 6.1. For all n ≥ 0, let A_{n} be the set of words of degree n with the property that they don’t contain any of the following sequences of two consecutive atoms:

(S1) τ̂_{i}τ̂_{j}, for 1 ≤ j ≤ i ≤ r; (S2) τ̂_{i}γ̂_{X}, for 1 ≤ pos X < i ≤ r;

(S3) γ̂_{Y_{i}} τ̂_{i}, for 1 ≤ i ≤ r;

(S4) γ̂_{X} γ̂_{Y}, for 1 ≤ pos Y < pos X ≤ r;

(S5) γ̂_{X} τ̂_{i}, for 1 ≤ i < pos X ≤ r.

We call pre-admissible the words w that can be written as a product w = w_{1}w_{2} … w_{r} where, for 1 ≤ i ≤ r, the word w_{i} is of type i. We call admissible the pre-admissible words that do not contain any of the sequences τ̂_{i}τ̂_{i} and γ̂_{Y_{i}} τ̂_{i}.

6.2. Proposition : Let w be a word of degree n. Then w belongs to the union of the A_{n}’s for all n if and only if it is admissible.

Proof : The forbidden sequences (S1) for i ≠ j, (S2), (S4) and (S5) imply that for j < i, the atoms τ̂_{i} and γ̂_{X} with pos X = i only appear after τ̂_{j} and γ̂_{Y} with pos Y = j. This gives a pre-admissible form w_{1}w_{2} … w_{r} for w. Moreover, in each w_{i}, the sequences τ̂_{i}τ̂_{i} and γ̂_{Y_{i}} τ̂_{i} are explicitly forbidden by (S1) for i = j and (S3).

6.3. Proposition : Let p be an odd prime and A be the graded k-algebra with generators

{τ̂_{i}|1 ≤ i ≤ r} in degree 1, and {γ̂_{X}|X ≤ G, |X| = p} in degree 2,

subject to the relations

(R1) τ̂_{i}τ̂_{i} = 0, if p ≥ 5, or

τ̂_{i}τ̂_{i} = − ∑_{X≰Ker φ_{i}} γ̂_{X} if p = 3.

(R2) τ̂_{i}τ̂_{j} + τ̂_{j}τ̂_{i} = 0 for 1 ≤ j ≤ i ≤ r, if p ≥ 5, or

τ̂_{i}τ̂_{j} + τ̂_{j}τ̂_{i} = ∑_{X≰Ker(φ_{i}+φ_{j})} γ̂_{X} − ∑_{X≰Ker φ_{i}} γ̂_{X} − ∑_{X≰Ker φ_{j}} γ̂_{X} if p = 3.

(R3) [τ̂_{i}, ∑_{X≰Ker φ_{i}} γ̂_{X}] = 0;

(R4) [φ_{j}(x)τ̂_{i} − φ_{i}(x)τ̂_{j}, γ̂_{⟨x⟩}] = 0, for 1 ≤ i < j ≤ 1, x ∈ G;

(R5) [γ̂_{X}, ∑_{Y <Q} γ̂_{Y}] = 0, for all X < Q ≤ G, |X| = p, |Q| = p^{2}.

Then for all n ≥ 1, A_{n} is a basis of the k-vector space of elements of degree n

in A.

6.4. Remark : In the case p = 3, Relation (R3) is implied by Relation (R1).

The relations (R1) to (R5) are rather technical. This comes from the need to have a finite, easy to order set of generators, appropriate for induction arguments. We can give a more intuitive presentation for the algebra A if we allow ourselves to increase the set of generators, by taking a generator τ̂_{φ} for every group homomorphism φ : G → 𝔽_{p}^{+}. Here we identify τ̂_{φ_{i}} with τ̂_{i}. The relations are then given by

(L1) τ̂_{φ} + τ̂_{ψ} = τ̂_{φ+ψ}.

(L2) τ̂_{φ}τ̂_{φ} = 0 and [τ̂_{φ}, ∑_{X≰Ker φ} γ̂_{X}] = 0, if p ≥ 5, or

τ̂_{φ}τ̂_{φ} = − ∑_{X≰Ker φ} γ̂_{X} if p = 3.

(L3) [τ̂_{φ}, γ̂_{X}] = 0, for X ≤ Ker φ, |X| = p.

(L4) [γ̂_{X}, ∑_{Y <Q} γ̂_{Y}] = 0, for all X < Q ≤ G, |X| = p, |Q| = p^{2}.

