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NON-ABELIAN COHOMOLOGY WITH COEFFICIENTS IN CROSSED BIMODULES

H. INASSARIDZE

Abstract. When the coefficients are crossed bimodules, Guin’s non-abelian 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]) fol-lowing Guin’s approach to non-abelian cohomology [2], [3]. The pointed sets of cohomologyHn_{(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]. H1_{(G,(A, µ)) is equipped with a} par-tial product and coincides with Guin’s cohomology group [3] when crossed G-modules are viewed as crossedG-G-bimodules. The pointed set of coho-mologyH2_{(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 crossedG-R-bimodules gives rise to a nine-term exact co-homology sequence and we recover the exact coco-homology sequence obtained in [1] when the coefficients are crossed modules. By analogy with the case n= 2 the definition of a pointed set of cohomologyHn_{(G,(A, µ)) of a group}

Gwith coefficients in a crossedG-R-bimodule (A, µ) is given for alln≥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 fromGto (A, µ).

LetG, RandAbe groups. (A, µ) is a crossedG-R-bimodule if: 1) (A, µ) is a crossedR-module,

2)Gacts onR andA,

3) the homomorphismµ:A−→Ris a homomorphism ofG-groups, 4)(g

r)_a=grg−1

aforg∈G,r∈R,a∈A.

1991Mathematics Subject Classification. 18G50, 18G55.

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

509

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 thatµα(x) =rx_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}

anya∈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 crossedG-R-bimodule. It will be said that a crossed homomorphismα:G−→Asatisfies condition(j) (resp. condition

(j′_{))if forc∈H}0_{(G, R) (resp. if forc∈H}0_{(G, R)such that there is b∈A}

with µ(b) =c) there existsa∈A such that c_{α(x) =a}−1_α(x)x_aforx∈G andµ(a) = 1. It will be said that an element(α, r)ofDer(G,(A, µ))satisfies condition (j) (resp. condition (j′_{))if αsatisfies this condition.}

It is obvious that any element of the form (α,1) satisfies condition (j′_{). If}

(A, µ) is a crossedG-R-bimodule induced by a surjective homomorphismf : G−→R, then every element (α, r)∈Der(G,(A, µ)) such thatα(kerf) = 1 satisfies condition (j). In effect, for c∈Z(R) =H0_{(G, R) we have z}

xdx=

xd, x∈ G, with f(d) = c and zx ∈ kerf. Thus, α(zx)zxα(dx) = α(xd),

whenceα(d)c_{α(x) =α(x)}x_{α(d) andµα(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 whenH}0_{(G, R)⊂Z(R). In effect, we} have α′_{(x) =b}−1_α(x)x_b,r′ _=µ(b)−1_{r tand}c_{α(x) =a}−1_α(x)x_{a,µ(a) = 1,}

wherec, t∈H0_{(G, R)⊂Z(R). Thus}

c_α′_{(x) =} c_b−1c_α(x)cx_b= c_b−1_a−1_α(x)x_acx_b=

= cb−1a−1α(x)x(acb) = cb−1a−1b α′(x)xb−1x(acb) =

= cb−1ba−1α′(x)x(ab−1cb)

with µ(a b−1c_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 iff : (A, µ)−→(B, λ) is a homomorphism of crossed G-R-bimodules and (α, r)∈Der(G,(A, µ)) satisfies condition (j), then (f α, 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)⇐⇒

(

∃a∈A:β(x) =a−1_α(x)x_a,

s=µ(a)−1_{r mod H}0_{(G, R)} .

Later we shall need the following assertion:

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

holds for anyx∈G,a∈A,r∈H0_{(G, R).} In effect,we have

rx_a= xx−1_rx

a = x(x− 1

r)_{a =} xr_a.

Proposition 2. The relation∼is an equivalence. AssumeH0_{(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_∈H0_{(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−1α′(x)xb, r′′=µ(b)−1r′z′,

wherez, z′_∈H0_{(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)) andc∈H0_{(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−1α(x)xa, x∈G,

1 =µ(a)−1z, z∈H0(G, R),

and

β′(x) =b−1β(x)xb, x∈G,

s′ =µ(b)−1sz′, z′ ∈H0(G, R).

