HIGHER NON-ABELIAN COHOMOLOGY OF GROUPS

HVEDRI INASSARIDZE

Abstract. The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substancially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.

Introduction

Our approach to non-abelian cohomology of groups follows Guin’s first non-abelian cohomology [5,6] which differs from the classical first non-abelian cohomology pointed set [10] and from the setting of various papers on non-abelian cohomology [4,2,3] extending the classical exact non-abelian cohomology sequence from lower dimensions [10] to higher dimensions.

Guin defined his first non-abelian cohomology group when the coefficient group is a crossed G-module and obtained a six term exact cohomology sequence for any short exact sequence of crossed G-modules.

A non-abelian cohomology of groups will be defined in any dimensions ≥ 1 extending Guin’s first non-abelian cohomology group and his exact cohomology sequence to nine term exact cohomology sequence. A substantially wider class of coefficients will be used consisting of partially crossed modules over a group P on which acts on the left a group G that will be called G-partially crossed modules over P. We describe the first non-abelian cohomology in terms of torsors and the second non-abelian cohomology in terms of extensions of groups. Moreover a close relation of non-abelian cohomology of groups with non-abelian right derived functors of the group of derivations will be established and for some particular cases of coefficients a long exact cohomology sequence will be obtained.

1991 Mathematics Subject Classification. 18G50, 18G55.

Key words and phrases. partially crossed module, higher cohomology, derivations, right derived functors.

All considered groups will be arbitrary (not necessarily commutative). An action of a group G on a group A means an action on the left of G on A by automorphisms and will be denoted by ^{g}a, g ∈ G, a ∈ A. We assume that G acts on itself by conjugation. The center of a group G will be denoted by Z(G). If the groups G and R act on a group A then the notation ^{gr}a means ^{g}(^{r}a), g ∈ G, r ∈ R, a ∈ A.

1. G-partially crossed P -modules and the group Der(G, (A, μ)) of derivations

A precrossed P -module (A, μ) consists of a group P acting on a group A and a homomorphism μ : A ⟶ P such that

μ(^{x}a) = xμ(a)x^{−1}, x ∈ P, a ∈ A.

If in addition we have

μ(a)a′ = aa′a^{−1}

for a, a′ ∈ A, then (A, μ) is a crossed P -module.

Definition 1.1. A partially crossed module μ : A ⟶ P over P is a precrossed module over P satisfying the equality

aa′a^{−1} = ^{μ(a)}a′ ,  (1.1)

for all a′ ∈ A and for all a ∈ A such that μ(a) is a commutator of P.

Note that the relation (1.1) is equivalent to the following relation:

a^{yx}a′ = ^{xy}a′a ,  (1.2)

for all a′ ∈ A and for all a ∈ A such that μ(a) = xyx^{−1}y^{−1}. In effect, (1.1) ⟹ (1.2) : take a′ =^{yx} b and (1.2) ⟹ (1.1) : take a′ =^{x^{−1}y^{−1}} b. Clearly any crossed module over P is a partially P -crossed module.

Let A be a metabelian (not abelian) group. Consider the precrossed module

A ⟶^{τ} A/[A, A] = P ,

where τ is the canonical surjection and P acts trivially on A. Then A ⟶^{τ} P is a partially crossed module over P which is not a crossed P -module.

Any precrossed module B ⟶^{μ} P induces in a natural way a partially crossed module over P as follows:

Consider the Peiffer commutators bb′b^{−1} ^{μ(b)}b^{′−1} for all b′ ∈ B subject to the relation: μ(b) is a commutator of P. Let N be the normal subgroup of B generated by these Peiffers commutators and take the quotient group B/N. One gets a precrossed module B/N ⟶^{μ′} P, μ′ being induced by μ. It is easy to check that in fact it is a partially crossed P -module. Moreover any morphism from B ⟶^{μ} P to a partially crossed module X ⟶^{ν} C induces in a natural way a unique morphism

\begin{matrix}B/N & ⟶^{μ′} & P \\ ↓ & & ↓ \\ X & ⟶^{ν} & C \end{matrix} .

It is obvious that if A ⟶^{μ} P is a partially P -crossed module, then Ker μ is contained in the center of A .

Definition 1.2. Let G, P and A be groups. It will be said that A ⟶^{μ} P is a G-precrossed module over P if

(1) (A, μ) is a precrossed P -module,

(2) G acts on P and A,

(3) μ : A ⟶ P is a homomorphism of G-groups,

(4) ^{(^{g}x)}a = ^{gxg^{−1}}a for g ∈ G, x ∈ P, a ∈ A (compatibility condition). If in addition (A, μ) is a crossed P -module, then (A, μ) is called G-

crossed P -module. If conditions (1)-(4) hold it will be said that the group G acts on the precrossed P -module (A, μ).

Definition 1.3. A G-precrossed P -module (A, μ) will be called G-partially crossed P -module if in addition the following condition holds:

aa′a^{−1} = ^{μ(a)}a′ or equivalently a^{yx}a′ = ^{xy}a′a ,

for all a′ ∈ A and a ∈ A such that μ(a) = xyx^{−1}y^{−1} for some x, y ∈ P.

It is clear that any precrossed (crossed) G-module is in a natural way a G-precrossed (G-crossed) G-module, G acting on itself by conjugation. A G-precrossed P -module was called in [8] as precrossed G − P - bimodule arising confusion with the notion of crossed bimodule defined in [9](see E.1.5.1) generalising the well-known notion of bimodule. If f : G′ ⟶ G is a homomorphism of groups then any G-precrossed P -module is a G′-precrossed P -module induced by f, G′ acting on A and P via f.

Note that if (A, μ) is a G-precrossed P -module the equality

rxa = ^{xr}a

holds for any x ∈ G, a ∈ A, r ∈ H^{0}(G, P). In effect, one has

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

Definition 1.4. Let (A, μ) be a G-partially crossed P -module. Denote by Der(G, (A, μ)) the set of pairs (α, r), α is a crossed homomorphism from G to A, i.e.

α(xy) = α(x) ^{x}α(y) , x, y ∈ G ,

and r is an element of P such that

μα(x) = r ^{x}r^{−1} , x ∈ G .

This set will be called the set of derivations from G to (A, μ).

For any (α, r) ∈ Der(G, (A, μ)) one has

α(x) ^{xr}a = ^{rx}αα(x) ,

for a ∈ A, x ∈ G.

We introduce in Der(G, (A, μ)) a product by

(α, r)(β, s) = (α ∗ β, rs) ,

where (α ∗ β)(x) = ^{r}β(x)α(x), x ∈ G.

Proposition 1.5. Under the aforementioned product Der(G, (A, μ)) is a group which coincides with the group Der_{G}(G, A) of Guin if (A, μ) is a crossed G-module viewed as a G-crossed G-module.

Proof. Clearly this product is associative. At first it will be shown that (α ∗ β, rs) belongs to Der(G, (A, μ)). Put γ = α ∗ β. We have

γ(xy) = ^{r}β(xy)α(xy) = ^{r}(β(x) ^{x}β(y))α(x) ^{x}α(y) = ^{r}β(x) ^{rx}β(y) · · α(x) ^{x}α(y).

On the other hand

γ(x)^{x}γ(y) =^{r} β(x)α(x)^{x}(^{r}β(y)α(y)) =^{r} β(x)α(x)^{xr}β(y)^{x}α(y) .

Using equality (1.1) one gets γ(xy) = γ(x)^{x}γ(y) showing that γ is a crossed homomorphism. Further, we have

μγ(x) = μ(^{r}β(x)α(x)) = ^{r}μβ(x)μα(x) = ^{r}(s ^{x}s^{−1})r ^{x}r^{−1} = ^{r}s^{r}(s^{−1})r

xr^{−1} = ^{r}sr^{x}s^{−1}^{x}r^{−1} = rs ^{x}(rs)^{−1} .

Therefore (α ∗ β, rs) ∈ Der(G, (A, μ)). It is obvious that (α_{0}, 1) is the unit of Der(G, (A, μ)) with α_{0}(x) = 1 for all x ∈ G.

