SYMMETRIC COHOMOLOGY OF GROUPS

arXiv:1706.01367v2 [] 14 Jun 2017

MARIAM PIRASHVILI

Abstract. We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first
two theories were introduced respectively by Staic and Zarelua. We show in particular, that there is a map from exterior
cohomology to symmetric cohomology which is a split monomorphism in general and an isomorphism in many cases, but not
always. We introduce two spectral sequences which help to explain the realtionship between these cohomology groups. As a
sample application we obtain that symmetric and classical cohomologies are isomorphic for torsion free groups.

AMS classification: 20J06 18G40.
1. Introduction
Let G be a group and M be a G-module. In order to better understand 3-algebras arising in lattice field theory [3],
Staic defined a variant of group cohomology, which he denoted by HS ∗ (G, M) and called symmetric cohomology of
groups [6]. Some aspects of this theory were later extended by Singh [5] and Todea [9]. There is an obvious natural
transformation from the symmetric cohomology to the classical Eilenberg-MacLane cohomology
αn : HS n (G, M) → H n (G, M), n ≥ 0.
According to [6],[7], αn is an isomorphism if n = 0, 1 and is a monomorphism for n = 2. By Corollary 2.3 in [7] we
know that α2 is an isomorphism if G has no elements of order two.
Ten years prior to this, Zarelua had also defined a version of group cohomology, denoted by Hλ∗ (G, M) and called
exterior cohomology of groups [10]. It also comes together with a natural transformation
βn : Hλn (G, M) → H n (G, M),
with similar properties. The exterior cohomology has the following striking property: If G is a finite group of order d,
then Hλi (G, M) = 0 for all i ≥ d.
The aim of this work is to obtain more information about homomorphisms α∗ and β∗ . We construct a natural
transformation γn : Hλn (G, M) → HS n (G, M) such that the following diagram commutes:
n

γ
/ HS n (G, M)
Hλn (G, M)
▼▼▼
♣
▼▼▼
♣♣♣
♣
♣
▼
♣♣ n
βn ▼▼▼
x♣♣♣ α
&
H n (G, M).

Our results in Section 3 show that the homomorphism γn : Hλn (G, M) → HS n (G, M) is a split monomorphism in
general, and an isomorphism in certain cases, namely if 0 ≤ n ≤ 4, or M has no elements of order two. In general, γ5
is not an isomorphism.
Our next results are related to the homomorphism βn : Hλn (G, M) → H n (G, A). We construct a spectral sequence for
which βn are edge homomorphisms, n ≥ 0. As any first quadrant spectral sequence, it gives a 5-term exact sequence
(see for example [8, Exercise 5.1.3]) which has the following form:
Y
β3
β2
0 → Hλ2 (G, M) −→ H 2 (G, M) →
H 2 (C2 , M) → Hλ3 (G, M) −→ H 3 (G, M).
C2 ⊂G

The research was supported by the EPSRC grant EP/N014189/1 Joining the dots: from data to insight.
1

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M. PIRASHVILI

Here the product is taken over all subgroups of order two. The exactness at H 2 (G, M) is an answer to Problem 25 by
Singh in [2]. At the very end of Section 4 in [6], Staic wondered about the injetivity of the map α3 under the assumption
that G has no elements of order 2. A trivial consequence of our spectral sequence says that, if G has no elements of
order two, then one has an exact sequence:
Y
β3
β4
H 3 (C3 , M) → Hλ4 (G, M) −→
0 → Hλ3 (G, M) −→ H 3 (G, M) →
C3 ⊂G

β

4

−→ H 4 (G, M) →

Y

β5

H 4 (C3 , M) → Hλ5 (G, M) −→ H 5 (G, M).

C3 ⊂G

In particular, if G has no elements of order two and three, then Hλi (G, M) = H i (G, M), for i = 0, 1, 2, 3, 4.
Among other consequences of our spectral sequence, we mention the following: if G is a torsion free group, then
βn : Hλn (G, M) → H n (G, M) is an isomorphism for all n ≥ 0.
The paper is organised as follows: In Section 2 we recall the definitions of the symmetric and exterior cohomologies.
In the next section we construct the transformation γ∗ and prove our first result, which shows that γn is quite often an
isomorphism, but not always. In the final section we construct a spectral sequence and we prove our main result
Theorem 4.2.
2. Preliminaries
2.1. Classical cohomology. Let G be a group and M be a G-module. One way to define the cohomology H ∗ (G, M) is
via cochains, as H ∗ (C ∗ (G, M)). The group of i-cochains of G with coefficients in M is the set of functions from Gi to
M:
n
o
C i (G, M) = φ : Gi → M .
The ith differential ∂i : C i (G, M) → C i+1 (G, M) is the map

∂i (φ)(g0, g1 , · · · , gi ) = g0 · φ(g1 , · · · , gi )
+

i
X

(−1) j φ(g0 , · · · , g j−2 , g j−1 g j , g j+1 , · · · , gi )

j=1

+ (−1)i+1 φ(g0 , · · · , gi−1 ).
Given a chain complex such as this one, one can define its normalised subcomplex. In each dimension n, define
NC n (G, M) to be the group of n-cochains which satisfy the normalisation condition
φ(g0 , · · · , gi−1 , 1, gi+1 , · · · , gn ) = 0,

i = 0, · · · , n.

