A DESCRIPTION OF THE SECOND MOD p COHOMOLOGY GROUP OF A p-GROUP

BRUNO KAHN AND PHAM ANH MINH

Contents

Introduction 1 1. Representability 4 2. Group-theoretic relations 7 3. p-filtered groups and mixed Lie algebras 8 4. Truncation 13 5. An example 20 6. Truncation, extensions and cohomology 21 7. Example: the case n = 1 25 8. The case n = 2 26 9. Proof of theorem 8.7 28 10. Enriching mixed Lie algebras? 30 References 35

Introduction

Let p be a prime number and G be a p-group. The Frattini filtration (also called lower p-series) of G is defined recursively as follows:

Φ_{1}G = G; Φ_{n+1}G = (Φ_{n}G)^{p}[G, Φ_{n}G] for n ≥ 1.  (1)

Let gr_{n}^{Φ} G = Φ_{n}G/Φ_{n+1}G be the associated graded of the Frattini filtration: this is a graded F_{p}-vector space, where F_{p} is the field with p elements. But gr^{Φ} G = ⊕ gr_{n}^{Φ} G is endowed with two extra structures:

• Bilinear mappings [,] : gr_{m}^{Φ} G × gr_{n}^{Φ} G → gr_{m+n}^{Φ} G, i, j ≥ 1, induced by the commutator map in G;

• a degree 1 operator q : gr_{n}^{Φ} G → gr_{n+1}^{Φ} G, induced by “raising to the p-th power” in G.

These data are subject to the following relations:

1. [,] is alternating and satisfies the Jacobi identity; 2. for x, y ∈ gr_{n}^{Φ} G,

Date: October 9, 1995.

q(x + y) = \begin{cases}q(x) + q(y) & if p > 2 or n > 1 \\ q(x) + q(y) + [x, y] & if p = 2 and n = 1; \end{cases}

3. for (x, y) ∈ gr_{m}^{Φ} G × gr_{n}^{Φ} G,

[q(x), y] = \begin{cases}q([x, y]) & if p > 2 or m > 1 \\ q([x, y]) + [x, [x, y]] & if p = 2 and m = 1. \end{cases}

In other terms, gr^{Φ} G defines a graded mixed Lie algebra in the sense of Lazard [13]. If f : G → H is a group homomorphism, f induces a homomorphism

gr^{Φ} f : gr^{Φ} G → gr^{Φ} H

which commutes with [,] and q.

Those mixed Lie algebras of the form gr^{Φ} G have one further property: they are generated in degree 1 (as algebras with operators). Denoting by (pgroups) the category of p-groups, by ML the category of finite dimensional mixed Lie algebras and by GML the full subcategory of mixed Lie algebras generated in degree 1, we therefore have a functor

\begin{matrix}gr^{Φ} : (p − groups) & → & GML \\ G & ↦ & gr^{Φ} G. \end{matrix}

This functor is of great interest in the study of the structure of p-groups. It is natural to ask:

Question. Is gr^{Φ} essentially surjective?

In other terms, given a (finite dimensional) mixed Lie algebra A generated in degree 1, is there a p-group G such that gr^{Φ} G is isomorphic to A?

The aim of this paper is to give a partial answer to this question. Let us say that a p-group G has p-class ≤ n if Φ_{n+1}G = {1}, and that a mixed Lie algebra A = (A_{n})_{n≥1} has length ≤ n if A_{i} = 0 for i > n. Let (p-groups)^{≤n} (resp. GML^{≤\}) be the full subcategory of (p-groups) (resp. GML) consisting of p-groups of p-class ≤ n (resp. algebras of length ≤ n). Then gr^{Φ} restricts to a functor

gr_{≤n}^{Φ} : (p − groups)^{≤n} → GML^{≤\}

for all n ≥ 1. Trivially, gr_{≤1}^{Φ} is essentially surjective (it is an equivalence of categories!) and it is a (nontrivial) folk result that gr_{≤2}^{Φ} is essentially surjective. Our main result is:

COHOMOLOGY OF A p-GROUP 3

Theorem. gr_{≤3}^{Φ} is essentially surjective.

Actually we prove a little more than this. As Lazard, we consider the category (F il) of p-filtered groups (definition 3.1) and prove that the cor-

responding functor (F il)^{≤3} ⟶^{gr^{≤3}} ML^{≤∋} is essentially surjective (theorem 8.5). This says that any mixed Lie algebra of length 3 (generated in degree 1 or not) is the associated graded of some p-filtered group.

This paper is organised as follows. Section 1 is devoted to some facts on categories and functors, that we use extensively in this paper. Categories and functors are for us both technically and conceptually convenient. In section 3, we essentially recall Lazard’s elementary theory of p-filtered groups and mixed Lie algebras, in the special case of integral filtrations. In section 4 we introduce our main tool: the functor s_{n}, left adjoint to the functor “truncation at order n” on mixed Lie algebras. In section 6 we relate s_{n} to group extensions, hence to degree-2 cohomology. In sections 8 and 9 we state and prove the main theorem. Its original proof by the second author was a difficult cohomological computation (cf remark 8.8); here it is replaced by a more elementary group-theoretic proof, whose heart is Lazard’s theorem that the Frattini algebra of a free group is free. Finally, in section 10, we examine how to ‘enrich’ the mixed algebra structure in order to refine a version of gr^{≤n} into an equivalence of categories, at least for n ≤ 3, using structures reminiscent of the secondary cohomology operations of algebraic topology; this section is somewhat inconclusive.

The proof of the key result of this paper, theorem 8.7, as well as certain constructions, are performed here by ad hoc methods, so to say “by hand”. It seems likely, however, that all this should be explained by relations between the (co)homology of p-groups and that of mixed Lie algebras. We hope to develop the latter theory and investigate these relations in a future paper.

Notation and terminology. 1) Throughout the paper, we fix a prime number p.

2) As in [7], [8], for a group G and g, h ∈ G, we write [g, h] = g^{−1}h^{−1}gh and g^{h} = h^{−1}gh.

3) There are two opposite conventions for the use of the term “extension”, in group theory and homological algebra. We choose the latter convention. So, in this paper, a group G is an extension of H by N if N is a normal subgroup of G with quotient H, or equivalently if there is a short exact sequence 1 → N → G → H → 1.

4) We denote by H^{∗}(G, A) the cohomology of a group G with coefficients in a G-module A. When A = Z/p with trivial action, we sometimes abbreviate this notation into H^{∗}(G) or H^{∗}G.

The reader should be careful that we use the word “complete” in two different contexts: complete categories (definition 1.1) and complete p-filtered groups (definition 3.3).

1. Representability

In this section, we recall some well-known and not-so-well-known facts about categories and functors. The main reference we use is Mac Lane’s book [14], except for terminology which is sometimes a little different and more relaxed. Notably we note Hom_{C}(A, B) rather than Mac Lane’s C(A, B) for the set of morphisms in a category C. We also prefer the more traditional ‘inverse limit’ to Mac Lane’s ‘limit’.

1.1 Definition. A category C is complete if all inverse limits (indexed by a small category) are representable in C.

1.2 Proposition ([14, cor. 2 to th. V.2.2]). A category C is complete if and only if products and equalisers of two arrows A ⇉ B are representable.

1.3 Definition. Let C and D be two categories and T : C → D a functor.

a) T is faithful if, for all X, Y ∈ C, the map Hom_{C}(X, Y) → Hom_{D}(T (X), T (Y)) is injective.

b) T is full if, for all X, Y ∈ C, the map Hom_{C}(X, Y) → Hom_{D}(T (X), T (Y)) is surjective.

c) T is essentially surjective if, for all Y ∈ D, there exists X ∈ C such that T (X) is isomorphic to Y.

d) T is an equivalence of categories if there exists U : D → C such that U ∘ T and T ∘ U are naturally isomorphic respectively to Id_{C} and Id_{D}.

Traditionally, we say that a functor is fully faithful (or faithfully full) if it is both full and faithful.

1.4 Proposition ([14, th. IV.4.1]). A functor is an equivalence of categories if and only if it is full, faithful and essentially surjective.

1.5 Definition. Let C be a category and X a class of objects of C. The full subcategory of C determined by X is the unique subcategory D of C such that:

• Ob(D) = X; • for X, Y ∈ X, Hom_{D}(X, Y) = Hom_{C}(X, Y). The inclusion C ↪ D is then fully faithful.

1.6 Definition. Let C and D be two categories and T : C → D, U : D → C be two functors.

a) T is left adjoint to U if there exists a natural transformation η : Id_{C} → UT such that, for all (X, Y) ∈ C × D, the composite map

Hom_{D}(T (X), Y) ⟶Hom_{C}(UT (X), U(Y)) ⟶^{η^{∗}} Hom_{C}(X, U(Y))

is bijective. We also say that U is right adjoint to T. This η is called the unit of the adjunction.

b) T is a section of U if UT is naturally equivalent to Id_{C}. We also say that U is a retraction of T.

1.7 Lemma ([14, th. IV.1.2]). T is left adjoint to U if and only if there exists a natural transformation ε : T U → Id_{D} such that the composite map

Hom_{C}(X, U(Y)) ⟶Hom_{D}(T (X), T U(Y)) ⟶^{ε_{∗}} Hom_{D}(T (X), Y)

is bijective. If this is the case, this bijection coincides with that of definition 1.6. The transformation ε is called the counit of the adjunction.

1.8 Definition. Let C, D, T, U be as in definition 1.6.

a) T is a left adjoint-right inverse of U if it is left adjoint to U and is a section of U through the unit of the adjunction.

a) T is a left adjoint-left inverse of U if it is left adjoint to U and is a retraction of U through the counit of the adjunction.

1.9 Definition ([14, p. 150]). A functor T : C → D reflects isomorphisms if, for any morphism u of C, u is an isomorphism ⟺ T (u) is an isomorphism.

1.10 Lemma. A full functor T : C → D reflects isomorphisms if and only if, for any X ∈ C, T^{−1}(Id_{T (X)}) ⊆ Aut_{C}(X).

Proof. Necessity is clear. To see sufficiency, let X, Y ∈ C and let u ∈ Hom_{C}(X, Y) be such that T (u) is an isomorphism. By the fullness of T, there exists v ∈ Hom_{C}(Y, X) such that uv ∈ T^{−1}(Id_{T (Y)}) and vu ∈ T^{−1}(Id_{T (X)}). By the hypothesis, uv and vu are isomorphisms. In particular, u is left and right-invertible, hence is an isomorphism. □

We finally state and prove a proposition which looks classical, but that we haven’t been able to trace in the literature. As the proof shows, the hypothesis that the base ring is a field is not necessary except for b), where we could restrict to projective modules in the general case. Compare [19] and [14, p. 193]. Let (Sets) (resp. (Ab)) denote the category of sets (resp. of abelian groups).