For (α, r) ∈ Der(G, (A, μ)) take the pair (\overline{α}, r^{−1}) with \overline{α}(x) = = ^{r^{−1}}α(x)^{−1}, x ∈ G. It will be shown that (\overline{α}, r^{−1}) ∈ Der(G, (A, μ)). For this it will be proved the equality

r^{−1}xa ·^{r^{−1}}α(x)^{−1} = ^{r^{−1}}α(x)^{−1} ·^{xr^{−1}}a, x ∈ G, a ∈ A. (1.3)

In effect, since μ(^{r^{−1}}α(x)^{−1}) = r^{−1}μα(x)^{−1}r = r^{−1}^{x}r, one gets

μ(^{r^{−1}}α(x)^{−1})(^{xr^{−1}}a) = ^{r^{−1} ^{x}r}(^{xr^{−1}}a) = ^{r^{−1}xrx^{−1}xr^{−1}}a = ^{r^{−1}x}a .

The required relation (1.3) follows now from the equality

μ(^{r^{−1}}α(x)^{−1})(^{xr^{−1}}a) = r^{−1}α(x) ^{xr^{−1}}a ^{r^{−1}}α(x) .

Therefore one has

\overline{α}(xy) = ^{r^{−1}}α(xy)^{−1} = ^{r^{−1}}(^{x}α(y)^{−1}α(x)^{−1}) = ^{r^{−1}x}α(y)^{−1} ^{r^{−1}}α(x)^{−1} =

r^{−1}α(x)^{−1} ^{xr^{−1}}α(y)^{−1} = \overline{α}(x) ^{x}\overline{α}(y) ,

i.e. \overline{α} is a crossed homomorphism. Moreover,

μ\overline{α}(x) = μ(^{r^{−1}}α(x)^{−1}) = r^{−1}μα(x)^{−1}r = r^{−1 x}rr^{−1}r = r^{−1} ^{x}r .

There follows that (\overline{α}, r^{−1}) ∈ Der(G, (A, μ)). It is easily checked that

(α, r)(\overline{α}, r^{−1}) = (\overline{α}, r^{−1})(α, r) = (α_{0}, 1) .

We conclude that Der(G, (A, μ)) is a group which coincides with the group Der_{G}(G, A) of derivations defined by Guin [6] when (A, μ) is a crossed G-module. □

A homomorphism f : (A, μ) ⟶ (B, λ) of G-partially crossed P - modules induces a homomorphism

f^{∗} : Der(G, (A, μ)) ⟶ Der(G, (B, λ))

given by (α, r) ↦ (αf, r).

There is an action of G on Der(G, (A, μ)) defined by

g(α, r) = (\overline{α},^{g}r) , g ∈ G , r ∈ P ,

with \overline{α}(x) = ^{g}α(^{g^{−1}}x), x ∈ G (see [6,8]). Moreover if P acts on G such that

(^{r}g)a = ^{rgr^{−1}}a , ^{(^{r}g)}r′ = ^{rgr^{−1}}r′ for r, r′ ∈ R, g ∈ G, a ∈ A ,

then there is also an action of P on Der(G, (A, μ)) given by ^{r}(α, s) = (\overline{α},^{r}s), \overline{α}(x) =^{r} α(^{r^{−1}}x), r ∈ P, x ∈ G [8].

It is well-known [1] that the groups G and P acting on each other and on themselves by conjugation are said to be acting compatibly if

(^{g}r)g′ = ^{grg^{−1}}g′, ^{(^{r}g)}r′ = ^{rgr^{−1}}r′ for g, g′ ∈ G, r, r′ ∈ R . (1.4)

Definition 1.6. It will be said that the groups G and P act on a group A compatibly if

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

Proposition 1.7 ([8] ). Let (A, μ) be a G-partially crossed P -module the groups G and P acting on each other and on A compatibly. Under the aforementioned actions of G and P on Der(G, (A, μ)) and the homomorphism γ : Der(G, (A, μ)) → P given by (α, r) ↦ r, the pair (Der(G, (A, μ)), γ) is a G-precrossed P -module.

2. The first non-abelian cohomology

Let (A, μ) be a G-partially crossed P -module. Define on the group Der(G, (A, μ)) an equivalence relation as follows:

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

Theorem 2.1. Let (A, μ) be a G-partially crossed P -module satisfying the following conditions:

1) H^{0}(G, P) is a normal subgroup of P,

2) for any c ∈ H^{0}(G, P) and (α, r) ∈ Der(G, (A, μ)) there exists a ∈ A such that μ(a) = 1 and ^{c}α(x) = a^{−1}α(x) ^{x}a, x ∈ G.

Then the group Der(G, (A, μ)) induces on the quotient set Der(G, (A, μ))/ ∼ a group structure and this quotient group will be called the first cohomology group H^{1}(G, (A, μ)) of the group G with coefficients in the G-partially crossed P -module (A, μ).

Proof. We have to show that the relation ∼ is a congruence, i.e. if (α, r) ∼ (α′, r′) and (β, s) ∼ (β′, s′), then (α, r)(β, s) ∼ (α′, r′)(β′, s′). For this it will be used Guin’s proof [6] which remains valid in our generalized case.

We first prove that

(α, r)(β, s) ∼ (α, rc)(β, s)

for any c ∈ H^{0}(G, P). For (β, s) and c ∈ H^{0}(G, P) there is a ∈ A such that μ(a) = 1 and a^{−1}β(x)^{x}a, x ∈ G. One gets

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

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

Use the equalities

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, P) and put

(α, rz)(β, s) = (γ, rzs)
and (α′, r′)(β′, s′) = (γ′, r′s′)

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

It will be shown that

(α, rz)(β, s) ∼ (α′, r′)(β′, s′) .

In effect,

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

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

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

There follows that (α, rz)(β, s) ∼ (α′, r′)(β′, s′). Therefore (α, r)(β, s) ∼ (α′, r′)(β′, s′) and the equivalence ∼ is a congruence. □

Clearly any partially crossed G-module viewed as a G-partially crossed G-module satisfies conditions of Theorem 2.1, in this case H^{0}(G, G) = Z(G) and for (α, g) ∈ Der(G, (A, μ)) = Der_{G}(G, A), c ∈ Z(G) the equality α(cx) = α(xc), x ∈ G, implies α(c) ^{c}α((x) = α(x) ^{x}α(c) and μ(α(c)) = gcg^{−1}c^{−1} = 1; we recover Guin’s first cohomology group of a group G with coefficients in a crossed G-module [6].

If f : (A, μ) → (B, λ) is a homomorphism of G-partially crossed P -modules satisfying conditions of Theorem 2.1, then f^{∗} induces a homomorphism f^{1} : H^{1}(G, (A, μ)) → H^{1}(G, (B, λ)). 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 .

By our next statement it will be shown that Guin’s first non-abelian cohomology group is closely related with torsors. A similar relationship of the first non-abelian pointed set cohomology with principal homogeneous spaces is well-known [1]. To this aim the notion of a G-torsor over a partially crossed G-module will be introduced.

Definition 2.2. A G-torsor over a partially crossed G-module (A, μ) is a pair (E, f) consisting of a non-empty G-set E with an action on the right of A on E denoted by xa (for x ∈ E, a ∈ A ) which is compatible with the action of G and such that for any x, y ∈ E there is a unique element b ∈ A with y = xb, and f is a map from E to G such that

1) for any x ∈ E, s ∈ G the following equality holds

μ(a) = f(x)sf(x)^{−1}s^{−1}

with ^{s}x = xa, a ∈ A;

2) if y = xb then

f(y) = μ(b^{−1})f(x) , x, y ∈ E , b ∈ A .

Definition 2.3. It will be said that G-torsors (E, f) and (E′, f′) over a partially crossed G-module (A, μ) are isomorphic if there is a bijection ϑ : E → E′ compatible with the actions of G and A such that

f(x) = f′(x) mod Z(G)

for any x ∈ E.

Denote by E(G, A) the set of classes of isomorphic G-torsors over the partially crossed G-module (A, μ).

It is introduced a product on the set E(G, A) as follows. Let [(E_{1}, f_{1})], [(E_{2}, f_{2})] ∈ E(G, A) and let x ∈ E_{1}, y ∈ E_{2}. Take A with a new action of G given by (^{s}a)′ = ^{f_{1}(x)}cb ^{s}a for any s ∈ G, a ∈ A with ^{s}x = xb, sy = yc. Denote so defined G-group by E and define the action of A on E by translation on the right. Define a map g : E → G given by

g(a) = μ(a^{−1})f_{1}(x)f_{2}(x) .

Then the pair (E, g) is a G-torsor over the partially crossed G-module (A, μ) and define the required product by

[(E_{1}, f_{1})] ∘ [(E_{2}, f_{2})] = [(E, g)] .

Theorem 2.4. If (A, μ) is a partially crossed G-module, there is a natural isomorphism between E(G, A) and H^{1}(G, A).