The canonical inclusion ι : NC ∗ (G, M) → C ∗ (G, M) is a chain equivalence [4].
Another way to define H ∗ (G, M) is via projective resolutions, as H∗ (K ∗ (G, M)). The standard projective resolution
of Z by G-modules is the sequence of G-module homomorphisms [1]
∂i−1

ǫ

· · · → Z[Gi+1 ] −−−
→ Z[Gi ] → · · · → Z[G] →
− Z,
where
∂i−1 (g0 , · · · , gi ) =

i
X

(−1) j(g0 , · · · , g j−1 , g j+1 , · · · , gi ),

j=0

and the mapping ǫ sends each generator (g) to 1 ∈ Z. An element of
K i (G, M) = HomG (Z[Gi+1 ], M)
is then a function f : Gi+1 → M such that
f (sg0 , sg1 , · · · , sgi ) = s · f (g0 , g1 , · · · , gi ).

SYMMETRIC COHOMOLOGY OF GROUPS

3

The maps
ψi

K i (G, M) −→ C i (G, M)
defined by
ψi ( f )(g1 , · · · , gi ) = f (1, g1 , g1 g2 , · · · , g1 g2 · · · gi )
induce an isomorphism of cochain complexes K ∗ (G, M) → C ∗ (G, M) [1]. Moreover, one has a commutative diagram
K ∗ (G,
O M)
NK ∗ (G, M)

ψ

ψ

/ C ∗ (G, M)
O
/ NC ∗ (G, M)

where the horizontal maps are isomorphisms and the vertical maps are inclusions and homotopy equivalences. Here
NK i (G, M) consists of such maps f ∈ K i (G, M) that
f (x0 , · · · , xi ) = 0, if x j = x j+1 , for 0 ≤ j < n.
Thus
H ∗ (G, M) = H ∗ (NC ∗ (G, M)) = H ∗ (C ∗ (G, M)) = H ∗ (K ∗ (G, M)) = H ∗ (NK ∗ (G, M)).
2.2. Symmetric cohomology. We now discuss a subcomplex of C ∗ (G, M) introduced by Staic in [6] and [7]. It is
based on an action of Σn+1 on C n (G, M) (for every n) compatible with the differential. In order to define this action, it
is enough to define how the transpositions τi = (i, i + 1), 1 ≤ i ≤ n act. For φ ∈ C n (G, M) one defines:



−g1 φ(g−1
if i = 1,

1 , g1 g2 , g3 , · · · , gn ),



−1
(τi φ)(g1, g2 , g3 , · · · , gn ) = 
−φ(g1 , · · · , gi−2 , gi−1 gi , gi , gi gi+1 , · · · , gn ), 1 < i < n,




−φ(g1 , g2 , g3 , · · · , gn−1 gn , g−1
if i = n.
n ),

Denote by CS n (G, M) the subgroup of the invariants of this action. That is, CS n (G, M) = C n (G, M)Σn+1 . Staic proved
that CS ∗ (G, M) is a subcomplex of C ∗ (G, M) [6], [7].
Definition 2.1. The homology of this subcomplex is called the symmetric cohomology of G with coefficients in M and
is denoted by HS n (G, M).
Remark 2.2. There is a natural map αn : HS n (G, M) → H n (G, M) induced by the inclusion CS ∗ (G, M) ֒→ C ∗ (G, M).
2.3. Exterior powers. In order to define the chain complex introduced by Zarelua [10] we need to recall some facts
about exterior powers.
Definition 2.3. The exterior algebra Λ∗ (A) of an abelian group A is a quotient algebra of the tensor algebra T ∗ (A)
with respect to the two-sided ideal generated by the elements of the form a ⊗ a ∈ T 2 (A) = A ⊗ A.
A weaker version of this, denoted by Λ̃∗ (A), can be defined as the quotient algebra of the tensor algebra T ∗ (A) with
respect to the two-sided ideal generated by the elements of the form a ⊗ b + b ⊗ a ∈ T 2 (A). Since
a ⊗ b + b ⊗ a = (a + b) ⊗ (a + b) − a ⊗ a − b ⊗ b,
it is clear that one has the canonical quotient maps
⊗n (A) ։ Λ̃n (A) ։ Λn (A).
Denote by ∆n (A) the kernel of the projection Λ̃n (A) ։ Λn (A). Thus we have a short exact sequence
0 → ∆n (A) → Λ̃n (A) ։ Λn (A) → 0.
Clearly Λ1 (A) = A = Λ̃1 (A). Hence
(2.3.1)

∆1 (A) = 0.

˜ n and a1 ∧ · · · ∧ an respectively.
The images of a1 ⊗ · · · ⊗ an ∈ ⊗n A in Λ̃n (A) and Λn (A) are denoted by a1 ∧˜ · · · ∧a
Recall that if A = Z[S ] is a free abelian group with a set S as basis, then ⊗n A is a free abelian group with basis elements

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M. PIRASHVILI

s1 ⊗ · · · ⊗ sn , where si ∈ S . It is also well-known that Λn (A) is a free abelian group with basis elements s1 ∧ · · · ∧ sn ,
where s1 < · · · < sn . Here < is a total order on S .
˜ = 0, which is a consequence of
In Λ̃n (A), A = Z[S ], things are a bit more complicated because of the relation 2a∧a
n
˜
˜
the relation a∧b + b∧a = 0. It implies that ∆ (A) is an F2 -vector space. The epimorphism Λ̃n (Z[S ]) → Λn (Z[S ]) has a
˜ n . Here s1 , · · · , sn are distinct elements in S . Thus
splitting given by s1 ∧ · · · ∧ sn 7→ s1 ∧˜ · · · ∧s
Λ̃n (Z[S ])  Λn (Z[S ]) ⊕ ∆n (Z[S ]),

(2.3.2)

˜ n , where s1 ≤ · · · ≤ sn , are canonical generators of Λ̃n (Z[S ]). Among these
Thus expressions of the form s1 ∧˜ · · · ∧s
elements, ones with strict inequalities s1 < · · · < sn form a basis of the summand corresponding to the free abelian
group part, while the rest form a basis of the F2 -vector space ∆n (Z[S ]).
2.4. Exterior cohomology of groups. We now discuss a subcomplex of K ∗ (G, M), denoted by Kλ∗ (G, M), introduced
by Zarelua in [10].
According to Lemma 3.1 in [10], there is a differential
∂ : Λn+1 (Z[G]) → Λn (Z[G])
in the exterior algebra generated by Z[G] given by
∂(g0 ∧ · · · ∧ gn ) =

n
X

(−1)i+1 g0 ∧ · · · ∧ ĝi ∧ · · · ∧ gn ,

i=0

where, as usual, the hat ˆ denotes a missing value. The group G acts on this chain complex by:
g(g1 ∧ g2 ∧ · · · ∧ gn ) = gg1 ∧ gg2 ∧ · · · ∧ ggn .
Definition 2.4. The homology groups of the cochain complex (denoted by Kλ∗ (G, M))
∂

∂

∂

∂

HomG (Λ1 Z[G], M) →
− HomG (Λ2 Z[G], M) →
− ···→
− HomG (Λn Z[G], M) →
− ···

are called the exterior cohomology groups of the group G with coefficients in M and are denoted by Hλn (G, M).
Therefore, Kλ∗ (G, M) is the subcomplex of K n (G, M) of all G-maps f ∈ K n (G, M) such that
f (g0 , · · · , gi , gi , · · · , gn ) = 0,
and
f (g0 , · · · , gi , gi+1 , · · · , gn ) = − f (g0 , · · · , gi+1 , gi , · · · , gn ),
for all 0 ≤ i < n.
Remark 2.5. There is a natural transformation βn : Hλn (G, M) → H n (G, M) induced by the inclusion Kλ∗ (G, M) ֒→
K ∗ (G, M).
Remark 2.6. Let G be a finite group of order d. Since Z[G] is a free abelian group of rank d, we have Λi Z[G] = 0, for
i > d and Hλn (G, M) = 0 for n ≥ d. On the other hand, as we will see later, the groups HS n (C2 , M) are nontrivial for
infinitely many n.
3. Comparison of symmetric and exterior cohomologies
3.1. Construction of the map γ. We need two more complexes: Cλ∗ (G, M) and KS ∗ (G, M). They are defined as
follows.
Definition 3.1. Let KS n (G, M) denote the subcomplex of K n (G, M) of all G-maps f ∈ K n (G, M) such that
(3.1.1)
for all 0 ≤ i < n.

f (g0 , · · · , gi , gi+1 , · · · , gn ) = − f (g0 , · · · , gi+1 , gi , · · · , gn )

SYMMETRIC COHOMOLOGY OF GROUPS

5

So we have the following subcomplexes:
Kλ∗ (G, M) ֒→ KS ∗ (G, M) ֒→ K ∗ (G, M).
Definition 3.2. Let Cλn (G, M) be the complex defined by
Cλ∗ (G, M) = CS n (G, M) ∩ NC ∗ (G, N)
Thus φ ∈ CS n (G, M) belongs to Cλn (G, M) if
φ(x1 , · · · , 1, · · · , xn ) = 0.
This subcomplex has already been considered by [9], who showed that if M has no elements of order 2, then
Cλn (G, M) = CS n (G, M) for all n. We will later prove the same fact in a different way.
We have the following subcomplexes:
Cλ∗ (G, M) ֒→ CS ∗ (G, M) ֒→ C ∗ (G, M).
In order to understand the relationship between all these complexes it is useful to rewrite them in terms of resolutions,
which we constructed in Lemma 3.3 below.
Since Z[Gi ] = Z[G]⊗i , the standard projective resolution can be rewritten as
· · · → Z[G]⊗i → Z[G]⊗i−1 → · · · → Z[G]⊗2 → Z[G].
If one replaces the tensor algebra by either version of the exterior algebra, one still obtains a resolution, though in
general no longer a projective one. This is the subject of the following lemma.
Lemma 3.3. One has a commutative diagram of resolutions of Z:
···

/ Z[G]⊗i

/ Z[G]⊗i−1

···


/ Λ̃i Z[G]


/ Λ̃i−1 Z[G]

···


/ Λi Z[G]


/ Λi−1 Z[G]

/ ···

/ Z[G]⊗2

/ Z[G]

/ ···


/ Λ̃2 Z[G]


/ Z[G]

/ ···


/ Λ2 Z[G]


/ Z[G]

Id

Id

One denotes these resolutions by (T ∗+1 (Z[G]), ∂), (Λ̃∗+1 (Z[G]), ∂) and (Λ∗+1 (Z[G]), ∂) respectively.
Proof. In this proof, take ∂−1 = ǫ : Z[G] → Z. We only present the proof for Λ∗ , as the proof for Λ̃∗ is similar. We
construct a homomorphism h : Λi Z[G] → Λi+1 Z[G] by the formula
h(g0 ∧ · · · ∧ gi ) = 1 ∧ g0 ∧ · · · ∧ gi .
To show that this is a contracting homotopy, we need to check that h ◦ ∂ + ∂ ◦ h = IdΛi Z[G] . Indeed, we have
∂i ◦ hi (g0 ∧ · · · ∧ gi ) = g0 ∧ · · · ∧ gi −

i
X

(−1) j1 ∧ g0 ∧ · · · ∧ gˆj ∧ · · · ∧ gi

j=0

= g0 ∧ · · · ∧ gi − hi−1 ◦ ∂i−1 (g0 ∧ · · · ∧ gi ).

Lemma 3.4. The differential ∂ : Λ̃n+1 (Z[G]) → Λ̃n (Z[G]) sends ∆n+1 (Z[G]) to ∆n (Z[G]). Moreover, it is compatible
with the decompostion (2.3.2). Hence
(Λ̃∗+1 (Z[G]), ∂)  (∆∗+1 (Z[G]), ∂) ⊕ (Λ∗+1 (Z[G]), ∂)
and H∗ (∆∗+1 (Z[G]), ∂) = 0.

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M. PIRASHVILI

Proof. By Lemma 3.3 the canonical projection Λ̃∗+1 (Z[G]) → Λ∗+1 (Z[G])) is a chain map, inducing an isomorphism
in homology, hence ∆∗+1 (Z[G]) is a chain subcomplex with trivial homology. To finish the proof, it suffices to note that
˜ n , g1 , · · · , gn ∈ G, commutes with differentials and hence defines a splitting of chain
the map g1 ∧ · · · ∧ gn 7→ g1 ∧˜ · · · ∧g
complexes.

Lemma 3.5. After applying the functor HomZ[G] (−, M) to the resolutions in Lemma 3.3 one obtains the following
diagram
HomZ[G] (Λ∗ (Z[G]), M)

/ HomZ[G] (Λ̃∗ (Z[G]), M)

=

/ HomZ[G] (T ∗ (Z[G]), M)
=

=


Kλ∗ (G, M)


/ KS ∗ (G, M)

ψ


/ K ∗ (G, M)
ψ

ψ


Cλ∗ (G, M)


/ CS ∗ (G, M)


/ C ∗ (G, M)

where all horizontal arrows are inclusions and vertical arrows are isomorphisms.
Proof. A key point is to show that restricting ψ on KS ∗ (G, M) yields an isomorphism between KS ∗ (G, M) and
CS ∗ (G, M). To this end, take φ ∈ CS n (G, M). Then
−1
−1
f (g0 , · · · , gn ) = (ψn )−1 (φ)(g0, g1 , · · · , gn ) = g0 · φ(g−1
0 g1 , g1 g2 , · · · , gn−1 gn ).

The equation φ(g1 , g2 , g3 , · · · , gn ) = −g1 φ(g−1
1 , g1 g2 , g3 , · · · , gn ) translates to
−1
f (g0 , · · · , gn ) = g0 (ψn )−1 (φ)(g−1
0 g1 , · · · , gn−1 gn )
−1
−1
−1 −1
n −1
−1
= −g0 · g−1
0 g1 · (ψ ) (φ)((g0 g1 ) , g0 g1 g1 g2 , · · · , gn−1 gn )
−1
= −g1 · (ψn )−1 (φ)(g1)−1 g0 , g−1
0 g2 , · · · , gn−1 gn )

= − f (g1 , g0 , · · · , gn ).
In a similar way, the other equations above give the condition 3.1.1 for n > 0.



If one passes to cohomology, one obtains the homomorphisms
/ H ∗ (G, M)
/ H ∗ (KS ∗ (G, M))
Hλ∗ (G, M)
PPP
♥7
♥
PPP
♥♥
♥
♥

P
♥♥
γ PPP
P'  ♥♥♥♥ α
HS ∗ (G, M)
and the commutativity of the diagram in Lemma 3.5 shows that β = αγ.
Proposition 3.6. The homomorphism γn : Hλn (G, M) → HS n (G, M) is a split monomorphism. Moreover, it is an
isomorphism provided M has no elements of order two.
Proof. The first part follows from Lemma 3.4. Assume M has no elements of order two. It suffices to show that
Kλ∗ (G, M) = KS ∗ (G, M). Take an element f ∈ KS n (G, M). Then we have
f (x0 , · · · , xi , xi+1 , · · · , xn ) = − f (x0 , · · · , xi+1 , xi , · · · , xn ),
for all 0 ≤ i < n. If xi = xi+1 , one obtains 2 f (x0 , · · · , xi , xi , · · · , xn ) = 0 and hence f (x0 , · · · , xi , xi , · · · , xn ) = 0. This
implies that f ∈ Kλn (G, M) and the proof is finished.


SYMMETRIC COHOMOLOGY OF GROUPS

7

3.2. δ-cohomology. In order to state the realtionship between the exterior and symmetric cohomology we need to
introduce new groups.
Definition 3.7. For a group G and a G-module M one defines the δ-homology Hδ∗ (G, M) by
Hδ∗ (G, M) = H ∗ (HomZ[G] (∆∗+1 (Z[G]), M)).
Since ∆n (Z[G]) is an F2 -vector space, it follows that the groups Hδn (G, M) are also F2 -vector spaces, n ≥ 0. The
importance of these groups comes from the fact that
HS n (G, M)  Hλn (G, M) ⊕ Hδn (G, M)

(3.2.1)

which is a trivial consequence of Lemma 3.4. It follows from Proposition 3.6 that if M has no elements of order two,
then Hδ∗ (G, M) = 0.
3.3. Preliminaries on spectral sequences. To state our main result of this section, let us recall the construction of the
hypercohomology spectral sequences. These spectral sequences will also play a prominent role in the next section.
Let G be a group and M be a left G-module. For any chain complex of left G-modules C∗ = (C0 ← C1 ← · · · )
one defines Ext∗Z[G] (C∗ , N) to be the homology of the total complex of the bicomplex HomZ[G] (C∗ , I ∗ ), where I ∗ is an
injective resolution of M.
There exist two spectral sequences. Both of them abut to the group Ext∗Z[G] (C∗ , M). They are:
q

pq

I1 = ExtZ[G] (C p , N)

=⇒

p+q

ExtZ[G] (C∗ , M),

p
II2pq = ExtZ[G]
(Hq (C∗ ), M) =⇒
We also need the following easy lemma on spectral sequences

p+q

ExtZ[G] (C∗ , M).

pq

Lemma 3.8. Assume a spectral sequence abuts to zero and E2 = 0 if q < 0 or p < k, where k is a fixed integer. Then
E2k 0 = 0 = E2k+1 0 .
3.4. Vanishing of δ-cohomology in low dimensions. Now we can state the main result of this section:
Theorem 3.9. Let G be a group and M be a G-module. Then
Hδi (G, M) = 0,
i

Hence γ :

Hλi (G,

for 0 ≤ i ≤ 4.

i

M) → HS (G, M) is an isomorphism for i = 0, 1, 2, 3, 4.

Proof. In the hypercohomology spectral sequence we take C∗ = (∆∗+1 (Z[G]), ∂). Since H∗ (C∗ ) = 0, the spectral
sequence II gives Ext∗Z[G] (C∗ , M) = 0. Thus, the spectral sequence I has the form
pq

q

E1 = ExtZ[G] (∆ p+1 (Z[G]), M)
Since

E1p0

=⇒ 0.

= HomZ[G] (∆ p+1 (Z[G]), M), we see that
E2∗0 = Hδ∗ (G, M).
pq

According to (2.3.1) we have E1 = 0 for p < 1. It follows from Lemma 3.8 that E2i 0 = Hδi (G, M) = 0 for i ≤ 2. Thus
by the same Lemma it suffices to show that E21,q = 0 = E22,q if q > 0.
One checks that the following diagram of G-modules commutes:
o
∆2 (Z[G])
O

∂1

∂2

ψ2

ψ1

F2 [G] o

o
∆3 (Z[G])
O

δ1

F2 [G × G] o

∆4 (Z[G])
O
ψ3

δ2

F2 [G × G]

where
˜
ψ1 (g) = g∧g,

˜ ∧h,
˜
ψ2 (g, h) = g∧g

˜ ∧g
˜ ∧h,
˜
ψ3 (g, h) = g∧g

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M. PIRASHVILI

δ1 (g, h) = g,

δ2 (g, h) = (g, h) − (g, g).

˜
˜ ∧t|s,
˜
Since the set of elements {s∧s|s
∈ G}, (resp. {s∧s
t ∈ G}) forms an F2 -basis of ∆2 (Z[G]) (resp. ∆3 (Z[G])), the
G-homomorphism ψ1 (resp. ψ2 ) is an isomorphism. In general, the G-homomorphism ψ3 is not an isomorphism, but
only a split monomorphism. Hence the projective resolutions
2

0 → Z[G] →
− Z[G] → F2 [G] → 0

2

and 0 → Z[G × G] →
− Z[G × G] → F2 [G × G] → 0

can be used to compute ExtqZ[G] (∆2 (Z[G]), M) and ExtiZ[G] (∆3 (Z[G]), M). In both cases
i
i
ExtZ[G]
(∆2 (Z[G]), M) = 0 = ExtZ[G]
(∆3 (Z[G]), M) if
1,q

i > 1.

2,q

Hence E1 = 0 = E1 if q > 1. The first projective resolution gives
1
E111 = ExtZ[G]
(∆2 (Z[G]), M) = N,

where N = M/2M. Since Z[G × G] = ⊕g∈G Z[G] as a G-module, the second projective resolution gives
1
(∆3 (Z[G]), M) = Maps(G, N).
E121 = ExtZ[G]
1
Moreover, it also shows that the group Maps(G, N) is a direct summand of ExtZ[G]
(∆4 (Z[G]), M). It follows that there
is an isomorphism of chain complexes

E101
O

∂0

/ E 11
1O

∂1

ψ∗1

0

δ0

/N

∂2

/ E 21
1O

/ E 31
1O

ψ∗2
δ1 /

ψ∗3

Maps(G, N)

δ2

/ X ⊕ Maps(G, N)

!
x
for some x and (δ′ (τ))(g) = τ(g) − τ(1). Since δ1 is a monomorphism, it
δ′
= 0. And as Ker(δ′ ) = Im(δ1 ), we obtain that E22,1 = 0 and the proof is finished.


for some X, where (δ1 (n))(g) = n, δ2 =
follows that E21,1

Now we give an example which shows that γn , n ≥ 5 is not an isomorphism in general.
3.5. The symmetric and exterior cohomologies of C2 . Let G = C2 = {1, t}, t2 = 1 be the cyclic group of order
two. In this section we compute both symmetric and exterior cohomologies of C2 . The computation of the exterior
cohomology is extremely easy. In fact, for G = C2 , the resolution (Λ∗ (Z[G]), ∂) has the following form:
···

/0

/ Λ3 (Z[C2 ])

∂




0

/ Λ2 (Z[C2 ])

∂

/ Z[G]




Z[C2 ]/(1 + t)

where ∂ = (1 − t). So,
Hλn (C2 ,




H n (C2 , M),
M) = 

0, else.

if n = 0, 1,

For the symmetric cohomology one has the following result:

Lemma 3.10. For G = C2 and M = F2 with trivial action of G on M, one has



if i = 0, or i ≡ 1 mod
F2 ,
i
HS (C2 , F2 ) = 

0,
else.
Thus, in general, Hλ∗ (G, M) , HS ∗ (G, M).

4,

SYMMETRIC COHOMOLOGY OF GROUPS

9

Proof. Consider the resolution
∂0

∂1

· · · → Λ̃3 Z[C2 ] −→ Λ̃2 Z[C2 ] −→ Z[C2 ].
Fix n > 0 and in Λ̃n Z[C2 ] consider the elements
· · · ∧˜}t,
αni = |
1∧˜ {z
· · · ∧˜}
1 ∧˜ t|∧˜ {z

0≤i<

i

n−i

n

β = 1|∧˜ {z
· · · ∧˜}
1 ∧˜ |
t∧˜ {z
· · · ∧˜}t,
m

n
,
2

n = 2m.

m

Then αni and βn generate Λ̃n Z[C2 ] as a C2 -module. More accurately, Z[C2 ] is a free Z[C2 ]-module with the generator
α10 . As a Z[C2 ]-module,
M
Λ̃2 Z[C2 ] = F2 [C2 ]
Z[C2 ]/(t + 1),

with α20 generating F2 [C2 ] and β2 generating Z[C2 ]/(t + 1). For odd n, n = 2m + 1 ≥ 3,
M M
Λ̃n Z[C2 ] = F2 [C2 ]
···
F2 [C2 ],

with αn0 , · · · , αnm generating each of the summands. Similarly to n = 2, for larger n = 2m ≥ 4 we have
M M
M
Λ̃n Z[C2 ] = F2 [C2 ]
···
F2 [C2 ]
F2 [C2 ]/(t − 1),
where the αn0 , · · · , αnm generate the F2 [C2 ] summands and βn generates F2 [C2 ]/(t − 1).
Beginning from ∂1 , the coboundary maps are given by the matrices



i f i is odd and j = i or
1,
(∂4k+1 )i j = 

0,
else,

j =i+1

where 1 ≤ i ≤ 2k + 2, 1 ≤ j ≤ 2k + 2,




1,
(∂4k+2 )i j = 

0,

i f j is
else,

even and

i = j or

i= j−1

where 1 ≤ i ≤ 2k + 2, 1 ≤ j ≤ 2k + 3,



1,




(∂4k+3 )i j = 
1,




0,

if i
if i
else,

is odd
is odd

and
and

j=i
j=i+1

where 1 ≤ i ≤ 2k + 3, 1 ≤ j ≤ 2k + 3,



1,
i f j is even and i = j or




(∂4k )i j = 
t − 1,
i f j = 2k + 2 and i = 2k + 1




0,
else,

and

i= j−1

i < 2k + 2

and

where 1 ≤ i ≤ 2k + 1, 1 ≤ j ≤ 2k + 2. Based on this the result easily follows.

j < 2k + 2



4. Relationship between exterior and classical cohomology
We start this section with the following easy and probably well-known fact. It will be used in the proof of Theorem
4.2 below.
Lemma 4.1. Let g ∈ G and ω = x1 ∧ · · · ∧ xn ∈ Λn Z[G], where x1 , · · · , xn are distinct elements in G. If gω = ±ω, then
the order of g divides n.

10

M. PIRASHVILI

Proof. If one forgets the sign, it follows from the assumption that the multiplication by g permutes the n elements
x1 , · · · , xn , meaning the cyclic group generated by g acts on the set {x1 , · · · , xn }. The action is free, because it is given
by the multiplication in G. Hence all orbits will have the same length equal to the order of g, dividing n.

Now we can state our main result.
Theorem 4.2. For any group G and any G-module M, there is a first quadrant spectral sequence
pq

E1 =⇒ H p+q (G, M)
with properties
(i)
(ii)
(iii)
(iv)

E2p,0 = Hλp (G, M) and the edge homomorphism E2p0 → H p (G, M) is precisely β p , p ≥ 0.
0q
If q > 0, then E1 = 0.
If q > 0, p > 0 and the equation x p+1 = 1 has only trivial solution in G, then E1pq = 0.
If ℓ is a prime number and q > 0, then
Q


 Cℓ ⊂G H q+1 (Cℓ , M), if ℓ = 2,
ℓ−1 q
E1
=
Q
q

Cℓ ⊂G H (Cℓ , M), if ℓ > 2.

Here the product is taken over all subgroups of order ℓ and for each such subgroup, the corresponding action
of Cℓ on M is induced by the inclusion.
pq

Remark 4.3. If p + 1 is not prime, then E1 , q > 0, p > 0 can be described as a product (usually of several copies) of
the group cohomology of subgroups of order k, where k|p + 1, but the exact formula is too complex to state here. From
pq
pq
this it is easy to deduce that E1 = E2 for all q > 0 (compare with the proof of the part i) of Corollary 4.4).
Proof. In the hypercohomology spectral sequence discussed in Section 3.3, we take R = Z[G], N = M and C ∗ =
(Λ∗+1 (Z[G]), ∂), which we denote simply by Λ∗+1 . This gives the spectral sequences
pq

q

I1 = ExtG (Λ p+1 , M)
II2pq = ExtGp (Hq (Λ∗+1 ), M)

=⇒
=⇒

p+q

ExtG (Λ∗+1 , M)
p+q

ExtG (Λ∗+1 , M).

Let us first consider the second spectral sequence. As Λ∗+1 is a resolution of Z, we have



for q = 0,
Z,
∗+1
Hq (Λ ) = 

0,
else.

Therefore, the second spectral sequence degenerates to the isomorphism
p

p

p

ExtG (Λ∗+1 , M) = ExtG (H0 (Λ∗+1 ), M) = ExtG (Z, M) = H p (G, M).

Substituting this value into the first spectral sequence, we obtain the spectral sequence
E1pq = ExtqZ[G] (Λ p+1 (Z[G]), M) =⇒ H p+q (G, M).
Since the differential d1 in the first page of the spectral sequence is induced by the boundary map in the resolution
p0
Λ∗+1 (Z[G]) → Z, it follows that for q = 0, the chain complex (E1 , d1 ) coincides with the Zarelua chain complex and
the statement (i) follows.
q
(Z[G], M) vanishes for q > 0. Hence E10q = 0 for q > 0, and the property (ii) holds.
If p = 0, then E1pq = ExtZ[G]

Next, the G-module Λq+1 (Z[G]) is free as an abelian group with a basis of the form x1 ∧· · ·∧ x p , where x1 < · · · < x p .
Here ≤ is any total order on G. If one ignores the sign, we see that G acts on the basis. Thus Λ p+1 (Z[G]) decomposes as
a direct sum of G-submodules corresponding to these orbits. In particular, summands corresponding to free orbits are
free G-modules. Now, if the assertion of (iii) holds, all orbits are free thanks to Lemma 4.1 and hence the Ext-group
vanishes and E1pq = 0 for q > 0. Thus the property (iii) is proved.