Thenβ′_(x)α′_{(x) =b}−1_β(x)x_ba−1_α(x)x_a=b−1_β(x)a−1_α(x)x_ax_{b= b}−1_a−1

µ(a)_β(x)α(x)x_ax_b=b−1_a−1_d−1_β(x)α(x)x_{(dab) ands}′_=µ(b)−1_µ(a)−1_zsz′₌

µ(b−1_a−1_d−1_)sz′′_z′_{whereβ(x) =d}−1_β(x)x_{d,µ(d) = 1 andz}′′_∈H0_{(G, R).} Therefore (α,1)(β, s)∼(α′_,1)(β′_{, s}′_).

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)

forc∈H0_{(G, R).}

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

rc_{β(x)α(x) =} r_(a−1_β(x)x_{a)α(x) =} r_a−1r_β(x)rx_{a α(x) =}

= ra−1rβ(x)α(x)rxa.

Since µ(r_a)−1 _{= (r µ(a)r}−1₎−1 _{= 1, one has rcs = µ(}r_a)−1_rsc′ _with

c′ _∈H0_{(G, R). Therefore, (α, r)(β, s)∼(α, rc)(β, s).} Further,we have

α′(x) =b−1α(x)xb, r′ =µ(b)−1rz,

andβ′_{(x) =d}−1_β(x)x_d,s′_=µ(d)−1_{st withz, 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−1β(x)xd)b−1α(x)xb=

= µ(b)−1r·zd−1µ(b)− 1_rz

β(x)µ(b)−1rzxd b−1α(x)xb=

= b−1r·z_d−1rz

β(x)r·x₍z_d)α(x)x_b=b−1r·z_d−1r·z

β(x)α(x)xrz_dx_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_{rzstwitht∈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

H1_{(G,(A, µ)) the quotient set Der(G,(A, µ))/ ∼ equipped with the}

afore-mentioned partial product and it will be called the first set of cohomology of

If (A, µ) is a crossed G-module viewed as a crossed G-G-bimodule then H0_{(G, G) = Z(G) and for (α, g) ∈ Der}

G(G, A) = Der(G,(A, µ)) and c ∈

Z(G) the equalityα(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 DerG(G, A) satisfies condition (j). It follows that if (A, µ)

is a crossedG-module we recover the groupH1_{(G, A) defined by Guin [3].} It is clear that the map H1_{(G, A) −→ H}1_{(G,(A,1)) given by [α] 7−→} [(α,1)] is an isomorphism where (A,1) is a crossedG-R-bimodule.

Proposition 4. Let (A, µ) be a crossed G-R-bimodule and assume

H0_{(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 ∈ H0_{(G, R). Then} c_{α(x) = b}−1_α(x)x_{b and µ(b) = 1. Since}

H0_{(G, R) is a normal subgroup ofR, there isc}′_∈H0_{(G, R) such that cr=} rc′_{. Forc}′ _{we have} c′

β(x) = d−1_β(x)x_{dand µ(d) = 1. Put (α, r)(β, s) =}

(γ, rs). Then

c_{γ(x) =} cr_β(x)c_{α(x) =} r_d−1r_β(x)rx_{d b}−1_α(x)x_b=

= r_d−1_b−1r_β(x)α(x)x_(br_d)

withµ(br_{d) =µ(b)r µ(d)r}−1_{= 1. Thus, (γ, rs) satisfies condition (j). Put} (α, r)−1 _{= (α, r}−1_{) whereα(x) =} r−1

α(x)−1_{, x∈G. Ifc ∈H}0_{(G, R) one} has

c_{α(x) =} cr−1

α(x)−1₌ r−1

c′

α(x)−1₌ r−1

(x_a−1_α(x)−1_a), wherecr−1_=r−1_c′_,c′ _∈H0_{(G, R) and}c′

α(x) =a−1_α(x)−1x_{a,µ(a) = 1.}

Hence

c_{α(x) =} r−1_x

a−1r−1α(x)−1r−1a= r−1α(x)−1xr−1a−1r−1a=

= r−1ar−1α(x)−1x(r−1a−1)

withµ(r−1

a−1_{) =r}−1_µ(a−1_{)r= 1.}

Therefore, (α, r−1_{) satisfies condition (j).}

Corollary 5. The subset ofH1_{(G,(A, µ))of all equivalence classes}

con-taining an element with condition (j)forms a group ifH0_{(G, R)is a normal}

subgroup ofR.

