GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 509-522

NON-ABELIAN COHOMOLOGY WITH COEFFICIENTS IN CROSSED BIMODULES

H. INASSARIDZE

Abstract. When the coefficients are crossed bimodules, Guin’s nonabelian cohomology [2], [3] is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.

We continue to study non-abelian cohomology of groups (see [1]) following Guin’s approach to non-abelian cohomology [2], [3]. The pointed sets of cohomology H^{n}(G, (A, μ)), n = 1, 2, will be defined when the group of coefficients (A, μ) is a crossed G-R-bimodule. The notion of a crossed bimodule has been introduced in [1]. H^{1}(G, (A, μ)) is equipped with a partial product and coincides with Guin’s cohomology group [3] when crossed G-modules are viewed as crossed G-G-bimodules. The pointed set of cohomology H^{2}(G, (A, μ)) coincides with the second pointed set of cohomology defined in [1] when the coefficients are crossed modules. A coefficient short exact sequence of crossed G-R-bimodules gives rise to a nine-term exact cohomology sequence and we recover the exact cohomology sequence obtained in [1] when the coefficients are crossed modules. By analogy with the case n = 2 the definition of a pointed set of cohomology H^{n}(G, (A, μ)) of a group G with coefficients in a crossed G-R-bimodule (A, μ) is given for all n ≥ 1.

The notation and diagrams of [1] will be used.

Recall the definitions of a crossed G-R-bimodule and the group Der(G, (A, μ)) of derivations from G to (A, μ).

Let G, R and A be groups. (A, μ) is a crossed G-R-bimodule if:

1) (A, μ) is a crossed R-module,

2) G acts on R and A,

3) the homomorphism μ : A ⟶ R is a homomorphism of G-groups,

4) ^{(^{g}r)}a = ^{grg^{−1}}a for g ∈ G, r ∈ R, a ∈ A.

509

1991 Mathematics Subject Classification. 18G50, 18G55.

Key words and phrases. Crossed bimodule, compactibility condition, free cotriple resolution.

1072-947X/97/1100-0509$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

The group Der(G, (A, μ)) is defined as follows. It consists of pairs (α, r) where α is a crossed homomorphism from G to A and r is an element of R such that μα(x) = r^{x}r^{−1} for all x ∈ G. A product in Der(G, (A, μ)) is given by (α, r)(β, s) = (α ∗ β, rs) where (α ∗ β)(x) = ^{x}β(x)α(x), x ∈ G. For any a ∈ A and (α, r) ∈ Der(G, (A, μ)) the following equality holds:

α(x)^{xr}a = ^{rx}a α(x) for all x ∈ G .

Definition 1. Let (A, μ) be a crossed G-R-bimodule. It will be said that a crossed homomorphism α : G ⟶ A satisfies condition (j) (resp. condition (j′)) if for c ∈ H^{0}(G, R) (resp. if for c ∈ H^{0}(G, R) such that there is b ∈ A with μ(b) = c) there exists a ∈ A such that ^{c}α(x) = a^{−1} α(x)^{x}a for x ∈ G and μ(a) = 1. It will be said that an element (α, r) of Der(G, (A, μ)) satisfies condition (j) (resp. condition (j′)) if α satisfies this condition.

It is obvious that any element of the form (α, 1) satisfies condition (j′). If (A, μ) is a crossed G-R-bimodule induced by a surjective homomorphism f : G ⟶ R, then every element (α, r) ∈ Der(G, (A, μ)) such that α(ker f) = 1 satisfies condition (j). In effect, for c ∈ Z(R) = H^{0}(G, R) we have z_{x}dx = xd, x ∈ G, with f(d) = c and z_{x} ∈ ker f. Thus, α(z_{x})^{z_{x}}α(dx) = α(xd), whence α(d)^{c}α(x) = α(x)^{x}α(d) and μα(d) = r f(d) r^{−1} f(d)^{−1} = 1.

Note that if (α, r) ∼ (α′, r′) (see below) and (α, r) satisfies condition (j) then (α′, r′) satisfies condition (j) too when H^{0}(G, R) ⊂ Z(R). In effect, we have α′(x) = b^{−1}α(x)^{x}b, r′ = μ(b)^{−1}r t and ^{c}α(x) = a^{−1}α(x)^{x}a, μ(a) = 1, where c, t ∈ H^{0}(G, R) ⊂ Z(R). Thus

cα′(x) = ^{c}b^{−1} ^{c}α(x)^{cx}b = ^{c}b^{−1} a^{−1}α(x)^{x}a^{cx}b =
= ^{c}b^{−1}a^{−1}α(x)^{x}(a^{c}b) = ^{c}b^{−1}a^{−1}b α′(x)^{x}b^{−1} ^{x}(a^{c}b) =
= ^{c}b^{−1}ba^{−1}α′(x)^{x}(ab^{−1} ^{c}b)

with μ(a b^{−1} ^{c}b)^{−1} = (μ(a) μ(b^{−1}) μ(^{c}b))^{−1} = μ(^{c}b^{−1}) μ(b) = c μ(b)^{−1}c^{−1} μ(b) = 1.

It is clear that if f : (A, μ) ⟶ (B, λ) is a homomorphism of crossed G-R-bimodules and (α, r) ∈ Der(G, (A, μ)) satisfies condition (j), then (fα, r) satisfies condition (j).