1.11 Proposition. Let F be a field and V ect_{F} the category of vector spaces over F.

a) Any additive functor T : V ect_{F} → (Ab) refines uniquely into an F -linear endofunctor of V ect_{F} followed by the forgetful functor V ect_{F} → (Ab).

b) Any F -linear endofunctor T of V ect_{F} is of the form V ↦ T_{0} ⊗_{F} V for some T_{0} ∈ V ect_{F}.

c) Let S, T be two F -linear endofunctors of V ect_{F}, given by S(V) = S_{0} ⊗_{F} V and T (V) = T_{0} ⊗_{F} V after b). Denote by S_{|(Sets)}, T_{|(Sets)} their composition with the forgetful functor V ect_{F} → (Sets). Then, in the sequence

Hom_{F} (S_{0}, T_{0}) → Hom_{V ect_{F}} (S, T) → Hom_{(Sets)}(S_{|(Sets)}, T_{|(Sets)}),

both maps are bijective.

d) Let T be an F -linear endofunctor of V ect_{F} and S a subfunctor of T_{|(Sets)} (i.e. S(V) is a subset of T (V) for all V), which commutes with finite products. Then S refines into a unique subfunctor of T (i.e. S(V) is a subvector space of T (V) for all V).

Proof. a) Let V ∈ V ect_{F}. Any λ ∈ F defines an endomorphism of V, hence by functoriality an endomorphism T (λ) of T (V). For λ, λ′ ∈ F, we clearly have T (λλ′) = T (λ)T (λ′) and (by additivity) T (λ + λ′) = T (λ) + T (λ′). Hence T (V) inherits a structure of an F -vector space, which is obviously natural in V.

b) Set T_{0} = T (F); let V ∈ V ect_{F}. Any v ∈ V defines an F -linear map

\begin{matrix}F & ⟶ & V \\ λ & ⟼ & λv \end{matrix}

hence an F -linear map

T_{0} ⟶T (V)

which is obviously linear in v. So we have defined a map

V → Hom_{F} (T_{0}, T (V))

which corresponds by adjunction to a map

T_{0} ⊗_{F} V → T (V).

For V = F, this map is the identity, hence, by additivity of T, it is an isomorphism for V = F^{(I)}, all I. Since every F -vector space has a basis, it is an isomorphism for all V.

c) The bijectivity of the first map is clear, as is the injectivity of the second one. To see its surjectivity, let η : S_{|(Sets)} → T_{|(Sets)} be a natural transformation. We want to show that, for all V, η_{V} is F -linear. It is enough to show that it is additive, as it commutes with the action of F by naturality. This follows from the commutative diagram

S(V ⊕ V) ⟶^{η_{V ⊕V}} T (V ⊕ V)
≀↓  ^{≀}↓
S(V) ⊕ S(V) ⟶^{η_{V} ×η_{V}} T (V) ⊕ T (V)
Σ_{S(V )}↓  ^{Σ_{T (V )}}↓
S(V)  ⟶^{η_{V}}  T (V).

where, for any W ∈ V ect_{F}, Σ_{W} : W ⊕ W → W is the addition map, by observing that the big and top square commute and that compositions of the two columns are respectively S(Σ_{V}) and T (Σ_{V}).

d) It is enough to show that, for all V, S(V) is closed under addition. This follows from the commutative diagram (analogous to the above):

S(V) × S(V)⟶T (V) ⊕ T (V)
≀↑  ^{≀}↑
S(V ⊕ V) ⟶ T (V ⊕ V)
S(Σ_{v})↓  ^{T (Σ_{v})}↓
S(V) ⟶ T (V).

1.12 Remark. Part of proposition 1.11 can be stated as follows: in the sequence

V ect_{F} ⟶^{α} Hom_{F} (V ect_{F} , V ect_{F}) ⟶^{β} Hom_{Z}(V ect_{F} , (Ab)) ⟶^{γ} Hom(V ect_{F} , (Sets))

where Hom_{F} (resp. Hom_{Z}) denotes F -linear (resp. additive) functors, α(T_{0}) = T_{0} ⊗_{F} − and β, γ are induced by the forgetful functors, α and β are equivalences of categories (a quasi-inverse of α being T ↦ T (F)) and γ is fully faithful.

2. Group-theoretic relations

In this section, we record four well-known relations that we shall use in the sequel:

\begin{aligned}[a, bc] & = & [a, c][a, b]^{c} \\ [b, a] & = & [a, b]^{−1} \end{aligned}  (2)

[7, Hilfsatz II.1.2]

\begin{aligned}(ab)^{p} & ≡ & a^{p}b^{p} (mod γ_{2}(H)^{p}γ_{p}(H)) \\ (ab)^{2} & = & a^{2}b^{2}[a, b^{−1}]^{b^{2}} \end{aligned}  (3)

where H =< a, b > and γ_{∗} is the descending central series [8, lemma VIII.1.1]

[[a, b^{−1}], c]^{b}[[b, c^{−1}], a]^{c}[[c, a^{−1}], b]^{a} = 1  (4)

(Witt’s identity) [7, Satz III.1.4]

\begin{aligned}[a, b^{p}] & ≡ & [a, b]^{p} (mod γ_{2}(H)^{p}γ_{p}(H)) \\ [a, b^{2}] & = & [a, b]^{2}[[a, b], b] \end{aligned}  (5)

where H is as above [8, lemma VIII.1.1].

3. p-filtered groups and mixed Lie algebras

3.1 Definition. 1. Let G be a group. A p-filtration on G is a nonincreasing sequence of normal subgroups (G^{(n)})_{n≥1} such that

a) G^{(1)} = G;

b) [G^{(m)}, G^{(n)}] ⊆ G^{(m+n)};

c) G^{(n)p} ⊆ G^{(n+1)}. A p-filtered group is a group provided with a p-filtration.

2. Let (G, (G^{(n)})) and (H, (H^{(n)})) be two p-filtered groups. A p-morphism between them is a group homomorphism f : G → H such that f(G^{(n)}) ⊆ H^{(n)} for all n ≥ 1. We denote by (F il) the category whose objects are p-filtered groups and morphisms are p-morphisms.

3.2 Remark. These definitions coincide with Lazard’s [13, II.1.1.1 and II.1.2.10], except that we only consider integral p-filtrations here.

3.3 Definition. A p-filtered group (G, (G^{(n)})) is complete if the natural homomorphism G → lim_{⟵} G/G^{(n)} is an isomorphism. We denote by (F ilc) the full subcategory of (F il) consisting of complete filtered groups.

3.4 Remark. The assignment G ↦ Ĝ = lim_{⟵} G/G^{(n)} defines a functor (F il) → (F ilc) (the completion functor). This functor is evidently a left adjoint-left inverse of the inclusion functor (F ilc) ⊂ (F il).

3.5 Example. For any group G, the Frattini filtration (1) of G (see introduction) is a p-filtration on G [8, th. VIII.1.5]. If G is finitely generated, it is complete if and only if it is a pro-p-group.

3.6. Let (groups) be the category of groups. There is a forgetful functor

\begin{matrix}(F il) & ⟶ & (groups) \\ (G, (G^{(n)})) & ⟼ & G \end{matrix}

3.7 Theorem. The forgetful functor of 3.6 has as a left adjoint-right inverse the functor Φ : G ↦ (G, (Φ_{n}G)), where Φ_{n}G is the Frattini filtration of example 3.5.

Proof. We have to check an equality, for G ∈ (groups) and (H, (H^{(n)})) ∈ (F il):

Hom_{(groups)}(G, H) = Hom_{(F il)}((G, (Φ_{n}G)), (H, (H^{(n)}))).

By forgetting p-filtrations, the right set is contained in the left one. Conversely, the axioms a), b) and c) of definition 3.1 and the definition of the Frattini filtration show that any homomorphism form G to H maps Φ_{n}G to H^{(n)} for all n ≥ 1. Finally, Φ obviously is also a right inverse of the forgetful functor.

3.8. Let G, H be two objects of (F il) and r an integer ≥ 0. We define an equivalence relation R_{∇} on Hom_{(F il)}(G, H) as follows:

u ≡ v (mod R_{∇}) ⟺ for all⟩ ≥ ∞ and} ∈ G^{(⟩)}, ⊑(})^{−∞}⊓(}) ∈ H^{(⟩+∇)}.

It is readily checked that this indeed defines an equivalence relation, which is compatible with composition of filtered morphisms.

Denote by (F il)_{r} the category whose objects are p-filtered groups and, for G, H two p-filtered groups, Hom_{(F il)_{r}}(G, H) = Hom_{(F il)}(G, H)/R_{∇}. We therefore have an inverse system of full “forgetful” functors:

… ⟶ (F il)_{r} ⟶ … (F il)_{r−1} ⟶ … ⟶ (F il)_{1} ⟶ (F il)_{0}

where (F il)_{0} is the trivial category where all Hom sets have one element. For æsthetical reasons, one should probably introduce the limit category

(F̂ il)

with the same objects and, for G, H two p-filtered groups,

Hom_{(F̂ il)}(G, H) = lim_{⟵} Hom_{(F il)_{r}}(G, H).

Note that the canonical functors (F il) ⟶ (F il)_{r} induce a ‘completion’ functor (F il) ⟶ (F̂ il) such that the composite (F ilc) ↪ (F il) → (F̂ il) is fully faithful. In practice this refinement will not be necessary here, as we shall almost only consider groups of finite p-length.

Proposition 3.9 shows that one does not lose too much information by passing from (F il) to (F il)_{1}:

3.9 Proposition. The functor (F ilc) → (F ilc)_{1} (hence also all functors (F ilc) → (F ilc)_{r} and (F ilc)_{r} → (F ilc)_{s}, r ≥ s) reflects isomorphisms.

Proof. Since the functor is full, it suffices by lemma 1.10 to show:

3.10 Lemma. Let G ∈ (F ilc) and u ∈ End_{(F ilc)}(G) be such that u ≡ Id_{G} (mod R_{∞}). Then u is an automorphism of G (as a filtered group).