Proposition 6. Let(A, µ)be a crossedG-R-bimodule such thatH0_(G,R)

is a normal subgroup ofR. If there is a mapη :H0_{(G, R)−→Z(G)such}

that Imη acts trivially on R andη(r)_a= r_{a, a∈A, then H}1_{(G,(A, µ))is}

Proof. We have to show that every element (α, r)∈Der(G,(A, µ)) satisfies condition (j). Ifc∈H0_{(G, R) take η(c) =d∈Z(G). Thenα(dx) =α(xd)} andα(d)d_{α(x) =α(x)}x_{α(d). Thus}c_{α(x) =α(d)}−1_α(x)x_{α(d) andµα(d) =}

rd_r−1_=rr−1_{= 1.}

Corollary 7. Let (A, µ) be either a crossed G-R-bimodule such that

H0_{(G, R) is a normal subgroup of R trivially acting on A or induced by}

a surjective homomorphism f :G−→R such thatf(Z(G)) =Z(R). Then

H1_{(G,(A, µ))is a group.}

Proof. In the first case takeη as the trivial map. In the second case take a mapη:Z(R)−→Z(G) such thatf η= 1Z(R).

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

f1:H1(G,(A, µ))−→H1(G,(B, λ))

which is a homomorphism in the following sense:

ifxy is defined forx, y ∈ H1_{(G,(A, µ)) then f}1_(x)f1_{(y) is defined and} f1_{(xy) =f}1_(x)f1_(y).

The above defined action ofGon Der(G,(A, µ)) induces an action ofG onH1_{(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−1α(x)xa, x∈G, this implies

α′₍g−1

x) =a−1_α(g−1 x)g−1

xg_{a, x∈G.}

Thus

g_α′₍g−1

x) = g_a−1g_α(g−1

x)xg_{a, x∈G.}

We also haver′ _=µ(a)−1_{rz,z ∈H}0_{(G, R), whence}g_r′ ₌ g_µ(a−1₎g_rg_z=

µ(g_a)−1g_rg_{z. Therefore}g_{(α, r)∼} g_(α′_{, r}′_).

In what follows if f is a map from a group Gto a group G′ _{then f}−1 _: G−→G′ _{denotes a map given byf}−1_{(x) =f(x)}−1_.

Let (A, µ) be a crossed G-R-bimodule. The definition of H2_{(G,(A, µ))} is similar to the case of (A, µ) being a crossedG-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 τ l0 and a crossed F-R-bimodule induced byτ. LetZf1_{(M,(A, µ)) be the subset of Der(M,(A, µ))} consisting of elements of the form (α,1).

Define, onZe1_{(M,(A, µ)), relation}

such that

(α′_{,1) = (βl0, h)(α,1)(βl1, 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 H2_{(G,(A, µ)) the quotient set Ze}1_{(M,(A, µ))} /∼. It will be called the second set of cohomology of Gwith coefficients in the crossedG-R-bimodule(A, µ).

It can be proved (as for a crossedG-module (A, µ) (see Proposition 8 [1])) that Zf1_{(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′:H2(G,kerµ)−→H2(G,(A, µ))

defined by the composite

[E] ϑ− 1

7−→[α]7−→[(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 ac-tion ofGonH2_{(G,(A, µ))such thatZ(G)acts trivially. IfRacts onGand}

satisfies the compatibility condition(3) of[1]then there is also an action of

R onH2_{(G,(A, µ)).}

Proof. The action of G on H2_{(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 MG

given by

r_(|g1|ǫ

· · · |gn|ǫ,|g1′|

ǫ · · · |g′m|

ǫ

) = (|rg1|ǫ· · · |rgn|ǫ,|rg1′|

ǫ

· · · |rgm′ | ǫ

)

and one gets an action ofRonDer(MG,(A, µ)) defined by

r_{(α, s) = (}

e

α,rs),

where α(m) =e r_α(r−1

m), r ∈ R, m ∈ MG. 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 crossedG-R-bimodule. Using (1) it can easily be shown that there is an action ofH0_{(G, R) onH}2_{(G,kerµ) given by}

r_{[α] = [}r_{α], r∈H}0_{(G, R),}

whereα:MG−→kerµ is a crossed homomorphism under the action ofG

If this action ofH0_{(G, R) is trivial and Der(F}

G,(A, µ)) = IDer(FG,(A, µ))

then the map

ϑ′_:H2_{(G,kerµ)−→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 ofH0_{(G, R)} onH2_{(G, A) is trivial then there is an action ofH}1_{(G,(C, λ)) onH}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

MG l0

−→ −→ l1

FG −→τG G

↓α

A −→ϕ B −→ψ C

(3)