Proof. Let [(E, f)] ∈ E(G, A) and take x ∈ E. For any s ∈ G one has sx = xa and an induced map α_{x} : G ⟶ A defined by α_{x}(s) = a that is a cocycle. Using 1) of Definition 2.2 we see that the pair (α_{x}, f(x)) is an element of Der(G, (G, A)) = Der_{G}(G, A).

Define a map α : E(G, A) ⟶ H^{1}(G, A) by α([(E, f)]) = [(α_{x}, f(x))]. We have to show the correctness of α.

If y ∈ E and y = xb, b ∈ A, then ^{s}y = ^{s}x ^{s}b = xa ^{s}b = xbb^{−1}a ^{s}b = yb^{−1}a ^{s}b. By 2) f(y) = μ(b^{−1})f(x). There follows that (α_{x}, f(x)) ∼ (α_{y}, f(y)). Let (E, f) be isomorphic to (E′, f′), i.e. there is a bijection ϑ : E ⟶ E′ with properties given in Definition 2.2. Take x′ ∈ E′ and ϑ(x) = x′, x ∈ E. Then ϑ(^{s}x) = ^{s}ϑ(x) = ^{s}x′ and ϑ(xa) = ϑ(x)a = x′a with ^{s}x = xa. Thus, α_{x} = α_{x′}. Since f(x) = f′(x′) mod Z(G), one deduces (α_{x}, f(x)) ∼ (α_{x′}, f′((x′)). Therefore, the map α is correctly defined.

Let [(α, g)] ∈ H^{1}(G, A). Take A with a new action of G given by (^{s}x)′ = α(s) ^{s}x, x ∈ A, s ∈ G, and with the action of A on itself by translation on the right. Denote so obtained G-set by P_{α}. Define a map f_{g} : P_{α} → G by f_{g}(x) = μ(x^{−1})g, x ∈ P_{α}, which verifies conditions 1) and 2) of Definition 2.2. In effect, if (^{s}x)′ = xa then α(s)^{s}x = xa. On the other hand μα(s) = gsg^{−1}s^{−1}. Thus, μα(s)μ(^{s}x) = μ(xa), gsg^{−1}s^{−1}sμ(x)s^{−1} = μ(x)μ(a). Whence

μ(a) = μ(x)^{−1}gsg^{−1}μ(x)s^{−1} = f_{g}(x)sf_{g}(x)^{−1}s^{−1}

and f_{g} verifies condition 1).

If y = xb then one has

f_{g}(y) = μ(y^{−1})g = μ(b^{−1}x^{−1})g = μ(b^{−1})μ(x^{−1})g = μ(b^{−1})f(x) .

Thus f_{g} satisfies condition 2) too. One gets a G-torsor (P_{α}, f_{g}) over the partially crossed G-module (A, μ) and define β : H^{1}(G, A) → E(G, A) by β([(α, g)]) = [(P_{α}, f_{g})].

If (α, g) ∼ (α′, g′) then α′(s) = b^{−1}α(s)^{s}b and g′ = μ(b^{−1})g mod Z(G). It will be shown that (P_{α}, f_{g}) is isomorphic (P_{α′}, f_{g′}).

Define ∂ : P_{α} ⟶ P_{α′} by ∂(x) = b^{−1}x, x ∈ P_{α}. Then ∂((^{s}x)′) = b^{−1}α(s) ^{s}x and (^{s}(∂(x)))′ = (^{s}(b^{−1}x))′ = α′(s) ^{s}(b^{−1}x) = b^{−1}α(s) ^{s}b sb^{−1} ^{s}x = b^{−1}α(s) ^{s}x. Thus, ∂((^{s}x)′) = (^{s}(∂(x)))′. It is obvious that ∂ preserves the action of A.

For x ∈ P_{α} one gets

f_{g′}(∂(x)) = f_{g′}(b^{−1}x) = μ(x^{−1}b)g′ = μ(x^{−1})μ(b)μ(b^{−1})g mod Z(G) =

μ(x^{−1})g mod Z(G) = f_{g}(x) mod Z(G) .

This implies that (P_{α}, f_{g}) is isomorphic to (P_{α′}, f_{g′}). Therefore the map β is correctly defined.

It is easily checked that α is a homomorphism and αβ = 1, βα = 1. □

3. The second non-abelian cohomology

Now the second cohomology H^{2}(G, (A, μ)) of a group G with coefficients in a G-partially crossed P -module (A, μ) will be defined.

Consider the diagram

M ⟶^{l_{0}} ; ⟶_{l_{1}}F ⟶^{τ} G  (3.1)

with F a free group, τ is a surjective homomorphism, M is a group consisting of pairs (x, y), x, y ∈ F, such that τ(x) = τ(y) and l_{0}, l_{1} are canonical projections, l_{0}(x, y) = x, l_{1}(x, y) = y. That means (M, l_{0}, l_{1}) is the simplicial kernel of τ. Put Δ = {(x, x), x ∈ F } ⊂ M.

Then (A, μ) can be viewed as a F -partially crossed P -module induced by τ and as a M-partially crossed P -module induced by τl_{0} (or by τl_{1}). Let Z̃^{1}(M, (A, μ)) be the subset of Der(M, (A, μ)) consisting of elements of the form (α, 1) satisfying the condition α(Δ) = 1, implying α(M) ⊂ Z(A). There follows that Z̃^{1}(M, (A, μ)) is an abelian subgroup of Der(M, (A, μ)).

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

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

such that the following equality holds

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

in the group Der(M, (A, μ)).

We see that if (α′, 1) ∼ (α, 1) one has

α′(x) = βl_{1}(x)^{−1} ^{h}α(x)βl_{0}(x) , x ∈ M .

Proposition 3.1. The relation ∼ defined on Z̃^{1}(M, (A, μ)) is an equivalence.

Proof. Clearly this relation is reflexive. If (α′, 1) ∼ (α, 1), i.e. (α′, 1) = (βl_{0}, h)(α, 1)(βl_{1}, 1)^{−1}, (β, h) ∈ Der(F, (A, μ)), then (α, 1) = (βl_{0}, h)^{−1}· ·(α′, 1)(βl_{1}, h) and (βl_{0}, h)^{−1} = (β̃l_{0}, h^{−1}), (βl_{1}, h) = (β̃l_{1}, h^{−1})^{−1} with (β̃, h^{−1}) = (β, h)^{−1} ∈ Der(F, (A, μ)). Thus the relation ∼ is symmetric. It remains to show the transitivity.

Let (α′, 1) ∼ (α, 1) and (α′′, 1) ∼ (α′, 1). Then one has

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

with (β, h), (β′, h′) ∈ Der(F, (A, μ)).

There follows that

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

with (β′ ∗ β, h′h) = (β′, h′)(β, h) ∈ Der(F, (A, μ)). Therefore (α′′, 1) ∼ (α, 1) and the relation ∼ is an equivalence. □

Proposition 3.2. The quotient set Z̃^{1}(M, (A, μ))/ ∼ is independent of the diagram (3.1) and is unique up to bijection.

We need the following Lemma 3.3. Let A be a G-group and let α : M → A be a crossed homomorphism such that α(Δ) = 1. Then there exists a map q : F → A such that

α(y) = ql_{1}(y)^{−1}ql_{0}(y) , y ∈ M .

Proof. Note that if (x, x′′), (x′, x′′) ∈ M, then α(x, x′′) = α(x′, x′′) · ·α(x, x′). In effect the equalities (x, x′′) = (1, x′′x^{′−1})(x, x′) and (x′, x′′) = (1, x′′x^{′−1})(x′, x′) imply α(x, x′′) = α(1, x′′x^{′−1})α(x, x′) and α(x′, x′′) = α(1, x′′x^{′−1})α(x′, x′) = α(1, x′′x^{′−1}) giving the required equality.

In particular, applying this equality for (x, x), (x′, x) ∈ M one gets α(x, x) = α(x′, x)α(x, x′). Therefore α(x′, x) = α(x, x′)^{−1} for any (x, x′) ∈ M.

Take a section η : G ⟶ F, τη = 1_{G} and define a map q : F ⟶ A by

q(x) = α(x, ητ(x)) , x ∈ F .

For (x, x′) ∈ M one has

ql_{1}(x, x′)^{−1}ql_{0}(x, x′) = q(x′)^{−1}q(x) = (α(x′, ητ(x′))^{−1}α(x, ητ(x)) = α(ητ(x′), x′)α(x, ητ(x)) .

