The low-degree cohomology of a k-linear G-representation

Let k be a commutative ring and G a group. This file gives simple expressions for the group cohomology of a k-linear G-representation A in degrees 0, 1 and 2.

In RepresentationTheory.GroupCohomology.Basic, we define the nth group cohomology of A to be the cohomology of a complex inhomogeneousCochains A, whose objects are (Fin n → G) → A; this is unnecessarily unwieldy in low degree. Moreover, cohomology of a complex is defined as an abstract cokernel, whereas the definitions here are explicit quotients of cocycles by coboundaries.

We also show that when the representation on A is trivial, H¹(G, A) ≃ Hom(G, A).

Given an additive or multiplicative abelian group A with an appropriate scalar action of G, we provide support for turning a function f : G → A satisfying the 1-cocycle identity into an element of the oneCocycles of the representation on A (or Additive A) corresponding to the scalar action. We also do this for 1-coboundaries, 2-cocycles and 2-coboundaries. The multiplicative case, starting with the section IsMulCocycle, just mirrors the additive case; unfortunately @[to_additive] can't deal with scalar actions.

The file also contains an identification between the definitions in RepresentationTheory.GroupCohomology.Basic, groupCohomology.cocycles A n and groupCohomology A n, and the nCocycles and Hn A in this file, for n = 0, 1, 2.

Main definitions

• groupCohomology.H0 A: the invariants Aᴳ of the G-representation on A.
• groupCohomology.H1 A: 1-cocycles (i.e. Z¹(G, A) := Ker(d¹ : Fun(G, A) → Fun(G², A)) modulo 1-coboundaries (i.e. B¹(G, A) := Im(d⁰: A → Fun(G, A))).
• groupCohomology.H2 A: 2-cocycles (i.e. Z²(G, A) := Ker(d² : Fun(G², A) → Fun(G³, A)) modulo 2-coboundaries (i.e. B²(G, A) := Im(d¹: Fun(G, A) → Fun(G², A))).
• groupCohomology.H1LequivOfIsTrivial: the isomorphism H¹(G, A) ≃ Hom(G, A) when the representation on A is trivial.
• groupCohomology.isoHn for n = 0, 1, 2: an isomorphism groupCohomology A n ≅ groupCohomology.Hn A.

TODO

• The relationship between H2 and group extensions
• The inflation-restriction exact sequence
• Nonabelian group cohomology

def groupCohomology.zeroCochainsLequiv {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ↑((inhomogeneousCochains A).X 0) ≃ₗ[k] ↑A.V

The 0th object in the complex of inhomogeneous cochains of A : Rep k G is isomorphic to A as a k-module.

Equations

• groupCohomology.zeroCochainsLequiv A = LinearEquiv.funUnique (Fin 0 → G) k ↑A.V

def groupCohomology.oneCochainsLequiv {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ↑((inhomogeneousCochains A).X 1) ≃ₗ[k] G → ↑A.V

The 1st object in the complex of inhomogeneous cochains of A : Rep k G is isomorphic to Fun(G, A) as a k-module.

Equations

• groupCohomology.oneCochainsLequiv A = LinearEquiv.funCongrLeft k (↑A.V) (Equiv.funUnique (Fin 1) G).symm

def groupCohomology.twoCochainsLequiv {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ↑((inhomogeneousCochains A).X 2) ≃ₗ[k] G × G → ↑A.V

The 2nd object in the complex of inhomogeneous cochains of A : Rep k G is isomorphic to Fun(G², A) as a k-module.

Equations

• groupCohomology.twoCochainsLequiv A = LinearEquiv.funCongrLeft k (↑A.V) (piFinTwoEquiv fun (x : Fin 2) => G).symm

def groupCohomology.threeCochainsLequiv {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ↑((inhomogeneousCochains A).X 3) ≃ₗ[k] G × G × G → ↑A.V

The 3rd object in the complex of inhomogeneous cochains of A : Rep k G is isomorphic to Fun(G³, A) as a k-module.