Let (A, μ) be a crossed G-R-bimodule. In the group Der(G, (A, μ)) we introduce a relation ∼ defined as follows:

(α, r) ∼ (β, s) ⟺ \{ \begin{matrix}∃ a ∈ A : β(x) = a^{−1}α(x)^{x}a, \\ s = μ(a)^{−1}r mod H^{0}(G, R) \end{matrix} .

Later we shall need the following assertion:

If (A, μ) is a precrossed G-R-bimodule the equality

rxa = ^{xr}a  (1)

holds for any x ∈ G, a ∈ A, r ∈ H^{0}(G, R).

In effect,we have

rxa = ^{xx^{−1}rx}a = ^{x(^{x^{−1}} r)}a = ^{xr}a.

Proposition 2. The relation ∼ is an equivalence. Assume H^{0}(G, R) is a normal subgroup of R; then the group Der(G, (A, μ)) induces on Der(G, (A, μ))/ ∼ a partial product defined by

[(α, 1)][(β, s)] = [(α ∗ β, s)]

if [(β, s)] contains an element satisfying condition (j′), and by

[(α, r)][(β, s)] = [(α ∗ β, rs)]

if [(β, s)] contains an element satisfying condition (j).

Proof. If (α, r) ∼ (α′, r′), i.e., α′(x) = a^{−1}α(x) ^{x}a, x ∈ G, and r′ = μ(a)^{−1}rz, z ∈ H^{0}(G, R), then α(x) = a α(x)^{x}a^{−1} and r = μ(a) r′z^{−1}, z^{−1} ∈ H^{0}(G, R). Thus, (α′, r′) ∼ (α, r).

If (α, r) ∼ (α′, r′) and (α′, r′) ∼ (α′′, r′′) we have

α′(x) = a^{−1}α(x)^{x}a, r′ = μ(a)^{−1}rz,
α′′(x) = b^{−1}α′(x)^{x}b, r′′ = μ(b)^{−1}r′z′,

where z, z′ ∈ H^{0}(G, R). This implies (α, r) ∼ (α′′, r′′) and the relation ∼ is an equivalence.

It is clear that if (α, r) ∈ Der(G, (A, μ)) and c ∈ H^{0}(G, R) then (α, r) ∼ (α, rc).

We have yet to show the correctness of the partial product.

Let (α, 1) ∼ (α′, 1), (β, s) ∼ (β′, s′) and (β, s) satisfy condition (j′). We will prove that (α, 1)(β, s) ∼ (α′, 1)(β′, s′). One has

α′(x) = a^{−1}α(x)^{x}a, x ∈ G,
1 = μ(a)^{−1}z, z ∈ H^{0}(G, R),

and

β′(x) = b^{−1}β(x)^{x}b, x ∈ G,
s′ = μ(b)^{−1}sz′, z′ ∈ H^{0}(G, R).

Then β′(x)α′(x) = b^{−1}β(x)^{x}ba^{−1}α(x)^{x}a=b^{−1}β(x)a^{−1}α(x)^{x}a ^{x}b= b^{−1}a^{−1} μ(a)β(x)α(x)^{x}a^{x}b=b^{−1}a^{−1}d^{−1}β(x)α(x)^{x}(dab) and s′ =μ(b)^{−1}μ(a)^{−1}zsz′ = μ(b^{−1}a^{−1}d^{−1}) sz′′z′ where β(x) = d^{−1}β(x)^{x}d, μ(d) = 1 and z′′ ∈ H^{0}(G, R). Therefore (α, 1)(β, s) ∼ (α′, 1)(β′, s′).

It is clear that the set of all elements of the form [(α, 1)] forms an abelian group under this product.

Finally, we will prove that if (α, r) ∼ (α′, r′), (β, s) ∼ (β′, s′) and (β, s) satisfies condition (j) then (α, r)(β, s) ∼ (α′, r′)(β′, s′) and we check Guin’s proof [3] in our case.

We first prove that

(α, r)(β, s) ∼ (α, rc)(β, s)

for c ∈ H^{0}(G, R).

Using condition (j) and equality (1) of [3] one gets

rcβ(x)α(x) = ^{r}(a^{−1}β(x)^{x}a) α(x) = ^{r}a^{−1} ^{r}β(x)^{rx}a α(x) =
= ^{r}a^{−1} ^{r}β(x)α(x)^{rx}a.

Since μ(^{r}a)^{−1} = (r μ(a) r^{−1})^{−1} = 1, one has rcs = μ (^{r}a)^{−1}rsc′ with c′ ∈ H^{0}(G, R). Therefore, (α, r)(β, s) ∼ (α, rc)(β, s).

Further,we have

α′(x) = b^{−1}α(x)^{x}b, r′ = μ(b)^{−1}rz,

and β′(x) = d^{−1}β(x)^{x}d, s′ = μ(d)^{−1}st with z, t ∈ H^{0}(G, R).

Put

(α, rz)(β, s) = (γ, rzs),

where γ(x) = ^{rz}β(x)α(x), x ∈ G, and (α′, r′)(β′, s′) = (γ′, r′s′), where γ′(x) = ^{r′}β′(x)α′(x), x ∈ G.

