Dept. of Math. Univ. of Oslo Pure Mathematics ISBN 82–553–1393–1 No. 25 ISSN 0806–2439 July 2003

THE MODULAR ISOMORPHISM PROBLEM

INGER CHRISTIN BORGE AND OLAV ARNFINN LAUDAL

Abstract. We show that the isomorphism class of a finite p-group G is determined by 𝔽_{p}G, the group algebra of G over the field of p elements, hence solving the modular isomorphism problem.

1. Introduction

The modular isomorphism problem goes back to the 1950s and asks whether the isomorphism class of a finite p-group G can be determined by its modular group algebra 𝔽_{p}G where 𝔽_{p} is the field of p elements. In [4] we considered this problem in the context of deformation theory, introducing generalized matric Massey products, see also [5], where these ideas were applied to deformations of schemes. Using the classical Massey product structure on H^{∗}(G, 𝔽_{p}), we could in [1] prove some partial results, and in [2] we discussed the limitations of the methods in [1].

This last work was inspired by the work done by Laudal in noncommutative geometry, [6] and [7]. Going back to this framework, and more specifically, to noncommutative deformation theory, we are now able to give a positive answer to the modular isomorphism problem. This is done in Section 4, where we show that the isomorphism class of a finite p-group G is in fact determined by its modular group algebra 𝔽_{p}G (Theorem 4.3).

Quillen showed in [9] that the restricted graded Lie algebra of a finite p-group G is determined by 𝔽_{p}G. The question is how does one get from the graded object to the corresponding filtered object? The idea is to use deformation theory. Both G and 𝔽_{p}G have a unique simple module, 𝔽_{p}. Also, both the group and the group algebra have an obstruction theory, and so in both cases we can work with the obstructions for deforming this simple, trivial module.

In Theorem 2.7, we give a detailed new proof of the main result in [4], proving that G is, in a sense, the formal moduli of 𝔽_{p} in the category of pro-p groups. The essential new element is the use of the Brauer-Jennings-Zassenhaus series, replacing the lower central p-series used in [4].

Date: July 3, 2003.

Section 3 treats the algebra case. The Generalized Burnside Theorem, see [6], implies that 𝔽_{p}G is the formal moduli of 𝔽_{p} in the category of complete local 𝔽_{p}-algebras. We choose to reprove this result as Theorem 3.7, insisting on the analogy with the proof of Theorem 2.7.

The properties of the Brauer-Jennings-Zassenhaus series now imply that the obstructions needed in the group construction can be made to coincide with the obstructions needed in the algebra construction, from which our main result, Theorem 4.3, readily follows.

2. Obstruction theory for pro-p groups

Any finite dimensional vector space over 𝔽_{p} will be called an elementary p-group. If G is a finite p-group and K an elementary p-group on which G acts trivially, we shall call K an elementary G-module.

The cohomology groups H^{n}(G, K) := Ext_{𝔽_{p}G}^{n}(𝔽_{p}, K) can be computed using the bar resolution and the corresponding cocomplex (B^{•}(G, K), δ) defined by

(2.1)  B^{n}(G, K) = Maps(G^{n}, K),

the set of maps from G^{n} = \underbrace{G × ⋯ × G}_{n} to K, and with differentials

\xymatrix{δ^{n}: B^{n}(G, K) \ar[r] & B^{n+1}(G, K)}

given for ϕ ∈ B^{n}(G, K) by

δ^{n}(ϕ) (g_{1}, …, g_{n+1})
= ϕ(g_{2}, …, g_{n+1})+
+ ∑_{i=1}^{n}(−1)^{i}ϕ(g_{1}, …, g_{i}g_{i+1}, …, g_{n+1})+
+(−1)^{n+1}ϕ(g_{1}, …, g_{n}).

In particular, for ϕ ∈ B^{1}(G, K),

δ^{1}(ϕ)(g_{1}, g_{2}) = ϕ(g_{2}) − ϕ(g_{1}g_{2}) + ϕ(g_{1}),

and for ϕ ∈ B^{2}(G, K),

(2.2) δ^{2}(ϕ)(g_{1}, g_{2}, g_{3}) = ϕ(g_{2}, g_{3}) − ϕ(g_{1}g_{2}, g_{3}) + ϕ(g_{1}, g_{2}g_{3}) − ϕ(g_{1}, g_{2}).

We have H^{n}(G, K) ≃ H^{n}(B^{•}(G, K), δ).

Our first lemma relates to the following lifting situation. Let

(2.3)  \xymatrix{D : 1 \ar[r] & K_{D} \ar[r]^{ι} & H_{1} \ar[r]^{κ} & H_{2} \ar[r] & 1 \\  &  &  & G \ar[u]_{ϕ}}

be a diagram where κ is a surjective homomorphism of groups and K_{D} = ker κ is a central elementary subgroup of H_{1}, i.e. ι(K_{D}) ⊂ Z(H_{1}) and K_{D} ≃ 𝔽_{p}^{m}, some m, but written multiplicatively.

Definition 2.1. Let D be the category of all such diagrams (2.3) for a fixed group G together with the obvious morphisms of such diagrams. An object D in D is thus a central extension of groups with elementary kernel K_{D} together with a homomorphism ϕ from G to the base group in the extension.

We have the following well-known result.

Lemma 2.2. To each diagram D in D one associates a cohomology class obs(D) ∈ H^{2}(G, K_{D}) such that obs(D) = 0 if and only if there exists a group homomorphism \xymatrix{ϕ′: G \ar[r] & H_{1}} with ϕ = κ ∘ ϕ′.

Proof. Let \xymatrix{σ: H_{2} \ar[r] & H_{1}} be a set-theoretic section of κ. For every pair g_{1}, g_{2} ∈ G, consider the element

(2.4)  ψ(g_{1}, g_{2}) = σ(ϕ(g_{1}g_{2}))σ(ϕ(g_{2}))^{−1}σ(ϕ(g_{1}))^{−1}

in H_{2}. Since κ(ψ(g_{1}, g_{2})) = 1, this defines an element ψ ∈ Maps(G×G, K_{D}), and using (2.2), we check that δ^{2}(ψ)(g_{1}, g_{2}, g_{3}) = 1.

Put obs(D) = [ψ] ∈ H^{2}(G, K_{D}), the cohomology class of the cocycle ψ. We note that a different choice σ̃ for the section σ in (2.4) will give a map ψ̃ with ψ−ψ̃ = δ^{1}(σ̃∘ϕ−σ∘ϕ), so the cohomology class obs(D) is well-defined.

Assume obs(D) = 0. Then ψ ∈ imδ^{1}, and so there exists α ∈ Maps(G, K_{D}) such that δ^{1}(α) = ψ. Define \xymatrix{ϕ′: G \ar[r] & H_{2}} by

ϕ′(g) = σ(ϕ(g))α(g), g ∈ G.

Then, since we have a central extension, ϕ′ is a group homomorphism, and ϕ = κ ∘ ϕ′.

Conversely, suppose there exists a ϕ′ such that ϕ = κ∘ϕ′. We need to show that obs(D) is 0 as an element of H^{2}(G, K_{D}). Let \xymatrix{γ: G \ar[r] & H_{2}} be defined by γ(g) = ϕ′(g)σ(ϕ(g))^{−1}. Then, for g ∈ G, we have that κ(γ(g)) = 1, so γ ∈ Maps(G, K_{D}), and

ψ(g_{1}, g_{2}) = σ(ϕ(g_{1}g_{2}))σ(ϕ(g_{2}))^{−1}σ(ϕ(g_{1}))^{−1}
= σ(ϕ(g_{1}g_{2}))ϕ′(g_{1}g_{2})^{−1}ϕ′(g_{1}g_{2})σ(ϕ(g_{2}))^{−1}σ(ϕ(g_{1}))^{−1}
= γ(g_{1}g_{2})^{−1}γ(g_{2})γ(g_{1})

since γ(g_{2}) commutes with ϕ′(g_{1}). Therefore,

ψ(g_{1}, g_{2}) = γ(g_{2})γ(g_{1}g_{2})^{−1}γ(g_{1}),

which proves the lemma. □

Consider a morphism ζ̄: \xymatrix{D \ar[r] & D′} in D. This is the same as a commutative diagram

\xymatrix{D′ : 1 \ar[r] & K_{D′} \ar[r]^{ι′} & H′_{1} \ar[r]^{κ′} & H′_{2} \ar[r] & 1 \\ D : 1 \ar[r] & K_{D} \ar[u]^{ζ} \ar[r]^{ι} & H_{1} \ar[u]^{η} \ar[r]^{κ} & H_{2} \ar[u]_{θ} \ar[r] & 1 \\  &  &  & G. \ar[uu] \ar[u]_{ϕ} \ar@{-->}[uul]_{ϕ′′} \ar@{-->}[ul]^{ϕ′}}

The morphism ζ̄ induces a map making H^{2}(G, −) a functor from the category D to the category of vector spaces over 𝔽_{p}. Moreover, to every diagram D in D, we have shown that there exists a cohomology class obs(D) ∈ H^{2}(G, K_{D}).

Corollary 2.3. The map \xymatrix{ζ_{∗}: H^{2}(G, K_{D}) \ar[r] & H^{2}(G, K_{D′})}, induced by ζ, maps obs(D) to obs(D′), i.e. the cohomology class obs(D) is functorial.

Proof. Let σ′ be a set-theoretic section of κ′. Referring to the proof of Lemma 2.2, obs(D′) = [ψ′], where ψ′ ∈ Maps(G × G, K_{D′}) is defined by

ψ′(g_{1}, g_{2}) = σ′(θ(ϕ(g_{1}g_{2})))σ′(θ(ϕ(g_{2})))^{−1}σ′(θ(ϕ(g_{1})))^{−1}.