Equations

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

def groupCohomology.dZero {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ↑A.V →ₗ[k] G → ↑A.V

The 0th differential in the complex of inhomogeneous cochains of A : Rep k G, as a k-linear map A → Fun(G, A). It sends (a, g) ↦ ρ_A(g)(a) - a.

Equations

• groupCohomology.dZero A = { toFun := fun (m : ↑A.V) (g : G) => (A.ρ g) m - m, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem groupCohomology.dZero_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (m : ↑A.V) (g : G) : (dZero A) m g = (A.ρ g) m - m

theorem groupCohomology.dZero_ker_eq_invariants {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : LinearMap.ker (dZero A) = A.ρ.invariants

@[simp]

theorem groupCohomology.dZero_eq_zero {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] : dZero A = 0

theorem groupCohomology.dZero_comp_subtype {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : dZero A ∘ₗ A.ρ.invariants.subtype = 0

def groupCohomology.dOne {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (G → ↑A.V) →ₗ[k] G × G → ↑A.V

The 1st differential in the complex of inhomogeneous cochains of A : Rep k G, as a k-linear map Fun(G, A) → Fun(G × G, A). It sends (f, (g₁, g₂)) ↦ ρ_A(g₁)(f(g₂)) - f(g₁g₂) + f(g₁).

Equations

• groupCohomology.dOne A = { toFun := fun (f : G → ↑A.V) (g : G × G) => (A.ρ g.1) (f g.2) - f (g.1 * g.2) + f g.1, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem groupCohomology.dOne_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (f : G → ↑A.V) (g : G × G) : (dOne A) f g = (A.ρ g.1) (f g.2) - f (g.1 * g.2) + f g.1

def groupCohomology.dTwo {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (G × G → ↑A.V) →ₗ[k] G × G × G → ↑A.V

The 2nd differential in the complex of inhomogeneous cochains of A : Rep k G, as a k-linear map Fun(G × G, A) → Fun(G × G × G, A). It sends (f, (g₁, g₂, g₃)) ↦ ρ_A(g₁)(f(g₂, g₃)) - f(g₁g₂, g₃) + f(g₁, g₂g₃) - f(g₁, g₂).

Equations

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

@[simp]

theorem groupCohomology.dTwo_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (f : G × G → ↑A.V) (g : G × G × G) : (dTwo A) f g = (A.ρ g.1) (f (g.2.1, g.2.2)) - f (g.1 * g.2.1, g.2.2) + f (g.1, g.2.1 * g.2.2) - f (g.1, g.2.1)

theorem groupCohomology.dZero_comp_eq {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : dZero A ∘ₗ ↑(zeroCochainsLequiv A) = ↑(oneCochainsLequiv A) ∘ₗ ModuleCat.Hom.hom ((inhomogeneousCochains A).d 0 1)

Let C(G, A) denote the complex of inhomogeneous cochains of A : Rep k G. This lemma says dZero gives a simpler expression for the 0th differential: that is, the following square commutes:

  C⁰(G, A) ---d⁰---> C¹(G, A)
  |                    |
  |                    |
  |                    |
  v                    v
  A ---- dZero ---> Fun(G, A)

where the vertical arrows are zeroCochainsLequiv and oneCochainsLequiv respectively.

theorem groupCohomology.dOne_comp_eq {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : dOne A ∘ₗ ↑(oneCochainsLequiv A) = ↑(twoCochainsLequiv A) ∘ₗ ModuleCat.Hom.hom ((inhomogeneousCochains A).d 1 2)

Let C(G, A) denote the complex of inhomogeneous cochains of A : Rep k G. This lemma says dOne gives a simpler expression for the 1st differential: that is, the following square commutes:

  C¹(G, A) ---d¹-----> C²(G, A)
    |                      |
    |                      |
    |                      |
    v                      v
  Fun(G, A) -dOne-> Fun(G × G, A)

where the vertical arrows are oneCochainsLequiv and twoCochainsLequiv respectively.