It is enough to see that the induced morphism u_{n} ∈ End(G/G^{(n)}) is bijective for any n. We do this by induction on n, the case n = 1 being trivial. Suppose u_{n} is surjective. Let g ∈ G/G^{(n+1)} and h ∈ G/G^{(n+1)} be such that u_{n+1}(h) ≡ g (mod G^{(n)}). By assumption on u, u_{n+1}(u_{n+1}(h)^{−1}g) = u_{n+1}(h)^{−1}g, hence g ∈ Im u_{n+1} and u_{n+1} is surjective. Let g ∈ G such that u(g) ∈ G^{(n+1)}. By induction on n, g ∈ G^{(n)}. By assumption on u, g^{−1}u(g) ∈ G^{(n+1)} and g ∈ G^{(n+1)}, so u_{n+1} is injective. □

We also have:

3.11 Theorem. The categories (F il), (F ilc), (F il)_{r} and (F ilc)_{r} (r ≥ 0) are all complete.

Proof. Clearly, arbitrary products are representable in all four categories; by proposition 1.2 it suffices to see that equalisers of two morphisms are also representable. Let C = (F⟩↕), (F⟩↕⌋), (F⟩↕)_{∇} or (F ilc)_{r} (r ≥ 0),

G, H ∈ C and G ⇉_{v}^{u} H be two morphisms in C. For i ≥ 1, define

K^{(i)} = \begin{cases}\{g ∈ G^{(i)} | v(g)^{−1}u(g) ∈ H^{(i+r)}\} & if C = (F⟩↕)_{∇} or (F ilc)_{r} \\ \{g ∈ G^{(i)} | v(g) = u(g)\} & if C = (F⟩↕) or (F ilc) \end{cases}

and K = K^{(1)}. If φ : M → G is a morphism in C, then u ∘ φ = v ∘ φ in C if and only if φ(M^{(i)}) ⊆ K^{(i)} for all i ≥ 1. To show that (K, (K^{(i)})) is an equaliser for (u, v), it is then sufficient to check that (K^{(i)}) is a pfiltration of K. We do it in the case of (F il)_{r} and (F ilc)_{r} (the case of (F il) and (F ilc) is similar and easier). To check [K^{(i)}, K^{(j)}] ⊆ K^{(i+j)}, take (g, h) ∈ K^{(i)} × K^{(j)}. Setting α = v(g)^{−1}u(g) and β = v(h)^{−1}u(h) we have α ∈ H^{(i+r)}, β ∈ H^{(j+r)} and, using (2):

u([g, h]) = [u(g), u(h)] = [v(g)α, v(h)β]
= [v(g)α, β][v(g), v(h)]^{βα}[α, v(h)]^{β}
= [v(g)α, β]v([g, h])^{βα}[α, v(h)]^{β}

hence

v([g, h])^{−1}u([g, h]) = [v(g)α, β]^{v([g,h])}[v([g, h]), βα][α, v(h)]^{β} ∈ H^{(i+j+r)}.

To prove that g^{p} ∈ K^{(i+1)}, we simply use the trivial estimate

(xy)^{p} ≡ x^{p}y^{p} (mod [A, B])

for (x, y) ∈ A × B in some ambiant group (with B normal, say). Applying this with (x, y, A, B) = (v(g), α, H^{(i)}, H^{(i+r)}), we get

u(g^{p}) = u(g)^{p} = (v(g)α)^{p} ≡ v(g)^{p}α^{p} (mod [H^{(i)}, H^{(i+r)}])

hence v(g^{p})^{−1}u(g^{p}) ∈ H^{(i+r+1)}, since [H^{(i)}, H^{(i+r)}] ⊆ H^{(2i+r)}, α^{p} ∈ H^{(i+r+1)} and i + r + 1 ≤ 2i + r.

To finish the proof in the case C = (F⟩↕⌋) or (F ilc)_{r}, we have to check that K is complete. Since K^{(i)} ⊆ G^{(i)} for all i, the filtration (K^{(i)}) is separated. Let (k_{i}) be a sequence of elements of K such that k_{i} ≡ k_{i+1} (mod K^{(i)}) for all i. Since G is complete, there exists g ∈ G such that g ≡ k_{i} (mod G^{(i)}) for all i. Taking i = r + 1, this implies

v(g)^{−1}u(g) = v(g_{r+1})^{−1}v(k_{r+1})^{−1}u(g_{r+1})u(k_{r+1})

= (v(k_{r+1})^{−1}u(k_{r+1}))^{v(g_{r+1})}v(g_{r+1})^{−1}u(g_{r+1})

where g_{r+1} = k_{r+1}^{−1}g ∈ G^{(r+1)}, hence g ∈ K. □

3.12 Remark. In spite of this result, the reader should bear in mind that the natural functors (F il) → (F il)_{r} and (F il)_{s} → (F il)_{r} (s ≥ r) are not continuous (do not commute with inverse limits), as the above proof shows. So computing an inverse limit in (F il)_{r} does not give in general the same result as in (F il). For example, there is an exact sequence

0 ⟶Z_{p} = lim ; ⟵_{(F il)} Z/p^{n} ⟶ lim ; ⟵_{(F il)_{1}} Z/p^{n} ⟶ ∏_{n≥1} Z/p^{n} ⟶0.

3.13 Definition (compare [13, II.1.2]). 1. A mixed Lie algebra is a graded F_{p}-Lie algebra A = (A_{n})_{n≥1} provided with a degree-1 operator q : A_{n} → A_{n+1} satisfying the identities:

a) for x, y ∈ A_{n},

q(x + y) = \begin{cases}q(x) + q(y) & if p > 2 or n > 1 \\ q(x) + q(y) + [x, y] & if p = 2 and n = 1; \end{cases}

b) for (x, y) ∈ A_{m} × A_{n},

[q(x), y] = \begin{cases}q([x, y]) & if p > 2 or m > 1 \\ q([x, y]) + [x, [x, y]] & if p = 2 and m = 1. \end{cases}

2. Let (A_{n}) and (B_{n}) be two mixed Lie algebras. A morphism f from A to B is a graded homomorphism (f_{n} : A_{n} → B_{n})_{n≥1} such that f_{m+n}([a, b]) = [f_{m}(a), f_{n}(b)] and f_{m+1}(q(a)) = q(f_{m}(a)) for (a, b) ∈ A_{m} × A_{n}. We denote by ML the category whose objects are mixed p-Lie algebras and morphisms are morphisms of mixed Lie algebras.

The following lemma follows immediately from the relations:

3.14 Lemma. Let A be a mixed Lie algebra and a, b two homogeneous elements of A. If [a, b] = 0, then [a, q(b)] = 0. In particular, [a, q(a)] = 0.

3.15 Proposition (compare [13, II.1.2.11]). Let G be a p-filtered group, and set A_{n} = G^{(n)}/G^{(n+1)}. Then, the commutator and p-power maps in G give A = (A_{n})_{n≥1} the structure of a mixed Lie algebra gr G. If f : G → H is a morphism of p-filtered groups, it induces a morphism gr f : gr G → gr H of mixed Lie algebras.

Proof. From the definition of a p-filtration, the A_{n} are p-elementary abelian, hence F_{p}-vector spaces. From (2) we deduce that the composite map

G^{(m)} × G^{(n)} ⟶^{[,]} G^{(m+n)} ⟶ A_{m+n}

induces a bilinear alternating map

A_{m} × A_{n} ⟶^{[,]} A_{m+n}.

Similarly, (3) shows that the composite

G^{(n)} ⟶^{p}G^{(n+1)} ⟶ A_{n+1}

factors as a map

A_{n} ⟶^{q}A_{n+1}

which satisfies the first identity of definition 3.13. Finally, the Jacobi identity (resp. the second identity of def. 3.13) follows from the Witt identity (4) (resp. from (5)).

3.16 Definition. a) Let A be a mixed Lie algebra and X be a subset of A. The subalgebra of A generated by X is the intersection of all mixed Lie subalgebras of A containing X.

b) We say that A is generated in degree 1 if the subalgebra of A generated by A_{1} is A.

3.17 Proposition. A mixed Lie algebra A is generated in degree 1 if and only if, for all n > 1, one has A_{n} = [A_{1}, A_{n−1}] + q(A_{n−1}).

Proof. The ‘if’ part is obvious, and the ‘only if’ part follows for example from [12, lemme 2] and the anticommutativity of the product.

3.18 Example. If G is provided with its Frattini filtration, we denote by gr^{Φ} G the associated mixed Lie algebra and call it the Frattini algebra of G. We have:

3.19 Corollary (to prop. 3.17). Let G be a p-filtered group. Then gr G is generated in degree 1 if and only if the filtration of G is its Frattini filtration. □

3.20. Proposition 3.15 defines a functor:

gr : (F il) ⟶ ML

We note that this functor factors as a faithful functor

gr : (F il)_{1} ⟶ ML

where (F il)_{1} has been defined in 3.8. This functor will be called the fundamental functor. We have the classical

3.21 Proposition. The restriction to (F ilc)_{1} of the fundamental functor reflects isomorphisms. □

4. Truncation

4.1. Let n ≥ 1. We define full subcategories (F il)^{≤n} and ML^{≤\} of (F il) and ML as follows:

An object (G, (G^{(m)})) of (F il) is in (F il)^{≤n} if G^{(n+1)} = 1; An object (A_{m}) of ML is in ML^{≤\} if A_{m} = 0 for m > n.

Note that any group G ∈ (F il)^{≤n} verifies G^{p^{n}} = 1; in particular, it is a p-group if finite. Also, any G ∈ (F il)^{≤n} is evidently complete.

4.2. Clearly, (F il)_{n}^{≤n} = (F il)^{≤n}. The following concrete description of (F il)_{n−1}^{≤n} is especially useful for n = 2.

4.3 Lemma. Let G, H ∈ (F il)^{≤n} and V = {u ∈ Hom_{(F il)}(G, H) | u ≡ 1 (mod R_{\−∞})}. Then V ≃ Hom(gr^{1} G, gr^{n} H) is naturally an F_{p}-vector space, acts freely on Hom_{(F il)}(G, H) by translation, and Hom_{(F il)_{n−1}}(G, H) ≃ Hom_{(F il)}(G, H)/V. □

4.4. The inclusion functor (F il)^{≤n} → (F il) (resp. ML^{≤\} → ML) has a left adjoint-left inverse: the truncation functor

(F il) ⟶^{T_{n}} (F il)^{≤n}
(G, (G^{(m)})) ↦ (G/G^{(n+1)}, (G^{(m)}G^{(n+1)}/G^{(n+1)}))

(resp.