We leave it as an easy exercise to the reader to verify that the two presentation lead to isomorphic algebras. These two presentations yield Theorem 1.3 and 1.2, respectively.

To prove Proposition 6.3, we need the following sequence of lemmas.

6.5. Lemma : Let l < m and let X and Y be subgroups of order p of G, such that pos X = m and pos Y = l. Then we have γ̂_{X} γ̂_{Y} = γ̂_{Y} γ̂_{X} +W where W is a sum of admissible words γ̂_{X′}γ̂_{X′′} of type m and degree 4.

Proof : Firstly we prove that all the subgroups of order p of ⟨X, Y ⟩, different from Y, have position m. Indeed, if x is a generator of X and y is a generator of Y, then denote by X_{c} the subgroup of ⟨X, Y ⟩ generated by x + cy for 1 ≤ c ≤ p − 1. As cy ∈ Y ≤ H_{m−1} and XH_{m−1} = H_{m} we get that X_{c}H_{m−1} = H_{m} so X_{c} has position m.

Secondly we use Relation (R5) with Q = ⟨X, Y ⟩ to get

γ̂_{X} γ̂_{Y} + ∑_{c=1}^{p−1} γ̂_{X} γ̂_{X_{c}} = ∑_{c=1}^{p−1} γ̂_{X_{c}} γ̂_{X} + γ̂_{Y} γ̂_{X}

Besides γ̂_{X} γ̂_{Y}, all the words appearing in the above equation are admissible and, excepting γ̂_{Y} γ̂_{X}, they are all of type m. The result follows.

6.6. Lemma : Let l < m and X of order p with pos X = m. Then γ̂_{X} τ̂_{l} = τ̂_{l}γ̂_{X} + W, where W is a linear combination of pre-admissible words of type m and degree 3.

Proof : Let x be a generator of X. If we set α := φ_{l}(x) and β := φ_{m}(x), relation (R4) gives that

γ̂_{X}(βτ̂_{l} − ατ̂_{m}) = (βτ̂_{l} − ατ̂_{m})γ̂_{X}

As τ̂_{l}γ̂_{X}, γ̂_{X} τ̂_{m} and τ̂_{m}γ̂_{X} are all pre-admissible words of degree 3, the latter two being of type m, and β ≠ 0 the result follows. The fact that we only get pre-admissibility (and not admissibility) comes from the fact that we might have X = Y_{m}.

6.7. Lemma : Every word in w ∈ A can be written as a finite sum of pre-admissible words, i.e. w = ∑_{j=1}^{s} w_{j1}w_{j2} … w_{jr}, with w_{ji} of type i for 1 ≤ i ≤ r (the w_{ji}’s are allowed to be the empty word). Moreover, if w is composed of atoms of position at least l, then the w_{ji}’s are empty for all i < l

and j = 1, 2, …, s.

Proof : We define a quadruple lexicographic order O on the set of words in A first by their degree then, for equal degrees, by the number of atoms of degree 1 (i.e. the τ̂_{i}’s) appearing in the word, then by the degree of the left term in the decomposition uv, where v is pre-admissible, of maximal possible degree, then, finally, by the number of atoms in v of higher position than the rightmost atom in u. We prove the lemma by induction on the order O. We’ll see in the proof that the order by the number of atoms of degree 1 appearing in the word is only needed for the case p = 3.

The lemma is trivial for words of degree 1 and straight forward, using Relations (R1) and (R2), for words of degree 2. Now take a word w ∈ A of degree d ≥ 3. Write w = uv, a concatenation, with v pre-admissible, of maximal possible degree. If deg u = 0, i.e. u is empty, then w is preadmissible and we are done. If the number of atoms in v of higher position than the rightmost atom in u is 0 or the position of the leftmost atom in v is n then w = v and we are again done. It is very important to remark that all the manipulations we make in the proof, using the relations in Proposition 6.3, are not increasing the number of atoms of order 1. Suppose now the lemma is true for all words smaller than w for the order O. Let a be the rightmost atom of u i.e. u = w′a. If u ≠ a then av is of degree smaller than w so by the induction hypothesis on the degree av = ∑_{j=1}^{s} w_{j} with w_{j} preadmissible. Moreover w′w_{j} is s maller than w with respect to the order O, for all j = 1, 2, …, s, so, by the induction hypothesis, w′w_{j} is a sum of preadmissible elements. Also by the induction hypothesis, the atoms appearing in the new words are of higher or equal position than the ones appearing in w. Thus we can suppose that u = a is an atom of position m so u = τ̂_{m} or u = γ̂_{X} with pos X = m.