On the other hand, ζ_{∗}(ψ) = ζ ∘ ψ. We may pick the section σ′ such that η ∘ σ = σ′ ∘ θ. Then

η(ψ(g_{1}, g_{2})) = η(σ(ϕ(g_{1}g_{2})))η(σ(ϕ(g_{2}))^{−1})η(σ(ϕ(g_{1}))^{−1})
= η(σ(ϕ(g_{1}g_{2})))η(σ(ϕ(g_{2})))^{−1}η(σ(ϕ(g_{1})))^{−1}
= σ′(θ(ϕ(g_{1}g_{2})))σ′(θ(ϕ(g_{2})))^{−1}σ′(θ(ϕ(g_{1})))^{−1}
= ψ′(g_{1}, g_{2}),

hence [ζ ∘ ψ] = [ψ′] and the result follows. □

Generalising the above, we make the following definition.

Definition 2.4. A map o defined on the set of objects of D mapping D ∈ ob(D) to an element o_{D} ∈ H^{2}(G, K_{D}) will be called an obstruction if it satisfies the following two conditions:

OG1:: o_{D} is functorial, i.e. if \xymatrix{D \ar[r] & D′} is a morphism in D, then

\xymatrix{H^{2}(G, K_{D}) \ar[r] & H^{2}(G, K_{D′})} maps o_{D} to o_{D′};

OG2:: o_{D} = 0 if and only if \xymatrix{ϕ: G \ar[r] & H_{2}} lifts to H_{1}.

In Lemma 2.2, we have constructed such an obstruction.

Now observe that functorially in D we have

H^{2}(G, K_{D}) ≃ H^{2}(G, 𝔽_{p}) ⊗_{𝔽_{p}} K_{D} ≃ Hom_{𝔽_{p}}(H^{2}(G, 𝔽_{p})^{∗}, K_{D}).

Moreover, we have the following easy lemma.

Lemma 2.5. Let o and o′ be two obstructions. For every D ∈ ob(D), consider o_{D}, o′_{D} ∈ H^{2}(G, K_{D}) ≃ Hom_{𝔽_{p}}(H^{2}(G, 𝔽_{p})^{∗}, K_{D}). Then imo_{D} = imo′_{D} as vector subspaces in K_{D}.

Proof. Consider a diagram D in D with K_{D}, H_{1} and H_{2} making up the central extension. Since o is an obstruction, we can lift ϕ to H_{1}/imo_{D}:

[figure: 1; K_{D}/imo_{D}; ῑ; H_{1}/imo_{D}; κ; H_{2}; 1; 1; K_{D}; ζ; ι; H_{1}; η; κ; H_{2}; 1; G; ϕ; ϕ′]

Now, o′_{D} is also an obstruction for lifting ϕ, and imo′_{D} is therefore, by functoriality, sent to 0 in K_{D}/imo_{D}. Hence imo′_{D} ⊆ imo_{D}, and by symmetry we get equality. □

From this result we see that the image imo_{D} ⊂ K_{D} of the obstruction is the interesting thing. This image will from now on be called the obstruction.

We introduce the notation ⟨−⟩ and \overline{⟨−⟩} to denote the normal subgroup generated by − and the closed normal subgroup generated by − respectively.

Let T^{i} be the free pro-p group on a dual basis for H^{i}(G, 𝔽_{p}), i = 1, 2. Since H^{1}(G, 𝔽_{p}) ≃ (G/⟨[G, G], G^{p}⟩)^{∗}, there is a surjective homomorphism \xymatrix{π: T^{1} \ar[r] & G}. In the next theorem we construct a morphism o_{G} from T^{2} to T^{1} with T^{1}/\overline{⟨imo_{G}⟩} ≃ G.

It is well-known that there exists such a morphism \xymatrix{o: T^{2} \ar[r] & T^{1}} with T^{1}/\overline{⟨imo⟩} ≃ G, see [10]. The interest of our result lies in the particular construction of o_{G}.

We construct the morphism o_{G} as the inverse limit of a certain coherent sequence of morphisms from T^{2} to quotients of T^{1} using the above obstruction calculus at each level. The quotients involve a filtration {F_{i}(T^{1})}_{i≥1} of T^{1}. This construction is written down in some detail in [4], but then using the lower central p-series as the filtration. However, for our purposes, we shall use the M-series, also known as the Brauer-Jennings-Zassenhaus series.

Definition 2.6. The M-series of a pro-p group G is defined inductively by

\begin{aligned}M_{1}(G) & = & G, \\ M_{i}(G) & = & \overline{⟨[M_{i−1}(G), G], M_{(i/p)}(G)^{(p)}⟩}, \end{aligned}

where (i/p) is the least integer ≥ i/p and M_{i}(G)^{(p)} denotes the set of p-th powers in M_{i}(G).

We note that T^{i} is the projective limit of the quotients T^{i}/M_{n}(T^{i}).

The successive quotients M_{i}(G)/M_{i+1}(G) are vector spaces over 𝔽_{p}. The series {M_{i}(G)}_{i≥1} is not the fastest descending series with this property, however, it is the fastest descending series satisfying

(2.5)  [M_{n}(G), M_{m}(G)] ≤ M_{n+m}(G),
(2.6)  M_{n}(G)^{p} ≤ M_{np}(G).

Hence the M-series is the “restricted mod p lower central series”, see [3] and [8, chapter 11].

Theorem 2.7. Let G be a finite p-group and let T^{i} be the free pro-p group on a dual basis for H^{i}(G, 𝔽_{p}), i = 1, 2. Then the above obstruction calculus induces an obstruction morphism \xymatrix{o_{G}: T^{2} \ar[r] & T^{1}} with T^{1}/\overline{⟨imo_{G}⟩} ≃ G.

Proof. We introduce the notation T_{n}^{1} := T^{1}/M_{n}(T^{1}) and G_{n} := G/M_{n}(G). Let \xymatrix{σ_{n}: G \ar[r] & G_{n}} denote the canonical homomorphism.

We know that dim_{𝔽_{p}}H^{1}(G, 𝔽_{p}) is the minimal number of generators for G. It is also known that dim_{𝔽_{p}}H^{2}(G, 𝔽_{p}) is the minimal number of pro-p relations for G, but we are not going to use this fact.

Fix a surjective homomorphism \xymatrix{π: T^{1} \ar[r] & G}. For the first step, we note that π induces an isomorphism

\xymatrix{θ_{2} : T_{2}^{1} \ar[r]^{≃} & G_{2}}

and consider the following commutative diagram

\xymatrix{D_{2} : 1 \ar[r] & M_{2}(T^{1})/M_{3}(T^{1}) \ar[r]^{ι} & T_{3}^{1} \ar[r]^{π_{2}^{3}} & T_{2}^{1} \ar[r] \ar[d]^{≃}_{θ_{2}} & 1 \\  &  &  & G_{2} \\  &  &  & G, \ar@{->>}[u]^{σ_{2}} \ar@{->>}[uu]^{ϕ̄_{1}}}

which is an object in the category D. By Lemma 2.2 we have a unique element obs(D_{2}) ∈ H^{2}(G, M_{2}(T^{1})/M_{3}(T^{1})) such that obs(D_{2}) = 0 if and only if there exists a lifting ϕ_{2} of ϕ̄_{1} to T_{3}^{1}.

Now,

H^{2}(G, M_{2}(T^{1})/M_{3}(T^{1})) ≃ H^{2}(G, 𝔽_{p}) ⊗_{𝔽_{p}} M_{2}(T^{1})/M_{3}(T^{1})
≃ Hom_{𝔽_{p}}(H^{2}(G, 𝔽_{p})^{∗}, M_{2}(T^{1})/M_{3}(T^{1}))
≃ Mor(T^{2}, M_{2}(T^{1})/M_{3}(T^{1}))

and

Mor(T^{2}, M_{2}(T^{1})/M_{3}(T^{1})) ⊆ Mor(T^{2}, T_{3}^{1}),

so let o_{2} be obs(D_{2}) as an element in Mor(T^{2}, T_{3}^{1}). We shall refer to imobs(D_{2}) as the obstruction on the first level, which lies in

K_{2} := M_{2}(T^{1})/M_{3}(T^{1}) = M_{2}(T_{3}^{1}).

Remark: By Lemma 2.2, ⟨imo_{2}⟩ is the smallest normal subgroup of T_{3}^{1} containing the elements that need to be reduced to 1 in order to get a lifting of ϕ̄_{1}. In other words, the universal functorial property of these obstructions implies that T_{3}^{1}/⟨imo_{2}⟩ is the largest quotient of T_{3}^{1} making a lifting ϕ_{2} of ϕ̄_{1} possible. So for all other quotients T_{3}^{1}/N and a map \xymatrix{ϕ̃_{2} : G \ar@{->>}[r] & T_{3}^{1}/N} lifting ϕ̄_{1} there exists a homomorphism \xymatrix{ρ: T_{3}^{1}/⟨imo_{2}⟩ \ar[r] & T_{3}^{1}/N} such that the following diagram commutes

\xymatrix{T_{3}^{1}/⟨imo_{2}⟩ \ar[d]_{ρ} \\ T_{3}^{1}/N & G. \ar@{->>}[l]^{ϕ̃_{2}} \ar@{->>}[ul]_{ϕ_{2}}}

Also note that π induces a commutative diagram of surjective homomorphisms,

\xymatrix{T_{3}^{1} \ar@{->>}[rr]^{α_{3}} \ar@{->>}[dr]_{π_{2}^{3}} &  & G_{3} \ar@{->>}[dl] \\  & T_{2}^{1} &  & G, \ar@{->>}[ll]_{ϕ̄_{1}} \ar@{->>}[ul]_{σ_{3}}}

and therefore a surjective homomorphism θ_{3} in the commutative diagram

\xymatrix{T_{3}^{1}/⟨imo_{2}⟩ \ar@{->>}[r]^{θ_{3}} & G_{3} \\ T^{1} \ar[u] \ar@{->>}[r]^{π} & G. \ar[u]_{σ_{3}}}

To show that θ_{3} is an isomorphism, consider the diagram

[figure: 1; 1; M_{2}(T_{3}^{1}); T_{3}^{1}; α_{3}; T_{2}^{1}; 1; T_{3}^{1}/⟨imo_{2}⟩; θ_{3}; ≃; M_{2}(T_{3}^{1})/⟨imo_{2}⟩; G,; ϕ_{2}; ϕ̄_{1}; σ_{3}; 1; G_{3}; ψ_{3}]

and observe that (any) ϕ_{2} lifting ϕ̄_{1} factorizes via G_{3} since ϕ_{2} maps M_{3}(G) to 1 in T_{3}^{1}/⟨imo_{2}⟩. Let \xymatrix{ψ_{3} : G_{3} \ar[r] & T_{3}^{1}/⟨imo_{2}⟩} be the induced homomorphism. Since ϕ_{2} must be surjective, ψ_{3} is surjective. Therefore G_{3} and T_{3}^{1}/⟨imo_{2}⟩ have the same order, thus θ_{3} (as well as ψ_{3}) is an isomorphism induced by π.