There is a crossed homomorphismβ:FG−→Bsuch thatψβ=ατG. Take

the product

(βl0, r)(ϕγ,1)(βl0, r)−1_{= (}

e

γ,1)

in the group Der(MG ,(B, µ)). Theneγ(x) =β(x)−1rϕγ(x)β(x) = rϕγ(x),

x∈ M. Therefore r_{γ : M}

G −→ A is a crossed homomorphism such that r_{γ(∆) = 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,

wheret∈H0_{(G, R). It follows that}

ϕ(r′γ(x)) = r′ϕγ(x) = λ(c)−1rtϕγ(x) = µ(b)−1rtϕγ(x) =

=b−1r·tϕγ(x)b= r·tϕγ(x) =ϕ(r·tγ(x)), x∈MG,

whereψ(b) =c. Hence we have

[r′γ] = [rtγ] = [rγ] proving the correctness of the action.

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

For [(α, r)]∈H1_{(G,(C, λ)) take a crossed homomorphism β:F}

G −→B

such thatψβ=α τG. Thus there is a crossed homomorphismγ:MG −→A

such that

ϕγ= (βl1)−1_βl0. It is clear thatγ(∆) = 1. Define

δ1([(α, r)]) = [γ].

We must prove the correctness ofδ1_{. If β}′ _{: F}

G −→ B withψβ′ =ατ,

thenψβ′_{=ψβ. Thus there is a crossed homomorphismσ:F}

G −→Asuch

thatβ′_{=βϕσ. Then we have}

ϕγ′_{= (β}′_l1)−1_β′_{l0= (βϕσ)l1}−1_{(βϕσ)l0=ϕσl}−1

1 βl−11βl0ϕσl0=

=βl−1

1 βl0ϕσl− 1

1 ϕσl0=ϕ(γσl− 1 1 σl0).

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).}

Takeβ′ _:F

G−→B such that

β′(x) =b−1β(x)xb

with ψ(b) = c where ψβ = ατ. Then (β′_l−1

1 β′l0)(y) = β′(x2)−1β′(x1) where y = (x1, x2) ∈ MG. Hence ϕγ′(y) = (β′l−11β′l0)(y) = (b−1β(x2)

x2_b)−1_b−1_β(x1)x1_{b =} x2_b−1_β(x2)−1_β(x1)x1_{b = β(x2)}−1_{β(x1) = ϕγ(y).}

Whenceγ′_=γ.

For any exact sequence (2) of crossed G-R-bimodules there is also an action of Der(F0,(C, λ)) onH3_{(G, A) defined as follows:}

(α,r)_{[f] = [}r_f],

wheref :F2−→A is a crossed homomorphism with 3

Q

i=0 (f l2

iτ3)ǫ= 1 where

ǫ= (−1)i_{(see diagram (7) of [1]) and (α, r)∈Der(F0,(C, λ)). The}

correct-ness of this action is proved similarly to the case of a short exact sequence of crossedG-modules (see [1]).

If either the aforementioned action of Der(F0,(C, λ)) onH3_{(G, A) is} triv-ial or Der(F0,(C, λ)) = IDer(F0,(C, λ)) and H0_{(G, R) acts trivially on} H2_{(G,kerλ), then a connecting map}

δ2:H2(G,(C, λ))−→H3(G, A)

is defined by

whereϕγ=βτ2 withβ= Q2

i=0 (βl1

i)ǫ,ǫ= (−1)i, and ψβ=ατ1 (see diagram

(7) of [1]). The correctness of δ2 _{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 mapδ2:H2(G, C)−→H3(G, A).

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

1−→H0_{(G, A)−→}ϕ0 _H0_{(G, B)−→}ψ0 _H0_{(G, C)} δ0 −→

δ0

−→H1_{(G, A)−→}ϕ1 _H1_{(G,(B, µ))−→}ψ1 _H1_{(G,(C, λ))} δ1

−→H2_{(G, A)−→}ϕ2

ϕ2

−→H2(G,(B, µ)) ψ 2

−→H2(G,(C, λ)),

whereϕ0_,ψ0_,δ0_,ϕ1_{are homomorphisms. IfH}0_{(G, R)is a normal subgroup}

ofR, thenψ1_andδ1_{are also homomorphisms. If in additionH}0_{(G, R)acts}

trivially onH2_{(G, A), thenδ}1_{is a crossed homomorphism under the action}

of H1_{(G,(C, λ)) onH}2_{(G, A)induced by the action of R on A. Moreover,}

if either the action ofDer(F0,(C, λ))onH3_{(G, A)is trivial(in particular if} R acts trivially on A) or Der(F0,(C, λ)) = IDer(F0,(C, λ)) andH0_{(G, R)}

acts trivially onH2_{(G,kerλ)then the sequence}

H2(G,(B, µ)) ψ 2

−→H2(G,(C, λ))−→δ2 H3(G, A)

is exact.