Case 1. Let u = γ̂_{X} with X of position m. We distinguish two sub-cases:

i) v = γ̂_{Y} v′ with Y of position l and l < m. Then, using Lemma 6.5, we have γ̂_{X}v = γ̂_{X} γ̂_{Y} v′ = γ̂_{Y} γ̂_{X}v′ + W v′. We have that γ̂_{X}v′ is smaller than w, hence, by the induction hypothesis, it is equal to a sum of preadmissible words, composed of atoms of position at most l. Thus the words obtained from these by multiplying to the left by γ̂_{Y} are still preadmissible. The expression W v′ is a sum of words of type γ̂_{X′}γ̂_{X′′}v′ with the first two atoms of position m. Again, by the induction hypothesis γ̂_{X′′}v′ is equal to a sum of pre-admissible words w_{i}, which, moreover, have less atoms of lower position than m than γ̂_{Y} v′. Thus γ̂_{X′}w_{i} is smaller than w for the order O and, by the induction hypothesis, the former is equal to a sum of pre-admissible words.

ii) v = τ̂_{l}v′ with l < m. Then, using Lemma 6.6, there exists a linear combination W of words of type m such that γ̂_{X}v = γ̂_{X} τ̂_{l}v′ = ατ̂_{l}γ̂_{X}v′+ W v′, with α ∈ k. Proceeding analogously to the part i) above we obtain that both τ̂_{l}γ̂_{X}v′ and W v′ are equal to linear combinations of pre-admissible words.

Case 2. Let u = τ̂_{m}. We distinguish two sub-cases.

i) v = γ̂_{Y} v′ with pos Y = l and l < m. Then Y ≤ Ker φ_{m} and, using Relation (R4), we have τ̂_{m}γ̂_{Y} v′ = γ̂_{Y} τ̂_{m}v′. Moreover τ̂_{m}v′ is of degree smaller than w so, by the induction hypothesis, τ̂_{m}v′ is a linear combination of pre-admissible words having only atoms of position greater or equal than l. So multiplying by γ̂_{Y} to the left still keeps these words pre-admissible.

ii) v = τ̂_{l}v′ with l < m. Then, for p ≥ 5, using Relation (R2) we have τ̂_{m}v = τ̂_{m}τ̂_{l}v′ = −τ̂_{l}τ̂_{m}v′ and the result as in part i) of Case 2. When p = 3 the proof is more difficult. Using also Relation (R2), we have

τ̂_{m}v = τ̂_{m}τ̂_{l}v′ = ∑_{X≰Ker(φ_{m}+φ_{l})} γ̂_{X}v′− ∑_{X≰Ker φ_{m}} γ̂_{X}v′− ∑_{X≰Ker φ_{l}} γ̂_{X}v′−τ̂_{l}τ̂_{m}v′.

Moreover pos X > l for all γ̂_{X} appearing in the right hand side term of this equality. Now τ_{m}v′ has smaller degree than w, so, by the induction hypothesis, it is equal to a linear combination of pre-admissible words. The other words appearing in the right hand side term have all a smaller number number of atoms of degree 1 than w, so, by the induction hypothesis, they are all equal to linear combinations of pre-admissible words. Remark here that Case 1 and Case 2 i) are complet ely solving the case of words having at most one atom of degree 1, without needing the Case 2 ii). So there is no problem in starting the induction process.

Proof : [of Proposition 6.3] Firstly we prove that A_{n} is a set of generators in degree n. Using Lemma 6.7 we have that any word of degree n can be written as a linear combination of pre-admissible words w_{1}w_{2} … w_{r}, i.e. with w_{i} of type i for 1 ≤ i ≤ r. Let w = w_{1}w_{2} … w_{r} be one of these words. By Lemma 6.2, it remains to show that, possibly using new linear combinations, we also eliminate the expressions τ_{i}τ_{i} and γ̂_{Y_{i}} τ̂_{i} that might appear in the words w_{i}. Let i be lowest index such that w_{i} has forbidden sequences, with the convention that i = r + 1 is w contains no forbidden sequence. We define a triple lexicographic order O on the words, given by decreasing order on the index i defined previously, then by the number of atoms of position i and, finally, by the decreasing order of the degree of the its admissible prefix of maximal degree. We proceed by induction on O. If the degree of the word is at most one or there are no atoms of degree 1 or the word is already admissible, then there is nothing to prove. Else, denote w_{<i} := w_{1}w_{2} … w_{i−1} and by w_{>i} := w_{i+1}w_{i+2} … w_{r}. Let f be the leftmost forbidden sequence in w_{i}. There are two kinds of forbidden sequences to consider.

