Cohomology of Finite Cyclic Groups

Executive summary

Period two, forever

For a cyclic group of order m generated by t, the elements t − 1 and the norm N = 1 + t + … + t^m−1 alternate as the differentials of a free resolution of period 2. Cohomology is therefore periodic with period 2 in positive degrees, alternating between the kernel of one map modulo the image of the other. This is the only infinite family that is completely computable by hand, and it is the standard testing ground.

Learning objectives

Write down the periodic resolution.
Compute the cohomology in each degree.
Identify the norm and difference maps.
Explain the periodicity and its relation to Tate cohomology.

Section 01The periodic resolution

… → ℤ[G] →^t−1 ℤ[G] →^N ℤ[G] →^t−1 ℤ[G] →^ε ℤ → 0

Exactness rests on two identities in ℤ[G]: N(t − 1) = 0, and the kernel of multiplication by t − 1 is generated by N, and conversely.

Why the period is 2

Two maps suffice to generate an exact sequence, and applying them alternately continues it indefinitely. Cyclic groups are the basic example of groups with periodic cohomology; the general characterisation is that a finite group has periodic cohomology exactly when every abelian subgroup is cyclic.

Section 02The computation

Applying Hom_ℤ[G](−, A) turns the resolution into A with the maps t − 1 and N acting alternately. So

Cohomology of a cyclic group of order m
Degree	Hⁿ(G, A)
0	A^G = ker(t − 1)
n ≥ 1 odd	ker N / im(t − 1)
n ≥ 2 even	A^G / N·A = ker(t − 1) / im N

With trivial coefficients A = ℤ, the maps are 0 and multiplication by m, giving

H⁰ = ℤ, H^odd = 0, H^{even ≥ 2} = ℤ/mℤ

Homology is the mirror image

H₀ = ℤ, H_odd = ℤ/mℤ, H_{even ≥ 2} = 0. The parity swap between homology and cohomology is a consequence of the universal coefficient theorem and its degree shift.

Section 03Tate cohomology

For a finite group the norm map N: A_G → A^G connects homology and cohomology, and splicing them through its kernel and cokernel gives Tate cohomology Ĥⁿ for all integers n.

Ĥ⁰ = A^G / NA, Ĥ⁻¹ = ker N / IA

Complete periodicity

For a cyclic group, Tate cohomology is periodic with period 2 in every degree, positive and negative. This is what makes cyclic groups the model case in class field theory, where the Herbrand quotient — the ratio of the orders of the two Tate groups — is a central computational tool.

ReferenceFrequently asked questions

Why does the resolution alternate between two maps?

Because the kernel of multiplication by t − 1 on the group ring is generated by N, and the kernel of N is generated by t − 1. Each map's kernel is the other's image, so alternating them produces an exact sequence indefinitely.

Which finite groups have periodic cohomology?

Exactly those in which every abelian subgroup is cyclic — equivalently, those acting freely on a sphere. Cyclic and generalised quaternion groups are the standard examples.

Is the Herbrand quotient useful outside class field theory?

It is used wherever a cyclic group acts and orders need comparing — notably in the arithmetic of number fields and in the study of units. Its multiplicativity in short exact sequences makes it a convenient invariant.

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