On the other hand, since α(x, x′) = α(1, x′x^{−1}) for all (x, x′) ∈ M, on gets α(ητ(x′), x′) = α(1, x′ητ(x′)^{−1}) and α(x, ητ(x)) = α(1, ητ(x)x^{−1}).

But (1, x′ητ(x′)^{−1})(1, ητ(x)x^{−1}) = (1, x′x^{−1}). Therefore, finally we obtain the equality

α(x, x′) = α(1, x′ητ(x′)^{−1})α(1, ητ(x)x^{−1}) = ql_{1}(x, x′)^{−1}ql_{0}(x, x′) .

□ Proof of Proposition 3.2 Consider the commutative diagram

\begin{matrix}M′ & ⟶^{l′_{0}} ; ⟶_{l′_{1}} & F′ & ⟶^{τ′} & G \\ \overline{γ_{1}} ↓ \overline{γ_{2}} & & ^{γ_{1}} ↓ ^{γ_{2}} \\ M & ⟶^{l_{0}} ; ⟶_{l_{1}} & F & ⟶^{τ} & G \end{matrix} ,

(M, l_{0}, l_{1}) and (M′, l′_{0}, l′_{1}) being simplicial kernels of τ and τ′ respectively, l_{i}\overline{γ_{1}} = γ_{1}l′_{i}, l_{i}\overline{γ_{2}} = γ_{2}l′_{i}, i = 0, 1, τγ_{1} = τγ_{2} = τ′.

The pair (γ_{i}, \overline{γ_{i}}) induces a homomorphism

Der(M, (A, μ)) ⟶ Der(M′, (A, μ))

given by (α, r) ↦ (α\overline{γ_{i}}, r), i = 1, 2.

If (α′, 1) ∼ (α, 1), i.e. (α′, 1) = (βl_{0}, h)(α, 1)(βl_{1}, h)^{−1} with (β, h) ∈ Der(F, (A, μ)), then

α′\overline{γ_{i}}(y) = βγ_{i}l′_{1}(y)^{−1} ^{h}α\overline{γ_{i}}(y)βγ_{i}l′_{0}(y) , y ∈ M′ .

Thus (α′\overline{γ_{i}}, 1) ∼ (α\overline{γ_{i}}, 1), i = 1, 2, and one gets a natural map

ε_{i} : Z̃^{1}(M, (A, μ))/ ∼⟶ Z̃^{1}(M′, (A, μ))/ ∼

induced by the pair (γ_{i}, γ̃_{i}) and given by [(α, 1)] ↦ [(αγ̃_{i}, 1)], i = 1, 2.

It will be shown that ε_{1} = ε_{2}. By Lemma 3.3 there is a map q : F → A such that

α(y) = ql_{1}(y)^{−1}ql_{0}(y) , y ∈ M .

Take the homomorphism s : F′ → M given by

s(x′) = (γ_{1}(x′), γ_{2}(x′)) , x′ ∈ F′ .

It is clear that (αs, 1) ∈ Der(F′, (A, μ)). Further one has

((αsl′_{1})^{−1}αγ̃_{2}αsl′_{0})(x′_{0}, x′_{1}) = αs((x′)^{−1}αγ̃_{2}(x′_{0}, x′_{1})αs(x′_{0}) =

α(γ_{1}(x′_{1}), γ_{2}(x′_{1}))^{−1}αγ̃_{2}(x′_{0}, x′_{1})α(γ_{1}(x′_{0}), γ_{2}(x′_{0})) = qγ_{1}(x′_{1})^{−1}qγ_{2}(x′_{1}) =

qγ_{2}(x′_{1})^{−1}qγ_{2}(x′_{0})qγ_{2}(x′_{0})^{−1}qγ_{1}(x′_{0}) = qγ_{1}(x′_{1})^{−1}qγ_{1}(x′_{0}) = αγ̃_{1}(x′_{0}, x′_{1})

for any (x′_{0}, x′_{1}) ∈ M′. Therefore (αγ̃_{1}, 1) ∼ (αγ̃_{2}, 1) with (αs, 1) ∈ Der(F′, (A, μ)) implying the required equality ε_{1} = ε_{2}.

Now the proof of the uniqueness is standard. □

It is easy to check that the quotient set Z̃^{1}(M, (A, μ))/ ∼ is naturally bijective to H^{2}(G, A) when A is a G-module viewed as a crossed Gmodule. That fact motivates the following

Definition 3.4. Let (A, μ) be a G-partially crossed P -module. The quotient set Z̃^{1}(M, (A, μ))/ ∼ will be called the second cohomology of G with coefficients in (A, μ) and denoted by H^{2}(G, (A, μ)).

A homomorphism of G-partially crossed P -modules f : (A, μ) → (B, λ) induces a map of pointed sets

f^{2} : H^{2}(G, (A, μ)) ⟶ H^{2}(G, (B, λ))

given by f^{2}([(α, 1)]) = [(fα, 1)]. It is easy to see that there is a canonical surjective map ϑ : H^{2}(G, Ker μ) → H^{2}(, (A, μ)) given by [α] ↦ (α, 1)].

Proposition 3.5. Let (A, μ) be a G-partially crossed P -module. There is an action of G on H^{2}(G, (A, μ)) such that Z(G) acts trivially. If P acts on Gand satisfies the compatibility condition (1.4), then there is also an action of P on H^{2}(G, (A, μ)).

Proof. Consider the diagram

M_{G} ⟶^{l_{0}} ; ⟶_{l_{1}}F_{G} ⟶^{τ_{G}} G

with F_{G} the free group generated by G, τ_{G} is the canonical homomorphism and (M_{G}, l_{0}, l_{1}) is the simplicial kernel of τ_{G}. There is an action of G on M_{G} defined as follows:

g(|g_{1}|^{ε} ⋯ |g_{n}|^{ε}) = |^{g}g_{1}|^{ε} ⋯ |^{g}g_{n}|^{ε} , g, g_{1}, …, g_{n} ∈ G ,

with ε = ±1. This action induces an action of G on M_{G} by

g(x, x′) = (^{g}x,^{g}x′) , g ∈ G , (x, x′) ∈ M_{G} .

Finally one gets an action of G on Der(M_{G}, (A, μ)) given by

g(α, r) = (α̃,^{g}r)

with α̃(m) = ^{g}α(^{g^{−1}}m), g ∈ G, m ∈ M_{G}, inducing an action of G on D̃er(M_{G}, (A, μ)) and on Z̃^{1}(M_{G}, (A, μ)) too. If (α, 1) ∼ (α′, 1) it is easy to see that ^{g}(α, 1) ∼ ^{g}(α′, 1), g ∈ G, defining an action of G on H^{2}(G, (A, μ)). Since the above defined surjective map ϑ : H^{2}(G, Ker μ) → H^{2}(G, (A, μ)) is a G-map and Z(G) acts trivially on H^{2}(G, Ker μ), there follows that Z(G) acts trivially on H^{2}(G, (A, μ)) too. □

Let (A, μ) be a G-partially crossed P -module. It can be shown easily that there is an action of H^{0}(G, P) on H^{2}(G, Ker μ) given by ^{r}[α] = [^{r}α], r ∈ H^{0}(G, P) with α : M_{G} → Ker μ a crossed homomorphism under the action of G on A such that α(Δ) = 1.

Let

1 ⟶ (A, 1) ⟶^{φ} (B, μ) ⟶^{ψ} (C, λ) ⟶ 1  (3.2)

be a short exact sequence of G-partially crossed P -modules. If the action of H^{0}(G, P) on H^{2}(G, A) is trivial then there is an action of H^{1}(G, (A, μ)) on H^{2}(G, A) given by

(α,r)[γ] = [^{r}γ] .

We have to show that ^{r}γ is a crossed homomorphism and this action is correctly defined. Consider the diagram

\begin{aligned}M_{G} & ⟶^{l_{0}} ; ⟶_{l_{1}} & F_{G} & ⟶^{τ_{G}} & G \\ & & & & ↓ ^{α} \\ A & ⟶^{φ} & B & ⟶^{ψ} & C \end{aligned} .  (3.3)

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) = βx)^{−1}^{r}φγ(x)β(x) = ^{r}φγ(x), x ∈ M_{G}. 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 with c ∈ C, t ∈ H^{0}(G, P). There follows that

φ(^{r′}γ(x)) = ^{r′}φγ(x) = ^{λ(c)^{−1}rt}φγ(x) = ^{μ(b)^{−1}rt}φγ(x) = b^{−1} ^{rt}φγ(x)b =

rtφγ(x) = φ(^{rt}γ(x)) , x ∈ M_{G} , with ψ(b) = c .