theorem groupCohomology.dTwo_comp_eq {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : dTwo A ∘ₗ ↑(twoCochainsLequiv A) = ↑(threeCochainsLequiv A) ∘ₗ ModuleCat.Hom.hom ((inhomogeneousCochains A).d 2 3)

Let C(G, A) denote the complex of inhomogeneous cochains of A : Rep k G. This lemma says dTwo gives a simpler expression for the 2nd differential: that is, the following square commutes:

      C²(G, A) -------d²-----> C³(G, A)
        |                         |
        |                         |
        |                         |
        v                         v
  Fun(G × G, A) --dTwo--> Fun(G × G × G, A)

where the vertical arrows are twoCochainsLequiv and threeCochainsLequiv respectively.

theorem groupCohomology.dOne_comp_dZero {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : dOne A ∘ₗ dZero A = 0

theorem groupCohomology.dTwo_comp_dOne {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : dTwo A ∘ₗ dOne A = 0

def groupCohomology.shortComplexH0 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.ShortComplex (ModuleCat k)

The (exact) short complex A.ρ.invariants ⟶ A ⟶ (G → A).

Equations

• groupCohomology.shortComplexH0 A = CategoryTheory.ShortComplex.moduleCatMk A.ρ.invariants.subtype (groupCohomology.dZero A) ⋯

def groupCohomology.shortComplexH1 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.ShortComplex (ModuleCat k)

The short complex A --dZero--> Fun(G, A) --dOne--> Fun(G × G, A).

Equations

• groupCohomology.shortComplexH1 A = CategoryTheory.ShortComplex.moduleCatMk (groupCohomology.dZero A) (groupCohomology.dOne A) ⋯

def groupCohomology.shortComplexH2 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.ShortComplex (ModuleCat k)

The short complex Fun(G, A) --dOne--> Fun(G × G, A) --dTwo--> Fun(G × G × G, A).

Equations

• groupCohomology.shortComplexH2 A = CategoryTheory.ShortComplex.moduleCatMk (groupCohomology.dOne A) (groupCohomology.dTwo A) ⋯

def groupCohomology.oneCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : Submodule k (G → ↑A.V)

The 1-cocycles Z¹(G, A) of A : Rep k G, defined as the kernel of the map Fun(G, A) → Fun(G × G, A) sending (f, (g₁, g₂)) ↦ ρ_A(g₁)(f(g₂)) - f(g₁g₂) + f(g₁).

Equations

• groupCohomology.oneCocycles A = LinearMap.ker (groupCohomology.dOne A)

def groupCohomology.twoCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : Submodule k (G × G → ↑A.V)

The 2-cocycles Z²(G, A) of A : Rep k G, defined as the kernel of the map Fun(G × G, A) → Fun(G × G × G, A) sending (f, (g₁, g₂, g₃)) ↦ ρ_A(g₁)(f(g₂, g₃)) - f(g₁g₂, g₃) + f(g₁, g₂g₃) - f(g₁, g₂).

Equations

• groupCohomology.twoCocycles A = LinearMap.ker (groupCohomology.dTwo A)

instance groupCohomology.instFunLikeSubtypeForallCarrierVModuleCatMemSubmoduleOneCocycles {k G : Type u} [CommRing k] [Group G] {A : Rep k G} : FunLike (↥(oneCocycles A)) G ↑A.V

Equations

• groupCohomology.instFunLikeSubtypeForallCarrierVModuleCatMemSubmoduleOneCocycles = { coe := Subtype.val, coe_injective' := ⋯ }

@[simp]

theorem groupCohomology.oneCocycles.coe_mk {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G → ↑A.V) (hf : f ∈ oneCocycles A) : ⇑⟨f, hf⟩ = f

