The group cohomology of a k-linear G-representation

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.

To talk about cohomology in low degree, please see the file RepresentationTheory.GroupCohomology.LowDegree, which gives simpler expressions for H⁰, H¹, H² than the definition groupCohomology in this file.

Main definitions

• groupCohomology.linearYonedaObjResolution A: a complex whose objects are the representation morphisms $\mathrm{Hom}(k[G^{n + 1}], A)$ and whose cohomology is the group cohomology $\mathrm{H}^n(G, A)$.
• groupCohomology.inhomogeneousCochains A: a complex whose objects are $\mathrm{Fun}(G^n, A)$ and whose cohomology is the group cohomology $\mathrm{H}^n(G, A).$
• groupCohomology.inhomogeneousCochainsIso A: an isomorphism between the above two complexes.
• groupCohomology A n: this is $\mathrm{H}^n(G, A),$ defined as the $n$th cohomology of the second complex, inhomogeneousCochains A.
• groupCohomologyIsoExt 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 inhomogeneousCochainsIso 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 (MonoidAlgebra 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 groupCohomologyIsoExt 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).

@[reducible, inline]

abbrev groupCohomology.linearYonedaObjResolution {k : Type u} {G : Type u} [CommRing k] [Monoid G] (A : Rep k G) : CochainComplex (ModuleCat k) ℕ

The complex Hom(P, A), where P is the standard resolution of k as a trivial k-linear G-representation.

Equations

• groupCohomology.linearYonedaObjResolution A = (groupCohomology.resolution k G).linearYonedaObj k A

theorem groupCohomology.linearYonedaObjResolution_d_apply {k : Type u} {G : Type u} [CommRing k] [Monoid G] {A : Rep k G} (i : ℕ) (j : ℕ) (x : (groupCohomology.resolution k G).X i ⟶ A) : ((groupCohomology.linearYonedaObjResolution A).d i j) x = CategoryTheory.CategoryStruct.comp ((groupCohomology.resolution k G).d j i) x

@[simp]

theorem inhomogeneousCochains.d_apply {k : Type u} {G : Type u} [CommRing k] [Monoid G] (n : ℕ) (A : Rep k G) (f : (Fin n → G) → CoeSort.coe A) (g : Fin (n + 1) → G) : (inhomogeneousCochains.d n A) f g = (A.ρ (g 0)) (f fun (i : Fin n) => g i.succ) + ∑ j : Fin (n + 1), (-1) ^ (↑j + 1) • f (j.contractNth (fun (x x_1 : G) => x * x_1) g)

def inhomogeneousCochains.d {k : Type u} {G : Type u} [CommRing k] [Monoid G] (n : ℕ) (A : Rep k G) : ((Fin n → G) → CoeSort.coe A) →ₗ[k] (Fin (n + 1) → G) → CoeSort.coe A

The differential in the complex of inhomogeneous cochains used to calculate group cohomology.

Equations

• One or more equations did not get rendered due to their size.

theorem inhomogeneousCochains.d_eq {k : Type u} {G : Type u} [CommRing k] (n : ℕ) [Group G] (A : Rep k G) : inhomogeneousCochains.d n A = CategoryTheory.CategoryStruct.comp (Rep.diagonalHomEquiv n A).toModuleIso.inv (CategoryTheory.CategoryStruct.comp ((groupCohomology.linearYonedaObjResolution A).d n (n + 1)) (Rep.diagonalHomEquiv (n + 1) A).toModuleIso.hom)

The theorem that our isomorphism Fun(Gⁿ, A) ≅ Hom(k[Gⁿ⁺¹], A) (where the righthand side is morphisms in Rep k G) commutes with the differentials in the complex of inhomogeneous cochains and the homogeneous linearYonedaObjResolution.

@[reducible, inline]

noncomputable abbrev groupCohomology.inhomogeneousCochains {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) : CochainComplex (ModuleCat k) ℕ

Given a k-linear G-representation A, this is the complex of inhomogeneous cochains $$0 \to \mathrm{Fun}(G^0, A) \to \mathrm{Fun}(G^1, A) \to \mathrm{Fun}(G^2, A) \to \dots$$ which calculates the group cohomology of A.

Equations

• groupCohomology.inhomogeneousCochains A = CochainComplex.of (fun (n : ℕ) => ModuleCat.of k ((Fin n → G) → CoeSort.coe A)) (fun (n : ℕ) => inhomogeneousCochains.d n A) ⋯

@[simp]

theorem groupCohomology.inhomogeneousCochains.d_def {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) (n : ℕ) : (groupCohomology.inhomogeneousCochains A).d n (n + 1) = inhomogeneousCochains.d n A

def groupCohomology.inhomogeneousCochainsIso {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) : groupCohomology.inhomogeneousCochains A ≅ groupCohomology.linearYonedaObjResolution A

Given a k-linear G-representation A, the complex of inhomogeneous cochains is isomorphic to Hom(P, A), where P is the standard resolution of k as a trivial G-representation.

Equations

• groupCohomology.inhomogeneousCochainsIso A = HomologicalComplex.Hom.isoOfComponents (fun (i : ℕ) => (Rep.diagonalHomEquiv i A).toModuleIso.symm) ⋯

@[reducible, inline]

abbrev groupCohomology.cocycles {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) (n : ℕ) : ModuleCat k

The n-cocycles Zⁿ(G, A) of a k-linear G-representation A, i.e. the kernel of the nth differential in the complex of inhomogeneous cochains.

Equations

• groupCohomology.cocycles A n = HomologicalComplex.cycles (groupCohomology.inhomogeneousCochains A) n

@[reducible, inline]

abbrev groupCohomology.iCocycles {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) (n : ℕ) : groupCohomology.cocycles A n ⟶ ModuleCat.of k ((Fin n → G) → CoeSort.coe A)

The natural inclusion of the n-cocycles Zⁿ(G, A) into the n-cochains Cⁿ(G, A).

Equations

• groupCohomology.iCocycles A n = HomologicalComplex.iCycles (groupCohomology.inhomogeneousCochains A) n

@[reducible, inline]

abbrev groupCohomology.toCocycles {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) (i : ℕ) (j : ℕ) : ModuleCat.of k ((Fin i → G) → CoeSort.coe A) ⟶ groupCohomology.cocycles A j

This is the map from i-cochains to j-cocycles induced by the differential in the complex of inhomogeneous cochains.

Equations

• groupCohomology.toCocycles A i j = HomologicalComplex.toCycles (groupCohomology.inhomogeneousCochains A) i j

def groupCohomology {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) (n : ℕ) : ModuleCat k

The group cohomology of a k-linear G-representation A, as the cohomology of its complex of inhomogeneous cochains.

Equations

• groupCohomology A n = HomologicalComplex.homology (groupCohomology.inhomogeneousCochains A) n

@[reducible, inline]

abbrev groupCohomologyπ {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) (n : ℕ) : groupCohomology.cocycles A n ⟶ groupCohomology A n

The natural map from n-cocycles to nth group cohomology for a k-linear G-representation A.

Equations

• groupCohomologyπ A n = HomologicalComplex.homologyπ (groupCohomology.inhomogeneousCochains A) n

def groupCohomologyIsoExt {k : Type u} {G : Type u} [CommRing k] [Group G] (A : Rep k G) (n : ℕ) : groupCohomology A n ≅ ((Ext k (Rep k G) n).obj (Opposite.op (Rep.trivial k G k))).obj A

The nth group cohomology of a k-linear G-representation A is isomorphic to Extⁿ(k, A) (taken in Rep k G), where k is a trivial k-linear G-representation.

Equations

• groupCohomologyIsoExt A n = isoOfQuasiIsoAt (HomotopyEquiv.ofIso (groupCohomology.inhomogeneousCochainsIso A)).hom n ≪≫ (groupCohomology.extIso k G A n).symm