Hence [^{r′}γ] = [^{rt}γ] = [^{r}γ] proving the correctness of the action.

Using diagram (3.3) for the short exact sequence (3.2) one defines as follows a connecting map

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

which is a crossed homomorphism when (G, (C, λ)) verifies conditions of Theorem 2.1.

For [(α, r)] ∈ H^{1}(G, (C, λ)) take a crossed homomorphism β : F_{G} → B such that ψβ = ατ_{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 show the correctness of δ^{1}. For another β′ : F_{G} → B with ψβ′ = ατ_{G}, one has ψβ′ = ψβ and there is a crossed homomorphism σ : F_{G} → A such that β′ = βψσ. Then one gets

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_{G} ,
r′ = λ(c)^{−1}rt , t ∈ H^{0}(G, P) .

Take β′ : F_{G} → B such that β′(x) = b^{−1}β(x) ^{x}b with ψ(b) = c and ψβ = ατ_{G}. Then (β′l_{1}^{−1}β′l_{0})(y) = β′(x_{2})^{−1}β′(x_{1}), 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. Therefore the connecting map δ^{1} is correctly defined.

For the short exact sequence 3.2 a connecting map δ^{2} : H^{2}(G, (C, λ)) → H^{3}(G, A) will be also defined. To this end consider the canonical free simplicial resolution of the group G in the category of groups acting on the abelian group A:

⋯ ⟶ ; ⋮ ; ⟶F_{3} ⟶^{τ_{3}} M_{2} ⟶^{l_{0}^{2}} ; ⋮ ; ⟶_{l_{3}^{2}}F_{2} ⟶^{τ_{2}} M_{1}⟶^{l_{0}^{1}} ; ⟶ ; ⟶_{l_{2}^{1}}F_{1} ⟶^{τ_{1}} M_{0} ⟶^{l_{0}^{0}} ; ⟶_{l_{1}^{0}}F_{0} ⟶^{τ_{0}} G

where F_{0} = F_{G}, F_{i} = F_{M_{i−1}}, i ≥ 1, τ_{i} is the canonical homomorphism and (M_{i}, l_{0}^{i}, ⋯, l_{i+1}^{i}) is the simplicial kernel of (l_{0}^{i−1}τ_{i}, ⋯, l_{i}^{i−1}τ_{i}), i ≥ 0 (see [7]). It will be used the equivalence of functors H^{n+1}(−, A) ≈ L_{n} Der(−, A), n ≥ 1, when A is a Z[G]-module. There is 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_{i}} = 1, ε_{i} = (−1)^{i}, 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 [8]).

For any G-partially crossed P -module (A, μ) denote by IDer(G, (A, μ)) a subgroup of Der(G, (A, μ)) consisting of elements of the form (α, r), r ∈ H^{0}(G, P).

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, P) acts trivially on H^{2}(G, Ker λ), then there is a connecting map δ^{2} : H^{2}(G, (C, λ)) → H^{3}(G, A) given by

δ^{2}([(α, r)]) = [γ] , (α, 1) ∈ Z̃^{1}(M_{G}, (C, λ)) ,

where φγ = \overline{β}τ_{2} with \overline{β} = ∏_{(}^{2} βl_{i}^{1})^{ε_{i}}, ε_{i} = (−)^{i}, and ψβ = ατ_{1}. The correctness of δ^{2} is proved similarly to the case of a short exact sequence of crossed G-modules (see [8]).

Theorem 3.6. Let (3.2) be a short exact sequence of G-partially crossed P -modules satisfying conditions of Theorem 2.1. Then there is an exact cohomology sequence

1 → H^{0}(G, A) ⟶^{φ^{0}} H^{0}(G, B) ⟶^{ψ0} H^{0}(G, C) ⟶^{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}} H^{2}(G, (B, μ))

⟶^{ψ^{2}} H^{2}(G, (C, λ)) ,

where φ^{0}, ψ^{0}, δ^{0}, φ^{1} and ψ^{1} are homomorphisms. If in addition H^{0}(G, P) 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 P on A. Moreover, if either the action of Der(F_{0}, (C, λ)) on H^{3}(G, A) is trivial (in particular if P acts trivially on A) or Der(F_{0}, (C, λ)) = IDer(F_{0}, (C, λ)) and H^{0}(G, P) 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 also 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 well known [10].

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. There follows that (α_{0}, 1) ∼ (φα, 1) where α_{0} is a trivial map, since φα(x) = b^{−1}α_{0} ^{x}b, x ∈ G, and μ(b) ∈ H^{0}(G, P) because μ(b) = λψ(b) 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, P). One has ψ(b^{−1}^{x}b) = ψφα(x) = 1. Thus ψ(b) = ψ(^{x}b) = ^{x}ψ(b), whence ψ(b) ∈ H^{0}(G, C). Clearly δ^{0}(ψ(b)) = [α]. Therefore Ker φ^{1} ⊂ Im δ^{0}. Obviously the composite ψ^{1}φ^{1} is the trivial map.

Let [(α, r) ∈ H^{1}(G, (B, μ)) such that (α_{0}, 1) ∼ (ψα, 1). Then ψα(x) = c^{−1} ^{x}c, c ∈ C, and r = λ(c)^{−1}t, t ∈ H^{0}(G, P). Let ψ(b) = λ(c) and r = μ(b)^{−1}t. Take α̃(x) = bα(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 the diagram ? and take the crossed homomorphism ατ_{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] with 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 one gets

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

implying a crossed homomorphism \overline{α} : G → B such that (φη)^{−1}β = \overline{α}τ_{G}. One has μβ(x) = λψβ(x) = λατ_{G} = 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 [8]. □

Clearly for a short exact sequence of crossed G-modules we recover the known exact cohomology sequence [6, 8]. Note also that Theorem 3.6 remains true for arbitrary G-partially crossed P -modules but in this case φ^{1}, ψ^{1} and δ^{1} are maps of pointed sets.

Now for any partially crossed G-module (A, μ) the second cohomology H^{2}(G, A) will be described in terms of extensions of groups.

Definition 3.7. An extension of G by a partially crossed G-module (A, μ) is a pair E = (1 → A ⟶^{σ} X ⟶^{ψ} G ⟶ 1, γ), where 1 ⟶ A ⟶^{σ} X ⟶^{ψ} G ⟶ 1 is a short exact sequence of groups, γ is a section of ψ, that is ψγ = 1_{G}, one has the equality

ga = σ^{−1}(γ(g)σ(a)γ(g)^{−1})

for a ∈ A, g ∈ G, and the following additional condition holds:

Ker \overline{ψ} ⊂ Z(A) ,

\overline{ψ} being the restriction of ψ on the subgroup of X generated by γ(G).

Example 3.8. Let A ◁ G be the semidirect product of A and G. Then one has an exact sequence of groups

1 ⟶ A ⟶^{σ_{0}} A ◁ G ⟶^{ψ_{0}} G ⟶ 1 ,

where σ_{0}(a) = (a, 1), ψ_{0}(a, g) = g. Take the canonical section γ_{0} of ψ_{0} given γ_{0}(g) = (1, g). It is easy to see that the pair (1 ⟶ A ⟶^{σ_{0}} A ◁ G ⟶^{ψ_{0}} G, γ_{0}) is an extension of G by (A, μ) called trivial.

Definition 3.9. It will be said that E = (1 → A ⟶^{σ} X ⟶^{ψ} G → 1, γ) is equivalent to E′ = (1 → A ⟶^{σ′} X′ ⟶^{ψ′} G → 1), γ′) if there exist a homomorphism ϑ : X → X′ and an element g ∈ G such that the diagram

1 ⟶ A ⟶^{σ} X ⟶^{ψ} G ⟶ 1
↓ ^{g} ↓ ^{ϑ} ||
1 ⟶ A ⟶^{σ′} X′ ⟶^{ψ′} G ⟶ 1

is commutative, g : A → A is the automorphism induced by the action of g on A, and for any element x ∈ G one has the equality

μ(ϑγ(x)γ′(x)^{−1}) = gxg^{−1}x^{−1} .

Clearly this relation ∼ is reflexive and symmetric. So for the relation ∼ to be an equivalence it remains to show the transitivity. Let E ∼ E′ and E′ ∼ E′′. Then the diagram

E = 1 ⟶ A ⟶^{σ} X  \underleftarrow{⟶^{ψ}} G ⟶ 1
↓ ^{g}  ↓ ^{ϑ}  ||
E′ = 1 ⟶ A ⟶^{σ′} X′  \underleftarrow{⟶^{ψ′}} G ⟶ 1
↓ ^{h}  ↓ ^{κ}  ||
E′′ = 1 ⟶ A ⟶^{σ′′} X′′  \underleftarrow{⟶^{ψ′′}} G ⟶ 1

is commutative and one has the equalities

μ(ϑγ(x) γ′(x)^{−1}) = gxg^{−1}x^{−1} ,
μ(κγ′(x)γ′′(x)^{−1}) = hxh^{−1}x^{−1} .

We shall show that μ(κϑγ(x) γ′′(x)^{−1}) = hgxg^{−1}h^{−1}x^{−1}. In effect, since κϑγ(x)γ′′(x)^{−1}(κγ′(x)γ′′(x)^{−1})^{−1} = κϑγ(x)κγ′(x)^{−1} = κ(ϑγ(x) · ·γ′(x)^{−1}, one gets

μ(κϑ(x)γ′′(x)^{−1}) = μκ(ϑγ(x)γ′(x)^{−1}μκγ′(x) γ′′(x)^{−1}) =
= h(gxg^{−1}x^{−1})h^{−1}hxh^{−1}x^{−1} = hgxg^{−1}h^{−1}x^{−1} .

Therefore the relation ∼ is an equivalence. Denote by E^{1}(G, A) the set of equivalence classes of extensions of G by the partially crossed G-module (A, μ).

Theorem 3.10. There is a natural bijection

η : H^{2}(G, A) ⟶^{≈} E^{1}(G, A) .

Proof. The map η is defined as follows. For [(α, 1)] ∈ H^{2}(G, A) consider the diagram

\begin{matrix}M_{G} & ⟶^{l_{0}} ; ⟶_{1} & F_{G} & ⟶^{τ} & G \\ ↓ _{α} \\ A \end{matrix} ,

take the semidirect product A ◁ F_{G}, F_{G} acting on A via G and introduce an equivalence relation:

(a, x) ∼ (a′, x′) ⟺^{ρ} τ(x) = τ(x′) and a = a′ · α(x, x′). In fact the

equivalence ρ is a congruence, since if (a, x) ∼ (a′, x′) and (b, y) ∼ (b′, y′), one has

a^{x}b = a′α(x, x′) ^{x}b′ ^{x}α(y, y′) = a′ ^{x}b′α(xy, x′y′) .

Denote C = A ◁ F_{G}/ρ. One gets an exact sequence of groups

1 ⟶ A ⟶^{σ} C ⟶^{ψ} G ⟶ 1 ,

where σ(a) = [(a, 1)], ψ([(a, x)]) = τ(x) and the following diagram

M_{G}  ⟶^{l_{0}} ; ⟶_{l_{1}} F_{G} ⟶^{τ} G
↓ ^{α}  ↓ ^{δ}  ||
A ⟶^{σ} C ⟶^{ψ} G

is commutative, δ(x) = [(1, x)], σα = δl_{1}δl_{0}^{−1}.

Take a section γ : G → C given by γ(g) = [(1, |g|), g ∈ G. It is easy to see that \overline{γ(G)} = Im δ. Therefore Ker(ψ |\overline{γ(G)}) = δ(Ker τ). The equality σα(1, x) = δ(x), x ∈ Ker τ, implies Ker(ψ |\overline{γ(G)}) ⊂ Z(A) and there follows that so constructed pair

E = (1 ⟶ A ⟶^{σ} C ⟶^{ψ} G ⟶ 1, γ)

is an extension of G by (A, μ).

Define η(([(α, 1)]) = [E]. We have to show the correctness of η. Let (α, 1) ∼ (α′, 1), that means

α′(x) = βl_{1}(x)^{−1}^{h}α(x)βl_{0}(x) , x ∈ M_{G} ,

for some (β, h) ∈ Der(F_{G}, (A, μ)) (see diagram 3.3) and let E′ = (1 ⟶ A ⟶^{σ′} C′ ⟶^{ψ′} G ⟶ 1, γ′) be the extension of G by (A, μ) corresponding to (α′, 1).

Define a map ν : A ◁ F_{G} ⟶ A ◁ G given by ν(a, x) = (^{h}aβ(x), x). So defined ν provides a homomorphism, since ν((a, x)(b, y)) = ν(a ^{x}b, xy) = (^{h}(a ^{x}b)β(xy), xy) and ν(a, x)ν(b, y) = (^{h}aβ(x), x)(^{h}bβ(y), y) = (^{h}aβ(x) ^{x}(^{h}bβ(y)), xy) = (^{h}a ^{hx}bβ(x) ^{x}β(y), xy) = (^{h}(a ^{x}b)β(xy), xy).

The homomorphism ν induces a homomorphism ν′ : A ◁ F_{G}/ρ → A ◁ F_{G}/ρ′ given by ν′([(a, x)] = [ν(a, x)]. In effect, let (a, x) ∼^{ρ} (a′, x′), that means a = a′α(x, x′) and τ(x) = τ(x′). We have to show that

haβ(x) ∼^{ρ′} ^{h}a′β(x′), where α′(x, x′) = β(x′)^{−1}^{h}α(x, x′)β(x).

One has

haβ(x) = ^{h}a′ ^{h}α(x, x′)β(x) = ^{h}a′β(x′)α′(x, x′) .

Thus ν′ is a correctly defined homomorphism and the diagram

A ◁ F_{G} ⟶ A ◁ F_{G}/ρ = C
↓ ^{ν}  ↓ ^{ν′}
A ◁ F_{G} ⟶ A ◁ F_{G}/ρ′ = C′

is commutative.

Now consider the following diagram

1 ⟶ A ⟶^{σ} C ⟶^{ψ} G ⟶ 1
↓ ^{h} ↓ ^{ν′} ||
1 ⟶ A ⟶^{σ′} C′ ⟶^{ψ′} G ⟶ 1

with sections γ : G → C, γ′ : G → C′ defined as above. Clearly σ′h = ν′σ and ν′γ(x)γ(x)^{−1} = ν′([(1, |x|))δ′(x) = [(β(|x|), |x|)][(1, |x|)]^{−1} = [(β|x|), 1)]. But μβ(|x|) = hxh^{−1}x^{−1}. There follows that [E] = [E′].

Conversely, define a map η′ : E^{1}(G, A) → H^{2}(G, A) as follows. Let [E] ∈ E^{1}(G, A) and E = (1 → A ⟶^{σ} C ⟶^{ψ} G → 1, γ). Then one gets a commutative diagram

M_{G}  ⟶^{l_{0}} ; ⟶_{l_{1}} F_{G} ⟶^{τ} G
↓ ^{α}  ↓ ^{δ}  ||
A ⟶^{σ} C ⟶^{ψ} G

with δ induced by γ and σα = δl_{1}δl_{0}^{−1}. Clearly α is a crossed homomorphism such that α(Δ) = 1 and Im α ⊂ Z(A). Define η′([E]) = [(α, 1)].

We have to show the correctness again. If δ′ is another homomorphism such that βδ′ = τ, then δ(y) δ′(y)^{−1} ∈ Z(A), y ∈ F_{G}. Thus δδ^{′−1} induces a crossed homomorphism β : F_{G} → Z(A) and it is obvious that (α′, 1) = (βl_{0}, 1)(α, 1)(βl_{1}, 1)^{−1} with (β, 1) ∈ Der(F_{G}, A).

Assume now that E is equivalent to E′ = (1 → A ⟶^{σ′} C′ ⟶^{ψ′} G → 1, γ′) by the pair (h : A → A, ϑ : C → C′) implying the equality μ(ϑγ(x)γ′(x)^{−1}) = hxh^{−1}x^{−1}, x ∈ G.

By using the equalities ψδ = ψ′δ′ = τ, σα = δl_{1}δl_{0}^{−1}, σ′α′ = δ′l_{1}δ′l_{0}^{−1}, σ′h = ϑσ, one gets ϑδ = σ′βδ′, ϑδl_{0} = σ′βl_{0}δ′l_{0}, ϑδl_{1} = σ′βl_{1}δ′l_{1}, where β : F_{G} ⟶ A is a crossed homomorphism induced by ϑδδ^{′−1}.

Clearly (β, h) is an element of Der(F_{G}, A). Further, σαδl_{0} = δl_{1}, so one has ϑσαϑδl_{0} = ϑδl_{1} and the following equalities

ϑσασ′βl_{0}δ′l_{0} = σ′βl_{1}δ′l_{1} ,
ϑσασ′βl_{0} = σ′βl_{1}δ′l_{1}δ′l_{0}^{−1} ,
σ′hασ′βl_{0} = σ′βl_{1}σ′α′ .

Finally α′ = βl_{1}^{−1}hαβl_{0}.

Therefore (α′, 1) is equivalent to (α, 1). It is easily checked that ηη′ and η′η are identity maps. □

4. Higher non-abelian cohomology

Let (A, μ) be a G-partially crossed P -module. Take the free cotriple resolution F_{∗}(G) of the group G:

⋯ ⟶ ; ⋮ ; ⟶F_{n+1} ⟶^{∂_{0}^{n+1}} ; ⋮ ; ⟶_{∂_{n+1}^{n+1}}F_{n} ⟶^{∂_{0}^{n}} ; ⋮ ; ⟶_{∂_{n}^{n}} ⋯ ⟶ ; ⋮ ; ⟶F_{2}⟶^{∂_{0}^{2}} ; ⟶ ; ⟶_{∂_{2}^{2}}F_{1} ⟶^{∂_{0}^{1}} ; ⟶_{∂_{1}^{1}}F_{0} ⟶ G (4.1)

with F_{n} = F^{n+1}(A), n ≥ 0, F_{0} = F (A) is the free group generated by A and F^{n+1}(A) = F (F^{n}(A)), ∂_{i}^{n} = F^{i}τF^{n−i}, s_{i}^{n} = F^{i}δF^{n−i}, where δ : F (A) ⟶ F^{2}(A) is induced by the canonical inclusion A → F (A). Clearly (A, μ) can be viewed as a F_{n}-partially crossed P -module induced by τ∂_{0}^{1}∂_{0}^{2} ⋯ ∂_{0}^{n}. Therefore the group Der(F_{n}, (A, μ)), n ≥ 0, is defined. 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) for n odd and of the form (α, r) for n even satisfying the condition

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

Since μα(x) = 1 for any x ∈ F_{n} and for n odd, in this case we have α(F_{n}) ⊂ Z(A), n ≥ 1. In the set Z̃^{1}(F_{n}, (A, μ)), n ≥ 1, a relation ∼ is introduced as follows:

(α′, 1) ∼ (α, 1) for n odd and (α′, r′) ∼ (α, r) for n even 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_{i}} , ε_{i} = (−1)^{i} , x ∈ F_{n} ,

and r′ = r for n even.

The homomorphism τ∂_{i_{n}}^{1}∂_{i_{n−1}}^{2} ⋯ ∂_{i_{2}}^{n−1}∂_{i_{1}}^{n} does not depend of the sequence (i_{1}, i_{2}, ⋯, i_{n−1}, i_{n}) implying

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

for 0 ≤ j, l ≤ n. There follows that the product ∏_{i=0}^{n}(β∂_{i}^{n}(x))^{ε_{i}}, ε_{i} = (−1)^{i}, does not depend on the order of the factors. Obviously the aforedefined relation is an equivalence.

Definition 4.1. The higher non-abelian cohomology of a group G with coefficients in a G-partially crossed P -module (A, μ) is given by

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

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

The map H^{n+1}(G, Ker μ) → H^{n+1}(G, (A, μ)) given by [f] ↦ [(f, 1)] is surjective and is bijective if μ : A → P is the trivial homomorphism (in this case A is abelian).

In order to express in terms of derived functors of the group of derivations with respect to the contravariant variable cohomotopy pointed sets of some cosimplicial groups will be introduced.

Let

G_{∗} : G_{0}\overrightarrow{⟶}G_{1}\begin{matrix}⟶ \\ ⟶ \\ ⟶ \end{matrix}G_{2}⟶ ; ⋮ ; ⟶⋯ ⟶ ; ⋮ ; ⟶G_{n}⟶ ; ⋮ ; ⟶G_{n+1}⟶ ; ⋮ ; ⟶⋯

be a cosimplicial group. Clearly ∂_{j}^{n}∂_{i}^{n−1} = ∂_{i}^{n}∂_{j−1}^{n−1}, i < j.

Assume that G_{∗} satisfies the following condition:

(a) Denote by L_{n+1} the subgroup of G_{n+1} generated by ⋃_{i=0}^{n+1}∂_{i}^{n}(G_{n}). Then for any element x ∈ G_{n}, n ≥ 0, the product ∂_{i}^{n}(x) ∂_{j}^{n}(x)^{−1}, 0 ≤ i, j ≤ n, commutes with every element of L_{n+1}.

In particular it follows that one has the equality ∂_{i}^{n}(x) ∂_{j}^{n}(x)^{−1} = ∂_{j}^{n}(x)^{−1} ∂_{i}^{n}(x), 0 ≤ i, j ≤ n, n ≥ 0.

Under this condition the cosimplicial group G_{∗} induces a group chain complex

1 ⟶ G_{0} ⟶^{d_{0}} G_{1} ⟶^{d_{1}} G_{2} ⟶^{d_{2}} ⋯ G_{n} ⟶^{d_{n}} G_{n+1} ⟶^{d_{n+1}} ⋯

with d_{n}(x) = ∏_{i=0}^{n+1}∂_{i}^{n}(x)^{ε_{i}}, ε_{i} = (−1)^{i}, n ≥ 0. It is easily checked that the maps d_{n}, n ≥ 0, are homomorphisms and d_{n}d_{n−1} = 0, n ≥ 1.

Definition 4.2. The right quotient sets Ker d_{n}/ Im d_{n−1} will be called cohomotopy sets π_{n}(G_{∗}), n ≥ 0, of the cosimplicial group G_{∗}.

It is obvious that for abelian cosimplicial groups we recover the well known homology groups.

Proposition 4.3. Let

1 ⟶ G′_{∗} ⟶ G_{∗} ⟶ G′′_{∗} ⟶ 1

be a short exact sequence of cosimplicial groups satisfying condition (a). Then there is a long exact sequence of pointed cohomotopy sets

1 ⟶ π_{0}(G′_{∗}) ⟶ π_{0}(G_{∗}) ⟶ π_{0}(G′′_{∗}) ⟶ π_{1}(G′_{∗}) ⟶ ⋯ ⟶ π_{n−1}(G′′_{∗}) ⟶ π_{n}(G′_{∗}) ⟶ π_{n}(G_{∗}) ⟶ π_{n}(G′′_{∗}) ⟶ π_{n+1}(G′_{∗}) ⟶ ⋯

Proof. Straightforward. □

This definition of cohomotopy pointed sets defined for cosimplicial groups satisfying the aforementioned condition allows us to define cotriple right derived functors of some group valued contravariant functors.

Let A be an arbitrary category and F = (F, τ, δ) be a cotriple in the category A. For any object A ∈ ob A take its cotriple resolution:

F_{∗}(A) ⟶^{τ} A .

similarly to the case of groups considered above.

Let T : A → Gr be a contravariant functor to the category of groups satisfying the following condition:

(b) the product T ∂_{i}^{n+1}(x) T ∂_{j}^{n+1}(x)^{−1}, 0 ≤ i, j ≤ n + 1, n ≥ 0, commutes with every element of the subgroup of T F_{n+1}(A) generated by ⋃_{i=0}^{n+2} Im T ∂_{i}^{n+1}, n ≥ 0.

Definition 4.4. The right derived functors of the contravariant functor T with respect to the cotriple F are the pointed sets

R_{F}^{n}T (A) = π_{n}(T F_{∗}(A)) , n ≥ 0 , A ∈ ob A .

If f : A → A′ is a morphism of the category A, then one gets a morphism T F_{∗}(f) : T F_{∗}(A′) → T F_{∗}(A) inducing maps of pointed sets R_{F}^{n}T (f) := π_{n}T F_{∗}(f) : R_{F}^{n}T (A′) → R_{F}^{n}T (A), n ≥ 0.

Remark 4.5. Similarly are defined the right derived functors R_{L}^{n}T with respect to a triple L in the category A for group valued covariant functors T satisfying the same condition.

As noted above the main application of these derived functors will be their close relationship with non-abelian cohomology of groups.

Let (A, μ) be a G-partially crossed P -module and consider the free cotriple resolution F_{∗}(G) of the group G (see (4.1)). In general the cosimplicial group Der F_{∗}(G) does not verify condition (a), in other words the contravariant functor Der(−, (A, μ)) from the category of groups acting on (A, μ) to the category of groups Gr does not verify condition (b). Below it will be shown that for a wide class of coefficients (A, μ) condition (b) holds for the functor Der(−, (A, μ)).

One gets a sequence of groups and maps

1 ⟶ Der(F_{0}, (A, μ)) ⟶^{d_{0}} Der(F_{1}, (A, μ)) ⟶^{d_{1}} ⋯ ⟶

⟶ Der(F_{n}, (A, μ) ⟶^{d_{n}} Der(F_{n+1}, (A, μ)) ⟶^{d_{n+1}} ⋯ (4.2)

with d_{n}((α, r))(x) = (γ, s), where γ = ∏_{i=0}^{n+1}(α∂_{i}^{n+1})^{ε_{i}}, ε_{i} = (−1)^{i}, (α, r) ∈ Der(F_{n}, (A, μ)), x ∈ F_{n+1}, s = 1 for n even and s = r for n odd.

Introduce in Ker d_{n}, n ≥ 0, an equivalence by (α′, r′) ∼ (α, r) ⟺ ∃ (β, h) ∈ Der(F_{n−1}, (A, μ)) such that α′(x) = ^{h}α(x) ∏_{i=0}^{n}β∂_{i}^{n}(x)^{ε_{i}}, ε_{i} = (−1)^{i}, r′ = r. Clearly the quotient sets coincide with H^{n+1}(G, (A, μ)), n ≥ 1. It is easily checked that one has Ker d_{0} ≈ Der(G, (A, μ)). Moreover, the composite d_{n}d_{n−1}, n ≥ 1, is the trivial map and the maps d_{n} are homomorphisms for n odd.

Proposition 4.6. Let (A, μ) be a G-partially crossed P -module such that P acts trivially on Ker μ. Then the maps d_{n} of the sequence (4.2) are homomorphisms for all n ≥ 0. Moreover, in this case Im d_{n} belongs to the center of Der(F_{n+1}, (A, μ)) for n even and Ker d_{n} belongs to the center of Der(F_{n}, (A, μ)) for n odd.

Proof. Easily to check using the equalities α∂_{i}^{n+1}(x) α∂_{j}^{n+1}(x)^{−1} = α∂_{j}^{n+1}(x)^{−1} α∂_{i}^{n+1}(x), 0 ≤ i, j ≤ n + 1, for (α, r) ∈ Der(F_{n}, (A, μ)) and the fact that in this case any element of the form (α, 1) ∈ Der(F_{n}, (A, μ)) belongs to the center. □

Theorem 4.7. Let (A, μ) be a G-partially crossed P -module with P acting trivially on Ker μ and let F be the free cotriple in the category of groups acting on (A, μ).

(i) One has isomorphisms

R_{F}^{0} Der(G, (A, μ)) ≈ Der(G, (A, μ)) ,
R_{F}^{n} Der(G, (A, μ)) ≈ H^{n+1}(G, (A, μ)) , n ≥ 1 ,

and H^{n+1}(G, (A, μ)) is an abelian group for n odd. (ii) Any short exact sequence of G-partially crossed P -modules

1 ⟶ (A′, μ′) ⟶^{φ} (A, μ) ⟶^{ψ} (A′′, μ′′) ⟶ 1

with P acting trivially on Ker μ′, Ker μ and Kerμ′′, induces a long exact cohomology sequence

1 ⟶ Der(G, (A′, μ′)) ⟶ Der(G, (A, μ)) ⟶ Der(G, (A′′, μ′′)) ⟶

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

⟶ H^{3}(G, (A′, μ′)) ⟶ ⋯ ⟶ H^{n−1}(G, (A′′, μ′′)) ⟶ ⟶ H^{n}(G, (A′, μ′)) ⟶ H^{n}(G, (A, μ)) ⟶ H^{n}(G, (A′′, μ′′)) ⟶

⟶ H^{n+1}(G, (A′, μ′)) ⟶ · · · .

Proof. Since the functor Der(−, (A, μ)) satisfies condition (b) with respect to the free cotriple F, the right derived functors of Der(−, (A, μ)) are well defined, the sequence (4.2) became a complex of non-abelian groups and the statement of (i) follows from Proposition 4.6.

The short exact coefficient sequence of (ii) induces a short exact sequence of cosimplicial groups

1 ⟶ Der(F_{∗}(G), (A′, μ′)) ⟶ Der(F_{∗}(G), (A, μ)) ⟶
Der(F_{∗}(G), (A′′, μ′′)) → 1

satisfying condition (a). It remains to apply Proposition 4.3 to get the required long exact cohomology sequence. □

The definition of non-abelian cohomology with coefficients in Gpartially crossed P -modules allows as to introduce the definition of higher non-abelian cohomology of a group G with coefficients in any G-group. It can be done as follows.

Let A be an arbitrary G-group, that means the group G acts on the left on the group A. Take the quotient group P = A/Z(A). Define an action of P on A and an action of G on P as follows:

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

Let μ_{A} : A ⟶ P be the canonical homomorphism. It is easily checked that the pair (A, μ_{A}) is a G-crossed P -module under the aforedefined actions.

Definition 4.8. The n-th cohomology H^{n}(G, A), n ≥ 0, of the group G with coefficients in a G-group A is given by

H^{n}(G, A) = H^{n}(G, (A, μ_{A})) , n ≥ 0 .

For n = 1 this cohomology differs from the first pointed set cohomology defined in [10].

Let

1 ⟶ A ⟶^{φ} B ⟶^{ψ} C ⟶ 1

be a central extension of G-groups. Then ψ induces an isomorphism of G-groups ϑ : B/Z(B) ⟶^{≈} C/ψ(Z(B)) and one gets a short exact sequence of G-crossed P -modules with P = B/Z(B)

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

where \overline{μ}_{C} is the composite of the canonical map C → C/ψ(Z(B)) and the isomorphism ϑ^{−1}.

Corollary 4.9. Any central extension of G-groups

1 ⟶ A ⟶^{φ} B ⟶^{ψ} C ⟶ 1

induces a long 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}}

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}}

H^{2}(G, (C, \overline{μ}_{C})) ⟶^{δ^{2}} H^{3}(G, (G, A) ⟶^{φ^{3}} ⋯ ⟶ H^{n−1}(G, (C, \overline{μ}_{C})) ⟶^{δ^{n−1}}

H^{n}(G, A) ⟶^{φ^{n}} H^{n}(G, B) ⟶^{ψ^{n}} H^{n}(G, (C, \overline{μ}_{C})) ⟶^{δ^{n}} H^{n}(G, A) ⟶ · · · .

Proof. Clearly in the induced short exact sequence of G-crossed P - modules the group P acts trivially on Ker μ_{B} and Ker \overline{μ}_{C}. So we can apply Theorem 4.7 giving together with Theorem 3.6 the required long exact cohomology sequence. □

Acknowledgements

The work was partially supported by INTAS Georgia grant No 213 and Nato Science Program No 975316.

References

[1] R.Brown, D.Johnson and E.F.Robertson, Some computations of the nonabelian tensor products of groups, J.Algebra 111 (1987), 177-202.

[2] M.Bullejos and A.Cegarra, A 3-dimentional cohomology of groups with applications to homotopy classification of continuous maps, J.Math. 43 (1991), 265-296.

[3] A.Cegarra and A.Garzon, A long exact sequence in non-abelian cohomology, Lecture Notes in Math., Springer-Verlag, 1488 (1991), 79-94.

[4] R.Dedecker, Cohomologie non abelienne, Seminaire de l’Institut Mathematique Lille, 1963-1964.

[5] D.Guin, Cohomologie et homologie non abeliennes des groupes, C. R. Acad. Sci. Paris 301 (1985), Serie 1, No 7.

[6] D.Guin, Cohomologie et homologie non abeliennes des groupes, J. Pure Appl. Algebra 50 (1988), 109-137.

[7] H.Inassaridze, Non-abelian homological algebra and its applications, Kluwer Academic Publishers, 1997, 265 pages.

[8] H.Inassaridze, Non-abelian cohomology of groups, Georgian Math. J. 4, No 4 (1997), 313-332.

[9] J.-L.Loday, Cyclic homology, Springer-Verlag, 1997, 513 pages.

[10] J.-P.Serre, Cohomologie galoisienne, Lecture Notes in Math., Springer-Verlag, 5 (1964).

A. Razmadze Matematical Institute Georgian Academy of Sciences

 E-mail address: hvedri@rmi.acnet.ge
 URL: http://www.rmi.acnet.ge/~hvedri/