ML ⟶^{t_{n}} ML^{≤\}
(A_{m}) ↦ (B_{m})

with B_{m} = \begin{cases}A_{m} & if m ≤ n \\ 0 & otherwise \end{cases}, the maps [,] and q on B being the same as

on A). It is clear that, for G ∈ (F il), the morphism

t_{n} gr G ⟶ gr(T_{n}G)  (6)

induced by G → T_{n}G is an isomorphism (although we don’t see a categorytheoretic reason for this fact!).

4.5 Theorem (compare [13, II.1.2.9]). a) Let (V gr) be the category of N − {0}-graded F_{p}-vector spaces. Then the forgetful functor ML → (V}∇) has a left adjoint F (“free algebra on a graded vector space”).

b) Let X be a set and F_{p}X the F_{p}-vector space with basis X. Consider F_{p}X as a graded F_{p}-vector space concentrated in degree 1. Let FX be the free group with basis X. Then there is a canonical isomorphism

F (F_{p}X) ⟶^{∼} gr^{Φ} FX .

Proof. a) It is convenient to introduce the category A of N − {0}-graded nonassociative F_{p}-algebras with one operator. An object of A consists of a graded vector space (A_{n})_{n≥1}, a bilinear graded product · : A_{i} × A_{j} → A_{i+j} and an operator of degree 1 q : A_{i} → A_{i+1}. One does not require the product nor the operator to have any other particular properties. A morphism between two objects of A is an F_{p}-linear map of degree 0 which commutes with the two products and qs. The forgetful functor of the theorem factors obviously as

ML ⟶^{α} A ⟶^{β} (V}∇).

We shall construct a left adjoint of the two intermediate forgetful functors α and β; the left adjoint F of the theorem will be their composite.

Left adjoint to β. Given a graded vector space V = (V_{n})_{n≥1} ∈ (V gr), we define an object A ∈ A by

A_{1} = V_{1};
for n > 1, A_{n} = V_{n} ⊕ ⨁_{i+j=n} A_{i} ⊗ A_{j} ⊕ A_{n−1}.

The product A_{i} ⊗A_{j} → A_{i+j} and the operator q : A_{n−1} → A_{n} are simply given by the inclusions of the relevant direct summands. There is a degree-0 map of F_{p}-graded vector spaces V → A given by the inclusion of the first summand. One checks immediately that this construction is functorial and indeed yields a left adjoint to β.

Left adjoint to α. Let A ∈ A. We associate to it the mixed Lie algebra defined as the quotient of A by the relations a · a = 0, a · (b · c) + b · (c · a) + c · (a · b) = 0 (a, b, c ∈ A) and the relations of definition 3.13. Again it is easy to check that this defines a left adjoint to α.

b) The inclusion in (V gr)

F_{p}X ≃ gr_{1}^{Φ} FX ⟶ gr^{Φ} FX

induces by adjunction a morphism in ML

F (F_{p}X) ⟶ gr^{Φ} FX .

This is the morphism of theorem 4.5 b). Since it is surjective in degree 1 and gr_{1}^{Φ} FX is generated in degree 1 (prop. 3.19), it is surjective. The difficulty is to show it is injective. This is done by Lazard [13, th. II.3.2.5] by using the Magnus algebra and the Poincaré-Birkhoff-Witt theorem for mixed Lie algebras (compare [1, § 5, th. 3] and its proof). □

The following theorem is the central tool of this paper.

4.6 Theorem. The truncation functor t_{n} : ML → ML^{≤\} has a left adjoint-right inverse s_{n}.

Proof. If A ∈ ML^{≤\} and F is the free functor of theorem 4.5 a), denote by \underline{a} the image of a ∈ A in F (A) under the unit (graded vector space-) homomorphism A → F (A). Define s_{n}(A) as the quotient of F (A) by the relations

[\underline{a}, \underline{b}] = \underline{[a, b]}
q(\underline{a}) = \underline{q(a)}

for (a, b) ∈ A_{m} × A_{r} with m + r ≤ n. We check that s_{n} is left adjoint to t_{n}. Indeed, let B ∈ ML and φ : A → t_{n}B be a morphism in ML^{≤\}. Forgetting the algebra structure of B and embedding t_{n}B into B as a graded subvector space, we extend φ into a (graded vector space-) homomorphism

φ′ : A → B

which gives by adjunction a mixed Lie algebra homomorphism

φ′′ : F (A) → B

where we consider again B as a mixed Lie algebra. For a, b ∈ A, we have

φ′′(\underline{[a, b]}) = φ([a, b]) = [φ(a), φ(b)] = [φ′′(\underline{a}), φ′′(\underline{b})] = φ′′([\underline{a}, \underline{b}])

if deg(a) + deg(b) ≤ n and similarly

φ′′(\underline{q(a)}) = φ′′(q(\underline{a}))

if deg(a) ≤ n−1. Hence φ′′ factors as a mixed Lie algebra homomorphism φ′′′ : s_{n}A → B. In other words, we have defined a map

Hom_{ML}_{≤\}(A, t_{n}B) → Hom_{ML}(s_{n}A, B).

On the other hand, the composition

A → F (A) → s_{n}A

defines a graded vector space homomorphism whose image sits into t_{n}s_{n}A ⊆ s_{n}A. By definition of s_{n}A the resulting linear homomorphism

A → t_{n}s_{n}A

is a mixed Lie algebra homomorphism (actually an isomorphism). Given now a mixed Lie algebra homomorphism s_{n}A → B, composition with the former gives a mixed Lie algebra homomorphism A → t_{n}B, hence a map in the other direction

Hom_{ML}(s_{n}A, B) → Hom_{ML}_{≤\}(A, t_{n}B).

It is readily checked that composing these maps both ways gives the identity, which completes the proof.

Note that s_{1} is a special case of F. □

4.7 Remarks.

1. The truncation functor T_{n} : (F il) → (F il)^{≤n} does not preserve all inverse limits (hence does not have a left adjoint). This can be seen by applying it to the pair or maps

G ⇉_{1}^{pr} G/G^{(n+1)}

for any p-filtered group G such that G^{(n+1)} ≠ 1 and G^{(n+2)} = 1. The same counterexample applies in the categories (F il)_{r}.

2. The right inverse property of s_{n} relative to t_{n} follows formally from the left inverse property of t_{n} with respect to the inclusion functor

ML^{≤\} ⟶^{⟩_{\}} ML.

3. One could give an a priori proof of the existence of s_{n} by applying Freyd’s theorem [14, th. V.6.2], i.e. noting that t_{n} commutes with all inverse limits and checking a “solution set condition”. The explicit construction of s_{n} has the advantage of being suited to computation.

4. In categorical terms, one can partially explain the construction of s_{n} by a naturally commutative diagram

ML^{≤\}  ⟶^{in} ; ⟵^{tn} ; ⟶^{sn} ML

U_{n}↓  F ↑↓U

(V gr)^{≤n} ⟶^{ιn} ; ⟵^{τn} ; ⟶^{ιn} (V gr)

Here U and U_{n} denote the forgetful functors and ι_{n}, τ_{n} respectively the inclusion and truncation functors on graded vector spaces. The two main points seem to be that (i) t_{n} and τ_{n} commute with U and U_{n} and (ii) ι_{n} is both left and right adjoint to t_{n}. However these remarks don’t completely enlighten the above construction.

5. From the construction of s_{n}, it follows that

• If m ≤ n, s_{n}(A)_{m} = A_{m} (right inverse property of s_{n}).

• (for n > 1)

s_{n}(A)_{n+1} = ( ⨁_{i+j=n+1} A_{i} ⊗ A_{j} ⊕ A_{n}) /R,

where R is the sub-vector space generated by the relations

(a ⊗ b, 0) = (−b ⊗ a, 0),  (a, b) ∈ A_{i} × A_{j},  i + j = n + 1
(a ⊗ a, 0) = 0,  a ∈ A_{\frac{n+1}{2}}  (if n + 1 even)
(a ⊗ [b, c] + b ⊗ [c, a] + c ⊗ [a, b], 0) = 0, (a, b, c) ∈ A_{i} × A_{j} × A_{k}, i + j + k = n + 1

and

(a⊗q(b), 0) = \begin{cases}(0, q([a, b]) & if p > 2 or i > 1 \\ (b ⊗ [a, b], q([a, b])) & if p = 2 and i = 1, \end{cases} (a, b) ∈ A_{i}×A_{j}, i+j = n.

• (for n = 1)

s_{1}(A)_{2} = \begin{cases}Λ^{2}A_{1} ⊕ A_{1} & if p > 2 \\ Γ^{2}(A_{1}) & if p = 2. \end{cases}

On the other hand, it becomes more and more complicated to describe s_{n}(A)_{n+i} in terms of A as i increases. In practice one tries to use e.g. proposition 4.14.

4.8. Instead of shuttling between ML and ML^{≤\}, we can shuttle between ML^{≤\} and ML^{≤⇕} for m ≤ n. It is clear that the left adjoint t_{m,n} of the inclusion ML^{≤⇕} ↪ ML^{≤\} is given by t_{m} restricted to ML^{≤\}. In view of the left adjoint property of t_{n}, we have:

4.9 Proposition. Let m ≤ n. The truncation functor t_{m,n} : ML^{≤\} → ML^{≤⇕} has a left adjoint s_{m,n}, given by t_{n} ∘ s_{m}. □

4.10 Corollary. Let B ∈ ML^{≤⇕} and A = s_{m}B. Then, for any n ≥ m, the counit morphism

s_{n}(t_{n}A) ⟶ A

is an isomorphism.

Proof. From the obvious isomorphism of functors

t_{m} = t_{m,n} ∘ t_{n}

we get by adjunction an isomorphism of functors

s_{m} = s_{n} ∘ s_{m,n} = s_{n} ∘ t_{n} ∘ s_{m}.□

4.11. Let G ∈ (F il). The inverse of the isomorphism (6)

gr T_{n}G ⟶^{∼} t_{n} gr G

yields by adjunction a morphism

s_{n}(gr T_{n}G) ⟶ gr G.  (7)

We have:

4.12 Proposition. For any group G (provided with its Frattini filtration) and any n ≥ 1, the homomorphism (7)

s_{n}(gr^{Φ} T_{n}G) ⟶ gr^{Φ} G

is surjective. In particular, for r > n, dim_{F_{p}} gr_{r}^{Φ} G ≤ dim_{F_{p}} s_{n}(gr^{Φ} T_{n}G)_{r}.

This follows from proposition 3.19 and the surjectivity of the map in degree 1.