i) f = τ̂_{i}τ̂_{i}. Write w = u_{1}fu_{2}. Using Relation (R1), we have f = 0 when p ≥ 5 and f = ∑_{X≰Ker φ_{i}} γ̂_{X} when p = 3. So, the case p ≥ 5 is trivial

,and in the following we study the case p = 3. Now for X ≰ Ker φ_{i} we have pos X ≥ i. If pos X = i then the word w_{<i}u_{1}γ̂_{X} doesn’t have any forbidden sequence, and, by induction hypothesis, γ̂_{X}u_{2}w_{>i} is equal to a linear combination of admissible words. Suppose now that pos X > i. Using Lemma 6.7, γ̂_{X}u_{2}w_{>i} is equal to a linear combination of preadmissible words with atoms of position at least i. Let v_{i} … v_{r} be one of these words. Then w_{<i}u_{1}v_{i} … v_{r} is a pre-admissible word with less atoms position i than w. By the induction hypothesis, w_{<i}u_{1}v_{i} … v_{r} is equal to a linear combination of admissible words.

ii) f = γ̂_{Y_{i}} τ̂_{i}. Write w = u_{1}fu_{2}. Relation (R3) gives

γ̂_{Y_{i}} τ̂_{i} = τ̂_{i} ∑_{X≰Ker φ_{i}} γ̂_{X} + ∑_{X≰Ker φ_{i}−{Y_{i}}} γ̂_{X} τ̂_{i} .

As H_{i−1} ≤ Ker φ_{i} we have that any X ≰ Ker φ_{i} is of position at least i. The words w_{<i}u_{1}τ̂_{i}γ̂_{Y_{i}}, w_{<i}u_{1}τ̂_{i}γ̂_{X} and w_{<i}u_{1}γ̂_{X} τ̂_{i} with X of position i contain no forbidden sequence and are of degree bigger than w_{<i}u_{1}, so, by the induction hypothesis these words multiplied by w_{>i} are equal to linear combinations of admissible words. In the case of the u_{1}τ̂_{i}γ̂_{X} and u_{1}γ̂_{X} τ̂_{i} with X of position greater than i, using Lemma 6.7, γ̂_{X} τ̂_{i}u_{2}w_{>i} and τ̂_{i}γ̂_{X}u_{2}w_{>i} are equal to a linear combination of pre-admissible words with atoms of position at least i. Let v_{i} … v_{r} be one of these words. Then w_{<i}u_{1}v_{i} … v_{r} has less atoms of position i than w, hence, by the induction hypothesis, it is equal to a linear combination of admissible words.

This proves that the algebra A is generated as k-module in degree n by A_{n}.

Secondly, to prove the k-linear independence of the words in A_{n}, we construct k-algebra homomorphism Θ from A onto E, by setting: Θ(τ̂_{i}) := τ_{φ_{i}}, Θ(γ̂_{X}) := γ_{X}. This is indeed a surjective algebra homomorphism given that the relations in A between the k-generators are all satisfied by their images through Θ inside Ext(S_{1}, S_{1}) as showed in Lemma 4.9, Lemma 4.10, Lemma 4.13 and Lemma 4.14 in this paper, and in [2, Proposition 12.9].

6.8. This gives a presentation by generators and relations of E as stated in Conjecture 3.7.

6.9. Theorem : The morphism Θ is an algebra isomorphism from A to

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

Proof : Θ : A → Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) is surjective and, in every degree n, the algebra Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}) has at least the dimension over k of A^{n}. Hence Ext_{M_{k}^{c}(G)}^{n}(S_{1}^{G}, S_{1}^{G}) and A^{n} have the same dimension over k and Θ is an isomorphism.

A. Arithmetics of extensions in an abelian category

This Appendix contain a series of classical results on computations in the graded algebra of extensions in an abelian category. We present the general framework and include the proofs of those results.

Let A be an abelian category. Consider following exact sequences in A

\xymatrix{0 \ar[r] & X \ar[r] & A_{1} \ar[r] & A_{2} \ar[r] & … \ar[r] & A_{n} \ar[r] & Y \ar[r] & 0}

and

\xymatrix{0 \ar[r] & X \ar[r] & B_{1} \ar[r] & B_{2} \ar[r] & … \ar[r] & B_{n} \ar[r] & Y \ar[r] & 0}

representing the elements η, ξ ∈ Ext_{A}^{n}(Y, X). Then η + ξ is represented by the exact sequence

\xymatrix{0 \ar[r] & X \ar[r] & C_{1} \ar[r] & A_{2} ⊕ B_{2} \ar[r] & … \ar[r] & A_{n−1} ⊕ B_{n−1} \ar[r] & C_{n} \ar[r] & Y \ar[r] & 0}

where C_{1}, respectively C_{n}, is pushout, respectively pullback, of the following diagrams.

\xymatrix{X ⊕ X \ar[r] \ar[d]_{π_{1}+π_{2}} & A_{1} ⊕ B_{1} \ar[d] \\ X \ar[r] & C_{1}} , \xymatrix{A_{n} ⊕ B_{n} \ar[r] & Y ⊕ Y \\ C_{n} \ar[r] \ar[u] & Y \ar[u]_{Δ}} .

When n = 1 there is an ambiguity of this construction, given by the order in which we take the pullback and the pushout. In fact, both choices lead to equivalent exact sequences. To show this we need the following technical lemma in abelian categories. Recall that if X is the pullback of the diagram

\xymatrix{X \ar[r] \ar[d] & Y \ar[d]^{f} \\ Z \ar[r]^{g} & T}

one also says that the square (X, Y, Z, T) is cartesian and we have that X is

isomorphic to the kernel of \xymatrix{Y ⊕ Z \ar[r]^{f−g} & T} . Analogously, if T is the pushout of the diagram

\xymatrix{X \ar[r]^{s} \ar[d]^{t} & Y \ar[d] \\ Z \ar[r] & T}

one also says that the square (X, Y, Z, T) is co-cartesian and and we have that T is isomorphic to the cokernel of \xymatrix{X \ar[r]^{s⊕t} & Y ⊕ Z} .

A.1. Lemma : Let A be an abelian category, and let

[figure: 0; A; a; α; B; b; β; C; c; 0; 0; D; δ; E; ε; F; 0; 0; L; φ; l; λ; M; m; ψ; μ; N; χ; n; 0; 0; P; τ; π; Q; σ; θ; R; ρ; 0]

be a commutative diagram with exact rows, such that the square (A, B, D, E) is co-cartesian, (Q, R, E, F) and (M, N, B, C) are cartesian, and the morphism n : N → R is an isomorphism.

Then there exists a unique morphism m : M → L such that nμ = θm and bψ = σm. Moreover, the square (L, M, P, Q) is commutative, and cocartesian, and the morphisms τ, φ and c are isomorphisms.

Proof : Via the universal properties of pullback/pushout and using that the horizontal sequences are exact, one gets that φ, τ and c are isomorphisms, (A, B, D, E) is cartesian and, (Q, R, E, F) and (M, N, B, C) are co-cartesian. Now set u = bψ and v = nμ. Then

εu = εbψ = cβψ = cχμ = ρnμ = ρv .

Since the square (Q, R, E, F) is cartesian, there exists a unique morphism m : M → Q such that

u = σm and v = θm .

Now setting s = mλ and t = πl, we have

σs = σmλ = uλ = bψλ = bαφ = δaφ = δτl = σπl = σt .

Moreover

θs = θmλ = vλ = nμλ = 0 ,

and

θt = θπl = 0 .

Thus θs = θt, and σs = σt. As the square (Q, R, E, F) is cartesian, this implies s = t, hence the square (L, M, P, Q) is commutative.

Now let X be any object of A, and let f : X → M and g : X → P be morphisms such that πg = mf. Setting w = ψf and r = τg, we have that

bw = bψf = σmf = σπg = δτg = δr .

As (A, B, D, E) is cartesian, there is a unique morphism y : X → A such that

w = αy and r = ay .

Setting z = φ^{−1}y : X → L, we have that

lz = lφ^{−1}y = τ^{−1}ay = τ^{−1}r = g .

Similarly

ψλz = ψλφ^{−1}y = αφφ^{−1}y = αy = w = ψf .

Moreover

μλz = 0 ,

and

nμf = θmf = θπg = 0 .

On the other hand

χμf = βψf = βαy = 0 .

As the morphism (χ, n) : N → C ⊕ R is an monomorphism, it follows that μf = 0 = μλz. Since moreover ψλz = ψf, and since (M, N, B, C) is cartesian, it follows that λz = f.

If there is another morphism z′ : X → L such that lz′ = g and λz′ = f, then the morphism x = z − z′ is such that lx = 0 and λx = 0. Then

αφx = ψλx = 0 ,

and

aφx = τlx = 0 .

Since (A, B, D, E) is cartesian, it follows that φx = 0, hence x = 0.

This shows that the square (L, M, P, Q) is cartesian. Thus, to show that this square is also co-cartesian it is enough to have that \xymatrix{P ⊕ M \ar[r]^{m+π} & Q} is an epimorphism. To prove this, suppose that we have Y ∈ A and f : Q → Y such that fπ = 0 and fm = 0. Then f factors through the cokernel of π so there exist g : R → Y such that f = gθ. Then 0 = fm = gθm = gnμ. As μ is an epimorphism and n is an isomorphism, the morphism nμ is also an epimorphism, so g = 0 and, thus f = 0.

When considering A as a module category we have an explicit description of the sum in Ext_{A}^{1}(Y, X).

A.2. Lemma : Let η, ξ ∈ Ext_{A}^{1}(Y, X). When constructing a representative for η + ξ both choices for the order pullback, pushout lead to the same construction. Moreover, if \xymatrix{0 \ar[r] & X \ar[r]^{i} & A \ar[r]^{s} & Y \ar[r] & 0} is a representative

for η and \xymatrix{0 \ar[r] & X \ar[r]^{j} & B \ar[r]^{t} & Y \ar[r] & 0} is a representative of ξ. Then a representative for η + ξ is given by \xymatrix{0 \ar[r] & X \ar[r] & C \ar[r] & Y \ar[r] & 0} with

C ≃ {(a, b, x) ∈ A⊕B⊕X|s(a) = t(b)}/{(i(x_{1}), j(x_{2}), −x_{1}−x_{2})|x_{1}, x_{2} ∈ X} .

Proof : To simplify notation, in the proof will be understood that a, b, x, y run through A, B, X, respectively Y. Also let

I = {(i(x_{1}), j(x_{2}), −x_{1} − x_{2})|x_{1}, x_{2} ∈ X} .

Consider \xymatrix{0 \ar[r] & X \ar[r] & C_{ob} \ar[r] & Y \ar[r] & 0} be the representative for η + ξ given by taking first the pushout and then the pullback. We have

\xymatrix{X ⊕ X \ar[r]^{i⊕j} \ar[d]_{π_{1}+π_{2}} & A ⊕ B \ar[d] \\ X \ar[r] & C′} , \xymatrix{C′ \ar[r]^{{ \atop s⊕t}} & Y ⊕ Y \\ C_{ob} \ar[r] \ar[u] & Y \ar[u]_{Δ}} .

with C′ ≃ A ⊕ B ⊕ X/I and, thus,

C_{ob} ≃ \{(c, y)|c ∈ C′, y ∈ Y, { \atop s ⊕ t} = Δ(y)\} ≃ \{(a, b, x)|s(a) = t(b)\}/I .

Similarly, consider \xymatrix{0 \ar[r] & X \ar[r] & C_{bo} \ar[r] & Y \ar[r] & 0} be the representative for η + ξ given by taking first the pullback and then pushout. We have

\xymatrix{A ⊕ B \ar[r]^{s⊕t} & Y ⊕ Y \\ C′′ \ar[r] \ar[u] & Y \ar[u]_{Δ}} , \xymatrix{X ⊕ X \ar[r]^{{ \atop i⊕j}} \ar[d]_{π_{1}+π_{2}} & C′′ \ar[d] \\ X \ar[r] & C_{bo}} .

with C′′ ≃ {(a, b, y)|s(a) = y = t(b)} ≃ {(a, b)|s(a) = t(b)} and, thus, C_{bo} ≃ {(a, b, x)|s(a) = t(b)}/I. Hence we obtain C_{ob} = C_{bo} and there is no ambiguity in the construction of a representative of η + ξ.

We introduce here a notation we extensively use in the paper. Let

\xymatrix{X \ar[r]^{f} & A \ar[r]^{g} & Y} be an exact sequence at A. Then we write A ≃ \binom{Coker f}{Ker g}.

In particular, we have a short exact sequence

\xymatrix{0 \ar[r] & X \ar[r] & \binom{Y}{X} \ar[r] & Y \ar[r] & 0} .

and a representative in Ext_{A}^{2}(Y, X) can be written as

\xymatrix{0 \ar[r] & X \ar[r] & \binom{Z}{X} \ar[r] & \binom{Y}{Z} \ar[r] & Y \ar[r] & 0} .

With this notation we have an easy way to compute the exact sequence representing the sum in Ext_{A}^{2}(Y, X). The computation is an easy consequence of the following technical lemma whose proofs are left to the reader.

A.3. Lemma : Let A be an abelian category, and let

\xymatrix{0 \ar[r] & X ⊕ X \ar[d]_{a} \ar[r]^{α} & \binom{A}{X} ⊕ \binom{B}{X} \ar[d]_{b} \ar[r]^{β} & A ⊕ B \ar[d]_{c} \ar[r] & 0 \\ 0 \ar[r] & X \ar[r]^{δ} & C \ar[r]^{ε} & A ⊕ B \ar[r] & 0}

be a commutative diagram with exact rows, such that the square on the left

is co-cartesian. Then c is an isomorphism and, hence, C ≃ \binom{A ⊕ B}{X}.

and its dual

A.4. Lemma : Let A be an abelian category, and let

\xymatrix{0 \ar[r] & A ⊕ B \ar[r]^{α} & \binom{Y}{A} ⊕ \binom{Y}{B} \ar[r]^{β} & Y ⊕ Y \ar[r] & 0 \\ 0 \ar[r] & A ⊕ B \ar[r]^{δ} \ar[u]_{f} & C′ \ar[r]^{ε} \ar[u]_{g} & Y \ar[u]_{h} \ar[r] & 0}

be a commutative diagram with exact rows, such that the square on the right

is cartesian. Then f is an isomorphism and, hence, C′ ≃ \binom{Y}{A ⊕ B}.

Remark that, in the above notation, \binom{A ⊕ B}{X} has \binom{A}{X} and \binom{B}{X} as subobjects and \binom{Y}{A ⊕ C} has \binom{Y}{A} and \binom{Y}{C} as factors. We conclude with

the computation of the exact sequence representing the sum in Ext_{A}^{2}(Y, X).

A.5. Lemma : Let η, ξ ∈ Ext_{A}^{2}(Y, X) be represented by

\xymatrix{0 \ar[r] & X \ar[r] & \binom{Z_{1}}{X} \ar[r] & \binom{Y}{Z_{1}} \ar[r] & Y \ar[r] & 0} ,

respectively by

\xymatrix{0 \ar[r] & X \ar[r] & \binom{Z_{2}}{X} \ar[r] & \binom{Y}{Z_{2}} \ar[r] & Y \ar[r] & 0} .

Then η + ξ is represented by

\xymatrix{0 \ar[r] & X \ar[r] & \binom{Z_{1} ⊕ Z_{2}}{X} \ar[r] & \binom{Y}{Z_{1} ⊕ Z_{2}} \ar[r] & Y \ar[r] & 0} .

Proof : Apply Lemma A.3 and Lemma A.4.

References

[1] M. Artin, A. Grothendieck, and J.-L. Verdier. Séminaire de Géométrie Algébrique du Bois-Marie, 1963-1964, Théorie des topos et cohomologie étale des schémas - (SGA 4) - Vol 1, volume 269. Springer, 1972.

[2] S. Bouc. Complexity and cohomology of cohomological Mackey functors. Adv. in Maths. 221: 983-1045, 2009.

[3] A. Dress. Contributions to the theory of induced representations, volume 342 of Lecture Notes in Mathematics, pages 183–240. Springer-Verlag, 1973.

[4] J. Green. Axiomatic representation theory for finite groups. J. Pure Appl. Algebra, 1:41–77, 1971.

[5] J. Thévenaz and P. Webb. The structure of Mackey functors. Trans. Amer. Math. Soc., 347(6):1865–1961, June 1995.

Serge Bouc - CNRS-LAMFA - Université de Picardie - 33, rue St Leu - 80039 - Amiens - France.

Email : serge.bouc@u-picardie.fr

Radu Stancu - LAMFA - Université de Picardie - 33, rue St Leu - 80039 - Amiens - France.

Email : radu.stancu@u-picardie.fr