@[simp]

theorem groupCohomology.oneCocycles.val_eq_coe {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(oneCocycles A)) : ↑f = ⇑f

theorem groupCohomology.oneCocycles_ext {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(oneCocycles A)} (h : ∀ (g : G), f₁ g = f₂ g) : f₁ = f₂

theorem groupCohomology.oneCocycles_ext_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(oneCocycles A)} : f₁ = f₂ ↔ ∀ (g : G), f₁ g = f₂ g

theorem groupCohomology.mem_oneCocycles_def {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G → ↑A.V) : f ∈ oneCocycles A ↔ ∀ (g h : G), (A.ρ g) (f h) - f (g * h) + f g = 0

theorem groupCohomology.mem_oneCocycles_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G → ↑A.V) : f ∈ oneCocycles A ↔ ∀ (g h : G), f (g * h) = (A.ρ g) (f h) + f g

@[simp]

theorem groupCohomology.oneCocycles_map_one {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(oneCocycles A)) : f 1 = 0

@[simp]

theorem groupCohomology.oneCocycles_map_inv {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(oneCocycles A)) (g : G) : (A.ρ g) (f g⁻¹) = -f g

theorem groupCohomology.dZero_apply_mem_oneCocycles {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (x : ↑A.V) : (dZero A) x ∈ oneCocycles A

theorem groupCohomology.oneCocycles_map_mul_of_isTrivial {k G : Type u} [CommRing k] [Group G] {A : Rep k G} [A.IsTrivial] (f : ↥(oneCocycles A)) (g h : G) : f (g * h) = f g + f h

theorem groupCohomology.mem_oneCocycles_of_addMonoidHom {k G : Type u} [CommRing k] [Group G] {A : Rep k G} [A.IsTrivial] (f : Additive G →+ ↑A.V) : ⇑f ∘ ⇑Additive.ofMul ∈ oneCocycles A

def groupCohomology.oneCocyclesLequivOfIsTrivial {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [hA : A.IsTrivial] : ↥(oneCocycles A) ≃ₗ[k] Additive G →+ ↑A.V

When A : Rep k G is a trivial representation of G, Z¹(G, A) is isomorphic to the group homs G → A.

Equations

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

@[simp]

theorem groupCohomology.oneCocyclesLequivOfIsTrivial_symm_apply_coe {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [hA : A.IsTrivial] (f : Additive G →+ ↑A.V) (a : Additive G) : ↑((oneCocyclesLequivOfIsTrivial A).symm f) a = f a

@[simp]

theorem groupCohomology.oneCocyclesLequivOfIsTrivial_apply_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [hA : A.IsTrivial] (f : ↥(oneCocycles A)) (a✝ : Additive G) : ((oneCocyclesLequivOfIsTrivial A) f) a✝ = (⇑f ∘ ⇑Additive.toMul) a✝

instance groupCohomology.instFunLikeSubtypeForallProdCarrierVModuleCatMemSubmoduleTwoCocycles {k G : Type u} [CommRing k] [Group G] {A : Rep k G} : FunLike (↥(twoCocycles A)) (G × G) ↑A.V

Equations

• groupCohomology.instFunLikeSubtypeForallProdCarrierVModuleCatMemSubmoduleTwoCocycles = { coe := Subtype.val, coe_injective' := ⋯ }

@[simp]

theorem groupCohomology.twoCocycles.coe_mk {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G × G → ↑A.V) (hf : f ∈ twoCocycles A) : ⇑⟨f, hf⟩ = f

@[simp]

theorem groupCohomology.twoCocycles.val_eq_coe {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(twoCocycles A)) : ↑f = ⇑f

theorem groupCohomology.twoCocycles_ext {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(twoCocycles A)} (h : ∀ (g h : G), f₁ (g, h) = f₂ (g, h)) : f₁ = f₂

theorem groupCohomology.twoCocycles_ext_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(twoCocycles A)} : f₁ = f₂ ↔ ∀ (g h : G), f₁ (g, h) = f₂ (g, h)