(Proposition 4.12 can be interpreted as follows: let \overline{G} be a group of plength n. For any G such that T_{n}G ≃ \overline{G}, the dimension of the higher Frattini quotients of G is bounded only in terms of s_{n}(gr^{Φ} \overline{G}).)

From theorem 4.5 b) and corollary 4.10, we also get:

4.13 Proposition. Let G be a free group. Then, for all n ≥ 1, the homomorphism (7)

s_{n}(gr^{Φ} T_{n}G) ⟶ gr^{Φ} G

is an isomorphism. □

The following proposition and corollary are important in practice.

4.14 Proposition. Let A ∈ ML^{≤\}, generated in degree 1. Then s_{n}(A) is generated in degree 1.

Proof. Let B be the mixed subalgebra of s_{n}(A) generated by s_{n}(A)_{1} = A_{1}. By remark 4.7 (5) and the hypothesis, B_{m} = s_{n}(A)_{m} = A_{m} for m ≤ n. In particular, t_{n}(B) = A. Hence the counit s_{n}t_{n} → Id yields a homomorphism

φ ∈ Hom_{ML}(s_{n}(A), B)

which is right inverse to the inclusion B ↪ s_{n}(A). Therefore this inclusion is surjective and B = s_{n}(A).

4.15 Corollary. Let A ∈ ML^{≤\}, generated in degree 1. Then

s_{n}(A)_{n+1} ≃ \frac{A_{1} ⊗ A_{n} ⊕ A_{n}}{R′}

with R′ = R ∩ (A_{1} ⊗ A_{n} ⊕ A_{n}), where R is as in remark 4.7 (5). The same holds for n = 2, unconditionally on A.

Proof. This follows from propositions 3.17 and 4.14 (the claim for n = 2 follows from the anticommutativity of the product).

5. An example

Let p = 2 and M be the group of 2-class 2 defined by generators and relations by

M =< n_{1}, n_{2}, e_{1}, e_{2}, f_{0}, f_{1}, f_{2} | [n_{i}, f_{j}] = [e_{i}, f_{j}] = f_{i}^{2} = 1,
[n_{1}, n_{2}] = [e_{1}, n_{2}] = [e_{2}, n_{1}] = 1, [e_{1}, e_{2}] = f_{0}f_{1}f_{2},
[e_{1}, n_{1}] = n_{1}^{2} = f_{0}f_{1}, [e_{2}, n_{2}] = n_{2}^{2} = f_{0}f_{2}, e_{1}^{2} = e_{2}^{2} = f_{0} > .

For a p-group G, let d(G) be the minimum number of generators of G. Recall [10, appendix] that G is called d-maximal if d(H) < d(G) for any proper subgroup H of G. One can show that, up to isomorphism, M is the only 2-class 2 d-maximal 2-group with d(M) = 4 such that s_{2}(gr^{Φ} M) ∉ ML^{≤∈} (compare [11], [15]). We have Φ(M) =< f_{0}, f_{1}, f_{2} >. Keeping the same notation for generators, gr^{Φ} M has the following presentation:

gr_{Φ}^{1} M =< e_{1}, e_{2}, n_{1}, n_{2} >;
gr_{Φ}^{2} M =< f_{1}, f_{2}, f_{3} >;
q(e_{1}) = q(e_{2}) = f_{0}, q(n_{1}) = f_{0} + f_{1}, q(n_{2}) = f_{0} + f_{2};
[e_{1}, e_{2}] = f_{0} + f_{1} + f_{2}, [e_{1}, n_{1}] = f_{0} + f_{1}, [e_{2}, n_{2}] = f_{0} + f_{2},
[e_{1}, n_{2}] = [e_{2}, n_{1}] = [n_{1}, n_{2}] = 0.

We shall prove:

5.1 Proposition ([11],[15]). One has s_{2}(gr^{Φ} M)_{3} ≃ Z/2 and s_{2}(gr^{Φ} M)_{i} = 0 for i > 3.

Proof. Let A = s_{2}(gr^{Φ} M), so that, as seen above, t_{2}(A) = gr^{Φ} M. Using if necessary lemma 3.14, we have the following relations in A:

[e_{1}, f_{0}] = [e_{1}, q(e_{1})] = 0
[e_{2}, f_{0}] = [e_{2}, q(e_{2})] = 0
[n_{1}, f_{0}] = [n_{1}, q(e_{2})] = 0
[n_{2}, f_{0}] = [n_{2}, q(e_{1})] = 0
[n_{1}, f_{1}] = [n_{1}, f_{0} + f_{1}] = [n_{1}, q(n_{1})] = 0
[n_{1}, f_{2}] = [n_{1}, f_{0} + f_{2}] = [n_{1}, q(n_{2})] = 0
[n_{2}, f_{1}] = [n_{2}, f_{0} + f_{1}] = [n_{2}, q(n_{1})] = 0
[n_{2}, f_{2}] = [n_{2}, f_{0} + f_{2}] = [n_{2}, q(n_{2})] = 0
[e_{1}, f_{2}] = [e_{1}, f_{0} + f_{2}] = [e_{1}, q(n_{2})] = 0
[e_{2}, f_{1}] = [e_{2}, f_{0} + f_{1}] = [e_{2}, q(n_{1})] = 0

Also:

[e_{1}, f_{1}] = [e_{1}, f_{0} + f_{1}] = [e_{1}, q(n_{1})] = q([e_{1}, n_{1}]) + [n_{1}, [e_{1}, n_{1}]] = q(f_{0}) + q(f_{1})
= [e_{1}, f_{0} + f_{1} + f_{2}] = [e_{1}, [e_{1}, e_{2}]] = q([e_{1}, e_{2}]) + [e_{2}, q(e_{1})]
= q([e_{1}, e_{2}]) + [e_{2}, q(e_{2})] = q(f_{0}) + q(f_{1}) + q(f_{2})

hence

q(f_{2}) = 0,

similarly

q(f_{1}) = 0,

and so

[e_{1}, f_{1}] = [e_{2}, f_{2}] = q(f_{0}).

This shows that A_{3} is generated by g = q(f_{0}), hence dim A_{3} ≤ 1. Moreover, it is clear by corollary 4.15 that the relations above generate all relations, so that A_{3} ≠ 0.

Finally, to see that A_{i} = 0 for i > 3, it suffices by propositions 3.17 and 4.14 to see that [A_{1}, A_{3}] = q(A_{3}) = 0. But, for any x ∈ A_{1} = gr_{1}^{Φ} G:

[x, g] = [x, q(f_{0})] = q([x, f_{0}]) = 0

and q(g) = q([e_{1}, f_{1}]) = [e_{1}, q(f_{1})] = 0. □

6. Truncation, extensions and cohomology

6.1. Let G ∈ (F il)^{≤n} and

1 → A → G̃ ⟶^{π} G → 1  (e)

be an extension. Endow G̃ with the filtration

G̃^{(m)} = \begin{cases}π^{−1}(G^{(m)}) & if m ≤ n \\ A & if m = n + 1 \\ 1 & if m > n + 1. \end{cases}

With this filtration, G̃ ∈ (F il)^{≤n+1} if and only if

\begin{cases}A^{p} = 1 \\ [G̃^{(i)}, G̃^{(j)}] = 1 for i + j = n + 2. \end{cases}

In particular, A must be central of exponent p. We call such an extension admissible.

6.2 Lemma. Let A be an elementary abelian p-group and H_{adm}^{2}(G, A) the subset of H^{2}(G, A) consisting of the classes of admissible extensions (for the p-filtered structure of G). Then H_{adm}^{2}(G, A) is a subgroup of H^{2}(G, A), a contravariant functor in G ∈ (F il)^{≤n} and a covariant functor in A ∈ elementary abelian p-groups.

Proof. The naturality in A and G is obvious. Fixing G, the fact that H_{adm}^{2}(G, A) is a subgroup of H^{2}(G, A) follows from proposition 1.11 d) provided we know it commutes with products. But given two admissible extensions

1 ⟶A ⟶G̃ ⟶G ⟶1
1 ⟶B ⟶G̃′ ⟶G ⟶1

their fibred product over G

1 ⟶A × B ⟶G̃′′ ⟶G ⟶1

is clearly admissible. □

6.3 Proposition. If G is provided with its Frattini filtration, then every central extension of G by an elementary abelian p-group A is admissible.

This follows from the Witt identity and the definition of Φ_{n}G (in terms of mixed Lie algebras, it follows from the relation [gr_{1}^{Φ} G, A] = 0, the Jacobi identity and proposition 3.17). □

6.4. Let G̃ be an admissible extension of G with kernel A, and c its class in H_{adm}^{2}(G, A). We have t_{n}(gr G̃) = gr G. The counit of the adjunction (s_{n}, t_{n}) yields a morphism

θ_{G,A,e} ∈ Hom_{ML}(s_{n}(gr G), gr G̃)

and in particular a homomorphism

θ_{G,A,e} : s_{n}(gr G)_{n+1} ⟶A.

Two isomorphic extensions clearly yield the same θ_{G,A,e}. We have therefore defined a map

\begin{matrix}H_{adm}^{2}(G, A) & ⟶^{θ_{G,A}} & Hom(s_{n}(gr G)_{n+1}, A) \\ e & ⟼ & θ_{G,A,e} \end{matrix}

which is covariant in A and contravariant in G (as a p-filtered group). From this and proposition 1.11, we deduce:

6.5 Proposition. For any A, θ_{G,A} is linear. In particular, for A = Z/p we get a canonical homomorphism

θ_{G} : H_{adm}^{2}(G, Z/p) ⟶ (s_{n}(gr G)_{n+1})^{∗}.

For any A, we have θ_{G,A} = θ_{G} ⊗ Id_{A} under the canonical isomorphisms H_{adm}^{2}(G, A) = H_{adm}^{2}(G, Z/p)⊗A and Hom(s_{n}(gr G)_{n+1}, A) = (s_{n}(gr G)_{n+1})^{∗}⊗ A. The homomorphism θ_{G} defines a natural transformation of H_{adm}^{2}(−, Z/p) to (s_{n}(gr −)_{n+1})^{∗} viewed as functors from (F il)^{≤n} to F_{p}-vector spaces. □

We agree to sometimes abbreviate the notation H_{adm}^{2}(G, Z/p) into H_{adm}^{2}(G) or H_{adm}^{2}G.

