The group cohomology of a k-linear G-representation #

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Let k be a commutative ring and G a group. This file defines the group cohomology of A : Rep k G to be the cohomology of the complex $$0 \to \mathrm{Fun}(G^0, A) \to \mathrm{Fun}(G^1, A) \to \mathrm{Fun}(G^2, A) \to \dots$$ with differential $d^n$ sending $f: G^n \to A$ to the function mapping $(g_0, \dots, g_n)$ to $$\rho(g_0)(f(g_1, \dots, g_n)) + \sum_{i = 0}^{n - 1} (-1)^{i + 1}\cdot f(g_0, \dots, g_ig_{i + 1}, \dots, g_n)$$ $$+ (-1)^{n + 1}\cdot f(g_0, \dots, g_{n - 1})$$ (where ρ is the representation attached to A).

We have a k-linear isomorphism $\mathrm{Fun}(G^n, A) \cong \mathrm{Hom}(k[G^{n + 1}], A)$, where the righthand side is morphisms in Rep k G, and the representation on $k[G^{n + 1}]$ is induced by the diagonal action of G. If we conjugate the $n$th differential in $\mathrm{Hom}(P, A)$ by this isomorphism, where P is the standard resolution of k as a trivial k-linear G-representation, then the resulting map agrees with the differential $d^n$ defined above, a fact we prove.

This gives us for free a proof that our $d^n$ squares to zero. It also gives us an isomorphism $\mathrm{H}^n(G, A) \cong \mathrm{Ext}^n(k, A),$ where $\mathrm{Ext}$ is taken in the category Rep k G.

Main definitions #

• group_cohomology.linear_yoneda_obj_resolution A: a complex whose objects are the representation morphisms $\mathrm{Hom}(k[G^{n + 1}], A)$ and whose cohomology is the group cohomology $\mathrm{H}^n(G, A)$.
• group_cohomology.inhomogeneous_cochains A: a complex whose objects are $\mathrm{Fun}(G^n, A)$ and whose cohomology is the group cohomology $\mathrm{H}^n(G, A).$
• group_cohomology.inhomogeneous_cochains_iso A: an isomorphism between the above two complexes.
• group_cohomology A n: this is $\mathrm{H}^n(G, A),$ defined as the $n$th cohomology of the second complex, inhomogeneous_cochains A.
• group_cohomology_iso_Ext A n: an isomorphism $\mathrm{H}^n(G, A) \cong \mathrm{Ext}^n(k, A)$ (where $\mathrm{Ext}$ is taken in the category Rep k G) induced by inhomogeneous_cochains_iso A.

Implementation notes #

Group cohomology is typically stated for G-modules, or equivalently modules over the group ring ℤ[G]. However, ℤ can be generalized to any commutative ring k, which is what we use. Moreover, we express k[G]-module structures on a module k-module A using the Rep definition. We avoid using instances module (monoid_algebra k G) A so that we do not run into possible scalar action diamonds.

TODO #

• API for cohomology in low degree: $\mathrm{H}^0, \mathrm{H}^1$ and $\mathrm{H}^2.$ For example, the inflation-restriction exact sequence.
• The long exact sequence in cohomology attached to a short exact sequence of representations.
• Upgrading group_cohomology_iso_Ext to an isomorphism of derived functors.
• Profinite cohomology.

Longer term:

• The Hochschild-Serre spectral sequence (this is perhaps a good toy example for the theory of spectral sequences in general).