theorem groupCohomology.mem_twoCocycles_def {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G × G → ↑A.V) : f ∈ twoCocycles A ↔ ∀ (g h j : G), (A.ρ g) (f (h, j)) - f (g * h, j) + f (g, h * j) - f (g, h) = 0

theorem groupCohomology.mem_twoCocycles_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G × G → ↑A.V) : f ∈ twoCocycles A ↔ ∀ (g h j : G), f (g * h, j) + f (g, h) = (A.ρ g) (f (h, j)) + f (g, h * j)

theorem groupCohomology.twoCocycles_map_one_fst {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(twoCocycles A)) (g : G) : f (1, g) = f (1, 1)

theorem groupCohomology.twoCocycles_map_one_snd {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(twoCocycles A)) (g : G) : f (g, 1) = (A.ρ g) (f (1, 1))

theorem groupCohomology.twoCocycles_ρ_map_inv_sub_map_inv {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(twoCocycles A)) (g : G) : (A.ρ g) (f (g⁻¹, g)) - f (g, g⁻¹) = f (1, 1) - f (g, 1)

theorem groupCohomology.dOne_apply_mem_twoCocycles {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (x : G → ↑A.V) : (dOne A) x ∈ twoCocycles A

def groupCohomology.oneCoboundaries {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : Submodule k (G → ↑A.V)

The 1-coboundaries B¹(G, A) of A : Rep k G, defined as the image of the map A → Fun(G, A) sending (a, g) ↦ ρ_A(g)(a) - a.

Equations

• groupCohomology.oneCoboundaries A = LinearMap.range (groupCohomology.dZero A)

def groupCohomology.twoCoboundaries {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : Submodule k (G × G → ↑A.V)

The 2-coboundaries B²(G, A) of A : Rep k G, defined as the image of the map Fun(G, A) → Fun(G × G, A) sending (f, (g₁, g₂)) ↦ ρ_A(g₁)(f(g₂)) - f(g₁g₂) + f(g₁).

Equations

• groupCohomology.twoCoboundaries A = LinearMap.range (groupCohomology.dOne A)

instance groupCohomology.instFunLikeSubtypeForallCarrierVModuleCatMemSubmoduleOneCoboundaries {k G : Type u} [CommRing k] [Group G] {A : Rep k G} : FunLike (↥(oneCoboundaries A)) G ↑A.V

Equations

• groupCohomology.instFunLikeSubtypeForallCarrierVModuleCatMemSubmoduleOneCoboundaries = { coe := Subtype.val, coe_injective' := ⋯ }

@[simp]

theorem groupCohomology.oneCoboundaries.coe_mk {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G → ↑A.V) (hf : f ∈ oneCoboundaries A) : ⇑⟨f, hf⟩ = f

@[simp]

theorem groupCohomology.oneCoboundaries.val_eq_coe {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(oneCoboundaries A)) : ↑f = ⇑f

theorem groupCohomology.oneCoboundaries_ext {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(oneCoboundaries A)} (h : ∀ (g : G), f₁ g = f₂ g) : f₁ = f₂

theorem groupCohomology.oneCoboundaries_ext_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(oneCoboundaries A)} : f₁ = f₂ ↔ ∀ (g : G), f₁ g = f₂ g

theorem groupCohomology.oneCoboundaries_le_oneCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : oneCoboundaries A ≤ oneCocycles A

@[reducible, inline]

abbrev groupCohomology.oneCoboundariesToOneCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ↥(oneCoboundaries A) →ₗ[k] ↥(oneCocycles A)

Natural inclusion B¹(G, A) →ₗ[k] Z¹(G, A).