6.6 Remark. Let G ∈ (F il). Any homomorphism gr^{n+1} G → Z/p defines by push-out an admissible central extension of T_{n}G by Z/p. Hence a canonical map

(gr^{n+1} G)^{∗} ⟶ H_{adm}^{2}(T_{n}G, Z/p)

which is seen to be a homomorphism by the same arguments as above. Composing it with θ_{T_{n}G}, we get a homomorphism

(gr^{n+1} G)^{∗} ⟶ (s_{n}(gr T_{n}G)_{n+1})^{∗}.

One checks immediately from the definitions that this is the (n + 1)-st component of the transpose of (7).

6.7 Proposition. Let G ∈ (F il)^{≤n}. Then the sequence

H_{adm}^{2}(G/G^{(n)}) ⟶^{Inf} H_{adm}^{2}(G) ⟶^{θ_{G}} (s_{n}(gr G)_{n+1})^{∗},

where G/G^{(n)} is considered as an object of (F il)^{≤n}, is exact.

Proof. Let e ∈ Ker θ_{G}, let 1 → Z/p → G̃ → G → 1 be the corresponding extension and let G̃^{(n)} be the inverse image of G^{(n)} in G̃. The assumption implies that G̃^{(n)} is elementary abelian and central in G̃. Let E be a complement of Z/p in G̃^{(n)}, and H = G̃/E. Then H is a central extension of G/G^{(n)} by Z/p, whose class lifts as e. Moreover, this extension is admissible since e was. This shows that Ker θ_{G} ⊆ Im Inf. The opposite inclusion is clear. □

As an application, we get a slight improvement of an upper bound due to P. M. Neumann [17, cor. 2.2] for |H^{2}| of a finite p-group (the reader should compare our proof with Neumann’s):

6.8 Corollary. Let G be a group of order p^{m} with d generators. Then:

dim_{F_{p}} H^{2}G ≤ dm − \frac{d(d − 1)}{2}.

Proof. By induction on the p-length n of G. For n = 1, G is elementary abelian and the inequality is an equality by the known cohomology of elementary abelian p-groups. Assume n ≥ 2. The central extension

1 → Φ_{n}G → G → G/Φ_{n}G → 1,

propositions 6.3 and 6.7 yield an exact sequence

0 ⟶H^{1}(Φ_{n}G) ⟶H^{2}(G/Φ_{n}G) ⟶^{Inf} H^{2}(G) ⟶^{θ_{G}} (s_{n}(gr^{Φ} G)_{n+1})^{∗},

where the extra injection comes from the Hochschild-Serre spectral sequence and the fact that Φ_{n}G ⊆ Φ(G). This gives the inequality:

dim H^{2}G ≤ dim H^{2}(G/Φ_{n}G) − dim H^{1}(Φ_{n}G) + dim s_{n}(gr^{Φ} G)_{n+1}.

Using corollary 4.15, we have dim s_{n}(gr^{Φ} G)_{n+1} ≤ d_{n}(d + 1), where d_{n} = dim Φ_{n}G. Moreover, we obviously have dim H^{1}Φ_{n}G = d_{n}. The corollary follows.

We also have:

6.9 Corollary (compare [11]). Let M be the 2-group of section 5. Up to isomorphism, there are at most 128 2-groups G such that T_{3}G ≃ M.

Proof. Let G be such that T_{3}G := G/Φ_{3}G ≃ M. By proposition 4.12, gr^{Φ} G is a quotient of s_{3}(gr^{Φ} M). By proposition 5.1, G is of 2-class ≤ 3 and Φ_{3}G has order 1 or 2, hence is represented by a class c ∈ H^{2}M. Arguing as above, we have an exact sequence:

0 ⟶H^{1}(Φ(G)) ⟶H^{2}(M/Φ(M)) ⟶^{Inf} H^{2}(M) ⟶^{θ_{M}} (s_{2}(gr^{Φ} M)_{3})^{∗}.

The class c, if it is ≠ 0, maps to the nontrivial element ε of (s_{2}(gr^{Φ} M)_{3})^{∗}. Since dim(M/Φ(M)) = 4 and dim Φ(M) = 3, we have dim H^{2}(M/Φ(M)) = 10 and dim H^{1}(Φ(G)) = 3. Hence θ_{M}^{−1}(ε) has cardinality either 0 or 2^{7} = 128 (theorem 8.7 below will show that this cardinality is indeed 128). □

6.10. Convention. From now on, we shall restrict to the full subcategories of (F il) (resp. ML) which consist of groups with gradually finite associated graded (resp. gradually finite algebras). We don’t change the notations (F il), ML, etc.

7. Example: the case n = 1

7.1 Theorem. The fundamental functor

gr : (F il)_{1} ⟶ ML

of 3.20 restricts to an equivalence of categories

gr^{≤2} : (F il)_{1}^{≤2} ⟶ ML^{≤∈}.

Proof. We already know gr^{≤2} is faithful; by proposition 1.4 we have to prove that it is full and essentially surjective. Essential surjectivity follows from

7.2 Proposition. Let E be an elementary abelian group, considered as an object of (F il)_{1}. Then the homomorphism θ_{E} : H^{2}E → (s_{1}(gr E)_{2})^{∗} is bijective.

This follows from propositions 6.3 and 6.7, the description of s_{1}(gr E)_{2} in remark 4.7 (5) and the known cohomology of an elementary abelian pgroup.

It remains to see that gr^{≤2} is full. Let G, H ∈ (F il)^{≤2} and f : gr G → gr H a homomorphism. We want to see that f comes from a filtered homomorphism from G to H. Let 1 → A_{2} → G → A_{1} → 1 and 1 → B_{2} → H → B_{1} → 1 be the central extensions corresponding to the pfiltered structures on G and H, c ∈ H^{2}(A_{1}, A_{2}) and d ∈ H^{2}(B_{1}, B_{2}) the corresponding cohomology classes and θ(c) ∈ Hom(s_{1}(A_{1})_{2}, A_{2}), θ(d) ∈ Hom(s_{1}(B_{1})_{2}, B_{2}) their images under θ_{A_{1},A_{2}} and θ_{B_{1},B_{2}}. To say that f is a morphism in ML^{≤∈} means that

θ(d) ∘ f_{1} = f_{2} ∘ θ(c) ∈ Hom(s_{1}(A_{1})_{2}, B_{2}),

where f_{1} : A_{1} → B_{1} and f_{2} : A_{2} → B_{2} are the components of f. The injectivity of θ_{A_{1},B_{2}} then implies that

f_{1}^{∗}d = f_{2∗}c ∈ H^{2}(A_{1}, B_{2}).

Let K (resp. L) be the push-out of G by f_{2} (resp. the pull-back of H by f_{1}). The equality f_{1}^{∗}d = f_{2∗}c means that there exists an isomorphism φ of K onto L which induces the identity on A_{1} and B_{2}. We then get a morphism f̃: G → H in (F il)^{≤2} inducing f by considering the composite:

\begin{aligned}1 & ⟶ & A_{2} & ⟶ & G & ⟶ & A_{1} & ⟶ & 1 \\ & & ^{f_{2}}↓ & & ↓ & & || \\ 1 & ⟶ & B_{2} & ⟶ & K & ⟶ & A_{1} & ⟶ & 1 \\ & & || & & ^{φ}↓ & & || \\ 1 & ⟶ & B_{2} & ⟶ & L & ⟶ & A_{1} & ⟶ & 1 \\ & & || & & ↓ & & ^{f_{1}}↓ \\ 1 & ⟶ & B_{2} & ⟶ & H & ⟶ & B_{1} & ⟶ & 1 & .□ \end{aligned}  (8)

7.3 Corollary. For G, H ∈ (F il)^{≤2}, G ≃ H if and only if gr G ≃ gr H.

This follows from propositions 3.9 and 3.21.

8. The case n = 2

8.1. For n > 1, the fundamental functor

gr^{≤n+1} : (F il)_{1}^{≤n+1} ⟶ ML^{≤\+∞}

is not full, even when restricted to abelian groups. The reason is simple: if G_{1}, G_{2} ∈ (F il)^{≤n+1} are abelian and u ∈ Hom(G_{1}, G_{2}), then gr u certainly respects Ker q. But it must also respect the kernel of “secondary operations”, like

q^{(2)} : Ker q ∩ gr^{1} G_{i} → gr^{3} G_{i}/q(gr^{2} G_{i})

coming from the finer structure of G_{1}, G_{2} (see 10.2), which is not detected by the functor gr. This makes it easy to construct a counterexample:

Counterexample 1. Let G_{1} = Z/p × Z/p × Z/p, with basis (e_{1}, e_{2}, e_{3}); provide G_{1} with the p-filtration G_{1}^{(1)} = G_{1}, G_{1}^{(2)} =< e_{2}, e_{3} >, G_{1}^{(3)} =< e_{3} >. Let G_{2} = Z/p × Z/p^{2}, with basis (e_{1}, e_{2}), provided with the pfiltration G_{2}^{(1)} = G_{2}, G_{2}^{(2)} =< e_{1}, e_{2}^{p} > and G_{2}^{(3)} =< e_{2}^{p} >. Then gr G_{1} and gr G_{2} are isomorphic as mixed Lie algebras. But an isomorphism between them cannot come from a homomorphism from G_{1} to G_{2}, since such a homomorphism would necessarily be an isomorphism by propositions 3.9 and 3.21. One could concoct similar nonabelian counterexamples.

8.2. Here is another argument showing that gr^{≤3} is not full; this argument will help us construct a counterexample with groups provided with their Frattini filtration. Let \overline{G} be of p-class 2, provided with its Frattini filtration. Let c ∈ H^{2}(\overline{G}), d ∈ H^{2}(\overline{G}/Φ(\overline{G})) and let G, H be the central extensions of \overline{G} by Z/p respectively defined by c and c + Inf(d). By proposition 6.7 we have gr^{Φ} G ≃ gr^{Φ} H, but in general G and H will not be isomorphic. More specifically, taking \overline{G} = Z/p × Z/p^{2}:

Counterexample 2. Let G = Z/p × Z/p^{3} and H =< e, f, c | [e, f] = f^{p^{2}} = c, e^{p} = c^{p} = [e, c] = [f, c] = 1 >. Then gr^{Φ} G ≃ gr^{Φ} H, but obviously G ≄ H.

8.3 Remark. If G and H are abelian (of any p-length), then Hom_{(F il)_{1}}(G, H) → Hom_{ML}(gr^{Φ} G, gr^{Φ} H) is bijective. This is easily seen by reducing to the case where G and H are cyclic (then both sets have p elements if |G| ≥ |H| and 1 otherwise). This shows that one cannot dispense with the noncommutativity of H in counterexample 2.