Proof. The exactness of the sequence

1−→H0(G, A) ϕ 0

−→H0(G, B) ψ 0

−→H0(G, C)−→δ0 H1(G, A)

is known [4].

If c ∈ H0_{(G, C) then δ}0_{(c) = [α] with α(x) = ϕ}−1_(b−1x_{b), x∈G and}

ψ(b) =c. It follows that (α0,1)∼(ϕα,1) whereα0 is the trivial map, since

ϕα(x) =b−1_α0x_{b, x∈G,}

and µ(b)∈ H0_{(G, R) because µ(b) = λψ(b) = λ(c) and} x_{λ(c) = λ(}x_{c) =}

λ(c),x∈G. Therefore Imδ0_⊂kerϕ1_.

Let [α] ∈ H1_{(G, A) such that (α0,1) ∼(ϕα,1). Then ϕα(x) = b}−1x_b,

x ∈ G and µ(b) ∈ H0_{(G, R). We have ψ(b}−1x_{b) = ψϕα(x) = 1. Thus}

ψ(b) =ψ(x_{b) =} x_{ψ(b), whenceψ(b)∈H}0_{(G, C). It is clear thatδ}0_{(ψ(b)) =} [α]. Therefore kerϕ1_⊂Imδ0_.

b α(x)x_b−1_{, x∈G. Since ψα(x) = 1,e x∈G, one hasϕ}−1_{αe:G−→A and} (α, r)∼(α,e 1). Thereforeϕ1_([ϕ−1_{α]) = [(α, r)].e}

Let [(α, r)] ∈ H1_{(G,(B, µ)). Then ψ}1_{([(α, r)]) = [(ψα, r)]. Consider} diagram (5) of [1] and takeα τG :FG −→B. Then ϕγ= (ατGl1)−1ατGl0

andδ1_ψ1_{([(α, r)]) = [γ]. Butγ=α0is the trivial map, sinceατ}

Gl0=ατGl1.

Therefore Imψ1_⊂kerδ1_.

Let [(α, r)]∈H1_{(G,(C, λ)) such thatδ}1_{([(α, r)]) = 1. Ifβ:F}

G −→B is

a crossed homomorphism such thatψβ=ατGthenδ1([(α, r)]) = [γ], where

ϕγ= (βl1)−1_{βl0.Thus there is a crossed homomorphismη:F}

G−→Asuch

thatγ= (ηl1)−1_{ηl0. Hence we have}

(βl1)−1_{βl0= (ϕηl1)}−1_{ϕηl0, (ϕη}−1_{β)l0= (ϕη}−1_β)l1.

Thus there is a crossed homomorphism α:G−→B such that (ϕη)−1_{β =} ατG. We have µβ(x) = λψβ(x) = λατG(x) = rτG(x)r−1, whence (β, r)∈

Der(FG,(B, µ)) and (α, r)∈Der(G,(B, µ)). Evidently,ψ1([(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 crossedG-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 casen= 1 we propose the following definition of the pointed set of cohomology Hn+1_{(G,(A, µ)) of a group Gwith coefficients} in a crossedG-R-bimodule (A, µ) (in particular, in crossedG-modules) for alln≥1.

Let (A, µ) be a crossedG-R-bimodule. Consider diagram (7) of [1] and the group Der(Fn,(A, µ)),n ≥1, where (A, µ) is viewed as a crossed Fn

-R-bimodule induced by τ0∂1

0∂20· · ·∂n− 1

0 ∂0n with ∂0i = li− 1

0 τi, i = 1, . . . , n.

Denote byZf1_(F

n,(A, µ)) the subset of Der(Fn,(A, µ)) consisting of all

ele-ments of the form (α,1) satisfying the condition

n_Y+1

j=0

(α∂_jn+1)ǫ_{= 1, ǫ= (−1)}i_.