We will show that

(α, rz)(β, s) ∼ (α′, r′)(β′, s′).

Using (1) and equality (1) of [1] one has

γ′(x) = ^{r′}(d^{−1}β(x)^{x}d) b^{−1}α(x)^{x}b =
= ^{μ(b)^{−1}r·z}d^{−1} ^{μ(b)^{−1}rz}β(x)^{μ(b)^{−1}rzx}d b^{−1}α(x)^{x}b =

= b^{−1} ^{r·z}d^{−1} ^{rz}β(x)^{r·x}(^{z}d) α(x)^{x}b = b^{−1} ^{r·z}d^{−1} ^{r·z}β(x)α(x)^{xrz}d^{x}b,

and μ(^{r·z}d b)^{−1} = μ(b)^{−1}rz μ(d)^{−1}z^{−1}r^{−1} = r′s′t^{−1}s^{−1}z^{−1}r^{−1}, r′s′ = μ(^{r·z}d b)^{−1}rzst with t ∈ H^{0}(G, R).

Therefore (α, rz)(β, s) ∼ (α′, r′)(β′, s′), whence (α, r)(β, s) ∼ (α′, r′) (β′, s′).

Definition 3. Let (A, μ) be a crossed G-R-bimodule. One denotes by H^{1}(G, (A, μ)) the quotient set Der(G, (A, μ))/ ∼ equipped with the aforementioned partial product and it will be called the first set of cohomology of G with coefficients in the crossed G-R-bimodule (A, μ).

If (A, μ) is a crossed G-module viewed as a crossed G-G-bimodule then H^{0}(G, G) = Z(G) and for (α, g) ∈ Der_{G}(G, A) = Der(G, (A, μ)) and c ∈ Z(G) the equality α(cx) = α(xc), x ∈ G, implies

α(c)^{c}α(x) = α(x)^{x}α(c),

whence ^{c}α(x) = α(c)^{−1}α(x)^{x}α(c) and μ(α(c)) = gcg^{−1}c^{−1} = 1. Therefore every element of Der_{G}(G, A) satisfies condition (j). It follows that if (A, μ) is a crossed G-module we recover the group H^{1}(G, A) defined by Guin [3].

It is clear that the map H^{1}(G, A) ⟶ H^{1}(G, (A, 1)) given by [α] ⟼ [(α, 1)] is an isomorphism where (A, 1) is a crossed G-R-bimodule.

Proposition 4. Let (A, μ) be a crossed G-R-bimodule and assume H^{0}(G, R) is a normal subgroup of R. If (α, r) and (β, s) satisfy condition (j) then (α, r)(β, s) and (α, r)^{−1} satisfy condition (j).

Proof. Let c ∈ H^{0}(G, R). Then ^{c}α(x) = b^{−1}α(x)^{x}b and μ(b) = 1. Since H^{0}(G, R) is a normal subgroup of R, there is c′ ∈ H^{0}(G, R) such that cr = rc′. For c′ we have ^{c′}β(x) = d^{−1}β(x)^{x}d and μ(d) = 1. Put (α, r)(β, s) = (γ, rs). Then

cγ(x) = ^{cr}β(x)^{c}α(x) = ^{r}d^{−1} ^{r}β(x)^{rx}d b^{−1}α(x)^{x}b =
= ^{r}d^{−1}b^{−1} ^{r}β(x)α(x)^{x}(b^{r}d)

with μ(b^{r}d) = μ(b) r μ(d) r^{−1} = 1. Thus, (γ, rs) satisfies condition (j). Put (α, r)^{−1} = (\overline{α}, r^{−1}) where \overline{α}(x) = ^{r^{−1}}α(x)^{−1}, x ∈ G. If c ∈ H^{0}(G, R) one has

c\overline{α}(x) = ^{cr^{−1}}α(x)^{−1} = ^{r^{−1}c′}α(x)^{−1} = ^{r^{−1}}(^{x}a^{−1}α(x)^{−1}a),

where cr^{−1} = r^{−1}c′, c′ ∈ H^{0}(G, R) and ^{c′}α(x) = a^{−1}α(x)^{−1} ^{x}a, μ(a) = 1. Hence

c\overline{α}(x) = ^{r^{−1}x}a^{−1} ^{r^{−1}}α(x)^{−1} ^{r^{−1}}a = ^{r^{−1}}α(x)^{−1} ^{xr^{−1}}a^{−1} ^{r^{−1}}a =
= ^{r^{−1}}a^{r^{−1}}α(x)^{−1} ^{x}(^{r^{−1}}a^{−1})

with μ(^{r^{−1}}a^{−1}) = r^{−1}μ(a^{−1}) r = 1.

Therefore, (\overline{α}, r^{−1}) satisfies condition (j).

Corollary 5. The subset of H^{1}(G, (A, μ)) of all equivalence classes containing an element with condition (j) forms a group if H^{0}(G, R) is a normal subgroup of R.

Proposition 6. Let (A, μ) be a crossed G-R-bimodule such that H^{0}(G,R) is a normal subgroup of R. If there is a map η : H^{0}(G, R) ⟶ Z(G) such that Im η acts trivially on R and ^{η(r)}a = ^{r}a, a ∈ A, then H^{1}(G, (A, μ)) is a group.