8.4. A solution to this defect would be to enrich the category of mixed Lie algebras with the extra structure alluded to above. We won’t do this here in general, but will give a taste of it in section 10. But first we shall prove:

8.5 Theorem. The fundamental functor gr^{≤3} : (F il)_{1}^{≤3} ⟶ ML^{≤∋} is essentially surjective.

8.6 Corollary. Let GML^{≤∋} be the full subcategory of ML^{≤∋} consisting of mixed Lie algebras generated in degree 1, (p − groups)^{≤3} the category of p-groups of p-length ≤ 3 and gr_{≤3}^{Φ} the composite functor

(p − groups)^{≤3} ⟶^{Φ} (F il)^{≤3} ⟶^{gr^{≤3}} ML^{≤∋}

viewed as a functor with values in GML^{≤∋} (by corollary 3.19). Then gr_{≤3}^{Φ} is essentially surjective.

This follows formally from theorem 8.5 and corollary 3.19; but it will actually be proven directly in section 9.

Theorem 8.5 follows from the more precise statement:

8.7 Theorem. Let G ∈ (F il)^{≤2}. Then the map θ_{G} : H_{adm}^{2}(G) → (s_{2}(gr G)_{3})^{∗} is surjective.

8.8 Remark. Theorem 8.7 is originally due to the second author for groups provided with their Frattini filtration. The proof we give here is completely different from the original one, that we now summarise for the convenience of the reader. Given a p-group G and an elementary abelian subgroup Φ of G which is central and contained in its Frattini subgroup Φ(G), there is an exact sequence [16]

H^{2}(K) ⟶^{Inf} H^{2}(G) ⟶^{Θ} (H^{1}(K) ⊗ H^{1}(Φ)) ⊕ βH^{1}(Φ) ⟶ H^{3}(K),

where K = G/Φ and β is the Bockstein homomorphism, refining earlier ones of Iwahori-Matsumoto [9], Thong [18] and Eckmann-Hilton-Stammbach [2], [3], Ganea [4], Gut [5] (in homology); compare especially [9, proposition 1.1], [5, theorem 2.3]. (The exactness at H^{2}(G) is used in a guise in the next section, in case 2 of the proof of theorem 8.7.) In the special case where G has p-class 2 and Φ = Φ(G), the map Θ factors as θ_{G} ∘ π^{∗}, where π^{∗} is the transpose of the surjection K ⊗ Φ ⊕ Φ ⟶^{π} s_{2}(gr^{Φ} G)_{3} of corollary 4.15. Considering the dual exact sequence

H_{3}(K) ⟶^{ρ}H_{1}(K) ⊗ H_{1}(Φ) ⊕ (βH^{1}(Φ))^{∗} ⟶ H_{2}(G) ⟶ H_{2}(K)

and identifying the second group to K ⊗ Φ ⊕ Φ, this means that Im ρ ⊇ Ker π, and the issue is to prove that Im ρ ⊆ Ker π. But K is elementary abelian, so its homology is ‘known’, and the latter is checked by an explicit computation (different for p > 2 and p = 2) on generators of H_{3}(K).

9. Proof of theorem 8.7

Let us introduce here the following terminology: if A is a mixed Lie algebra of length n and V is an F_{p}-vector space, we say that a mixed Lie algebra B is a central extension of A by V [n+1] if t_{n}B = A and B_{n+1} = V (this is purely for convenience: it is not our intention to develop here the theory of extensions of mixed Lie algebras at large).

The proof proceeds in three steps:

1. The free case. Let G be a free group, provided with its Frattini filtration. By proposition 4.13 the homomorphism from (7)

s_{2}(gr^{Φ} T_{2}G)_{3} ⟶ gr_{3}^{Φ} G

is an isomorphism. Dually, the transposed homomorphism

(gr_{3}^{Φ} G)^{∗} ⟶ (s_{2}(gr^{Φ} T_{2}G)_{3})^{∗}

is bijective, in particular surjective. By remark 6.6, this implies that θ_{T_{2}G} is surjective (in fact it is bijective but we don’t need that).

2. The Frattini case. Let G be a p-class 2 p-group, provided with its Frattini filtration. Let E = G/Φ(G), X a basis of E, FX the free group with basis X, F = FX /Φ_{∋}FX and α : F → G the natural surjection. Let A = gr^{Φ} G, Ã a central extension of A by Z/p[3] and B the pull-back of Ã by gr^{Φ} α (a central extension of gr^{Φ} F). By case 1, there is a central extension H of F by Z/p such that gr^{Φ} H ≃ B. Let F = Ker α: this is a subgroup of Φ(F), which is hence central and elementary abelian. As checked inside B, its inverse image F̃ in H is still central and elementary abelian. Let F′ be a complement of Z/p = Ker(H → F) in F̃ and G̃ = H/F′. Then G̃ is a central extension of G by Z/p such that gr^{Φ} G̃ ≃ Ã.

0 ⟶ Z/p[3] ⟶ B ⟶ gr^{Φ} F ⟶ 0
||  ↓  ^{gr^{Φ} α}↓
0 ⟶ Z/p[3] ⟶ Ã ⟶ A ⟶ 0

F′_{∩}

1 ⟶ Z/p ⟶ F̃ ⟶ F ⟶ 1
||  ↓  ↓
1 ⟶ Z/p ⟶ H ⟶ F ⟶ 1
||  ↓  ^{α}↓
1 ⟶ Z/p ⟶ G̃ ⟶ G ⟶ 1

H/F′^{||}

3. The general case. Let G ∈ (F il)^{≤2}, E = G/G^{(2)} and F be a complement of Φ(G) in G^{(2)}. We can find a subgroup H of G such that G = H ×F (direct product); then H/Φ(H) ≃ E and Φ(H) = Φ(G). Let A = gr G and B = gr^{Φ} H (a subalgebra of A). We have:

\begin{matrix}A_{1} = B_{1} = E \\ A_{2} = Φ(G) ⊕ F, B_{2} = Φ(G). \end{matrix}

Let Ã be an extension of A by Z/p[3]. By restriction to B, this defines an extension B̃ of B. By case 2, there is a central extension H̃ of H by Z/p such that gr^{Φ} H̃ ≃ B. On the other hand, the mixed algebra structure of Ã gives

• a pairing [,] : E × (Φ(G) ⊕ F) → Z/p; • a linear map q : Φ(G) ⊕ F → Z/p.

Let K be an (abelian) extension of F by Z/p given by the restriction of q to F. The restriction of [,] to E × F defines an action of E, hence of H̃, on K. Let C = Z/p × Z/p be the kernel of the reduction map

H̃ ⋉ K ⟶ H × F = G

and Δ its diagonal subgroup. Then the group G̃ = H̃ ⋉ K/Δ is a central extension of G by Z/p such that gr G̃ ≃ Ã. □

9.1 Remark. In cohomological terms, the proof of case 3 can be interpreted as follows. By the Künneth formula, H^{2}G ≃ H^{2}H ⊕ H^{1}H ⊗ H^{1}F ⊕ H^{2}F, and H_{adm}^{2}G is the subgroup H^{2}H ⊕ H^{1}H ⊗ H^{1}F ⊕ βH^{1}F ≃ H^{2}H ⊕ E^{∗} ⊗ F^{∗} ⊕ F^{∗}, where β is the Bockstein map (because the restriction of an admissible extension to F is abelian). On the other hand, s_{2}(A)_{3} ≃

\frac{E ⊗ (Φ(G) ⊕ F) ⊕ Φ(G) ⊕ F}{R} for R as in remark 4.7 (5), and this is easily

seen to be \frac{E ⊗ Φ(G) ⊕ Φ(G)}{R} ⊕E ⊗F ⊕F ≃ s_{2}(B)_{3} ⊕E ⊗F ⊕F. Details

are left to the reader.

10. Enriching mixed Lie algebras?

10.1. We have seen that the functor (F il)_{1}^{≤3} ⟶^{gr^{≤3}} ML^{≤∋} is faithful and essentially surjective. This result may be viewed as interesting but worth an improvement (another example of a faithful, essentially surjective functor is the forgetful functor (groups) ⟶(sets)!). It would be nice to refine gr^{≤3} into an equivalence of categories g̃r^{≤3}

M̃L^{≤3}
g̃r^{≤3} ↗  ↓
(F il)_{1}^{≤3} ⟶^{gr^{≤3}} ML^{≤∋}

where the category M̃L^{≤3} would be an enrichment of ML^{≤3}, still of a multilinear nature, and the vertical functor a forgetful functor.

At present we are not able to define a category M̃L^{≤3} such that the above conditions are satisfied. The aim of this section, however, is to give ideas on the structures which should presumably be involved in the definition of such a category.

10.2. Let G be a p-filtered group and A = gr G. Let a ∈ A_{1} be such that q(a) = 0 ∈ A_{2}. If ã is a lift of a in G, we have ã^{p} ∈ G^{(3)}. The image of ã^{p} in A_{3} does not depend only on a, but also on the choice of the lifting ã. However, changing ã into ãb with b ∈ G^{(2)} changes ã^{p} into

(ãb)^{p} ≡ ã^{p}b^{p} (mod γ_{2}(H)^{p}γ_{p}(H)),

(H =< ã, b >, cf (3)) at least for p > 2, so that (in this case) the image of ã^{p} in A_{3}/q(A_{2}) does not depend on the choice of ã. In other words, we have defined for p > 2 a map

q^{(2)} : Ker q ∩ A_{1} ⟶A_{3}/q(A_{2}).

This is an example of an “enrichment” on the structure of the mixed Lie algebra A = gr G. Similarly, let a, b ∈ A_{1} be such that [a, b] = 0 ∈ A_{2}. One sees in the same way as above that, if ã, b̃ are lifts of a, b in G, the image of the commutator [ã, b̃] ∈ G^{(3)} in A_{3}/[A_{1}, A_{2}] only depends on a and b.

We can go further in this direction. Let g_{1}, h_{1}, …, g_{m}, h_{m}, k_{1}, …, k_{r} ∈ G be such that

∏_{i=1}^{m}[g_{i}, h_{i}] ∏_{j=1}^{r} k_{j}^{p} ∈ G^{(3)}.

