Invariants, derivations, abelianisation
In degree 0 cohomology is the invariants and homology is the coinvariants. In degree 1, cohomology is derivations modulo principal derivations — which reduces to Hom(G, A) when the action is trivial — and homology with trivial integer coefficients is the abelianisation. These low-degree identifications are what make the theory interpretable, and they are the base cases for every dimension-shifting induction.
Learning objectives
- Identify H0 and H0 explicitly.
- Describe H1 as derivations modulo principal derivations.
- Specialise to trivial coefficients.
- State the identification of H1 with the abelianisation.
Section 01Degree zero
With trivial action both reduce to A itself. The content of degree 0 is entirely in the action, and it is the point at which the derived functor recovers the functor being derived.
Section 02Degree one in cohomology
A derivation, or crossed homomorphism, is a map d: G → A with
It is principal when d(g) = g·a − a for some fixed a. Then
If G acts trivially, the derivation condition becomes d(gh) = d(g) + d(h) — an ordinary homomorphism — and every principal derivation is zero. So H1(G, A) = Hom(G, A) = Hom(Gab, A).
| Setting | H1 classifies |
|---|---|
| Trivial action | Homomorphisms G → A |
| General action | Splittings of the semidirect product, up to conjugacy |
| Galois cohomology | Twisted forms — Hilbert 90 is the vanishing statement |
| Coefficients in a G-module of an extension | Complements to the normal subgroup, up to conjugacy |
Section 03Degree one in homology
The proof uses the fundamental sequence: applying − ⊗ℤ[G] ℤ and identifying I/I² with the abelianisation.
This mirrors the topological Hurewicz theorem: the first homology of a space is the abelianisation of its fundamental group. Since H*(G) is the homology of a K(G, 1), the two statements are the same theorem read in two languages.
ReferenceFrequently asked questions
Why are derivations the right notion in degree 1?
Because the bar resolution in degree 1 produces exactly the cocycle condition d(gh) = d(g) + g·d(h), and the coboundaries are exactly the principal derivations. The interpretation is read off the explicit resolution, not imposed.
What does H<sup>1</sup> = 0 mean?
That every derivation is principal, equivalently that all complements to the normal subgroup in the corresponding semidirect product are conjugate. Hilbert's Theorem 90 is precisely such a vanishing statement in Galois cohomology.
Is H<sub>1</sub> ever non-abelian?
No — homology groups are abelian by construction. The abelianisation is exactly what H1 sees, and all non-abelian information about G is invisible in degree 1, appearing instead in degree 2 and above.
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