Proof. We have to show that every element (α, r) ∈ Der(G, (A, μ)) satisfies condition (j). If c ∈ H^{0}(G, R) take η(c) = d ∈ Z(G). Then α(dx) = α(xd) and α(d) ^{d}α(x) = α(x)^{x}α(d). Thus ^{c}α(x) = α(d)^{−1}α(x)^{x}α(d) and μα(d) = r^{d}r^{−1} = rr^{−1} = 1.

Corollary 7. Let (A, μ) be either a crossed G-R-bimodule such that H^{0}(G, R) is a normal subgroup of R trivially acting on A or induced by a surjective homomorphism f : G ⟶ R such that f(Z(G)) = Z(R). Then H^{1}(G, (A, μ)) is a group.

Proof. In the first case take η as the trivial map. In the second case take a map η : Z(R) ⟶ Z(G) such that fη = 1_{Z(R)}.

If f : (A, μ) ⟶ (B, λ) is a homomorphism of crossed G-R-bimodules then f induces a natural map

f^{1} : H^{1}(G, (A, μ)) ⟶ H^{1}(G, (B, λ))

which is a homomorphism in the following sense:

if xy is defined for x, y ∈ H^{1}(G, (A, μ)) then f^{1}(x)f^{1}(y) is defined and f^{1}(xy) = f^{1}(x)f^{1}(y).

The above defined action of G on Der(G, (A, μ)) induces an action of G on H^{1}(G, (A, μ)) given by

g[(α, r)] = [^{g}(α, r)], g ∈ G.

We have to show that if (α, r) ∼ (α′, r′) then ^{g}(α, r) ∼ ^{g}(α′, r′). In effect, since

α′(x) = a^{−1}α(x)^{x}a, x ∈ G,

this implies

α′(^{g^{−1}}x) = a^{−1}α(^{g^{−1}}x)^{g^{−1}xg}a, x ∈ G.

Thus

gα′(^{g^{−1}}x) = ^{g}a^{−1} ^{g}α(^{g^{−1}}x)^{xg}a, x ∈ G.

We also have r′ = μ(a)^{−1}rz, z ∈ H^{0}(G, R), whence ^{g}r′ = ^{g}μ(a^{−1})^{g}r^{g}z = μ(^{g}a)^{−1} ^{g}r^{g}z. Therefore ^{g}(α, r) ∼ ^{g}(α′, r′).

In what follows if f is a map from a group G to a group G′ then f^{−1} : G ⟶ G′ denotes a map given by f^{−1}(x) = f(x)^{−1}.

Let (A, μ) be a crossed G-R-bimodule. The definition of H^{2}(G, (A, μ)) is similar to the case of (A, μ) being a crossed G-module (see [1]).

Consider diagram (4) of [1] and the group Der(M, (A, μ)) where (A, μ) is viewed as a crossed M-R-bimodule induced by τl_{0} and a crossed F -R-bimodule induced by τ. Let Z̃^{1}(M, (A, μ)) be the subset of Der(M, (A, μ)) consisting of elements of the form (α, 1).

Define, on Z̃^{1}(M, (A, μ)), relation

(α′, 1) ∼ (α, 1) ⟺ (β, h) ∈ Der(F, (A, μ))

such that

(α′, 1) = (βl_{0}, h)(α, 1)(βl_{1}, h)^{−1}

in the group Der(M, (A, μ)).

Definition 8. Let (A, μ) be a crossed G-R-bimodule. The relation ∼ is an equivalence. Denote by H^{2}(G,(A, μ)) the quotient set Z̃^{1}(M,(A, μ)) / ∼. It will be called the second set of cohomology of G with coefficients in the crossed G-R-bimodule (A, μ).

It can be proved (as for a crossed G-module (A, μ) (see Proposition 8 [1])) that Z̃^{1}(M, (A, μ)) / ∼ is independent of diagram (4) of [1] and is unique up to bijection.

Let (A, μ) be a crossed G-R-bimodule. Then there is a canonical map

ϑ′ : H^{2}(G, ker μ) ⟶ H^{2}(G, (A, μ))

defined by the composite

[E] ⟼^{ϑ^{−1}} [α] ⟼ [(α, 1)].

This map is surjective and was defined when (A, μ) is a crossed G-module [3].

Proposition 9. Let (A, μ) be a crossed G-R-bimodule. There is an action of G on H^{2}(G, (A, μ)) such that Z(G) acts trivially. If R acts on G and satisfies the compatibility condition (3) of [1] then there is also an action of R on H^{2}(G, (A, μ)).

Proof. The action of G on H^{2}(G, (A, μ)) is defined exactly in the same manner as for a crossed G-module (A, μ) (see Proposition 12 [1]). The action of R is defined similarly. Namely, we have an action of R on M_{G} given by

r(|g_{1}|^{ϵ} ⋯ |g_{n}|^{ϵ} , |g′_{1}|^{ϵ} ⋯ |g′_{m}|^{ϵ}) = (|^{r}g_{1}|^{ϵ} ⋯ |^{r}g_{n}|^{ϵ} , |^{r}g′_{1}|^{ϵ} ⋯ |^{r}g′_{m}|^{ϵ})

and one gets an action of R on Der(M_{G}, (A, μ)) defined by

r(α, s) = (α̃,^{r} s),

where α̃(m) = ^{r}α(^{r^{−1}}m), r ∈ R, m ∈ M_{G}. Define ^{r}[(α, 1)] = [^{r}(α, 1)], r ∈ R. If (α, 1) ∼ (α′, 1) it is easy to see that ^{r}(α, 1) ∼ ^{r}(α′, 1).