We have now seen most of the first non-trivial step in the construction of o_{G}: So far we have

1): a surjective homomorphism \xymatrix{π: T^{1} \ar[r] & G} and

2): a homomorphism \xymatrix{o_{2} : T^{2} \ar[r] & T_{3}^{1}} inducing the isomorphism θ_{3} in : the commutative diagram

\xymatrix{T^{1} \ar[r]^{π} \ar[d] & G \ar[d]^{σ_{3}} \\ T_{3}^{1}/⟨imo_{2}⟩ \ar[r]^{θ_{3}}_{≃} & G_{3}.}

Our aim is to find a coherent sequence of homomorphisms

\{\xymatrix{o_{n}: T^{2} \ar[r] & T_{n+1}^{1}} \}_{n≥1},

o_{1} being trivial, inducing an isomorphism θ_{n+1} in the commutative diagram

\xymatrix{T^{1} \ar[r]^{π} \ar[d] & G \ar[d]^{σ_{n+1}} \\ T_{n+1}^{1}/⟨imo_{n}⟩ \ar[r]^{θ_{n+1}}_{≃} & G_{n+1}.}

Let \xymatrix{ϕ̄_{n}: G \ar[r] & T_{n+1}^{1}/⟨imo_{n}⟩} be the obvious composition of σ_{n+1} and (θ_{n+1})^{−1} for n ≥ 2.

To finish the first step of the construction, we need to see the coherence between the first and second step. So, before we do the general step, let us see what happens when we try to lift ϕ̄_{2}.

We start with the diagram

\xymatrix{1 \ar[r] & ⟨M_{3}(T^{1}), L_{2}⟩/M_{4}(T^{1}) \ar[r] & T_{4}^{1} \ar[r] & T_{3}^{1}/⟨imo_{2}⟩ \ar[r] & 1 \\  &  &  & G, \ar@{->>}[u]_{ϕ̄_{2}}}

where L_{2} is the preimage of ⟨imo_{2}⟩ in T^{1} (so L_{2} ⊲ T^{1}).

However, this is not a diagram to which we can apply Lemma 2.2. We want the kernel to be both central and elementary, hence we must consider the quotient diagram

\xymatrix{D_{3} : 1 \ar[r] & K_{3} \ar[r] & T^{1}/N_{4} \ar[r] & T_{3}^{1}/⟨imo_{2}⟩ \ar[r] & 1 \\  &  &  & G, \ar@{->>}[u]_{ϕ̄_{2}}}

where N_{4} := \overline{⟨M_{4}(T^{1}), [T^{1}, L_{2}], (L_{2})^{p}⟩} and K_{3} := ⟨M_{3}(T^{1}), L_{2}⟩/N_{4}. From Lemma 2.2 we get a unique cohomology class

obs(D_{3}) ∈ H^{2}(G, K_{3}) ≃ Mor(T^{2}, K_{3}) ⊂ Mor(T^{2}, T^{1}/N_{4}),

which has to be 0 in order to get a lifting of ϕ̄_{2}. Hence the obstruction imobs(D_{3}) on the second level lies in K_{3}.

Let o(ϕ̄_{2}) be obs(D_{3}) as an element in Mor(T^{2}, T^{1}/N_{4}) and consider the following diagram where the four arrows [figure]denote two short-exact sequences

(2.7) [figure: T_{2}^{1}; T^{2}; o(ϕ̄_{2}); o_{2}; o_{3}; K_{2}; T_{3}^{1}; π_{2}^{3}; T_{4}^{1}; π_{3}^{4}; ρ_{4}; T_{3}^{1}/⟨imo_{2}⟩; K_{3}; μ̄; T^{1}/N_{4}; μ; _{\overline{π}_{4}_{3}}; (T^{1}/N_{4})/imobs(D_{3}); θ_{4}; T_{4}^{1}/⟨imo_{3}⟩; B_{4}; =; G_{4}; ψ_{4}; G; ϕ̄_{1}; ϕ̄_{2}; ϕ_{3}; σ_{4}; T^{1}.; π; π_{2}]

Recall that ⟨imo_{2}⟩ lies in M_{2}(T^{1})/M_{3}(T^{1}) and that L_{2} lies in M_{2}(T^{1}), so ⟨[T^{1}, L_{2}], (L_{2})^{p}⟩ ⊆ M_{3}(T^{1}). Therefore there exists a homomorphism μ which factors π_{3}^{4}. It also factors \overline{π}_{3}^{4} and induces a homomorphism

\xymatrix{\overline{μ}: K_{3} \ar[r] & K_{2}} ,

which again induces a linear map

\xymatrix{\overline{μ}_{∗}: H^{2}(G, K_{3}) \ar[r] & H^{2}(G, K_{2})}.

At the same time we see that L_{2} ⊂ T^{1} is mapped to 1 in G_{3}, therefore into M_{3}(G) in G. But then π maps ⟨[T^{1}, L_{2}], (L_{2})^{p}⟩ into M_{3p}(G) ⊆ M_{4}(G). Therefore the homomorphism \xymatrix{T_{4}^{1} \ar[r] & G_{4}} induced by π factorizes through a surjection \xymatrix{T^{1}/N_{4} \ar@{->>}[r] & G_{4}} . By functoriality of the obstructions (see Corollary 2.3), it also factorizes via the surjection

\xymatrix{θ_{4} : (T^{1}/N_{4})/imobs(D_{3}) \ar@{->>}[r] & G_{4}} .

As above we prove that (any) lifting ϕ_{3} of ϕ̄_{2} must be surjective and must map M_{4}(G) to 1, inducing a surjection

\xymatrix{ψ_{4} : G_{4} \ar@{->>}[r] & (T^{1}/N_{4})/imobs(D_{3})}.

Therefore θ_{4} is an isomorphism induced by π.

Pick an o_{3} such that ρ_{4} ∘ o_{3} = o(ϕ̄_{2}). By construction,

π_{3}^{4} ∘ o_{3} = μ ∘ ρ_{4} ∘ o_{3} = μ ∘ o(ϕ̄_{2}) = \overline{μ}_{∗}(obs(D_{3})),

which as an element of Mor(T^{2}, K_{2}) ⊂ Mor(T^{2}, T_{3}^{1}) is equal to o_{2}, since by functoriality the map \overline{μ}_{∗} sends obs(D_{3}) to obs(D_{2}). Let L_{3} be the preimage of ⟨imo_{3}⟩ in T^{1}. Then, by the properties (2.5) and (2.6) of the M-series, we find that

⟨[T^{1}, L_{2}], (L_{2})^{p}⟩ ⊆ L_{3}.

Hence there is a canonical surjection \xymatrix{T^{1}/N_{4} \ar@{->>}[r] & T_{4}^{1}/⟨imo_{3}⟩}. Moreover, ⟨imo_{3}⟩ maps onto imobs(D_{3}) in K_{3}, inducing the canonical isomorphism B_{4} which identifies T_{4}^{1}/⟨imo_{3}⟩ and (T^{1}/N_{4})/imobs(D_{3}). Put

\xymatrix{ϕ̄_{3} = (θ_{4})^{−1} ∘ σ_{4} : G \ar[r] & T_{4}^{1}/⟨imo_{3}⟩}.

We have now seen the first non-trivial step of the construction of o_{G} in detail. The construction continues: Assume, by induction, that we have constructed o_{n−1} such that there exists an isomorphism

\xymatrix{θ_{n}: T_{n}^{1}/⟨imo_{n−1}⟩ \ar[r]^{≃} & G_{n}}

induced by π. Put ϕ̄_{n−1} = (θ_{n})^{−1} ∘ σ_{n}. Then, for lifting ϕ̄_{n−1}, we have the diagram (2.8)

\xymatrix{1 \ar[r] & ⟨M_{n}(T^{1}), L_{n−1}⟩/M_{n+1}(T^{1}) \ar[r] & T_{n+1}^{1} \ar[r] & T_{n}^{1}/⟨imo_{n−1}⟩ \ar[r] & 1 \\  &  &  & G, \ar@{->>}[u]_{ϕ̄_{n−1}}}

where L_{n−1} is the preimage of ⟨imo_{n−1}⟩ in T^{1}.

Define N_{n+1} by

(2.9)  N_{n+1} := \overline{⟨M_{n+1}(T^{1}), [T^{1}, L_{n−1}], (L_{n−1})^{p}⟩}

and consider the central elementary extension, i.e. the diagram

\xymatrix{D_{n} : 1 \ar[r] & K_{n} \ar[r] & T^{1}/N_{n+1} \ar[r] & T_{n}^{1}/⟨imo_{n−1}⟩ \ar[r] & 1 \\  &  &  & G, \ar[u]_{ϕ̄_{n−1}}}

where

(2.10)  K_{n} := ⟨M_{n}(T^{1}), L_{n−1}⟩/N_{n+1}.

As before, we get the obstruction imobs(D_{n}) ⊆ K_{n} and a lifting

\xymatrix{ϕ_{n}: G \ar[r] & (T^{1}/N_{n+1})/imobs(D_{n}),}

inducing an isomorphism

\xymatrix{θ_{n+1} : (T^{1}/N_{n+1})/imobs(D_{n}) \ar[r]^{≃} & G_{n+1}}

commuting with π.