Note that sinceµα(x) = 1,x∈Fn, we haveα(Fn)⊂Z(A). InZf1(Fn,(A, µ))

we introduce a relation∼as follows: (α′_{,1)∼(α,1) if there is an element}

(β, h)∈Der(Fn−1,(A, µ)) such that

α′(x) = hα(x)

n

Y

i=0

whereǫ= (−1)i_{. Since the homomorphismτ0∂}1

in∂

2

in−1· · ·∂

n−1

i2 ∂

n

i1 does not depend on the sequence (i1, i2, . . . , in−1, in), we have

β∂n

j(x)(β∂nl(x))−1= (β∂ln(x))−1β∂nj(x)∈kerµ, x∈Fn,

forj even andl odd. It follows that the product Qn

i=0 (β∂n

i(x))ǫ in (4) does

not depend on the order of the factors. Note that ifnis even thenβ(Fn)⊂

kerµ⊂Z(A).

Similarly to the case n = 1 it can be shown that the relation ∼ is an equivalence, the quotient set Zf1_(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 :Hn+1(G,kerµ)−→Zf1(Fn,(A, µ))/∼, n≥1,

given by [α]7−→[(α,1)] which is bijective if (A, µ) is a crossedG-G-bimodule and eitherµis the trivial map ornis even.

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

Hn+1_{(G,(A, µ)) =Zf}1_(F

n,(A, µ))/∼, n≥1.

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

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

a groupGwith coefficients in aG-groupA.

Consider the quotient groupA=A/Z(A) and define an action of Aon Aand an action ofGonA as follows:

[a′_] a= a′

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

Let µA : A −→ A be the canonical homomorphism. Then (A, µA) is a

crossedG-A-bimodule and we define

Hn(G, A) =Hn(G,(A, µA)), n≤2.

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

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-B-bimodules

1−→(A,1)−→ϕ (B, µB) ψ

−→(C, µC)−→1,

where µC is the composite of the canonical map τ : C −→ C/ψ(Z(B))

and the isomorphism ϑ−1_{. SinceB acts trivially on A, from Theorem 10} immediately follows the exact cohomology sequence

1−→H0(G, A) ϕ 0

−→H0(G, B) ψ 0

−→H0(G, C)−→δ0 H1(G, A) ϕ 1

−→

ϕ1

−→H1(G, B) ψ 1

−→H1_{(G,(C, µ}

C)) δ1

−→H2_{(G, A)−→}ϕ2 _H2_{(G, B)−→}ψ2

ψ2

−→H2_{(G,(C, µ}

C)) δ2

−→H3_{(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 H3(G,(A, µ)) of Gwith coefficients in a crossed G-R-bimodule (A, µ). To this end consider the commutative diagram

M1 G ϕ0 −→ −→ ϕ1 QG ηG

−→ MG

q1↓↓q0 l1↓↓l0

F2_(G) F(τG)

−→ F(G)

↓τF(G) ↓τG

F(G) τG

−→ G

,

where F(G) =FG, F2(G) =F(F(G)),τG and τF(G) are canonical surjec-tions,ηGis induced byF(τG), and (MG, l0, l1), (QG, q0, q1), (MG1, ϕ0, ϕ1) are

the simplicial kernels ofτG,τF(G)andηG, respectively. It is clear that (A, µ)

is a crossedQG-G-bimodule induced byτGl0ηG. LetDer(Qg G,(A, µ)) be the

subgroup of Der(QG,(A, µ)) consisting of elements (β, g) such thatβ(∆Q) =

1, where ∆Q={(x, x), x∈F2(G)}. Consider the setZf1(MG1,(A, µ)) of all

crossed homomorphismsα:M1

G−→Awithα(∆) = 1 where ∆ ={(y, y), y∈

QG} and M1

G acts on A via τGl0ηGϕ0. Introduce, in Zf1(MG1,(A, µ)), a

relation of equivalence as follows:

α′_{∼α if ∃(β, g)∈Der(Qg}

G,(A, µ))

such that α′_{(x) =βϕ1(x)}−1_{α(x)βϕ0(x), x∈M}1

G. DefineH3(G,(A, µ)) =

f

Z1_(M1

G,(A, µ))/ ∼. Then H3(G,(A, µ)) is a covariant functor from the

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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2. D. Guin, Cohomologie et homologie non abeliennes des groupes. C. R. Acad. Sci. Paris301(1985), Serie l,No. 7, 337–340.

3. D. Guin, Cohomologie et homologie non abeliennes des groupes. J. Pure Appl. Algebra50(1988) 109–137.

4. J.-P. Serre, Cohomologie galoisienne. Lecture Notes in Math. 5,

Springer, Berlin,1964.

(Received 02.06.1995)

Author’s address: