The Group Ring and the Augmentation Ideal

Executive summary

Group cohomology is Ext over the group ring

The group ring ℤ[G] has the elements of G as a basis, with multiplication extending that of G. A module over it is exactly a group representation. The augmentation map sends every group element to 1, and its kernel — the augmentation ideal — is generated by the elements g − 1. That ideal carries the homological content: its abelianisation is H₁(G), and the whole theory is derived functors over ℤ[G].

Learning objectives

Define the group ring and identify modules over it with representations.
Define the augmentation map and its kernel.
Show that the augmentation ideal is free on the elements g − 1 for g ≠ 1.
Relate I/I² to the abelianisation of G.

Section 01The group ring

ℤ[G] = { ∑_{g ∈ G} n_g g : n_g ∈ ℤ, finitely many non-zero }

The dictionary
Group theory	Module theory over ℤ[G]
Representation of G on A	ℤ[G]-module structure on A
Trivial action	The module ℤ with g acting as identity
G-equivariant map	ℤ[G]-module homomorphism
Invariants A^G	Hom_ℤ[G](ℤ, A)
Coinvariants A_G	ℤ ⊗_ℤ[G] A
Subgroup H ≤ G	The subring ℤ[H] ⊆ ℤ[G]

Invariants and coinvariants are the functors to derive

Hⁿ(G, A) is the n-th right derived functor of invariants; H_n(G, A) is the n-th left derived functor of coinvariants. Everything in this stream follows from that one sentence together with the general theory of derived functors.

Section 02The augmentation ideal

The augmentation ε: ℤ[G] → ℤ sends ∑n_gg to ∑n_g. Its kernel IG is the augmentation ideal, and

0 → IG → ℤ[G] →^ε ℤ → 0

is the fundamental short exact sequence of the theory. IG is free as an abelian group on the elements g − 1 for g ≠ 1, and as a ℤ[G]-module it is generated by those elements.

The first homology

I/I² ≅ G^ab, the abelianisation. Combined with the fundamental sequence this gives H₁(G, ℤ) ≅ G^ab — the first non-trivial computation in the subject, and the reason group homology is a genuine generalisation of abelianisation.

Section 03Dimension shifting with the fundamental sequence

AlgorithmReducing degree using I Gin: Hⁿ(G, A) → out: a lower-degree computation

Take the fundamental sequence 0 → IG → ℤ[G] → ℤ → 0.
ℤ[G] is free, hence projective, so its higher cohomology vanishes.
The long exact sequence gives Hⁿ(G, A) ≅ Hⁿ⁻¹(G, Hom(IG, A)) for n ≥ 2. Degree drops by one at the cost of changing the coefficients.
Dually for homology, with Tor and tensor.
Iterating reduces any degree to degree 1, where the interpretation via derivations applies.

ReferenceFrequently asked questions

Why use ℤ[G] rather than a field?

Because integral coefficients retain torsion information that field coefficients destroy. Over a field of characteristic not dividing |G| the group algebra is semisimple by Maschke's theorem and all higher cohomology vanishes, so the interesting cases are integral or modular.

Is the group ring commutative?

Only when G is abelian. For non-abelian G it is a genuinely non-commutative ring, which is why left and right modules must be distinguished and why induction and coinduction differ.

What does the augmentation ideal generate?

As a ℤ[G]-module it is generated by g − 1 for g in any generating set of G. This is why a presentation of G translates directly into a partial free resolution of ℤ, which is the route to Hopf's formula.

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