Cohomologies of finite abelian groups
We construct a simplified resolution for the trivial G-module Z, where G is a finite abelian group, and compare it with the standard resolution. We use it to calculate cohomologies of irreducible G-lattices and their duals.
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Drozd, Yu.A. Plakosh, A.I. 2019-06-18T10:25:42Z 2019-06-18T10:25:42Z 2017 Cohomologies of finite abelian groups / Yu.A. Drozd, A.I. Plakosh // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 144-157. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC:20J06, 18G10, 20K01. https://nasplib.isofts.kiev.ua/handle/123456789/156259 We construct a simplified resolution for the trivial G-module Z, where G is a finite abelian group, and compare it with the standard resolution. We use it to calculate cohomologies of irreducible G-lattices and their duals. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Cohomologies of finite abelian groups Article published earlier |
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Cohomologies of finite abelian groups |
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Cohomologies of finite abelian groups Drozd, Yu.A. Plakosh, A.I. |
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Cohomologies of finite abelian groups |
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Cohomologies of finite abelian groups |
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Cohomologies of finite abelian groups |
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Cohomologies of finite abelian groups |
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cohomologies of finite abelian groups |
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Drozd, Yu.A. Plakosh, A.I. |
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Drozd, Yu.A. Plakosh, A.I. |
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Algebra and Discrete Mathematics |
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We construct a simplified resolution for the trivial G-module Z, where G is a finite abelian group, and compare it with the standard resolution. We use it to calculate cohomologies of irreducible G-lattices and their duals.
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Cohomologies of finite abelian groups / Yu.A. Drozd, A.I. Plakosh // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 144-157. — Бібліогр.: 7 назв. — англ. |
| work_keys_str_mv |
AT drozdyua cohomologiesoffiniteabeliangroups AT plakoshai cohomologiesoffiniteabeliangroups |
| first_indexed |
2025-11-24T20:43:37Z |
| last_indexed |
2025-11-24T20:43:37Z |
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1850496123732492288 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 24 (2017). Number 1, pp. 144–157
c© Journal “Algebra and Discrete Mathematics”
Cohomologies of finite abelian groups
Yuriy A. Drozd and Andriana I. Plakosh
Abstract. We construct a simplified resolution for the trivial
G-module Z, where G is a finite abelian group, and compare it with
the standard resolution. We use it to calculate cohomologies of
irreducible G-lattices and their duals.
Introduction
The theory of cohomologies of groups was inspired by the works of
Hurewicz on cohomologies of acyclic spaces and was founded in 1940’s
by Eilenberg–MacLane, Eckmann, Hopf and others. It was one of the
origins of the homological algebra. It was also related to the theory of
group extensions and projective representations, where cohomologies arise
as factor sets. This theory is widely used in topology, number theory,
algebraic geometry and other branches of mathematics. Thus it is actively
studied by plenty of mathematicians. In particular, there is a lot of
papers devoted to the calculation of cohomologies of concrete groups
and their classes. In these investigations one often needs special sorts of
resolutions, which are simpler and more convenient than the standard one.
For instance, Takahashi [7] proposed a new approach to the calculation of
cohomologies of finite abelian groups and gave applications of his method
to the cohomologies of the trivial module and of some Galois groups.
The aim of our paper is to describe a rather simple resolution for finite
abelian groups (Section 1) and to use it for calculation of cohomologies
of irreducible G-lattices and their duals (Sections 4 and 5). Our approach
2010 MSC: 20J06, 18G10, 20K01.
Key words and phrases: cohomologies, finite abelian groups, resolution,
G-lattice.
Yu. Drozd, A. Plakosh 145
is close to that of Takahashi, though it seems more explicit. We also
compare our resolution with the standard one (Section 2) and prove
some facts concerning duality for cohomologies of G-lattices (Section 3).
The results about the second cohomologies can be useful in the study of
crystallographic groups and of Chernikov groups.
1. Resolution
For a periodic element a of a group G we denote by o(a) the order
of a, sa =
∑o(a)−1
i=0 ai. Let G =
∏s
i=1 Gi be a direct product of finite
cyclic groups Gi = 〈 ai | aoi
i = 1 〉 of orders oi = o(ai), R = ZG, P =
R[x1, x2, . . . , xs] and Pn be the set of homogeneous polynomials from P
of degree n (including 0). We define a differential d : Pn → Pn−1 by the
rule
dn(xk1
1 xk2
2 . . . xks
s ) =
s
∑
i=1
(−1)KiCix
k1
1 . . . xki−1
i . . . xks
s ,
where Ki =
∑i−1
j=1 kj and
Ci =
ai − 1 if ki is odd,
sai
if ki > 0 is even,
0 if ki = 0.