SYMMETRIC COHOMOLOGY OF GROUPS

11

If ℓ is prime and Cℓ = {1, g, · · · , g p−1 } is a cyclic subgroup of G, then for the basis element ω = 1 ∧ g ∧ · · · ∧ gℓ−1 one
has gω = ω for odd ℓ, and gω = −ω for ℓ = 2. Thus ω determines a non free summand of Λ p (Z[G]). This summand is
isomorphic to Z[G]/(g − 1) for odd ℓ and Z[G]/(g + 1) for ℓ = 2. This summand has an obvious projective resolution
g−1

1+g+···gℓ−1

0 ← Z[G]/(g−1) ← Z[G] ←−−− Z[G] ←−−−−−−−− · · ·
if ℓ is odd and
g+1

g−1

0 ← Z[G]/(g+1) ← Z[G] ←−−− Z[G] ←−−− · · ·
if ℓ = 2. From this it follows that this summand of Λℓ (Z[G]) contributes the factor H i (Cℓ , M) (resp. H i+1 (Cℓ , M)) in
m
ExtZ[G]
(Λ p+1 (Z[G]), M) for odd ℓ (resp. ℓ = 2). By Lemma 4.1 all non-free summands of Λℓ (Z[G]) arise in this way
and hence E1ℓ−1 m has the form described in (iv).

Thus the first plane/page of the spectral sequence is:

q

0

Y

H q+1 (C2 , M)

C2 ⊂G

..
.

..
.

2

0

0

..
.

..
.

Y

H 3 (C2 , M)

···

Extq (Λ p+1 Z[G], M)

···

..

..
.

···

Y

H 2 (C2 , M)

.

Y

H 2 (C3 , M)

···

Ext2 (Λ p+1 Z[G], M)

···

Y

H 1 (C3 , M)

···

Ext1 (Λ p+1 Z[G], M)

···

C3 ⊂G

C2 ⊂G

0

H q (C3 , M)

C3 ⊂G

C2 ⊂G

1

Y

C3 ⊂G

p

Hλ0 (G, M)

Hλ1 (G, M)

Hλ2 (G, M)

···

Hλ (G, M)

0

1

2

···

p

···

As an immediate consequence of Theorem 4.2 one obtains the following corollary.
Corollary 4.4.
(i) For any group G and any G-module M, the homomorphism βi : Hλi (G, M) → H i (G, M) is an
isomorphism for i = 0 and i = 1, while β2 and β3 can be fit in an exact sequence:
Y
β3
β2
H 2 (C2 , M) → Hλ3 (G, M) −→ H 3 (G, M).
0 → Hλ2 (G, M) −→ H 2 (G, M) →
C2 ⊂G

(ii) If G has no elements of order two, then for any G-module M, the homomorphism β2 is an isomorphism, while
β3 , β4 and β5 can be fit in an exact sequence:
Y
β4
β3
H 3 (C3 , M) → Hλ4 (G, M) −→
0 → Hλ3 (G, M) −→ H 3 (G, M) →
C3 ⊂G

12

M. PIRASHVILI
β4

−→ H 4 (G, M) →

Y

β5

H 4 (C3 , M) → Hλ5 (G, M) −→ H 5 (G, M).

C3 ⊂G

(iii) If all nontrivial elements of G are of infinite order, then βi : Hλi (G, M) → H i (G, M) is an isomorphism for all
i ≥ 0.
(i) We first show that if q > 0, the differential E11q → E12q vanishes. In fact, by part (iv) of Theorem 4.2 the
1q
1q
group E1 is annihilated by the multiplication by 2, while the group E1 is annihilated by the multiplication
1q
1q
by 3 and hence the corresponding map is zero. This fact implies that E2 = E1 for all q > 0. The rest is a
consequence of the 5-term exact sequence, which we have in any first quadrant spectral sequence.
(ii) Assume q > 0. By part (iii) of Theorem 4.2 and the fact that G does not contain an element of order two, we
pq
pq
pq
have E0 = 0, if q > 0 and p + 1 is a power of two. It follows that E2 = E1 , for p = 2 and hence the result.
pq
(iii) By part (iii) of Theorem 4.2 we have E1 = 0 for all q > 0. Hence the spectral sequence degenerates and in
particular, the edge homomorphism is an isomorphism.

Proof.


Example. Let ℓ be a prime number and G = Cℓ be a cyclic group of order ℓ. Then



H i (Cℓ , M), if i ≤ ℓ − 1,
i
Hλ (Cℓ , M) = 

0, if i ≥ ℓ.

In fact, the case when i ≥ ℓ follows from Remark 2.6, while the case i ≤ ℓ − 1 follows from part (iii) of Theorem 4.2.
Acknowledgements
The paper was written during the author’s postdoctoral fellowship at the University of Southampton. The author
would like to thank the staff of the School of Mathematics, especiallly Prof. J. Brodzki, for providing me with excellent
conditions to work.
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[3] R.J. Lawrence. Algebras and triangle relations. J. Pure and Appl. Algebras. 100(1995), 43-72.
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[5] M. Singh. Symmetric continuous cohomology of topological groups. Homology, Homotopy and Appl. 15(2013), 279-302.
[6] M. D. Staic. From 3-algebras to ∆-groups and symmetric cohomology. J. of Algebra 332(2009), 1360-1376.
[7] M. D. Staic. Symmetric cohomology of groups in low dimensions. Arch. Math.93(2009), 205-211.
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Department of Mathematics, University of Southampton, University Road, Southampton, SO17 1BJ
E-mail address: mp2m15@soton.ac.uk

