Extensions of simple cohomological Mackey functors

Serge Bouc

Abstract: This is a report on some recent joint work with Radu Stancu, to appear in [4]. It is an expanded version of a talk given at the RIMS workshop Cohomology of finite groups and related topics, February 18-20, 2015.

1. Cohomological Mackey functors

1.1. Let G be a finite group, and k be a commutative ring. There are many equivalent definitions of Mackey functors for G over k. For the “naive” one, this is an assignment H ↦ M(H) of a k-module M(H) to any subgroup H of G, together with k-linear maps

M(H) ⟶^{t_{H}^{K}} M(K) ⟶^{r_{H}^{K}} M(H), M(H) ⟶^{c_{x,H}} M(^{x}H)

whenever H ≤ K ≤ G and x ∈ G, subject to a list of compatibility conditions, e.g. transitivity of transfers and restrictions, or the Mackey formula (see [6] for details).

A Mackey functor M is called cohomological if

∀H ≤ K ≤ G, t_{H}^{K} ∘ r_{H}^{K} = |K : H|Id_{M(K)} .

The cohomological Mackey functors for G over k form a category M_{k}^{c}(G). 1.2. Examples :

• Let V be a kG-module. The fixed points functor F P_{V} is defined by M(H) = V^{H}, for any H ≤ G, and by

∀H ≤ K ≤ G, r_{H}^{K} : V^{K} ↪ V^{H}, t_{H}^{K} = Tr_{H}^{K} : V^{H} → V^{K} ,

and by c_{x,H}(v) = x · v, for x ∈ G.

More generally, for n ∈ ℕ, the cohomology functor H^{n}(−, V) is a cohomological Mackey functor.

• Let k be a field of characteristic p, let G be a finite p-group. The simple cohomological Mackey functors for G over k are the functors S_{X} = S_{X}^{G}, where X ≤ G (up to G-conjugation), defined by

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

1.3. Yoshida’s Theorem

• Let perm_{k}(G) denote the full subcategory of kG-Mod consisting of finitely generated permutation kG-modules.

• Let Fun_{k}(G) denote the category of (contravariant) k-linear functors from perm_{k}(G) to k-Mod.

• If M ∈ M_{k}^{c}(G), the functor M̃ : V ↦ Hom_{M_{k}^{c}(G)}(F P_{V} , M) is an object of Fun_{k}(G).

1.4. Theorem [Yoshida [7]] : The functor M ↦ M̃ is an equivalence of categories from M_{k}^{c}(G) to Fun_{k}(G).

1.5. The (cohomological) Mackey algebra

• [Thévenaz-Webb [6]] The (cohomological) Mackey functors for G over k are exactly the modules over the (cohomological) Mackey algebra.

• Consider the Hecke algebra Y_{k}(G) = End_{kG}( ⊕_{H≤G} kG/H). This k-

algebra is called the Yoshida algebra of G over k. It is isomorphic to the cohomological Mackey algebra. In other words, the category M_{k}^{c}(G) is equivalent to Y_{k}(G)-Mod.

• The algebra Y_{k}(G) is a free k-module of rank ∑_{H,K≤G} |H\G/K|. In

particular, when k is a field, the algebra Y_{k}(G) is a finite dimensional k-algebra.

2. Complexity

Let k be a field, and A be a finite dimensional k-algebra. Then every finitely generated A-module M admits a resolution

⋯ → P_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → M → 0

by finitely generated projective A-modules. 2.1. Definition : The module M has polynomial growth if there exists such a resolution and numbers c, d, e such that ∀n ∈ ℕ, dim_{k} P_{n} ≤ cn^{d} + e. The lower bound of such d’s is called the complexity of M.

The module M has exponential growth if for any such resolution, there exist numbers c > 0, d > 1, and e such that ∀n ∈ ℕ, dim_{k} P_{n} ≥ cd^{n} + e.

The module M has intermediate growth in all other cases.

2.2. Lemma [Link with extensions] : 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 of representatives of isomorphism classes of simple A-modules, and P_{S} denotes a projective cover of S.

2. In particular M has polynomial growth ⟺ ∀S ∈ Irr(A), ∃(c, d, e) such that ∀n ∈ ℕ, dim_{k} Ext_{A}^{n}(M, S) ≤ c n^{d} + e .

3. Let 0 → L → M → N → 0 be a short exact sequence of finitely generated A-modules. If any two of L, M, N have polynomial growth, so does the third.

2.3. Definition [Poco groups] : Let k be a field of positive characteristic p. A finite group G is called a poco group over k if any finitely generated cohomological Mackey functor for G over k has polynomial growth.

2.4. Theorem [B. [3]] : Let G be a finite group, and k be be a field of characteristic p > 0. The following conditions are equivalent:

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

2. Let S be a Sylow p-subgroup of G. Then :

• If p > 2, the group S is cyclic.

• If p = 2, the group S has sectional rank at most 2.

2.5. Remark : A 2-group has sectional rank at most 2 if and only if it is cyclic or metacyclic (Blackburn [2], Andersen-Oliver-Ventura [1]).

3. Construction of functors

3.1. Let k be a field of characteristic p > 0, and G be a finite group. By Yoshida’s equivalence M_{k}^{c}(G) ≅ Fun_{k}(G), cohomological Mackey functors for G over k can be viewed as functors

perm_{k}(G) ⟶ k-Mod .

When H is another finite group, any k-linear functor

F : perm_{k}(H) ⟶ perm_{k}(G)

induces a functor

M_{k}^{c}(G) ≅ Fun_{k}(G) ⟶ Fun_{k}(H) ≅ M_{k}^{c}(H) .

3.2. In particular, when U is a (finite) (G, H)-biset, the functor

t_{U} : W ∈ perm_{k}(H) ↦ kU ⊗_{kH} W ∈ perm_{k}(G)

induces a functor L_{U} : M_{k}^{c}(G) → M_{k}^{c}(H). Similarly, the functor

h_{U} : W′ ∈ perm_{k}(G) ↦ Hom_{kG}(kU, W′) ∈ perm_{k}(H)

induces a functor R_{U} : M_{k}^{c}(H) → M_{k}^{c}(G).

3.3. Properties • The functors L_{U} and R_{U} are exact.

• As t_{U} is left adjoint to h_{U}, the functor L_{U} is left adjoint to R_{U}.

• Let U′ be another finite (G, H)-biset. Then

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

• Let Id_{G} denote the identity (G, G)-biset. Then L_{Id_{G}} and R_{Id_{G}} are isomorphic to the identity functor.

• If K is another finite group, and V is an (H, K)-biset, then

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

3.4. Examples

• Let H be a subgroup of G, and U denote the set G, as a (G, H)-biset. Then L_{U} ≅ Res_{H}^{G}, and R_{U} ≅ Ind_{H}^{G}.

• Let H be a subgroup of G, and U denote the set G, as an (H, G)-biset. Then L_{U} ≅ Ind_{H}^{G}, and R_{U} ≅ Res_{H}^{G}.

• Let N ⊴ G, let H = G/N, and let U denote the set H, as a (G, H)- biset. Then L_{U} = ρ_{G/N}^{G}, and R_{U} = ȷ_{G/N}^{G}.

• Let N ⊴ G, let H = G/N, and let U denote the set H, as an (H, G)- biset. Then L_{U} = ı_{G/N}^{G}, and R_{U} = ρ_{G/N}^{G}.

• Let f : G → H be a group isomorphism, and U denote the set H, as a (G, H)-biset. Then L_{U} ≅ Iso(f) and R_{U} ≅ Iso(f^{−1}).

3.5. Sketch of proof of Theorem 2.4

Recall that k is a field of characteristic p > 0, that G is a finite group, and S is a Sylow p-subgroup of G.

• Use the functors Ind_{S}^{G} and Res_{S}^{G} to reduce to the case where G = S is a p-group.

• Let (B, A) be a section of G (i.e. A ⊴ B ≤ G). The set G/A is a (G, B/A)-biset, and the set A\G is a (B/A, G)-biset. The corresponding functors L_{G/A}, R_{G/A}, L_{A\G} and R_{A\G} allow for a reduction to the case where G is elementary abelian.

• The case of cyclic groups and Klein four group was settled by M. Samy Modeliar ([5]). In particular, these groups are poco groups.

• Describe the subfunctor structure of Ind_{H}^{G}S_{1}^{H}, leading to long exact sequences of Ext groups. These sequences show that the functor S_{1}^{G} has exponential growth if G ≅ (C_{p})^{m}, when p > 2 and m ≥ 2, or p = 2 and m ≥ 3.

• Use induction on the order of a 2-group G, to complete the case p = 2.

4. Presentation of some Ext algebras

Let p be a prime number, and G ≅ (C_{p})^{n}, n ≥ 1.

• Let X ≤ G with |X| = p. Then there exists a unique non split extension α_{X}^{G} : 0 → S_{1}^{G} → \binom{S_{X}^{G}}{S_{1}^{G}} → S_{X}^{G} → 0 in M_{𝔽_{p}}^{c}(G).

Let γ_{X}^{G} ∈ Ext_{M_{𝔽p}^{c}(G)}^{2}(S_{1}^{G}, S_{1}^{G}) denote the class of the splice

α_{X}^{G}(α_{X}^{G})^{∗} : 0 → S_{1}^{G} → \binom{S_{X}^{G}}{S_{1}^{G}} → \binom{S_{1}^{G}}{S_{X}^{G}} → S_{1}^{G} → 0 .

• When p > 2 and φ : G → 𝔽_{p} is a group homomorphism, let U_{φ}^{G} be the vector space 𝔽_{p} ⊕ 𝔽_{p}, on which g ∈ G acts by g(x, y) = (x + φ(g)y, y). There is a unique (cohomological) Mackey functor T_{φ}^{G} for G over 𝔽_{p} such that T_{φ}(H) = {0} if 1 < H ≤ G, and T_{φ}^{G}(1) ≅ U_{φ}^{G}. It fits in an extension

0 → S_{1}^{G} → U_{φ}^{G} → S_{1}^{G} → 0

in M_{𝔽_{p}}^{c}(G). Let τ_{φ}^{G} ∈ Ext_{M_{𝔽p}^{c}(G)}^{1}(S_{1}^{G}, S_{1}^{G}) denote the class of this extension.

4.1. The algebra E_{k} = Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G})

4.2. Theorem [B. Stancu [4]] : Let k be a field of characteristic p > 0, and G ≅ (C_{p})^{n}. Let E_{k} denote the algebra Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}). Then:

1. The extension of scalars from 𝔽_{p} to k induces an isomorphism of kalgebras E_{k} ≅ k ⊗_{𝔽_{p}} E_{𝔽_{p}}.

2. The algebra E_{𝔽_{p}} is generated by the elements γ_{X}^{G}, where X ≤ G with |X| = p, together, when p > 2, with the elements τ_{φ}^{G}, where

φ : G → 𝔽_{p}^{+}.

4.3. Presentation of E_{k} for p = 2

4.4. Theorem [B. [3]] : Let k be a field of characteristic 2, and G ≅ (C_{2})^{m}. Then the graded algebra E_{k} = Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) admits the following presentation:

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

• The relations are the following:

1. If H < G with |G : H| = 2, then ∑_{x∉H} γ_{x} = 0.

2. If x and y are distinct elements of G − {0}, then

[γ_{x} + γ_{y}, γ_{x+y}] = 0 .

4.5. Presentation of E_{k}, for p > 2

4.6. Theorem [B. Stancu [4]] : Let k be a field of characteristic p > 2, and G ≅ (C_{p})^{m}. Then the graded algebra E_{k} = Ext_{M_{k}^{c}(G)}^{∗}(S_{1}^{G}, S_{1}^{G}) admits the following presentation:

1. The generators are the elements γ_{X} in degree 2, for X ≤ G such that |X| = p, and the elements τ_{φ} in degree 1, for φ ∈ Hom(G, 𝔽_{p}^{+}).

2. The relations are the following:

(a) τ_{φ+ψ} = τ_{φ} + τ_{ψ}, for any φ, ψ in Hom(G, 𝔽_{p}^{+}).

(b) • If p ≥ 5, then τ_{φ}^{2} = 0 and [τ_{φ}, ∑_{X≰Kerφ} γ_{X}] = 0, for any φ in

Hom(G, 𝔽_{p}^{+}).

• If p = 3, then τ_{φ}^{2} = − ∑_{X≰Kerφ} γ_{X}, for any φ ∈ Hom(G, 𝔽_{p}^{+}).

(c) [γ_{X}, τ_{φ}] = 0, for any φ ∈ Hom(G, 𝔽_{p}^{+}), for any X ≤ Kerφ with

|X| = p.

(d) [γ_{X}, ∑_{{Y ≤Q \atop |Y |=p}} γ_{Y}] = 0, for any Q ≤ G with |Q| = p^{2} and any X ≤ Q

with |X| = p.

4.7. Corollary : The Poincaré series of E_{k} is equal to

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

when p = 2, and 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})}

when p is odd.

4.8. Corollary : Let k be a field of characteristic p > 0. When G is an elementary abelian p-group, one can compute explicitly all the extension groups between any two simple cohomological Mackey functors for G over k.

Acknowledgements: I wish to thank Professor Fumihito Oda for his invitation, and Kinki University, Osaka for support. I also thank Professor Akihiko Hida for the opportunity of giving a talk at the RIMS workshop Cohomology of finite groups and related topics, February 18-20, 2015.

References

[1] K. Andersen, B. Oliver, and J. Ventura. Reduced fusion systems over 2-groups. Preprint, 2008.

[2] N. Blackburn. Generalizations of certain elementary theorems on pgroups. Proc. London Math. Soc., 11:1–22, 1961.

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

[4] S. Bouc and R. Stancu. The extension algebra of some cohomological Mackey functors. Preprint, 2015, Adv. in Maths., to appear, 2015.

[5] M. Samy Modeliar. Certaines constructions liées aux foncteurs de Mackey cohomologiques. PhD thesis, Université Paris 7-Denis Diderot, 2005.

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

[7] T. Yoshida. On G-functors (II): Hecke operators and G-functors. J. Math. Soc. Japan, 35:179–190, 1983.

Serge Bouc, CNRS-LAMFA, 33 rue St Leu, 80039 Amiens Cedex 01, France.

email:  serge.bouc@u-picardie.fr