When speaking of the G-module Z, we always suppose that the ele-
ments of G act trivially.
Theorem 1.1. P = (Pn, dn) is a free resolution of the G-module Z.
Proof. If s = 1, it is well-known. If Ri = ZGi and Pi denotes such
resolution for the group Gi, then R =
⊗s
i=1 Ri and P is the tensor
product of complexes
⊗s
i=1 P
i. As all groups of cycles and boundaries in
the complexes Pi are free abelian, the claim follows from the Künneth
relations [3, Theorem VI.3.1].
2. Correspondence with standard resolution
To apply Theorem 1.1, for instance, to extensions of groups, we have
to compare it with the standard resolution, which is usually used for this
purpose [2, 3]. So, in what follows, S denotes the normalized standard
146 Cohomologies of finite abelian groups
resolution for Z as R-module, { [g1, g2, . . . , gn] | gi ∈ G \ {1} } is the usual
basis of Sn such that the standard differential ds is defined as
ds
n[g1, g2, . . . , gn] = g1[g2, . . . , gn] +
n
∑
i=1
(−1)i[g1, . . . , gigi+1, . . . , gn]+
+ (−1)n[g1, g2, . . . , gn−1],
setting [g1, g2, . . . , gn] = 0 if some gi = 1. Note that P0 = S0 = R.
We denote a{i} = 1 + a + a2 + . . . ai−1. Then sa = a{o(a)},
a{i+k} = a{i} + aia{k}, (2.1)
in particular,
a{m+o(a)} = a{m} + amsa.
Theorem 2.1. There is a quasi-isomorphism σ : S → P such that
σ0 = id,
σ1[ak1
1 ak2
2 . . . aks
s ] =
s
∑
i=1
(
i−1
∏
j=1
a
kj
j
)
a
{ki}
i xi,
σ2[ak1
1 ak2
2 . . . aks
s , al1
1 al2
2 . . . als
s ]=
s
∑
i=1
i
∑
j=1
(
i−1
∏
q=1
akq
q
j−1
∏
r=1
alr
r
)
σ2[aki
i , a
lj
j ],
where σ2[ak
i , al
j ] =
[(k + l)/oi]x
2
i if i = j,
0 if i < j,
a
{l}
j a
{k}
i xjxi if i > j
(2.2)
Since S and P are free resolutions of Z, σ induces isomoprhisms
of cohomologies Hn(HomR(S, M)) → Hn(HomR(P, M)). In particular,
combining σ2 with cocycles from HomR(P2, M), we obtain the “usual”
presentation of cocycles from H2(G, M).
Proof. Actually, we have to show that the diagram
S2
ds
2
//
σ2
��
S1
σ1
��
ds
1
// S0
P2
d2
// P1
d1
// P0.
Yu. Drozd, A. Plakosh 147
is commutative. Then the set of homomorphisms {σ0, σ1, σ2} extends to
a quasi-isomorphism σ : S → P.
Note that gh − 1 = (g − 1) + g(h − 1) and ak − 1 = a{k}(a − 1).
Therefore,
ds
1[ak1
1 ak2
2 . . . aks
s ] = ak1
1 ak2
2 . . . aks
s − 1 =
s
∑
i=1
(
i−1
∏
j=1
a
kj
j
)
(aki
i − 1)
=
s
∑
i=1
(
i−1
∏
j=1
a
kj
j
)
a
{ki}
i (ai − 1) =
s
∑
i=1
(
i−1
∏
j=1
a
kj
j
)
a
{ki}
i d1xi,
hence d1σ1 = ds
1.
Set (r)i = res(r, oi), the residue of r modulo oi. Then, for 0 6 k < oi,
0 6 l < oi,
ds
2[ak
i , al
i] = ak
i [al
i] − [ak+l
i ] + [ak
i ],
thus
σ1ds
2[ak
i , al
i] = (ak
i a
{l}
i − a
{(k+l)i}
i + a
{l}
i )xi
= (ak
i a
{l}
i − a
{k+l}
i + a
{l}
i + [(k + l)/oi]sai
)xi
= [(k + l)/oi]sai
xi = d2([(k + l)/oi]x
2
i ),
so, if we set σ2[ak
i , al
i] = [(k + l)/oi]x
2
i , we have
d2σ2[ak
i , al
i] = σ1ds
2[ak
i , al
i].
In the same way,
ds
2[ak
i , al
j ] = ak
i [al
j ] − [ak
i al
j ] + [ak
i ],
thus, if i < j,
σ1ds
2[ak
i , al
j ] = ak
i a
{l}
j xj − a
{k}
i xi − ak
i a
{l}
j xj + a
{k}
i xi = 0,
while if i > j
σ1ds
2[ak
i , al
j ] = ak
i a
{l}
j xj − a
{l}
j xj − al
ja
{k}
i xi + a
{k}
i xi
= (ak
i − 1)a
{l}
j xj − (al
j − 1)a
{k}
i xi = −d2(a
{l}
j a
{k}
i xjxi).
So, if we set
σ2[ak
i , al
j ] =
{
0 if i < j,
a
{l}
j a
{k}
i xjxi if i > j
148 Cohomologies of finite abelian groups
we have
d2σ2[ak
i , al
j ] = σ1ds
2[ak
i , al
j ] for i 6= j.
Let now σ2 is defined by the rule (2.2). We check that d2σ2 = σ1ds
2
for s = 3. The general case is analogous, though a bit cumbersome. We
write a, b, c instead of a1, a2, a3 and x, y, z instead of x1, x2, x3. Then
σ1ds
2[aibjcr, akblcs] = σ1(aibjcr[akblcs] − [ai+kbj+lcr+s] + [aibjcr])
= aibjcr(a{k}x + akb{l}y + akblc{s}z) − a{i+k}x − ai+kb{j+l}y
− ai+kbj+lc{r+s}z + [(i + k)/oa]sax + ai+k[(j + l)/ob]sby
+ ai+kbj+l[(r + s)/oc]scx + a{i}x + aib{j}y + aibjc{r}z
= (aibjcra{k} − a{i+k} + a{i} + [(i + k)/oa]sa)x
+ ai(akbjcrb{l} − akb{j+l} + b{j} + ak[(j + l)/ob]sb)y
+ aibj(akblcrc{s} − akblc{r+s} + c{r} + akbl[(r + s)/oc]sc)z,
while
d2σ2[aibjcr, akblcs]=d2
(
−aia{k}b{j}xy−aibja{k}c{s}xz−ai+kbjb{l}c{r}yz
+ [(i + k)/oa]x2 + ai+k[(j + l)/ob]y
2 + ai+kbj+l[(r + s)/oc]z
2)
= −ai(ak − 1)b{j}y + ai(bj − 1)a{k}x − ai(ak − 1)bjc{r}z
+ aibj(cr − 1)a{k}x − ai+kbj(bl − 1)c{r}z + ai+kbj(cr − 1)bk}y,
+ [(i + k)/oa]sax + ai+k[(j + l)/ob]sby + ai+kbj+l[(r + s)/oc]scx
= (−aia{k} + aibjcra{k} + [(i + k)/oa]sa)x
+ ai(−akb{j} + b{j} + akbjcrb{l} − akbjb{l} + ak[(j + l)/ob]sb)y
+ aibj(c{r} − akblc{r} + akbl[(r + s)/oc]sc)z.
Relations (2.1) immediately imply that both results are equal.
3. Cohomologies of G-lattices
In this section G denotes a finite group, R = ZG. Recall that a G-
lattice (or an integral representation of G) is a G-module M such that
its abelian group is free of finite rank. They also say that M is a lattice
in the QG-module M̃ = Q ⊗Z M . Two G-lattices M, N are said to be
of the same genus if Mp ≃ Np for each prime p, where Mp = Zp ⊗Z M
(Zp = { r/z | r ∈ Z, s ∈ Z \ pZ }). Then they write M ∨ N . We also set
M∗ = HomZ(M,Z), where G acts by the rule gf(u) = f(g−1u).
Yu. Drozd, A. Plakosh 149
We denote by Ĥn(G, M) the Tate cohomologies of G with coefficients
in M [2, 3]. Let
F : · · · → Fn
dn−→ Fn−1
dn−1
−−−→ . . .
d2−→ F1
d1−→ F0 → 0
be a free resolution of Z, where all modules Fn are finitely generated,
F∗ : 0 → F∗
0
d∗
1−→ F∗
1
d∗
2−→ . . .
d∗
n−1
−−−→ F∗
n−1
d∗
n−→ F∗
n → . . .
be the dual complex, d0 : F0 → F∗
0 be the composition of the maps
F0 → coker d1 ≃ Z ≃ ker d∗
0 → F∗
0. Set F−n = F∗
n−1, d−n = d∗
n. The
sequence
F+ : . . . → Fn
dn−→ Fn−1
dn−1
−−−→ . . .
d2−→ F1
d1−→ F0
d0−→
d0−→ F−1
d−1
−−→ F−2
d−2
−−→ . . .
d−n
−−→ F∗
−n
d−n
−−→ F−n−1 → . . .
is called a complete resolution for the group G. Then Ĥn(G, M) are
just the cohomologies of the complex HomR(F+, M). If F0 = R and the
surjection F0 → Z maps g to 1, then F−1 ≃ R and d0 is just the trace,
i.e. the multiplication by trG =
∑
x∈G x. It is the case for the resolutions
F and S.
Proposition 3.1. Let G be a finite group, M, N be G-lattices such that
M ∨ N . Then Ĥn(G, M) ≃ Ĥn(G, N) for all n.
Proof. It is known that all groups Ĥn(G, M) (n > 0) are periodic of
period #(G), hence Ĥn(G, M) ≃
⊕
p|#(G) Ĥn(G, M)p. Moreover, as Zp
is flat over Z, Ĥn(G, M)p ≃ Ĥn(G, Mp). It implies he claim.
We denote by DM the dual G-module DM = HomZ(M,T), where
T = Q/Z.
Proposition 3.2. Let M be a G-lattice. Then
Ĥn−1(G, DM) ≃ DĤ−n(G, M), (3.1)
Ĥn(G, DM) ≃ Ĥn+1(G, M∗), (3.2)
Ĥn(G, M∗) ≃ DĤ−n(G, M). (3.3)
If M = Z, (3.3) coincides with [3, Theorem XII.6.6].
150 Cohomologies of finite abelian groups
Proof. (3.1) follows from [3, Corollary XII.6.5].
Consider the exact sequence 0 → Z → Q → T → 0. As M is free
abelian, it gives the exact sequence of G-modules
0 → M∗ → HomZ(M,Q) → DM → 0.
Ĥn(G, HomZ(M,Q)) = 0 for all n, since the multiplication by #(G) is
an automorphism of HomZ(M,Q), whence we obtain (3.2).
(3.3) follows from (3.1) and (3.2).
We also need some information on cohomologies of direct products.
Proposition 3.3. Let N be a normal subgroup of G, F = G/N and
gcd(#(N), #(F )) = 1. For every G-module M and all n
Ĥn(G, M) ≃ Ĥn(N, M)F ⊕ Ĥn(F, MN ). (3.4)
Proof. As #(G) annihilates all Hn(G, M) if n > 0 and the same is true
for N and F , in the Hochschild–Serre spectral sequence
Hp(F, Hq(N, M)) =⇒ Hn(G, M)
all terms with p > 0 and q > 0 are zero. Hence, if n > 0,
Ĥn(G, M) ≃ H0(F, Ĥn(N, M)) ⊕ Ĥn(F, H0(N, M))
= Ĥn(N, M)F ⊕ Ĥn(F, MN ).
Suppose now that the claim holds for Ĥn. Choose an exact sequence
0 → L → P → M → 0, where P is a free ZG-module. Then
Ĥn−1(G, M) ≃ Ĥn(G, L) ≃ Ĥn(N, L)F ⊕ Ĥn(F, LN ).
As P is also free as ZN -module, Ĥn(N, L) ≃ Ĥn−1(N, M). On the other
hand, there are exact sequences
0 → LN → P N → M ′ → 0
and
0 → M ′ → MN → MN /M ′ → 0,
where M ′ is the image of the map P N → MN . Obviously, M ′ ⊇ trN M ,
thus #(N)(MN /M ′) = 0, whence Ĥn(F, MN /M ′) = 0. Therefore,
Ĥn−1(F, MN ) ≃ Ĥn−1(F, M ′) ≃ Ĥn(F, LN ),
since P N is a free ZF -module. So the isomorphism (3.4) holds for Ĥn−1,
hence for all values of n.
Yu. Drozd, A. Plakosh 151
Corollary 3.4. Let G = G1 × G2 with gcd(#(G1), #(G2)) = 1, M =
M1 ⊗Z M2, where Mi is a Gi-lattice (i = 1, 2). Then
Ĥn(G, M) ≃ Ĥn(G1, M1) ⊗Z MG2
2 ⊕ MG1
1 ⊗Z Ĥn(G2, M2).
Proof. As Mi are free abelian, ⊗ZMi is an exact functor and MGi =
MGi
i ⊗Z Mj (j 6= i). Hence Ĥn(Gi, M) ≃ Ĥn(Gi, Mi) ⊗Z Mj , where j 6= i.
So the claim is just a reformulation of Proposition 3.3 for this special
case.
4. Cohomologies of irreducible G-lattices
A G-lattice M is called irreducible if there are no submodules 0 6= N ⊂
M such that M/N is torsion free (i.e. again a G-lattice). Equivalently,
M̃ = Q ⊗Z M is a simple QG-module. If G is a finite abelian group, then
any simple QG-module is defined by a group homomorphism ρ : G → K
×,
where K is a cyclotomic field and the image of ρ generates the ring of
integers of K. Therefore, any two G-lattices in K are of the same genus
[4], so have the same cohomologies. In particular, if M is a G-lattice in
K, so is M∗, hence M∗ ∨ M and
Ĥn(G, M) ≃ Ĥn(G, M∗) ≃ DĤ−n(G, M) ≃ DĤn−1(G, DM). (4.1)
The subgroup of periodic elements of K is cyclic and generated by a
primitive root of unity ζ. Hence, there is an element a ∈ G such that
ρ(a) = ζ. Let G =
∏s
i=1 Ci, where Ci = 〈 ai | aoi
i = 1 〉 are cyclic groups.
We can suppose that a1 = a. Set o = o1. Changing he generators ai, we
can make ρ(ai) = 1 for i 6= 1. Let G′ = 〈 a2, a3, . . . , as 〉, so G = C1 × G′.
Then M ≃ M1 ⊗Z Z, where M1 is M considered as C1-module and Z is
the trivial G′-module. Note that MG = 0, as ζv = v implies v = 0. Hence
Ĥ0(G, M) = 0. Consider the trace T =
∑
g∈G g = (
∑o−1
k=0 ak)(
∑
g∈G′ g).
Obviously,
∑o−1
k=0 ζk = 0, hence TM = 0. It implies that Ĥ−1(G, M) =
H0(G, M) = M/(ζ − 1)M . If o = pm fore some m, then also o(ζ) = pk for
some k, whence NK/Q(1−ζ) = p [1] and Ĥ−1(G, M) = H0(G, M) ≃ Z/pZ.
If o(ζ) is not a degree of a prime number, then NK/Q(1 − ζ) = 1 and
Ĥ−1(G, M) = H0(G, M) = 0 (it also follows from Corollary 3.4)..
Let a finite abelian group G be a direct product G1 × G2 and the
orders of G1 and G2 be coprime. If Ki (i = 1, 2) is a cyclotomic field
arising from a simple QGi-module, then K = K1 ⊗Q K2 is again a
field, hence a simple QG-module, and all simple QG-modules arise in this
152 Cohomologies of finite abelian groups
way. If Mi (i = 1, 2) is a Gi-lattice in Ki, then M = M1 ⊗Z M2 is a G-
lattice in K, unique up to genus. Corollary 3.4 shows that Ĥn(G, M) = 0
if neither M1 nor M2 is trivial. If M1 is non-trivial and M2 is trivial,
then Ĥn(G, M) ≃ Ĥn(G1, M1), and if both M1 and M2 are trivial, then
Ĥn(G, M) ≃ Ĥn(G1,Z) ⊕ Ĥn(G2,Z). Thus we only need to consider the
case of p-groups. Note also that T =
⊕
p Tp and Tp is the quasicyclic
p-group, i.e. the direct limit lim
−→m
Z/pmZ with respect to the natural
embeddings Z/pm Z → Z/pm+1Z . Hence, if M is finitely generated,
DM ≃
⊕
p DMp, where DpM = HomZ(M,Tp). If M is a lattice, the
additive group of DpM is a direct product of several copies of Tp. Moreover,
if G is a p-group, Ĥn(G, DqM) = 0 and DqĤn(G, M) = 0 for q 6= p, so
we can always replace D by Dp in all formulae from Proposition 3.2.
So, let G =
∏s
k=1 Gk, where Gk is a cyclic group of order pmk . We
calculate cohomologies of a non-trivial irreducible G-lattices. Actually, it
is easier to calculate homologies.
Theorem 4.1. Let M be a non-trivial irreducible G-lattice. Then
Hn(G, M) ≃ (Z/pZ)ν(n,s), where
ν(n, s) = (−1)n
n
∑
i=0
(
−s
i
)
. (4.2)
Note that for fixed n the value of ν(n, s) is a polynomial of degree n
with respect to s with the leading coefficient (n!)−1. For instance,
ν(0, s) = 1, ν(1, s) = s − 1,
ν(2, s) =
s2 + s + 2
2
, ν(3, s) =
s3 + 5s − 6
6
.
Proof. We consider G as a direct product G′ × Gs, where G′ =
∏s−1
i=1 Gi,
and suppose that Gs acts trivially on M . Then M can be considered as the
outer tensor product M ′ ×Z Z, where M ′ = M considered as G′-module
and Z is considered as trivial Gs-module. Then we can use the Künneth
formula [2, Corollary V.5.8]:
Hn(G, M) ≃
(
n
⊕
i=0
Hi(G
′, M ′) ⊗Z Hn−i(Gs,Z)
)
⊕
(
n−1
⊕
i=0
TorZ1 (Hi(G
′, M ′), Hn−i−1(Gs,Z))
)
. (4.3)
Yu. Drozd, A. Plakosh 153
Recall that, for a cyclic group C = Z/pmZ,
H0(C,Z) = Z;
Hn(C,Z) =
{
Z/pmZ if n is odd,
0 if n is even;
while for a non-trivial irreducible lattice M
Hn(C, M) =
{
Z/pZ if n is even,
0 if n is odd,
that is,
ν(n, 1) =
{
1 if n is even,
0 if n is odd.
Moreover,
H0(G, M) = Z/pZ,
that is,
ν(0, s) = 1.
Thus (4.1) is valid for n = 0 and for s = 1, the minimal values of n and s.
Therefore, the Künneth formula implies that Hn(G, M) ≃ (Z/pZ)ν(n,s)
for some ν(n, s). Moreover, it implies that
ν(n, s) =
n
∑
k=0
ν(n, s − 1) = ν(n, s − 1) + ν(n − 1, s)
Hence we can prove (4.1) by induction, supposing that it is true for
ν(n, s − 1) and ν(n − 1, s). Then we have
ν(n, s) = ν(n, s − 1) + ν(n − 1, s)
= (−1)n
n
∑
i=0
(
−s + 1
i
)
− (−1)n
n−1
∑
i=0
(
−s
i
)
= (−1)n
n
∑
i=0
((
−s + 1
i
)
−
(
−s
i − 1
))
= (−1)n
n
∑
i=0
(
−s
i
)
.
Note that in this case Ĥ−1(G, M) = H0(G, M) and Ĥ0(G, M) = 0.
The formulae (4.1) and (4.2) give the following result.
154 Cohomologies of finite abelian groups
Corollary 4.2. If M is a non-trivial irreducible G-lattice, then
Ĥn(G, M) ≃ Ĥn−1(G, DM) ≃ (Z/pZ)ν(|n|−1,s).
Analogous calculations give the known result for the trivial G-module
Z (cf. [6, 7]).
Theorem 4.3. If n 6= 0 and m1 > m2 > · · · > ms, then
Ĥn(G,Z) ≃
⊕s
k=1(Z/pmkZ)ν(|n|−1,k)+(−1)n). (4.4)
Recall that Ĥ0(G,Z) ≃ Z/pmZ, where m =
∑s
k=1 mk.
Proof. First of all, the Künneth formula (4.3) implies that Hn(G,Z) is a
direct sum of µ(n, s) cyclic groups so that
µ(n, s) =
n
∑
i=1
µ(i, s − 1) + ε,
where
ε =
{
1 if n is odd,
0 if n is even,
whence
µ(n, s) = µ(n, s − 1) + µ(n − 1, s) + (−1)n−1.
Using induction by s, we obtain that
µ(n, s) = ν(n, s) − (−1)n,
hence
µ(n, s) = µ(n, s − 1) + ν(n − 1, s).
Note that all groups H i(Gs,Z) are of period pms . Therefore, by (4.3),
Hn(G,Z) ≃ Hn(G′,Z) ⊕ (Z/pmsZ)r
for some r. Together with the formula for µ(n, s), it gives that
Hn(G,Z) ≃ Hn(G′,Z) ⊕ (Z/pmsZ)ν(n,s)−(−1)n
.
By induction, we obtain that
Hn(G,Z) ≃
s
⊕
k=1
(Z/pmkZ)ν(n,k)−(−1)n
.
In view of (4.1), it is just the formula (4.4).
Yu. Drozd, A. Plakosh 155
5. Explicit formulae
In this section we find explicit formulae for crossed homomoprhisms
(elements of H1(G, M)) and cocycles (elements of H2(G, M)) for irre-
ducible latticies and their duals (the latter are important, for instance,
in study of Chernikov groups see [5]). We use the resolution defined in
Section 1.
Let G =
∏ms
i=1 Gi, where Gi = 〈 ai | apmi
i = 1 〉 is a cyclic group of
order oi = pmi . We set si = sai
. For a cochain µ : Pn → M we denote by
∂µ its coboundary, that is the composition µdn+1 : Pn+1 → M . Then, if
ξ : P1 → M , i < j,
∂ξ(x2
i ) = siξ(xi),
∂ξ(xixj) = (ai − 1)ξ(xj) − (aj − 1)ξ(xi).
(5.1)
Thus ξ is a cocycle if and only if
siξ(xi) = 0 for all i,
(ai − 1)ξ(xj) = (aj − 1)ξ(xi) for all i 6= j.
(5.2)
If γ : P2 → M , i < j < k, then
∂γ(x3
i ) = (ai − 1)γ(x2) = 0,
∂γ(x2
i xj) = siγ(xixj) + (aj − 1)γ(x2
i ),
∂γ(xix
2
j ) = (ai − 1)γ(x2
j ) − sjγ(xixj),
∂γ(xixjxk) = (ai − 1)γ(xjxk) − (aj − 1)γ(xixk) + (ak − 1)γ(xixj).
Thus γ is a cocycle if and only if
(ai − 1)γ(x2
i ) = 0 for all i,
siγ(xixj) = −(aj − 1)γ(x2
i ), sjγ(xjxi) = (ai − 1)γ(x2
j ),
(aj − 1)γ(xixk) = (ai − 1)γ(xjxk) + (ak − 1)γ(xixj).
(5.3)
Finally, if we identify an element u ∈ M with the homomorphism
P0 → M which maps a to au, then ∂u(xi) = (ai − 1)u.
First suppose that M = Z. Then the element si acts on M as pmi
and the formulae (5.2) show that H1(G,Z) = 0. As ai − 1 acts as 0, the
formulae (5.3) mean that γ is a cocycle if and only if γ(xixj) = 0. The
formulae (5.1) imply that, adding a coboundary, we can reduce γ(x2
i )
modulo pmi . Therefore, H2(G,Z) ≃
⊕ms
i=1 Z/pmiZ and generators of this
156 Cohomologies of finite abelian groups
group can be chosen as the cohomology classes of the cocycles γk : P2 → Z
such that γk(xixj) = 0 for all i, j and γk(x2
i ) = δik.
For the dual module DpZ = Tp, the formulae (5.2) mean that ξ is
a cocylce if and only if pmiξ(xi) = 0. Hence H1(G,Tp) ≃
⊕ms
i=1 Tmi
,
where Tmi
= { u ∈ Tp | pmiu = 0 } (it is a cyclic group of order pmi). As
Tp is divisible, the formulae (5.1) imply that, adding a coboundary to
a 2-dimensional cocycle γ, one can always make γ(x2
i ) = 0. Then the
formulae (5.3) mean that pmij γxixj
= 0, where mij = min{mi, mj}. Hence
H2(G,Tp) ≃
⊕
i<j Tmij
≃
⊕
i<j Z/pmijZ, and generators of this group
are the classes of cocycles γkl (1 6 k < l 6 s) such that γkl(x
2
i ) = 0 for
all i, while γkl(xixj) = δkiδljukl, where ukl is a fixed element of Tp of
order pmkl .
Let now M be a lattice in a cyclotomic field K of order pm such that
a1 acts as the multiplication by the primitive root ζ of unity of order
pm and all ai (i > 1) act trivially. As we can choose any lattice in the
same genus, we can suppose that M = Z[ζ]. Therefore, the formulae (5.2)
show that ξ is a cocycle if and only if ξ(xi) = 0 for i > 1. As ζ − 1 is a
prime element in Z[ζ] with the norm p [1], M/(ζ − 1)M ≃ Z/pZ. Hence,
adding a coboundary ∂u to ξ, one can make ξ(x1) = λ, where λ ∈ Z
is defined modulo p. Thus H1(G, M) ≃ Z/pZ. The formulae (5.3) show
that γ is a cocycle if and only if γ(x2
1) = 0, γ(xixj) = 0 if 1 < i < j
and pmiγ(x1xi) = (ζ − 1)γ(x2
i ). The formulae (5.1) imply that, adding a
coboundary, one can make γ(x1xi) = λi, where λi ∈ Z is defined modulo
p. Then γ(x2
i ) is uniquely defined. Thus H2(G, M) ≃ (Z/pZ)s−1. The
generators of this group are the classes of cocycles γk (1 < k 6 s) such
that γk(x2
1) = γ(xixj) = 0 for all 1 < i < j, γk(x1xi) = δik, γk(xi)
2 = 0 if
i 6= k and (1 − ζ)γk(xk) = pmk .
Consider the dual module DpM . As the multiplication by ζ − 1 is
injective on M , it is surjective on DpM . On the other hand, the subgroup
{ u ∈ DpM | (ζ − 1)u = 0 } is dual to M/(ζ − 1)M , so it is generated by
one element u0 of period p. Thus, adding a couboundary ∂u to a 1-cocycle
ξ, one can make ξ(x1) = 0. Then (ζ − 1)ξ(xi) = 0 if i > 1, whence ξ(xi) =
λiu0, where λi ∈ Z/pZ. Hence H1(G, DpM) ≃ P
s−1
1 ≃ (Z/pZ)s−1. In the
same way, adding a coboundary to a 2-cocycle γ, we can make γ(x1xi) = 0
for i > 1. Then the conditions (5.3) give (ζ − 1)γ(x2
i ) = 0 for all i, whence
γ(x2
i ) = λiu0 (λi ∈ Z/pZ), and (ζ − 1)γ(xixj) = 0 for 1 < i < j, whence
γ(xixj) = λiju0 (λij ∈ Z/pZ). Therefore H2(G, DpM) ≃ T
(s2−s+2)/2
1 ≃
(Z/pZ)(s2−s+2)/2. Th generators of this group are cocycles γk (1 6 k 6 s)
Yu. Drozd, A. Plakosh 157
and γkl (1 < k < l 6 s) such that γk(x1xi) = γkl(x1xi) for i > 1, γk(x2
i ) =
δiku0, γk(xixj) = 0 for i 6= j, γkl(x
2
i = 0 for all i and γkl(xixj) = δikδjlu0.
References
[1] Borevich Z. I., Shafarevich I. R. Number Theory. Nauka, Moscow, 1985.
[2] Brown K. S. Cohomologies of Groups. Springer–Verlag, 1982.
[3] Cartan H., Eilenberg S. Homological Algebra. Princeton Univ. Press, 1956.
[4] Curtis Ch. W., Reiner I. Methods of Representation Theory with Applications to
Finite Groups and Orders, vol. 1. Wiley Interscience Publications, 1981.
[5] Gudivok P. M, Shapochka I. V. On the Chernikov p-groups. Ukr. Mat. Zh. 51,
No. 3, 1999, pp. 291–304.
[6] Lyndon R. C. The cohomology theory of group extensions. Duke Math. J. 15,
No. 1, 1948, pp. 271–292.
[7] Takahashi Sh. Cohomology groups of finite abelian groups. Tohoku Math. J. 4,
No. 3, 1952, pp. 294-302.
Contact information
Yu. Drozd,
A. Plakosh
Institute of Mathematics, National Academy of
Sciences of Ukraine, Tereschenkivska 3, 01601
Kyiv, Ukraine
E-Mail(s): y.a.drozd@gmail.com,
drozd@imath.kiev.ua,
andrianaplakoshmail@gmail.com
Web-page(s): www.imath.kiev.ua/∼drozd
Received by the editors: 26.08.2017.
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