Let (A, μ) be a crossed G-R-bimodule. Using (1) it can easily be shown that there is an action of H^{0}(G, R) on H^{2}(G, ker μ) given by

r[α] = [^{r}α], r ∈ H^{0}(G, R),

where α : M_{G} ⟶ ker μ is a crossed homomorphism under the action of G on A (see diagram (5) of [1]) such that α(Δ) = 1.

If this action of H^{0}(G, R) is trivial and Der(F_{G},(A, μ))=IDer(F_{G},(A, μ)) then the map

ϑ′ : H^{2}(G, ker μ) ⟶ H^{2}(G, (A, μ))

is a bijection.

Let

1 ⟶ (A, 1) ⟶^{φ} (B, μ) ⟶^{ψ} (C, λ) ⟶ 1  (2)

be an exact sequence of crossed G-R-bimodules. If the action of H^{0}(G, R) on H^{2}(G, A) is trivial then there is an action of H^{1}(G, (C, λ)) on H^{2}(G, A) given by

[(α,r)][γ] = [^{r}γ].

We have to show that ^{r}γ is a crossed homomorphism and the correctness of the action.

Consider the diagram

\begin{aligned}M_{G} & ⟶_{l_{1}}^{⟶}^{l_{0}} & F_{G} & ⟶^{τ_{G}} & G \\ & & & & ↓ α \\ A & ⟶^{φ} & B & ⟶^{ψ} & C \end{aligned}  (3)

There is a crossed homomorphism β : F_{G} ⟶ B such that ψβ = ατ_{G}. Take the product

(βl_{0}, r)(φγ, 1)(βl_{0}, r)^{−1} = (γ̃, 1)

in the group Der(M_{G} , (B, μ)). Then γ̃(x) = β(x)^{−1} ^{r}φγ(x) β(x) = ^{r}φγ(x), x ∈ M. Therefore ^{r}γ : M_{G} ⟶ A is a crossed homomorphism such that rγ(Δ) = 1.

If (α′, r′) ∈ [(α, r)] ∈ H^{1}(G, (C, λ)), i.e., (α, r) ∼ (α′, r′), then

α′(x) = c^{−1}α(x)^{x}c and r′ = λ(c)^{−1}rt,

where t ∈ H^{0}(G, R). It follows that

φ(^{r′}γ(x)) = ^{r′}φγ(x) = ^{λ(c)^{−1}rt}φγ(x) = ^{μ(b)^{−1}rt}φγ(x) =
= b^{−1} ^{r·t}φγ(x) b = ^{r·t}φγ(x) = φ(^{r·t}γ(x)), x ∈ M_{G},

where ψ(b) = c.

Hence we have

[^{r′}γ] = [^{rt}γ] = [^{r}γ]

proving the correctness of the action.

Using diagram (3) for the exact sequence (2) one defines a connecting map

δ^{1} : H^{1}(G, (C, λ)) ⟶ H^{2}(G, A)

as follows.

For [(α, r)] ∈ H^{1}(G, (C, λ)) take a crossed homomorphism β : F_{G} ⟶ B such that ψβ = α τ_{G}. Thus there is a crossed homomorphism γ : M_{G} ⟶ A such that

φγ = (βl_{1})^{−1}βl_{0}.

It is clear that γ(Δ) = 1. Define

δ^{1}([(α, r)]) = [γ].

We must prove the correctness of δ^{1}. If β′ : F_{G} ⟶ B with ψβ′ = ατ, then ψβ′ = ψβ. Thus there is a crossed homomorphism σ : F_{G} ⟶ A such that β′ = βφσ. Then we have

φγ′ = (β′l_{1})^{−1} β′l_{0} = (βφσ)l_{1}^{−1} (βφσ)l_{0} = φσl_{1}^{−1} βl_{1}^{−1} βl_{0} φσl_{0} =
= βl_{1}^{−1} βl_{0}φσl_{1}^{−1}φσl_{0} = φ(γσl_{1}^{−1}σl_{0}).

Hence [γ′] = [γ].

If (α, r) ∼ (α′, r′) then

α′(y) = c^{−1}α(y)^{y}c, c ∈ C, y ∈ M,
r′ = λ(c)^{−1}rt, t ∈ H^{0}(G, R).

Take β′ : F_{G} ⟶ B such that

β′(x) = b^{−1}β(x)^{x}b

with ψ(b) = c where ψβ = ατ. Then (β′l_{1}^{−1}β′l_{0})(y) = β′(x_{2})^{−1}β′(x_{1}) where y = (x_{1}, x_{2}) ∈ M_{G}. Hence φγ′(y) = (β′l_{1}^{−1}β′l_{0})(y) = (b^{−1}β(x_{2}) x_{2}b)^{−1}b^{−1}β(x_{1})^{x_{1}}b = ^{x_{2}}b^{−1}β(x_{2})^{−1}β(x_{1})^{x_{1}}b = β(x_{2})^{−1}β(x_{1}) = φγ(y). Whence γ′ = γ.

For any exact sequence (2) of crossed G-R-bimodules there is also an action of Der(F_{0}, (C, λ)) on H^{3}(G, A) defined as follows:

(α,r)[f] = [^{r}f],

where f : F_{2} ⟶ A is a crossed homomorphism with ∏_{i=0}^{3}(fl_{i}^{2}τ_{3})^{ϵ} = 1 where

ϵ = (−1)^{i} (see diagram (7) of [1]) and (α, r) ∈ Der(F_{0}, (C, λ)). The correctness of this action is proved similarly to the case of a short exact sequence of crossed G-modules (see [1]).

If either the aforementioned action of Der(F_{0}, (C, λ)) on H^{3}(G, A) is trivial or Der(F_{0}, (C, λ)) = IDer(F_{0}, (C, λ)) and H^{0}(G, R) acts trivially on H^{2}(G, ker λ), then a connecting map

δ^{2} : H^{2}(G, (C, λ)) ⟶ H^{3}(G, A)

is defined by

δ^{2}([(α, 1)]) = [γ], (α, 1) ∈ D̃er(M_{0}, (C, λ)),

where φγ = \overline{β}τ_{2} with \overline{β} = ∏_{i=0}^{2}(βl_{i}^{1})^{ϵ}, ϵ = (−1)^{i}, and ψβ = ατ_{1} (see diagram

(7) of [1]). The correctness of δ^{2} is proved similarly to the case of crossed G-modules [1], and if (2) is an exact sequence of crossed G-modules we recover the above-defined connecting map δ^{2} : H^{2}(G, C) ⟶ H^{3}(G, A).

Theorem 10. Let (2) be an exact sequence of crossed G-R-bimodules. Then there is an exact sequence

1 ⟶ H^{0}(G, A) ⟶^{φ^{0}} H^{0}(G, B) ⟶^{ψ^{0}} H^{0}(G, C) ⟶^{δ^{0}}

⟶^{δ^{0}} H^{1}(G, A) ⟶^{φ^{1}} H^{1}(G, (B, μ)) ⟶^{ψ^{1}} H^{1}(G, (C, λ)) ⟶^{δ^{1}} H^{2}(G, A) ⟶^{φ^{2}}

⟶^{φ^{2}} H^{2}(G, (B, μ)) ⟶^{ψ^{2}} H^{2}(G, (C, λ)),

where φ^{0}, ψ^{0}, δ^{0}, φ^{1} are homomorphisms. If H^{0}(G, R) is a normal subgroup of R, then ψ^{1} and δ^{1} are also homomorphisms. If in addition H^{0}(G, R) acts trivially on H^{2}(G, A), then δ^{1} is a crossed homomorphism under the action of H^{1}(G, (C, λ)) on H^{2}(G, A) induced by the action of R on A. Moreover, if either the action of Der(F_{0}, (C, λ)) on H^{3}(G, A) is trivial (in particular if R acts trivially on A) or Der(F_{0}, (C, λ)) = IDer(F_{0}, (C, λ)) and H^{0}(G, R) acts trivially on H^{2}(G, ker λ) then the sequence

H^{2}(G, (B, μ)) ⟶^{ψ^{2}} H^{2}(G, (C, λ)) ⟶^{δ^{2}} H^{3}(G, A)

is exact.

Proof. The exactness of the sequence

1 ⟶ H^{0}(G, A) ⟶^{φ^{0}} H^{0}(G, B) ⟶^{ψ^{0}} H^{0}(G, C) ⟶^{δ^{0}} H^{1}(G, A)

is known [4].

If c ∈ H^{0}(G, C) then δ^{0}(c) = [α] with α(x) = φ^{−1}(b^{−1} ^{x}b), x ∈ G and ψ(b) = c. It follows that (α_{0}, 1) ∼ (φα, 1) where α_{0} is the trivial map, since

φα(x) = b^{−1}α_{0} ^{x}b, x ∈ G,

and μ(b) ∈ H^{0}(G, R) because μ(b) = λψ(b) = λ(c) and ^{x}λ(c) = λ(^{x}c) = λ(c), x ∈ G. Therefore Im δ^{0} ⊂ ker φ^{1}.

Let [α] ∈ H^{1}(G, A) such that (α_{0}, 1) ∼ (φα, 1). Then φα(x) = b^{−1} ^{x}b, x ∈ G and μ(b) ∈ H^{0}(G, R). We have ψ(b^{−1} ^{x}b) = ψφα(x) = 1. Thus ψ(b) = ψ(^{x}b) = ^{x}ψ(b), whence ψ(b) ∈ H^{0}(G, C). It is clear that δ^{0}(ψ(b)) = [α]. Therefore ker φ^{1} ⊂ Im δ^{0}.

Clearly, ψ^{1}φ^{1} is the trivial map. Let [(α, r)] ∈ H^{1}(G, (B, μ)) such that (α_{0}, 1) ∼ (φα, 1). Then ψα(x) = c^{−1} ^{x}c, c ∈ C, and r = λ(c)^{−1}t, t ∈ H^{0}(G, R). Let ψ(b) = c. Then μ(b) = λ(c) and r = μ(b)^{−1}t. Take α̃(x) = b α(x)^{x}b^{−1}, x ∈ G. Since ψα̃(x) = 1, x ∈ G, one has φ^{−1}α̃ : G ⟶ A and (α, r) ∼ (α̃, 1). Therefore φ^{1}([φ^{−1}α̃]) = [(α, r)].

Let [(α, r)] ∈ H^{1}(G, (B, μ)). Then ψ^{1}([(α, r)]) = [(ψα, r)]. Consider diagram (5) of [1] and take α τ_{G} : F_{G} ⟶ B. Then φγ = (ατ_{G}l_{1})^{−1}ατ_{G}l_{0} and δ^{1}ψ^{1}([(α, r)]) = [γ]. But γ = α_{0} is the trivial map, since ατ_{G}l_{0} = ατ_{G}l_{1}. Therefore Im ψ^{1} ⊂ ker δ^{1}.

Let [(α, r)] ∈ H^{1}(G, (C, λ)) such that δ^{1}([(α, r)]) = 1. If β : F_{G} ⟶ B is a crossed homomorphism such that ψβ = ατ_{G} then δ^{1}([(α, r)]) = [γ], where φγ = (βl_{1})^{−1}βl_{0} . Thus there is a crossed homomorphism η: F_{G} ⟶ A such that γ = (ηl_{1})^{−1}ηl_{0}. Hence we have

(βl_{1})^{−1}βl_{0} = (φηl_{1})^{−1}φηl_{0}, (φη^{−1} β)l_{0} = (φη^{−1} β)l_{1}.

Thus there is a crossed homomorphism \overline{α} : G ⟶ B such that (φη)^{−1}β = \overline{α}τ_{G}. We have μβ(x) = λψβ(x) = λατ_{G}(x) = r^{τ_{G}(x)}r^{−1}, whence (β, r) ∈ Der(F_{G}, (B, μ)) and (\overline{α}, r)∈Der(G, (B, μ)). Evidently, ψ^{1}([(\overline{α}, r)])=[(α, r)].

The rest of the proof repeats with minor modifications the proof of the exactness of the cohomology sequence for a coefficient short exact sequence of crossed G-modules (see Theorems 13 and 15 of [1]).

It is clear that when (2) is an exact sequence of crossed G-modules, Theorem 10 implies Theorems 13 and 15 of [1].

By analogy with the case n = 1 we propose the following definition of the pointed set of cohomology H^{n+1}(G, (A, μ)) of a group G with coefficients in a crossed G-R-bimodule (A, μ) (in particular, in crossed G-modules) for all n ≥ 1.

Let (A, μ) be a crossed G-R-bimodule. Consider diagram (7) of [1] and the group Der(F_{n}, (A, μ)), n ≥ 1, where (A, μ) is viewed as a crossed F_{n}- R-bimodule induced by τ_{0}∂_{0}^{1}∂_{0}^{2} ⋯ ∂_{0}^{n−1}∂_{0}^{n} with ∂_{0}^{i} = l_{0}^{i−1}τ_{i}, i = 1, …, n. Denote by Z̃^{1}(F_{n}, (A, μ)) the subset of Der(F_{n}, (A, μ)) consisting of all elements of the form (α, 1) satisfying the condition

∏_{j=0}^{n+1}(α∂_{j}^{n+1})^{ϵ} = 1, ϵ = (−1)^{i}.

Note that since μα(x)=1, x ∈ F_{n}, we have α(F_{n}) ⊂ Z(A). In Z̃^{1}(F_{n}, (A, μ)) we introduce a relation ∼ as follows: (α′, 1) ∼ (α, 1) if there is an element (β, h) ∈ Der(F_{n−1}, (A, μ)) such that

α′(x) = ^{h}α(x) ∏_{i=0}^{n}(β∂_{i}^{n}(x))^{ϵ}, x ∈ F_{n},  (4)

where ϵ = (−1)^{i}. Since the homomorphism τ_{0}∂_{i_{n}}^{1}∂_{i_{n−1}}^{2} ⋯ ∂_{i_{2}}^{n−1}∂_{i_{1}}^{n} does not depend on the sequence (i_{1}, i_{2}, …, i_{n−1}, i_{n}), we have

β∂_{j}^{n}(x)(β∂_{l}^{n}(x))^{−1} = (β∂_{l}^{n}(x))^{−1}β∂_{j}^{n}(x) ∈ ker μ, x ∈ F_{n},

for j even and l odd. It follows that the product ∏_{i=0}^{n}(β∂_{i}^{n}(x))^{ϵ} in (4) does

not depend on the order of the factors. Note that if n is even then β(F_{n}) ⊂ ker μ ⊂ Z(A).

Similarly to the case n = 1 it can be shown that the relation ∼ is an equivalence, the quotient set Z̃^{1}(F_{n}, (A, μ))/ ∼ is independent of diagram (7) of [1] (for instance, we can take the free cotriple resolution of the group G), and there is a surjective map

ϑ′_{n} : H^{n+1}(G, ker μ) ⟶ Z̃^{1}(F_{n}, (A, μ))/ ∼, n ≥ 1,

given by [α] ⟼ [(α, 1)] which is bijective if (A, μ) is a crossed G-G-bimodule and either μ is the trivial map or n is even.

Definition 11. Let (A, μ) be a crossed G-R-bimodule. Define

H^{n+1}(G, (A, μ)) = Z̃^{1}(F_{n}, (A, μ))/ ∼, n ≥ 1.

It is clear that for n = 1 we recover the second set of cohomology of G with coefficients in (A, μ).

Remark 1. Using the above-defined cohomology with coefficients in crossed bimodules it is possible to define a cohomology H^{n}(G, A), n ≤ 2, of a group G with coefficients in a G-group A.

Consider the quotient group \overline{A} = A/Z(A) and define an action of \overline{A} on A and an action of G on \overline{A} as follows:

[a′]a = ^{a′}a, a, a′ ∈ A,
g[a] = [^{g}a], g ∈ G, a ∈ A.

Let μ_{A} : A ⟶ \overline{A} be the canonical homomorphism. Then (A, μ_{A}) is a crossed G-A-bimodule and we define

H^{n}(G, A) = H^{n}(G, (A, μ_{A})), n ≤ 2.

For n = 1 this cohomology differs from the pointed set of cohomology defined in [4]. If

1 ⟶ A ⟶^{φ} B ⟶^{ψ} C ⟶ 1

is a central extension of G-groups then ψ induces an isomorphism ϑ : B/Z(B) ⟶^{≈} C/ψ(Z(B)) and one gets a short exact sequence of crossed G-\overline{B}-bimodules

1 ⟶ (A, 1) ⟶^{φ} (B, μ_{B}) ⟶^{ψ} (C, \overline{μ}_{C}) ⟶ 1,

where \overline{μ}_{C} is the composite of the canonical map τ : C ⟶ C/ψ(Z(B)) and the isomorphism ϑ^{−1}. Since \overline{B} acts trivially on A, from Theorem 10 immediately follows the exact cohomology sequence

1 ⟶ H^{0}(G, A) ⟶^{φ^{0}} H^{0}(G, B) ⟶^{ψ^{0}} H^{0}(G, C) ⟶^{δ^{0}} H^{1}(G, A) ⟶^{φ^{1}}

⟶^{φ^{1}} H^{1}(G, B) ⟶^{ψ^{1}} H^{1}(G, (C, \overline{μ}_{C})) ⟶^{δ^{1}} H^{2}(G, A) ⟶^{φ^{2}} H^{2}(G, B) ⟶^{ψ^{2}}

⟶^{ψ^{2}} H^{2}(G, (C, \overline{μ}_{C})) ⟶^{δ^{2}} H^{3}(G, A).

Remark 2. As for the case n = 2 (see Remark of [1]) it is possible to give an alternative more non-abelian definition of the third cohomology \overline{H}^{3}(G, (A, μ)) of G with coefficients in a crossed G-R-bimodule (A, μ). To this end consider the commutative diagram

\begin{aligned}M_{G}^{1} & ⟶_{φ_{1}}^{⟶}^{φ_{0}} & Q_{G} & ⟶^{η_{G}} & M_{G} \\ & & q_{1} ↓↓ q_{0} & & l_{1} ↓↓ l_{0} \\ & & F^{2}(G) & ⟶^{F (τ_{G})} & F (G) \\ & & ↓ τ_{F (G)} & & ↓ τ_{G} \\ & & F (G) & ⟶^{τ_{G}} & G \end{aligned} ,

where F (G) = F_{G}, F^{2}(G) = F (F (G)), τ_{G} and τ_{F (G)} are canonical surjections, η_{G} is induced by F (τ_{G}), and (M_{G}, l_{0}, l_{1}), (Q_{G}, q_{0}, q_{1}), (M_{G}^{1}, φ_{0}, φ_{1}) are the simplicial kernels of τ_{G}, τ_{F (G)} and η_{G}, respectively. It is clear that (A, μ) is a crossed Q_{G}-G-bimodule induced by τ_{G}l_{0}η_{G}. Let D̃er(Q_{G}, (A, μ)) be the subgroup of Der(Q_{G}, (A, μ)) consisting of elements (β, g) such that β(Δ_{Q}) = 1, where Δ_{Q} = {(x, x), x ∈ F^{2}(G)}. Consider the set Z̃^{1}(M_{G}^{1}, (A, μ)) of all crossed homomorphisms α : M_{G}^{1} ⟶ A with α(Δ)=1 where Δ={(y, y), y ∈ Q_{G}} and M_{G}^{1} acts on A via τ_{G}l_{0}η_{G}φ_{0}. Introduce, in Z̃^{1}(M_{G}^{1}, (A, μ)), a relation of equivalence as follows:

α′ ∼ α if ∃(β, g) ∈ D̃er(Q_{G}, (A, μ))

such that α′(x) = βφ_{1}(x)^{−1}α(x) βφ_{0}(x), x ∈ M_{G}^{1}. Define \overline{H^{3}}(G, (A, μ)) = Z̃^{1}(M_{G}^{1}, (A, μ))/ ∼. Then \overline{H^{3}}(G, (A, μ)) is a covariant functor from the category of crossed G-R-bimodules to the category of pointed sets. It can be proved that \overline{H^{3}}(G, (A, 1)) is isomorphic to the classical third cohomology group H^{3}(G, A) if A is a G-module.

Acknowledgement

The research described in this publication was made possible in part by Grant MXH200 from the International Science Foundation and by INTAS Grant No 93-2618.

References

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(Received 02.06.1995)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia