Two functors, two families of derived functors
Invariants AG is left exact; coinvariants AG is right exact. Deriving them gives cohomology and homology respectively, and both are computed by resolving the trivial module ℤ over ℤ[G] — so Hn(G, A) = Extn(ℤ, A) and Hn(G, A) = Torn(ℤ, A). One resolution serves both.
Learning objectives
- Define invariants and coinvariants and state their exactness.
- Express group (co)homology as Ext and Tor over the group ring.
- State the values in degree 0.
- Write the long exact sequences in the coefficient variable.
Section 01The two functors
Left exact. Right derived functors give cohomology Hn(G, A) = Extnℤ[G](ℤ, A).
Right exact. Left derived functors give homology Hn(G, A) = Torℤ[G]n(ℤ, A).
| Degree | Cohomology | Homology |
|---|---|---|
| 0 | AG, the invariants | AG, the coinvariants |
| 1, trivial coefficients | Hom(G, A) — homomorphisms G → A | Gab ⊗ A for A = ℤ: Gab |
| 2, trivial coefficients | Classifies central extensions | The Schur multiplier for A = ℤ |
Take a projective resolution of the trivial module ℤ over ℤ[G]. Applying Homℤ[G](−, A) gives cohomology; applying − ⊗ℤ[G] A gives homology. The same resolution serves both, which is why they are always developed together.
Section 02Long exact sequences
A short exact sequence of coefficient modules gives long exact sequences in both theories:
The first connecting map is the obstruction to lifting an invariant element of C to an invariant element of B — the prototypical use of H1 as an obstruction group.
Invariants of a quotient are not the quotient of the invariants. The failure is measured by H1, and this single observation is what makes group cohomology useful in Galois theory, in arithmetic and in the descent arguments of algebraic geometry.
Section 03Basic properties
Finite groups
|G| annihilates Hn(G, A) for n ≥ 1. So cohomology of a finite group is torsion in positive degrees, and vanishes when |G| is invertible in A.
Free groups
Hn = 0 for n ≥ 2. Free groups have cohomological dimension 1, matching the fact that their classifying space is a graph.
Trivial group
H0(1, A) = A and Hn = 0 for n ≥ 1.
Induced coefficients
Shapiro's lemma reduces cohomology with induced coefficients to a subgroup computation.
Direct products
Künneth applies, since ℤ[G × H] = ℤ[G] ⊗ ℤ[H].
Topological interpretation
H*(G, ℤ) is the cohomology of the classifying space BG, an Eilenberg–MacLane space K(G, 1).
ReferenceFrequently asked questions
Why does |G| annihilate positive-degree cohomology?
Because the composite of restriction to the trivial subgroup and corestriction back is multiplication by |G|, and the middle term vanishes in positive degrees. The transfer argument is short and is the standard proof.
Is group cohomology the cohomology of a space?
Yes — of the classifying space BG, which is an Eilenberg–MacLane space K(G, 1). The algebraic and topological definitions agree, and each supplies techniques the other lacks.
Do homology and cohomology determine each other?
Via a universal coefficient theorem, up to an extension involving Ext. For finite groups there are additional duality relations, and Tate cohomology unifies the two into a single ℤ-graded theory.
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