Now we need to check that there exists an o_{n} and a commutative diagram

\xymatrix{T^{2} \ar[dr]_{o(ϕ̄_{n−1})} \ar@{-->}[r]^{o_{n}} \ar[rr]^{o_{n−1}} & T_{n+1}^{1} \ar[d]^{ρ_{n+1}} \ar[r] & T_{n}^{1} \ar[d] \\  & T^{1}/N_{n+1} \ar[r] & T^{1}/N_{n}}

with the properties we want, in particular, the canonical isomorphism

\xymatrix{B_{n+1} : T_{n+1}^{1}/⟨imo_{n}⟩ \ar[r]^{≃} & (T^{1}/N_{n+1})/imobs(D_{n}),}

identifying the two groups.

We denote by L_{n} the preimage of ⟨imo_{n}⟩ in T^{1} under the map \xymatrix{T^{1} \ar@{->>}[r] & T_{n+1}^{1}} . Observe that, by induction, L_{n−1} ⊆ L_{n−2}, implying

(2.11)  ⟨[T^{1}, L_{n−1}], (L_{n−1})^{p}⟩ ⊆ ⟨[T^{1}, L_{n−2}], (L_{n−2})^{p}⟩.

Note that (2.11) is an equality modulo M_{n+1}(T^{1}).

Recall that we defined N_{n} = \overline{⟨M_{n}(T^{1}), [T^{1}, L_{n−2}], (L_{n−2})^{p}⟩} and that N_{n+1} = \overline{⟨M_{n+1}(T^{1}), [T^{1}, L_{n−1}], (L_{n−1})^{p}⟩}. From (2.11) it now follows that we have a surjection

(2.12)  \xymatrix{M_{n}(T^{1})/M_{n+1}(T^{1}) \ar@{->>}[r] & N_{n}/N_{n+1}} .

Consider the diagram

\xymatrix{T^{2} \ar[dr]_{o(ϕ̄_{n−1})} \ar[drr]_{o(ϕ̄_{n−2})} \ar@{-->}[r]^{o_{n}} \ar[rr]^{o_{n−1}} & T_{n+1}^{1} \ar@{->>}[d]^{ρ_{n+1}} \ar@{->>}[r]^{π_{n}^{n+1}} & T_{n}^{1} \ar@{->>}[d]^{ρ_{n}} \\  & T^{1}/N_{n+1} \ar@{->>}[r]^{γ} & T^{1}/N_{n}.}

We need to show that there exists an o_{n} such that π_{n}^{n+1} ∘ o_{n} = o_{n−1} and ρ_{n+1} ∘o_{n} = o(ϕ̄_{n−1}). By definition, T^{2} is the free pro-p group on a dual basis for H^{2}(G, 𝔽_{p}). Fix generators y_{1}, …, y_{r} for T^{2} corresponding to a basis for H^{2}(G, 𝔽_{p})^{∗}. Let y := y_{i} ∈ T^{2} for an i ∈ {1, …, r} and pick y′ ∈ T_{n+1}^{1} such that π_{n}^{n+1}(y′) = o_{n−1}(y). We have

γ((ρ_{n+1}(y′))^{−1} · o(ϕ̄_{n−1})(y)) = (ρ_{n}(π_{n}^{n+1}(y′)))^{−1} · o(ϕ̄_{n−2})(y)
= (ρ_{n}(o_{n−1}(y)))^{−1} · o(ϕ̄_{n−2})(y)
= 1,

hence (ρ_{n+1}(y′))^{−1} · o(ϕ̄_{n−1})(y) ∈ ker γ = N_{n}/N_{n+1}. By (2.12), there exists an element a ∈ ker π_{n}^{n+1} = M_{n}(T^{1})/M_{n+1}(T^{1}) such that ρ_{n+1}(a) = (ρ_{n+1}(y′))^{−1}·o(ϕ̄_{n−1})(y). Hence if we let o_{n}(y) = y′·a, we get π_{n}^{n+1}(o_{n}(y)) = o_{n−1}(y) and ρ_{n+1}(o_{n}(y)) = o(ϕ̄_{n−1})(y), which is what we needed to show.

By construction of o_{n}, there is a canonical surjection

\xymatrix{B_{n+1} : T_{n+1}^{1}/⟨imo_{n}⟩ \ar@{->>}[r] & (T^{1}/N_{n+1})/imobs(D_{n}).}

By definition, L_{n} is the preimage in T^{1} of ⟨imo_{n}⟩, and T^{1}/L_{n} = T_{n+1}^{1}/⟨imo_{n}⟩, therefore M_{n+1}(T^{1}) ⊆ L_{n}.

Now, N_{n+1} = \overline{⟨M_{n+1}(T^{1}), [T^{1}, L_{n−1}], (L_{n−1})^{p}⟩} is contained in L_{n}, since ⟨[T^{1}, L_{n−1}], (L_{n−1})^{p}⟩ ⊆ L_{n}. Therefore we have an obvious commutative diagram

\xymatrix{T_{n+1}^{1} \ar[r]^{ρ_{n+1}} \ar[d] & T^{1}/N_{n+1} \ar@{-->}[dl] \ar[d] \\ T_{n+1}^{1}/⟨imo_{n}⟩ \ar[r]^{B_{n+1}} & (T^{1}/N_{n+1})/imobs(D_{n})}

By construction, ρ_{n+1}(⟨imo_{n}⟩) = imobs(D_{n}), hence B_{n+1} is an equality.

As before, we let \xymatrix{ϕ̄_{n} = (θ_{n+1})^{−1} ∘ σ_{n+1} : G \ar[r] & T_{n+1}^{1}/⟨imo_{n}⟩}, completing the induction process.

To finish, the map \xymatrix{o_{G}: T^{2} \ar[r] & T^{1}} is now the inverse limit of the system \{\xymatrix{o_{n}: T^{2} \ar[r] & T_{n+1}^{1}} \}_{n≥1}. Moreover, the liftings \{ϕ̄_{n}\}_{n≥1} induce isomorphisms T_{n+1}^{1}/⟨imo_{n}⟩ ≃ G_{n+1} for all n ≥ 1, and so T^{1}/\overline{⟨imo_{G}⟩} ≃ G. □

3. Obstruction theory for complete local 𝔽_{p}-algebras

There exists a noncommutative deformation theory for families of modules over k-algebras (k a field), see [6]. In particular, the theory will apply to the case where the family is reduced to 𝔽_{p}, the only simple 𝔽_{p}G-module for a finite p-group G.

Recall that in the previous section, T^{i} denoted the free pro-p group on a dual basis for H^{i}(G, 𝔽_{p}), i = 1, 2. Now let T_{𝔽_{p}G}^{i} denote the completed tensor algebra of the dual vector space of Ext_{𝔽_{p}G}^{i}(𝔽_{p}, 𝔽_{p}) over 𝔽_{p}, i = 1, 2. Therefore, if we put E_{i} = Ext_{𝔽_{p}G}^{i}(𝔽_{p}, 𝔽_{p}), we have

T_{𝔽_{p}G}^{i} ≃ 𝔽_{p} × E_{i}^{∗} × (E_{i}^{∗} ⊗ E_{i}^{∗}) × (E_{i}^{∗} ⊗ E_{i}^{∗} ⊗ E_{i}^{∗}) × ⋯

Let (−) denote the two-sided ideal generated by −, and let \overline{I} denote the closure of the two-sided ideal I in the complete algebra. The closure-notation will sometimes be omitted.

General theory provides us with the existence of an obstruction morphism

\xymatrix{o_{𝔽_{p}G}: T_{𝔽_{p}G}^{2} \ar[r] & T_{𝔽_{p}G}^{1}}

with T_{𝔽_{p}G}^{1}/\overline{(imo_{𝔽_{p}G})} ≃ 𝔽_{p}G.

In the language of deformation theory, this says that 𝔽_{p}G is “the formal moduli” of the trivial module 𝔽_{p}. Similarly, we have seen in Theorem 2.7 that G is “the formal moduli” of 𝔽_{p} when we are in the category of prop groups. Whereas the construction of the obstruction morphism in the previous section used a filtration of T^{1} by its M-series, we will now filter T_{𝔽_{p}G}^{1} by powers of its maximal ideal.

To make it clear that the construction of o_{𝔽_{p}G} just depends on the 𝔽_{p}- algebra structure of 𝔽_{p}G and not on the group structure, we shall construct the obstruction morphism for any complete associative local 𝔽_{p}-algebra A with residue field 𝔽_{p}.

Since Ext_{A}^{1}(𝔽_{p}, 𝔽_{p}) ≃ (\underline{m}/\underline{m}^{2})^{∗} where \underline{m} is the maximal ideal of A, there is a surjective homomorphism \xymatrix{π_{A}: T^{1} \ar[r] & A}.

We shall need an algebra version of Lemma 2.2. In order to formulate this, we have to give some easy (and well-known) results on the Hochschild cohomology of A.

Definition 3.1. Let A be an associative k-algebra, k a field, and let Q be an A-bimodule. The Hochschild cocomplex (C^{•}(A, Q), ∂) is given by

\xymatrix{⋯ \ar[r] & Hom_{k}(A^{⊗n}, Q) \ar[r]^{∂^{n}} & Hom_{k}(A^{⊗(n+1)}, Q) \ar[r] & ⋯}

where the differential ∂ is given by the formula

∂^{n}(ϕ) (a_{1} ⊗ ⋯ ⊗ a_{n+1})
= a_{1}ϕ(a_{2} ⊗ ⋯ ⊗ a_{n+1})+
+ ∑_{i=1}^{n}(−1)^{i}ϕ(a_{1} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1})+
+(−1)^{n+1}ϕ(a_{1} ⊗ ⋯ ⊗ a_{n})a_{n+1}

for ϕ ∈ C^{n}(A, Q), a_{1}, …, a_{n+1} ∈ A. The cohomology groups of this cocomplex are the Hochschild cohomology groups, denoted HH^{n}(A, Q).

We note that if M and N are right A-modules, Hom_{k}(M, N) is an Abimodule, and we have the isomorphism

(3.1)  Ext_{A}^{n}(M, N) ≃ HH^{n}(A, Hom_{k}(M, N))

for n ≥ 0.

Consider the diagram

(3.2)  \xymatrix{E : 0 \ar[r] & V_{E} \ar[r]^{ι} & B_{1} \ar[r]^{κ} & B_{2} \ar[r] & 0 \\  &  &  & A, \ar[u]_{ϕ}}

where κ is a small surjective homomorphism of finite dimensional local algebras over 𝔽_{p} and V_{E} = ker κ, therefore \underline{m}_{1}V_{E} = V_{E}\underline{m}_{1} = 0 where \underline{m}_{1} is the maximal ideal of B_{1}. Thus V_{E} is a vector space over 𝔽_{p} as a B_{1}-module.

Definition 3.2. Let E denote the category of all such diagrams (3.2) for a fixed associative algebra A together with the obvious morphisms of such diagrams. An object E in E is thus a small extension of finite dimensional 𝔽_{p}-algebras with kernel V_{E} and a homomorphism ϕ from A to the base algebra in the extension.

Lemma 3.3. To each diagram E in E one associates a cohomology class obs(E) ∈ Ext_{A}^{2}(𝔽_{p}, V_{E}) such that obs(E) = 0 if and only if there exists an algebra homomorphism \xymatrix{ϕ′: A \ar[r] & B_{1}} with ϕ = κ ∘ ϕ′.

Proof. Let σ be a section from B_{2} to B_{1} as vector spaces over 𝔽_{p}. For every pair a_{1}, a_{2} ∈ A, consider the element

ψ(a_{1} ⊗ a_{2}) = σ(ϕ(a_{1}a_{2})) − σ(ϕ(a_{1}))σ(ϕ(a_{2}))

in B_{2}. Then ψ ∈ Hom_{𝔽_{p}}(A ⊗ A, V_{E}) since ϕ is a homomorphism. We check that (∂^{2}ψ)(a_{1} ⊗ a_{2} ⊗ a_{3}) = 0 and let obs(E) = [ψ] ∈ HH^{2}(A, V_{E}). Note that this cohomology class is independent of the choice of σ.

Assume obs(E) = 0. Then ψ ∈ im∂^{1}, and so there exists α ∈ Hom_{𝔽_{p}}(A, V_{E}) such that ∂^{1}α = ψ. Define \xymatrix{ϕ′: A \ar[r] & B_{2}} by

ϕ′(a) = σ(ϕ(a)) + α(a), a ∈ A.

Then, since we have a small extension, ϕ′ is an algebra homomorphism, and ϕ = κ ∘ ϕ′.

Conversely, suppose there exists a ϕ′ such that ϕ = κ ∘ ϕ′. We need to show that obs(E) is 0 as an element in Ext_{A}^{2}(𝔽_{p}, V_{E}). Let \xymatrix{γ: A \ar[r] & B_{2}} be defined by γ(a) = ϕ′(a) − σ(ϕ(a)). Then we have κ(γ(a)) = 0, hence γ ∈ Hom_{𝔽_{p}}(A, V_{E}). Furthermore, ψ(a_{1} ⊗ a_{2}) = γ(a_{1}) − γ(a_{1}a_{2}) + γ(a_{2}), which proves the lemma. □

We make a similar definition of an obstruction as in the previous section. The functoriality and universal property will also be similar.

Consider a morphism \xymatrix{ζ̄: E \ar[r] & E′} in E, i.e. a commutative diagram

\xymatrix{E′ : 0 \ar[r] & V_{E′} \ar[r]^{ι′} & B′_{1} \ar[r]^{κ′} & B′_{2} \ar[r] & 0 \\ E : 0 \ar[r] & V_{E} \ar[u]^{ζ} \ar[r]^{ι} & B_{1} \ar[u]^{η} \ar[r]^{κ} & B_{2} \ar[u]_{θ} \ar[r] & 0 \\  &  &  & A. \ar[uu] \ar[u]_{ϕ} \ar@{-->}[uul]_{ϕ′′} \ar@{-->}[ul]^{ϕ′}}

The morphism ζ̄ induces a map making Ext_{A}^{2}(𝔽_{p}, −) a functor from the category E to the category of vector spaces over 𝔽_{p}. Moreover, to every diagram E in E, we have shown that there exists a cohomology class obs(E) ∈ Ext_{A}^{2}(𝔽_{p}, V_{E}).

Corollary 3.4. The map \xymatrix{ζ_{∗}: Ext_{A}^{2}(𝔽_{p}, V_{E}) \ar[r] & Ext_{A}^{2}(𝔽_{p}, V_{E′})}, induced by ζ, maps obs(E) to obs(E′), i.e. the cohomology class obs(E) is functorial.

Proof. Similar to the proof of Corollary 2.3. □

As in the previous section, we generalize the above and make the following definition.

Definition 3.5. A map o defined on the set of objects of E mapping E ∈ ob(E) to an element o_{E} ∈ Ext_{A}^{2}(𝔽_{p}, V_{E}) will be called an obstruction if is satisfies the following two conditions:

OA1:: o_{E} is functorial, i.e. if \xymatrix{E \ar[r] & E′} is a morphism in E, then

\xymatrix{Ext_{A}^{2}(𝔽_{p}, V_{E}) \ar[r] & Ext_{A}^{2}(𝔽_{p}, V_{E′})} maps o_{E} to o_{E′};

OA2:: o_{E} = 0 if and only if \xymatrix{ϕ: A \ar[r] & B_{2}} lifts to B_{1}.

In Lemma 3.3, we have constructed such an obstruction.

Now observe that functorially in E we have

Ext_{A}^{2}(𝔽_{p}, V_{E}) ≃ Ext_{A}^{2}(𝔽_{p}, 𝔽_{p}) ⊗_{𝔽_{p}} V_{E} ≃ Hom_{𝔽_{p}}(Ext_{A}^{2}(𝔽_{p}, 𝔽_{p})^{∗}, V_{E}).

Lemma 3.6. Let o and o′ be two obstructions. For every E ∈ ob(E) consider o_{E}, o′_{E} ∈ Ext_{A}^{2}(𝔽_{p}, V_{E}) ≃ Hom_{𝔽_{p}}(Ext_{A}^{2}(𝔽_{p}, 𝔽_{p})^{∗}, V_{E}). Then imo_{E} = imo′_{E} as vector subspaces in V_{E}.

Proof. Similar to the proof of Lemma 2.5. □

Again, we see that the image of the obstruction is the interesting thing. As in the group situation, this image will from now on be called the obstruction.

We are now ready to state and prove the algebra version of Theorem 2.7.

Theorem 3.7. Let A be a complete associative local algebra over 𝔽_{p} with residue field 𝔽_{p}. Let T_{A}^{i} be the completed tensor algebra of the dual vector space of Ext_{A}^{i}(𝔽_{p}, 𝔽_{p}), i = 1, 2, over 𝔽_{p}. Then the above obstruction calculus induces an obstruction morphism \xymatrix{o_{A}: T_{A}^{2} \ar[r] & T_{A}^{1}} with T_{A}^{1}/\overline{(imo_{A})} ≃ A.

Proof. In the proof we will use T^{i} for T_{A}^{i}, \underline{m} and \underline{m}_{A} will denote the maximal ideals of T^{1} and A respectively, and T_{n}^{1} and A_{n} will be short for T^{1}/\underline{m}^{n} and A/\underline{m}_{A}^{n} respectively.

We note that Ext_{A}^{1}(𝔽_{p}, 𝔽_{p}) ≃ (\underline{m}_{A}/\underline{m}_{A}^{2})^{∗} and that A/\underline{m}_{A}^{2} ≃ T^{1}/\underline{m}^{2}. Moreover, T^{1} is the completed tensor algebra, and so T_{n}^{1} is always the universal extension of T_{n−1}^{1} with (Ext_{A}^{1}(𝔽_{p}, 𝔽_{p})^{∗})^{⊗(n−1)}, in particular, T^{1}/\underline{m}^{2} ≃ 𝔽_{p} ⊕ Ext_{A}^{1}(𝔽_{p}, 𝔽_{p})^{∗}. Fix a surjective homomorphism \xymatrix{π_{A}: T^{1} \ar[r] & A}. Let \xymatrix{σ_{n}: A \ar[r] & A_{n}} be the canonical homomorphism.

We want to lift algebra homomorphisms from A to quotients of T^{1} involving powers of \underline{m}. For the first step of the construction we note that π_{A} induces an isomorphism \xymatrix{θ_{2} : T_{2}^{1} \ar[r]^{≃} & A_{2}} and consider the following commutative diagram

\xymatrix{E_{2} : 0 \ar[r] & \underline{m}^{2}/\underline{m}^{3} \ar[r]^{ι} & T_{3}^{1} \ar[r]^{π_{2}^{3}} & T_{2}^{1} \ar[r] \ar[d]^{≃}_{θ_{2}} & 0 \\  &  &  & A_{2} \\  &  &  & A. \ar@{->>}[u]^{σ_{2}} \ar@{->>}[uu]^{ϕ̄_{1}}}

For the analogy with the proof of Theorem 2.7, it helps to keep the following “correspondence” in mind: Let T_{G}^{1} be T^{1} in the proof of Theorem 2.7. Then M_{n}(T_{G}^{1}) corresponds to \underline{m}^{n} for n ≥ 2.

Now, E_{2} is an object in the category E. By Lemma 3.3 we have a unique cohomology class obs(E_{2}) ∈ Ext_{A}^{2}(𝔽_{p}, \underline{m}^{2}/\underline{m}^{3}) such that obs(E_{2}) = 0 if and only if there exists a lifting ϕ_{2} of ϕ̄_{1} to T_{3}^{1}.

Furthermore,

Ext_{A}^{2}(𝔽_{p}, \underline{m}^{2}/\underline{m}^{3}) ≃ Ext_{A}^{2}(𝔽_{p}, 𝔽_{p}) ⊗_{𝔽_{p}} \underline{m}^{2}/\underline{m}^{3}
≃ Hom_{𝔽_{p}}(Ext_{A}^{2}(𝔽_{p}, 𝔽_{p})^{∗}, \underline{m}^{2}/\underline{m}^{3})
≃ Mor(T^{2}, \underline{m}^{2}/\underline{m}^{3}),

and

Mor(T^{2}, \underline{m}^{2}/\underline{m}^{3}) ⊆ Mor(T^{2}, T_{3}^{1}).

Let o_{2} be obs(E_{2}) as an element in Mor(T^{2}, T_{3}^{1}). Again, we shall refer to imobs(E_{2}) as the obstruction on the first level, which lies in V_{2} := \underline{m}^{2}/\underline{m}^{3} = \underline{m}^{2}(T_{3}^{1}), the square of the maximal ideal of T_{3}^{1}.

In analogy with the group situation we shall write (imo_{n}) for the two-sided ideal generated by o_{n}(\underline{m}_{T^{2}}).

By Lemma 3.3, (imo_{2}) is the smallest ideal of T_{3}^{1} containing the elements that need to be reduced to 0 in order to get a lifting of ϕ̄_{1}. In other words, the universal property of these obstructions implies that T_{3}^{1}/(imo_{2}) is the largest quotient of T_{3}^{1} making a lifting ϕ_{2} of ϕ̄_{1} possible. So for all other quotients T_{3}^{1}/I and a map \xymatrix{ϕ̃_{2} : A \ar@{->>}[r] & T_{3}^{1}/I} lifting ϕ̄_{1}, there exists a homomorphism \xymatrix{ρ: T_{3}^{1}/(imo_{2}) \ar[r] & T_{3}^{1}/I} such that the following diagram commutes

\xymatrix{T_{3}^{1}/(imo_{2}) \ar[d]_{ρ} \\ T_{3}^{1}/I & A. \ar@{->>}[l]^{ϕ̃_{2}} \ar@{->>}[ul]_{ϕ_{2}}}

Also note that π_{A} induces a commutative diagram of surjective homomorphisms,

\xymatrix{T_{3}^{1} \ar@{->>}[dr]_{π_{2}^{3}} \ar@{->>}[rr]^{α_{3}} &  & A_{3} \ar@{->>}[dl] \\  & T_{2}^{1} &  & A, \ar@{->>}[ll]_{ϕ̄_{1}} \ar@{->>}[ul]_{σ_{3}}}

and therefore a surjective homomorphism θ_{3} in the commutative diagram

\xymatrix{T_{3}^{1}/(imo_{2}) \ar@{->>}[r]^{θ_{3}} & A_{3} \\ T^{1} \ar[u] \ar@{->>}[r]^{π_{A}} & A. \ar[u]_{σ_{3}}}

To show that θ_{3} is an isomorphism, consider the diagram

[figure: 0; 0; \underline{m}^{2}(T_{3}^{1}); T_{3}^{1}; α_{3}; T_{2}^{1}; 0; T_{3}^{1}/(imo_{2}); ≃; θ_{3}; \underline{m}^{2}(T_{3}^{1})/(imo_{2}); A.; ϕ_{2}; ϕ̄_{1}; σ_{3}; 0; A_{3}; ψ_{3}]

Observe that (any) ϕ_{2} lifting ϕ̄_{1} factorizes via A_{3} since ϕ_{2} maps \underline{m}_{A}^{3} to 0 in T_{3}^{1}/(imo_{2}). Let \xymatrix{ψ_{3} : A_{3} \ar[r] & T_{3}^{1}/(imo_{2})} be the induced homomorphism.

Since ϕ_{2} is surjective, ψ_{3} is surjective. Therefore A_{3} and T_{3}^{1}/(imo_{2}) have the same dimension as 𝔽_{p}-vector spaces, thus θ_{3} (and ψ_{3}) is an isomorphism induced by π_{A}.

We have now seen most of the first non-trivial step in the construction of o_{A}: So far we have

1): a surjective homomorphism \xymatrix{π_{A}: T^{1} \ar[r] & A} and

2): a homomorphism \xymatrix{o_{2} : T^{2} \ar[r] & T_{3}^{1}} inducing the isomorphism θ_{3} in : the commutative diagram

\xymatrix{T^{1} \ar@{->>}[r]^{π_{A}} \ar[d] & A \ar[d]^{σ_{3}} \\ T_{3}^{1}/(imo_{2}) \ar[r]^{θ_{3}}_{≃} & A_{3}.}

Our aim is to find a coherent sequence of homomorphisms

\{\xymatrix{o_{n}: T^{2} \ar[r] & T_{n+1}^{1}} \}_{n≥1},

o_{1} being trivial, inducing an isomorphism θ_{n+1} in the commutative diagram

\xymatrix{T^{1} \ar@{->>}[r]^{π_{A}} \ar[d] & A \ar[d]^{σ_{n+1}} \\ T_{n+1}^{1}/(imo_{n}) \ar[r]^{θ_{n+1}}_{≃} & A_{n+1}.}

Let \xymatrix{ϕ̄_{n}: A \ar[r] & T_{n+1}^{1}/(imo_{n})} be the obvious composition of σ_{n+1} and (θ_{n+1})^{−1} for n ≥ 2.

To see the coherence between the first and second step, let us see what happens when we try to lift ϕ̄_{2}.

We start with the diagram

\xymatrix{0 \ar[r] & (\underline{m}^{3} + I_{2})/\underline{m}^{4} \ar[r] & T_{4}^{1} \ar[r]^{κ′} & T_{3}^{1}/(imo_{2}) \ar[r] & 0 \\  &  &  & A, \ar@{->>}[u]_{ϕ̄_{2}}}

where I_{2} is the preimage of (imo_{2}) in T^{1}. However, this is not a diagram to which we can apply Lemma 3.3. We want κ′ to be small such that the kernel becomes elementary. Hence we must consider the diagram

\xymatrix{E_{3} : 0 \ar[r] & V_{3} \ar[r] & T^{1}/J_{4} \ar[r] & T_{3}^{1}/(imo_{2}) \ar[r] & 0 \\  &  &  & A, \ar@{->>}[u]_{ϕ̄_{2}}}

where J_{4} := \overline{(\underline{m}^{4} + \underline{m} · I_{2})} and V_{3} := (\underline{m}^{3} + I_{2})/(\underline{m}^{4} + \underline{m} · I_{2}). From Lemma 3.3 we get a unique cohomology class obs(E_{3}) ∈ Ext_{A}^{2}(𝔽_{p}, V_{3}) ≃ Mor(T^{2}, V_{3}) ⊂ Mor(T^{2}, T^{1}/J_{4}) which has to be 0 in order to get a lifting of ϕ̄_{2}. Hence the obstruction imobs(E_{3}) on the second level lies in V_{3}.

Let o(ϕ̄_{2}) be obs(E_{3}) as an element in Mor(T^{2}, T^{1}/J_{4}) and consider the following diagram where the four arrows [figure]denote two short-exact sequences

(3.3) [figure: T_{2}^{1}; T^{2}; o(ϕ̄_{2}); o_{2}; o_{3}; V_{2}; T_{3}^{1}; π_{2}^{3}; T_{4}^{1}; π_{3}^{4}; ρ_{4}; T_{3}^{1}/(imo_{2}); V_{3}; \overline{μ}; T^{1}/J_{4}; μ; _{\overline{π}_{4}_{3}}; (T^{1}/J_{4})/imobs(E_{3}); θ_{4}; T_{4}^{1}/(imo_{3}); B_{4}; =; A_{4}; ψ_{4}; A; ϕ̄_{1}; ϕ̄_{2}; ϕ_{3}; σ_{4}; T^{1}.; π_{A}; π_{2}]

Recall that (imo_{2}) lies in \underline{m}^{2}/\underline{m}^{3} and I_{2} lies in \underline{m}^{2}, so \underline{m}·I_{2} ⊆ \underline{m}^{3}. Therefore there exists a homomorphism μ which factors π_{3}^{4}. It also factors \overline{π}_{3}^{4} and induces a homomorphism \xymatrix{\overline{μ}: V_{3} \ar[r] & V_{2}} , which again induces a linear map

\xymatrix{\overline{μ}_{∗}: Ext_{A}^{2}(𝔽_{p}, V_{3}) \ar[r] & Ext_{A}^{2}(𝔽_{p}, V_{2})}.

At the same time we see that I_{2} ⊂ T^{1} is mapped to 0 in A_{3}, therefore into \underline{m}_{A}^{3} in A. But then π_{A} maps \underline{m} · I_{2} into \underline{m}_{A}^{4}. Therefore the homomorphism \xymatrix{T_{4}^{1} \ar[r] & A_{4}} induced by π_{A} factorizes through a surjection \xymatrix{T^{1}/J_{4} \ar@{->>}[r] & A_{4}} . By functoriality of the obstructions (see Corollary 3.4), it also factorizes via the surjection

\xymatrix{θ_{4} : (T^{1}/J_{4})/imobs(E_{3}) \ar@{->>}[r] & A_{4}.}

As above we prove that (any) lifting ϕ_{3} of ϕ̄_{2} must be surjective and must map \underline{m}_{A}^{4} to 0, inducing a surjection

\xymatrix{ψ_{4} : A_{4} \ar@{->>}[r] & (T^{1}/J_{4})/imobs(E_{3}).}

Therefore θ_{4} is an isomorphism induced by π_{A}.

Pick an o_{3} such that ρ_{4} ∘ o_{3} = o(ϕ̄_{2}). By construction,

π_{3}^{4} ∘ o_{3} = μ ∘ ρ_{4} ∘ o_{3} = μ ∘ o(ϕ̄_{2}) = \overline{μ}_{∗}(obs(E_{3})),

which as an element of Mor(T^{2}, V_{2}) ⊂ Mor(T^{2}, T_{3}^{1}) is equal to o_{2}, since by functoriality the map \overline{μ}_{∗} sends obs(E_{3}) to obs(E_{2}). Let I_{3} be the preimage of (imo_{3}) in T^{1}. Then \underline{m} · I_{2} ⊆ I_{3}, so there is a canonical surjection \xymatrix{T^{1}/J_{4} \ar@{->>}[r] & T_{4}^{1}/(imo_{3})}. Moreover, (imo_{3}) maps onto imobs(E_{3}) in V_{3}, inducing the canonical isomorphism B_{4} which identifies T_{4}^{1}/(imo_{3}) and

(T^{1}/J_{4})/imobs(E_{3}).

Put

\xymatrix{ϕ̄_{3} = (θ_{4})^{−1} ∘ σ_{4} : A \ar[r] & T_{4}^{1}/(imo_{3})}.

We have now seen the first non-trivial step of the construction of o_{A} in detail. The construction continues: Assume, by induction, that we have constructed o_{n−1} such that there exists an isomorphism

\xymatrix{θ_{n}: T_{n}^{1}/(imo_{n−1}) \ar[r]^{≃} & A_{n}}

induced by π_{A}. Put ϕ̄_{n−1} = (θ_{n})^{−1} ∘ σ_{n}. Then, for lifting ϕ̄_{n−1}, we have the diagram

\xymatrix{0 \ar[r] & (\underline{m}^{n} + I_{n−1})/\underline{m}^{n+1} \ar[r] & T_{n+1}^{1} \ar[r] & T_{n}^{1}/(imo_{n−1}) \ar[r] & 0 \\  &  &  & A, \ar@{->>}[u]_{ϕ̄_{n−1}}}

where I_{n−1} is the preimage of (imo_{n−1}) in T^{1}.

Put J_{n+1} := \overline{(\underline{m}^{n+1} + \underline{m} · I_{n−1})} and consider the small extension, i.e. the diagram

\xymatrix{E_{n} : 0 \ar[r] & V_{n} \ar[r] & T^{1}/J_{n+1} \ar[r] & T_{n}^{1}/(imo_{n−1}) \ar[r] & 0 \\  &  &  & A, \ar[u]_{ϕ̄_{n−1}}}

where V_{n} := (\underline{m}^{n} + I_{n−1})/J_{n+1}.

As before, we get the obstruction imobs(E_{n}) ⊆ V_{n} and a lifting

\xymatrix{ϕ_{n}: A \ar[r] & (T^{1}/J_{n+1})/imobs(E_{n}),}

inducing an isomorphism \xymatrix{θ_{n+1} : (T^{1}/J_{n+1})/imobs(E_{n}) \ar[r]^{≃} & A_{n+1}} commuting with π_{A}.

Now we need to check that there exists an o_{n} and a commutative diagram

\xymatrix{T^{2} \ar[dr]_{o(ϕ̄_{n−1})} \ar@{-->}[r]^{o_{n}} \ar[rr]^{o_{n−1}} & T_{n+1}^{1} \ar[d]^{ρ_{n+1}} \ar[r] & T_{n}^{1} \ar[d] \\  & T^{1}/J_{n+1} \ar[r] & T^{1}/J_{n}}

with the properties we want, in particular, the canonical isomorphism

\xymatrix{B_{n+1} : T_{n+1}^{1}/(imo_{n}) \ar[r]^{≃} & (T^{1}/J_{n+1})/imobs(E_{n}),}

identifying the two algebras.

We denote by I_{n} the preimage of (imo_{n}) in T^{1} under the map \xymatrix{T^{1} \ar@{->>}[r] & T_{n+1}^{1}} . Observe that, by induction, I_{n−1} ⊂ I_{n−2}, which implies

(3.4)  \underline{m} · I_{n−1} ⊂ \underline{m} · I_{n−2}.

Note that (3.4) is an equality modulo \underline{m}^{n+1}.

Recall that we defined J_{n} = \overline{(\underline{m}^{n} + \underline{m} · I_{n−2})}. From (3.4) it now follows that we have a surjection

(3.5)  \xymatrix{\underline{m}^{n}/\underline{m}^{n+1} \ar@{->>}[r] & J_{n}/J_{n+1}} .

Consider the diagram

\xymatrix{T^{2} \ar[dr]_{o(ϕ̄_{n−1})} \ar[drr]_{o(ϕ̄_{n−2})} \ar@{-->}[r]^{o_{n}} \ar[rr]^{o_{n−1}} & T_{n+1}^{1} \ar@{->>}[d]^{ρ_{n+1}} \ar@{->>}[r]^{π_{n}^{n+1}} & T_{n}^{1} \ar@{->>}[d]^{ρ_{n}} \\  & T^{1}/J_{n+1} \ar@{->>}[r]^{γ} & T^{1}/J_{n}.}

We need to show that there exists an o_{n} such that π_{n}^{n+1} ∘ o_{n} = o_{n−1} and ρ_{n+1} ∘ o_{n} = o(ϕ̄_{n−1}). By definition, T^{2} is the completed tensor algebra of the dual vector space of Ext_{𝔽_{p}G}^{2}(𝔽_{p}, 𝔽_{p}). Fix generators y_{1}, …, y_{r} for T^{2} corresponding to a basis for Ext_{𝔽_{p}G}^{2}(𝔽_{p}, 𝔽_{p})^{∗}. Let y := y_{i} ∈ T^{2} for an i ∈ {1, …, r} and pick y′ ∈ T_{n+1}^{1} such that π_{n}^{n+1}(y′) = o_{n−1}(y). We have

γ(−ρ_{n+1}(y′) + o(ϕ̄_{n−1})(y)) = −ρ_{n}(π_{n}^{n+1}(y′)) + o(ϕ̄_{n−2})(y)
= −ρ_{n}(o_{n−1}(y)) + o(ϕ̄_{n−2})(y)
= 0,

hence −ρ_{n+1}(y′) + o(ϕ̄_{n−1})(y) ∈ ker γ = J_{n}/J_{n+1}. By (3.5), there exists an element a ∈ ker π_{n}^{n+1} = \underline{m}^{n}/\underline{m}^{n+1} such that ρ_{n+1}(a) = −ρ_{n+1}(y′) + o(ϕ̄_{n−1})(y). Hence if we let o_{n}(y) = y′ + a, we get π_{n}^{n+1}(o_{n}(y)) = o_{n−1}(y) and ρ_{n+1}(o_{n}(y)) = o(ϕ̄_{n−1})(y), which is what we needed to show.

By construction of o_{n}, there is a canonical surjection

\xymatrix{B_{n+1} : T_{n+1}^{1}/(imo_{n}) \ar@{->>}[r] & (T^{1}/J_{n+1})/imobs(E_{n})}.

By definition, I_{n} is the preimage in T^{1} of (imo_{n}), and T^{1}/I_{n} = T_{n+1}^{1}/(imo_{n}), therefore \underline{m}^{n+1} ⊆ I_{n}. Now, J_{n+1} = \overline{(\underline{m}^{n+1} + \underline{m} · I_{n−1})} is contained in I_{n}, since \underline{m} · I_{n−1} ⊆ I_{n}. Therefore we have an obvious commutative diagram

\xymatrix{T_{n+1}^{1} \ar[r]^{ρ_{n+1}} \ar[d] & T^{1}/J_{n+1} \ar@{-->}[dl] \ar[d] \\ T_{n+1}^{1}/(imo_{n}) \ar[r]^{B_{n+1}} & (T^{1}/J_{n+1})/imobs(E_{n}).}

By construction, ρ_{n+1}((imo_{n})) = imobs(E_{n}), hence B_{n+1} is an equality. As before, we let \xymatrix{ϕ̄_{n} = (θ_{n+1})^{−1} ∘ σ_{n+1} : A \ar[r] & T_{n+1}^{1}/(imo_{n})}, completing

the induction process.

To finish, the map \xymatrix{o_{A}: T^{2} \ar[r] & T^{1}} will be the inverse limit of the system \{\xymatrix{o_{n}: T^{2} \ar[r] & T_{n+1}^{1}} \}_{n≥1}. Moreover, the liftings \{ϕ̄_{n}\}_{n≥1} induce isomorphisms T_{n+1}^{1}/(imo_{n}) ≃ A_{n+1} for all n ≥ 1, and so for a complete local 𝔽_{p}-algebra A, T^{1}/\overline{(imo_{A})} ≃ A. □

4. The main theorem

In [3], Jennings proves that the M-series is the same as the modular dimension subgroup-series, i.e.

(4.1)  g ∈ M_{i}(G) if and only if (g − 1) ∈ (IG)^{i},

where IG is the augmentation ideal of 𝔽_{p}G. We know that 𝔽_{p}G is local with IG being the maximal ideal, so we get the following lemma.

Lemma 4.1. Let G be a finite p-group and let \underline{m} be the maximal ideal of 𝔽_{p}G. Then there is an injection

[figure: ι: G_{i} := G/M_{i}(G); 𝔽_{p}G/\underline{m}^{i}]

inducing an isomorphism \xymatrix{β_{i}: 𝔽_{p}[G_{i}]/\underline{m}^{i} \ar[r]^{≃} & 𝔽_{p}G/\underline{m}^{i}}. Moreover, the restriction of ι, \xymatrix{M_{i−1}(G)/M_{i}(G) \ar[r] & \underline{m}^{i−1}/\underline{m}^{i}} is a linear map for all i ≥ 2.

Proof. Using Jennings’ result (4.1), let g ∈ G. Then

ι(gM_{i}(G)) = (g − 1) + \underline{m}^{i}.

Since gh − 1 = (g − 1) + (h − 1) + (g − 1)(h − 1), the lemma follows. □

Our goal is to prove that the obstruction calculus for G (Theorem 2.7) is determined by the obstruction calculus for 𝔽_{p}G (Theorem 3.7).

Let us start with the following lemma.

Lemma 4.2. Let G be a finite p-group and let A = 𝔽_{p}G. Choose an embedding \xymatrix{ψ: G \ar@{^{(}->}[r] & A^{∗}}. Then a diagram in E induces a diagram in D.

Proof. Let

\xymatrix{E : 0 \ar[r] & V \ar[r] & B_{1} \ar[r]^{π} & B_{2} \ar[r] & 0 \\  &  &  & A \ar[u]_{ϕ}}

be a diagram in E. Then ψ induces a diagram

\xymatrix{D : 1 \ar[r] & V′ \ar[r] & π^{−1}(ψ′(G)) \ar[r]^{π} & ψ′(G) \ar[r] & 1 \\  &  &  & G \ar@{->>}[u]_{ψ′}}

in D, where V′ := V + 1 and ψ′ = ϕ ∘ ψ. □

Theorem 4.3. The isomorphism class of a finite p-group G is determined by its modular group algebra 𝔽_{p}G.

Proof. We let A denote 𝔽_{p}G. As before, we use A_{n}, G_{n} and T_{n}^{1} for A/\underline{m}^{n}, G/M_{n}(G) and T^{1}/M_{n}(T^{1}) respectively, where \underline{m} is the maximal ideal of A and T^{1} is the free pro-p group on a dual basis for H^{1}(G, 𝔽_{p}).

We know that G_{2} = T_{2}^{1} depends only on H^{1}(G, 𝔽_{p}) = Ext_{A}^{1}(𝔽_{p}, 𝔽_{p}). Moreover, there is a surjective homomorphism \xymatrix{A \ar@{->>}[r] & 𝔽_{p}[G_{2}]/\underline{m}^{2} = A_{2}} . By induction, we will show that G_{n}, n ≥ 3, can be constructed using obstructions depending only on A.

Assume that G_{n−1} has been constructed as in the proof of Theorem 2.7. Then the following diagram of exact sequences is given (see diagrams (2.7) and (2.8))

\xymatrix{ & 1 \ar[r] & K′_{n−1} \ar[dd] \ar[r] & T_{n}^{1} \ar@{->>}[d]_{ρ_{n}} \ar[r] & G_{n−1} \ar[r] & 1 \\  &  &  & T^{1}/N_{n} \ar@{-->}[ur] \\ \overline{D}_{n−1} : & 1 \ar[r] & K_{n−1} \ar[ur]}

with N_{n} and K_{n−1} defined as in (2.9) and (2.10) respectively.

We put G′_{n} := T^{1}/N_{n}. Then we have K_{n−1} ⊂ M_{n−1}(G′_{n}) and M_{n−1}(G′_{n}) ⊃ M_{n}(G′_{n}) = {1}, M_{2}(K_{n−1}) = {1}.

Note that G_{n−1} = G′_{n}/M_{n−1}(G′_{n}). Consider the following diagram which, by induction, only depends on the structure of A (4.2)

[figure: \overline{D}_{n−1} : 1; K_{n−1}; G′_{n}; ι; κ_{G}; G_{n−1}; ι_{n−1}; 1; \overline{E′} : 0; V_{n−1}; 𝔽_{p}[G′_{n}]/\underline{m}^{n}; κ_{A}; 𝔽_{p}[G_{n−1}]/\underline{m}^{n−1}; 0; 0; V_{n−1}; B′_{n}; :=; B_{n−1}; :=; 0.]

The injectivity of ι follows from Lemma 4.1. Here \overline{D}_{n−1} is an exact sequence of groups, and \overline{E′} an exact sequence corresponding to a surjection of 𝔽_{p}- algebras. Since K_{n−1} ⊂ M_{n−1}(G′_{n}) and K_{n−1} ⊂ Z(G′_{n}), and since V_{n−1} is the ideal generated by \{g − 1|g ∈ K_{n−1}\} ⊂ \underline{m}^{n−1} in 𝔽_{p}[G′_{n}], we have that \underline{m} · V_{n−1} = 0 in B′_{n}, i.e. κ_{A} is small.

Clearly, ι induces an injective linear map \xymatrix{τ: K_{n−1} \ar[r] & V_{n−1}} where τ(g) = g − 1.

Consider now the following object in the category D

\xymatrix{D_{n−1} : 1 \ar[r] & K_{n−1} \ar[r] & G′_{n} \ar[r] & G_{n−1} \ar[r] & 1 \\  &  &  & G, \ar@{->>}[u]_{ϕ_{n−1}}}

where \xymatrix{ϕ_{n−1} = σ_{n−1}^{G}: G \ar[r] & G_{n−1}} is the canonical homomorphism. Pick any embedding \xymatrix{ψ: G \ar@{^{(}->}[r] & A^{∗}} inducing an isomorphism from 𝔽_{p}G to A. By Lemma 4.1, ϕ_{n−1} therefore induces an isomorphism

\xymatrix{β_{n−1} : B_{n−1} \ar[r]^{≃} & A_{n−1}}

and the following diagram

[figure: E : 0; V_{n−1}; B′_{n}; A_{n−1}; 0; E′ : 0; V_{n−1}; B′_{n}; κ_{A}; B_{n−1}; β_{n−1}; ≃; 0; A,; ϕ_{n−1}^{A}; σ_{n−1}^{A}]

where \xymatrix{σ_{n−1}^{A} = β_{n−1} ∘ ϕ_{n−1}^{A}: A \ar[r] & A_{n−1}} is the canonical homomorphism and ϕ_{n−1}^{A} is the algebra homomorphism induced by ϕ_{n−1}. By functoriality, the obstructions imo_{E} and imo_{E′} are equal as vector subspaces of V_{n−1}.

Now, by Lemma 4.2, E′ induces a diagram D′ in the category D

\xymatrix{D′ : 1 \ar[r] & V′_{n−1} \ar[r] & κ_{A}^{−1}(G_{n−1}) \ar[r] & G_{n−1} \ar[r] & 1 \\  &  &  & G, \ar@{->>}[u]_{ϕ_{n−1}}}

where V′_{n−1} := V_{n−1} + 1, since (ϕ_{n−1}^{A} ∘ ψ)(G) = G_{n−1}.

Let \xymatrix{τ′: K_{n−1} \ar[r] & V′_{n−1}} and \xymatrix{μ: V_{n−1} \ar[r] & V′_{n−1}} be the obvious injections. The obstruction imo_{D_{n−1}} is now mapped to imo_{D′}, hence

(4.3)  τ′(imo_{D_{n−1}}) = imo_{D′}.

The morphism \xymatrix{ϕ_{n−1}^{A}: A \ar[r] & B_{n−1}} can be (minimally) lifted to

\xymatrix{A \ar[r] & B′_{n}/imo_{E′}} ,

therefore the morphism \xymatrix{ϕ_{n−1} : G \ar[r] & G_{n−1}} can be lifted to

\xymatrix{G \ar[r] & κ_{A}^{−1}(G_{n−1})/⟨imo_{E′} + 1⟩},

which implies

(4.4)  imo_{D′} ⊆ μ(imo_{E′}).

Now, the morphism \xymatrix{ϕ_{n−1} : G \ar[r] & G_{n−1}} can be lifted to

\xymatrix{G \ar[r] & G′_{n}/imo_{D_{n−1}}} ,

therefore the morphism \xymatrix{ϕ_{n−1}^{A}: A \ar[r] & B_{n−1}} can be lifted to

\xymatrix{A \ar[r] & B′_{n}/(τ(imo_{D_{n−1}}))},

hence

(4.5)  imo_{E′} ⊆ τ(imo_{D_{n−1}}).

From (4.5), (4.3) and (4.4) it follows that

μ(imo_{E′}) ⊆ μ(τ(imo_{D_{n−1}})) = imo_{D′} ⊆ μ(imo_{E′}).

Therefore, τ(imo_{D_{n−1}}) = imo_{E′}, so imo_{D_{n−1}} is (entirely) determined by the obstruction imo_{E′} = imo_{E}. Since, from Section 2, we know that

G_{n} ≃ G′_{n}/imo_{D_{n−1}},

the theorem follows. □

We note that \underleftarrow{lim}_{n} B_{n} ≃ A. As a final remark we note that the last proof also gives us a criterion for

when a local (complete) 𝔽_{p}-algebra A with 𝔽_{p} as the only simple module is the group algebra 𝔽_{p}G for a p-group G: The necessary and sufficient conditions are that, inductively, the map \xymatrix{τ: K_{n} \ar[r] & V_{n}} in the diagram (4.2) is injective, and that imo_{E} ⊂ τ(K_{n}) ⊂ V_{n}.

Further consequences will be treated in a forthcoming paper.

Acknowledgement

The authors would like to thank Peter Kropholler for useful discussions.

References

[1] Borge, I. C., A cohomological approach to the modular isomorphism problem, Preprint in Pure Mathematics 2002 No. 15, Matematisk Institutt, Universitetet i Oslo. Submitted to Journal of Pure and Appl. Alg.

http://www.math.uio.no/eprint/pure_math/2002/15-02.html

[2] Borge, I. C., A cohomological approach to the classification of p-groups, D. Phil. thesis 2001, Mathematical Institute, Oxford University.

http://www.maths.ox.ac.uk/research/theses/

[3] Jennings, S. A., The structure of the group ring of a p-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175-185.

[4] Laudal, O. A., p-groups and Massey products, Preprint series 1975/76 No. 30, Matematisk Institut, Aarhus Universitet.

[5] Laudal, O. A., Formal moduli of algebraic structures, Lecture Notes in Mathematics 754, Springer 1979.

[6] Laudal, O. A., Noncommutative deformations of modules, Homology, Homotopy and Applications, vol.4 (2) 2002, 357-396. (Special Issue in Honor of Jan-Erik Roos. Ed. Hvedri Inassaridze. International Press (2002)).

[7] Laudal, O. A., Noncommutative algebraic geometry, Preprint Series 2000 (115), Max-Planck-Institut für Mathematik, Bonn. See also Noncommutative algebraic geometry, Rev. Mat. Ibero Americana 19 (2003), 1-72.

[8] Passman, D., The algebraic structure of group rings, Wiley, 1965.

[9] Quillen, D., On the associated graded ring of a group ring, J. Alg. 10 (1968), 411-418.

[10] Shafarevich, I. R., Algebraic number fields (in Russian), Proc. ICM Stockholm 1962, 163-176.

28 INGER CHRISTIN BORGE AND OLAV ARNFINN LAUDAL

Matematisk institutt, Universitetet i Oslo

E-mail address: ingerbo@math.uio.no,arnfinnl@math.uio.no