It is easily checked that the image of this element in A_{3}/[A_{1}, A_{2}] + q(A_{2}) only depends on

∑_{i=1}^{m} \overline{g}_{i} ∧ \overline{h}_{i} + ∑_{j=1}^{r} \overline{k}_{j} ∈ Ker(Λ^{2}A_{1} ⊕ A_{1} ⟶^{([,],q)} A_{2}),

and depends linearly on it (if p > 2: when p = 2 one should use squaring and Ker(Γ^{2}(A_{1}) ⟶^{q}A_{2})). So we get a well-defined “secondary” homomorphism

Ker(s_{1}(t_{1}A)_{2} ⟶A_{2}) ⟶^{q^{(2)}} Coker(s_{2}(t_{2}A)_{3} ⟶A_{3}).

Although suggestive, this enriched structure is not enough to define a good category M̃L^{≤3}. For example, for a group G of p-class 3 provided with its Frattini filtration, the target group of q^{(2)} is 0 by proposition 3.19, so this extra structure will certainly not separate isomorphism classes! The answer seems to be that, given an element x ∈ Ker(s_{1}(t_{1}A)_{2} ⟶ A_{2}), the element q^{(2)}(x) is in general well-defined in a larger quotient of A_{3}, which depends on x: if we represent x as above, it is enough to quotient A_{3} by

V = ∑_{i=1}^{m}[A_{2}, \overline{g}_{i}] + ∑_{i=1}^{m}[A_{2}, \overline{h}_{i}] + q(A_{2}),

at least for p > 2. This subgroup of A_{3} depends on the particular way x is represented in Ker(s_{1}(t_{1}A)_{2} ⟶A_{2}), so one could replace it by the intersection of all V s corresponding to such representations (we don’t know of an intrinsic description of this intersection!). One should investigate, however, the additivity properties of this collection of elements, so that it is not clear what the exact definition of M̃L^{≤3} should be.

We shall see, nevertheless, that this line of investigation goes in the right direction at least if we restrict to semi-trivial mixed Lie algebras of length 3. Let

ML_{st}^{≤∋}

be the full subcategory of ML^{≤∋} consisting of those algebras A = (A_{1}, A_{2}, A_{3})

such that the maps A_{1} × A_{2} ⟶^{[,]} A_{3} and A_{2} ⟶^{q} A_{3} are identically 0, and let (F il)_{1,st}^{≤3} be the full subcategory of (F il)_{1}^{≤3} consisting of those G such that gr G ∈ ML_{st}^{≤∋}. Objects of (F il)_{1,st}^{≤3} and ML_{st}^{≤∋} are called semi-trivial.

10.3 Definition. 1) We define a category M̃L_{st}^{≤3} (enriched semi-trivial length 3 mixed Lie algebras) as follows:

a) An object of M̃L_{st}^{≤3} is a semi-trivial mixed Lie algebra A of length 3, provided with a homomorphism

Ker(s_{1}(t_{1}A)_{2} ⟶A_{2}) ⟶^{q^{(2)}} A_{3}.

b) A morphism of M̃L_{st}^{≤3} is a morphism of semi-trivial mixed Lie algebras A ⟶^{f} B such that the diagram

Ker(s_{1}(t_{1}A)_{2} → A_{2}) ⟶^{q^{(2)}} A_{3}
f_{∗}↓  ^{f_{3}}↓
Ker(s_{1}(t_{1}B)_{2} → B_{2}) ⟶^{q^{(2)}} B_{3}

commutes, where f_{∗} is the homomorphism induced by f_{1} and f_{2}.

2) We define a functor

(F il)_{1,st}^{≤3} ⟶^{g̃r_{st}^{≤3}} M̃L_{st}^{≤3}

by associating to a semi-trivial length 3 filtered group G its mixed Lie algebra gr G provided with the enriched structure defined by the p-th power and commutator maps as in 10.2.

10.4 Theorem. The functor g̃r_{st}^{≤3} is an equivalence of categories.

Proof. The faithfulness of g̃r_{st}^{≤3} follows from that of gr^{≤3}; we have to prove fullness and essential surjectivity.

Let \overline{G} be a p-filtered group of length 2 and A an elementary abelian pgroup. The functor g̃r_{st}^{≤3} induces a map

ρ_{Ḡ,A} : Ker(H_{adm}^{2}(Ḡ, A) ⟶^{θ_{Ḡ,A}} Hom(s_{2}(gr Ḡ)_{3}, A)) → Hom(Ker(s_{1}(t_{1}A)_{2} → A_{2}), A)

(9)

which is natural in A, hence additive by proposition 1.11. Let A_{i} = gr^{i} Ḡ (i = 1, 2).

10.5 Lemma. 1) The diagram

H^{1}(A_{2}, A) ⟶^{δ} H^{2}(A_{1}, A)
↓^{≀}  ^{θ_{A_{1},A}}↓^{≀}
Hom(A_{2}, A) ⟶^{ε_{A_{1}}^{∗}} Hom(s_{1}(A_{1})_{2}, A)

where δ is the transgression map in the Hochschild-Serre spectral sequence for the extension 1 → A_{2} → Ḡ → A_{1} → 1, ε_{A_{1}} : s_{1}(A_{1})_{2} → A_{2} is the counit of the adjunction (s_{1}, t_{1}) and the left vertical map is the obvious map, anticommutes.

2) The map ρ_{Ḡ,A} of (9) is bijective.

Proof. 1) The explicit description of δ (see [6, pp. 128-129]) shows that it can be computed as follows: let f ∈ H^{1}(A_{2}, A), viewed as a homomorphism from A_{2} to A. Then δ(f) = −f_{∗}c, where c ∈ H^{2}(A_{1}, A_{2}) is the class of the extension G. It is then enough to check 1) over Id_{A_{2}} ∈ H^{1}(A_{2}, A_{2}) (in the case A = A_{2}), in which case it follows tautologically from the definition of θ_{A_{1},A_{2}}. This yields a ±-commutative diagram with exact rows

H^{1}(A_{2}, A) ⟶^{δ} H^{2}(A_{1}, A) ⟶  Ker θ_{Ḡ,A}  ⟶ 0
↓^{≀}  ^{θ_{A_{1},A}}↓^{≀}  ^{ρ_{Ḡ,A}}↓
Hom(A_{2}, A) ⟶^{ε_{A_{1}}∗} Hom(s_{1}(A_{1})_{2}, A) ⟶ Hom(Ker(s_{1}(A_{1})_{2} → A_{2}), A) ⟶ 0

hence 2) follows from 1).

We now prove fullness and essential surjectivity in theorem 10.4 just as in section 7. Essential surjectivity follows from the essential surjectivity of gr^{≤2} (theorem 7.1) and the surjectivity of ρ_{Ḡ,A}. Similarly, let G, H ∈ (F il)_{1,st}^{≤3}, A = g̃rG, B = g̃rH and u : A → B a morphism

in M̃L_{st}^{≤3}. Let Ḡ = G/A_{3} and H̄ = H/B_{3}. By theorem 7.1, t_{2}(u) comes from a homomorphism of filtered groups ū : Ḡ → H̄. Let c ∈ H_{adm}^{2}(Ḡ, A_{3}), c′ ∈ H_{adm}^{2}(H̄, B_{3}) be the classes of the extensions G and H: they actually sit respectively in Ker θ_{Ḡ,A_{3}} and Ker θ_{H̄,B_{3}}. The hypothesis

that u is a morphism in M̃L_{st}^{≤3} implies that ρ_{Ḡ,B_{3}}(ū^{∗}c′) = ρ_{Ḡ,B_{3}}(u_{3∗}c) ∈ Hom(Ker(s_{1}(t_{1}A)_{2} → A_{2}), B_{3}). The injectivity of ρ_{Ḡ,B_{3}} now implies that ū^{∗}c′ = u_{3∗}c ∈ Ker θ_{Ḡ,B_{3}}, hence a commutative diagram similar to (8). □

References

[1] N. Bourbaki Groupes et algèbres de Lie, Chapitre II: algèbres de Lie libres, Hermann, Paris, 1972.

[2] B. Eckmann, P. J. Hilton and U. Stammbach On the homology theory of central group extensions; I- The commutator map and stem extensions, Comment. Math. Helv. 47 (1972), 102–122.

[3] B. Eckmann, P. J. Hilton and U. Stammbach On the homology theory of central group extensions; II- The extended exact sequence in the general case, Comment. Math. Helv. 47 (1972), 171–178.

[4] T. Ganea Homologie et extensions centrales de groupes, C. R. Acad. Sci. Paris 266 (1968), 556–558.

[5] A. Gut A ten-term exact sequence in the homology of a group extension, J. Pure Appl. Alg. 8 (1976), 243–260.

[6] G. Hochschild and J. P. Serre Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110–134.

[7] B. Huppert Endlische Gruppen, I, Grundl. math. Wiss. 134, Springer, Berlin, 1967.

[8] B. Huppert and N. Blackburn Finite groups, II, Grundl. math. Wiss. 242, Springer, Berlin, 1982.

[9] N. Iwahori and H. Matsumoto Several remarks on projective representations of finite groups, J. Fac. Sci. Univ. Tokyo 10 (1963), 129–146.

[10] B. Kahn The total Stiefel-Whitney class of a regular representation, J. Alg. 144 (1991), 214–247.

[11] B. Kahn Erratum: The total Stiefel-Whitney class of a regular representation, J. Alg. 152 (1992), 258.

[12] B. Kahn Fonction de Hilbert-Samuel de certaines algèbres graduées, C. R. Acad. Sci. Paris 316 (1993), 1129–1134.

[13] M. Lazard Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965).

[14] S. Mac Lane Categories for the working mathematician, Grad. Texts in Math. 5, Springer, New York, 1971.

[15] P. A. Minh d-maximal p-groups and Stiefel-Whitney classes of a regular representation, J. Alg. 179 (1996), 483–500.

[16] P. A. Minh On a conjecture of Kahn on Stiefel-Whitney classes of a regular representation, preprint, 1995.

[17] P. M. Neumann On an enumeration theorem for finite groups, Quart. J. Math. Oxford 20 (1969), 395–401.

[18] N. Q. D. Thong Sur le module des relations d’un p-groupe et la suite exacte d’inflation - restriction, J. Reine Angew. Math. 316 (1980), 110–121.

[19] C. E. Watts Intrinsic characterization of some additive functors, Proc. Amer. Math. Soc. 11 (1960), 5–8.

Institut de Mathématiques de Jussieu, Equipe Théories Géométriques, Université Paris 7, Case 7012, 75251 Paris Cedex 05, France

E-mail address: kahn@mathp7.jussieu.fr

Department of Mathematics, University of Hue, Dai hoc Tong hop Hue, Hue, Vietnam