Equations

• groupCohomology.oneCoboundariesToOneCocycles A = Submodule.inclusion ⋯

@[simp]

theorem groupCohomology.oneCoboundariesToOneCocycles_apply {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (x : ↥(oneCoboundaries A)) : ⇑((oneCoboundariesToOneCocycles A) x) = ↑x

theorem groupCohomology.oneCoboundaries_eq_bot_of_isTrivial {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] : oneCoboundaries A = ⊥

instance groupCohomology.instFunLikeSubtypeForallProdCarrierVModuleCatMemSubmoduleTwoCoboundaries {k G : Type u} [CommRing k] [Group G] {A : Rep k G} : FunLike (↥(twoCoboundaries A)) (G × G) ↑A.V

Equations

• groupCohomology.instFunLikeSubtypeForallProdCarrierVModuleCatMemSubmoduleTwoCoboundaries = { coe := Subtype.val, coe_injective' := ⋯ }

@[simp]

theorem groupCohomology.twoCoboundaries.coe_mk {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G × G → ↑A.V) (hf : f ∈ twoCoboundaries A) : ⇑⟨f, hf⟩ = f

@[simp]

theorem groupCohomology.twoCoboundaries.val_eq_coe {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(twoCoboundaries A)) : ↑f = ⇑f

theorem groupCohomology.twoCoboundaries_ext {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(twoCoboundaries A)} (h : ∀ (g h : G), f₁ (g, h) = f₂ (g, h)) : f₁ = f₂

theorem groupCohomology.twoCoboundaries_ext_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f₁ f₂ : ↥(twoCoboundaries A)} : f₁ = f₂ ↔ ∀ (g h : G), f₁ (g, h) = f₂ (g, h)

theorem groupCohomology.twoCoboundaries_le_twoCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : twoCoboundaries A ≤ twoCocycles A

@[reducible, inline]

abbrev groupCohomology.twoCoboundariesToTwoCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ↥(twoCoboundaries A) →ₗ[k] ↥(twoCocycles A)

Natural inclusion B²(G, A) →ₗ[k] Z²(G, A).

Equations

• groupCohomology.twoCoboundariesToTwoCocycles A = Submodule.inclusion ⋯

@[simp]

theorem groupCohomology.twoCoboundariesToTwoCocycles_apply {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (x : ↥(twoCoboundaries A)) : ⇑((twoCoboundariesToTwoCocycles A) x) = ↑x

def groupCohomology.IsOneCocycle {G : Type u_1} {A : Type u_2} [Mul G] [AddCommGroup A] [SMul G A] (f : G → A) : Prop

A function f : G → A satisfies the 1-cocycle condition if f(gh) = g • f(h) + f(g) for all g, h : G.

Equations

• groupCohomology.IsOneCocycle f = ∀ (g h : G), f (g * h) = g • f h + f g

def groupCohomology.IsTwoCocycle {G : Type u_1} {A : Type u_2} [Mul G] [AddCommGroup A] [SMul G A] (f : G × G → A) : Prop

A function f : G × G → A satisfies the 2-cocycle condition if f(gh, j) + f(g, h) = g • f(h, j) + f(g, hj) for all g, h : G.

Equations

• groupCohomology.IsTwoCocycle f = ∀ (g h j : G), f (g * h, j) + f (g, h) = g • f (h, j) + f (g, h * j)

theorem groupCohomology.map_one_of_isOneCocycle {G : Type u_1} {A : Type u_2} [Monoid G] [AddCommGroup A] [MulAction G A] {f : G → A} (hf : IsOneCocycle f) : f 1 = 0

theorem groupCohomology.map_one_fst_of_isTwoCocycle {G : Type u_1} {A : Type u_2} [Monoid G] [AddCommGroup A] [MulAction G A] {f : G × G → A} (hf : IsTwoCocycle f) (g : G) : f (1, g) = f (1, 1)

theorem groupCohomology.map_one_snd_of_isTwoCocycle {G : Type u_1} {A : Type u_2} [Monoid G] [AddCommGroup A] [MulAction G A] {f : G × G → A} (hf : IsTwoCocycle f) (g : G) : f (g, 1) = g • f (1, 1)

theorem groupCohomology.map_inv_of_isOneCocycle {G : Type u_1} {A : Type u_2} [Group G] [AddCommGroup A] [MulAction G A] {f : G → A} (hf : IsOneCocycle f) (g : G) : g • f g⁻¹ = -f g

theorem groupCohomology.smul_map_inv_sub_map_inv_of_isTwoCocycle {G : Type u_1} {A : Type u_2} [Group G] [AddCommGroup A] [MulAction G A] {f : G × G → A} (hf : IsTwoCocycle f) (g : G) : g • f (g⁻¹, g) - f (g, g⁻¹) = f (1, 1) - f (g, 1)

def groupCohomology.IsOneCoboundary {G : Type u_1} {A : Type u_2} [AddCommGroup A] [SMul G A] (f : G → A) : Prop

A function f : G → A satisfies the 1-coboundary condition if there's x : A such that g • x - x = f(g) for all g : G.

Equations

• groupCohomology.IsOneCoboundary f = ∃ (x : A), ∀ (g : G), g • x - x = f g

def groupCohomology.IsTwoCoboundary {G : Type u_1} {A : Type u_2} [Mul G] [AddCommGroup A] [SMul G A] (f : G × G → A) : Prop

A function f : G × G → A satisfies the 2-coboundary condition if there's x : G → A such that g • x(h) - x(gh) + x(g) = f(g, h) for all g, h : G.

Equations

• groupCohomology.IsTwoCoboundary f = ∃ (x : G → A), ∀ (g h : G), g • x h - x (g * h) + x g = f (g, h)

def groupCohomology.oneCocyclesOfIsOneCocycle {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G → A} (hf : IsOneCocycle f) : ↥(oneCocycles (Rep.ofDistribMulAction k G A))

Given a k-module A with a compatible DistribMulAction of G, and a function f : G → A satisfying the 1-cocycle condition, produces a 1-cocycle for the representation on A induced by the DistribMulAction.

Equations

• groupCohomology.oneCocyclesOfIsOneCocycle hf = ⟨f, ⋯⟩

@[simp]

theorem groupCohomology.oneCocyclesOfIsOneCocycle_coe {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G → A} (hf : IsOneCocycle f) (a✝ : G) : ↑(oneCocyclesOfIsOneCocycle hf) a✝ = f a✝

theorem groupCohomology.isOneCocycle_of_mem_oneCocycles {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] (f : G → A) (hf : f ∈ oneCocycles (Rep.ofDistribMulAction k G A)) : IsOneCocycle f

def groupCohomology.oneCoboundariesOfIsOneCoboundary {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G → A} (hf : IsOneCoboundary f) : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))

Given a k-module A with a compatible DistribMulAction of G, and a function f : G → A satisfying the 1-coboundary condition, produces a 1-coboundary for the representation on A induced by the DistribMulAction.

Equations

• groupCohomology.oneCoboundariesOfIsOneCoboundary hf = ⟨f, ⋯⟩

@[simp]

theorem groupCohomology.oneCoboundariesOfIsOneCoboundary_coe {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G → A} (hf : IsOneCoboundary f) (a✝ : G) : ↑(oneCoboundariesOfIsOneCoboundary hf) a✝ = f a✝

theorem groupCohomology.isOneCoboundary_of_mem_oneCoboundaries {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] (f : G → A) (hf : f ∈ oneCoboundaries (Rep.ofDistribMulAction k G A)) : IsOneCoboundary f

def groupCohomology.twoCocyclesOfIsTwoCocycle {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G × G → A} (hf : IsTwoCocycle f) : ↥(twoCocycles (Rep.ofDistribMulAction k G A))

Given a k-module A with a compatible DistribMulAction of G, and a function f : G × G → A satisfying the 2-cocycle condition, produces a 2-cocycle for the representation on A induced by the DistribMulAction.

Equations

• groupCohomology.twoCocyclesOfIsTwoCocycle hf = ⟨f, ⋯⟩

@[simp]

theorem groupCohomology.twoCocyclesOfIsTwoCocycle_coe {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G × G → A} (hf : IsTwoCocycle f) (a✝ : G × G) : ↑(twoCocyclesOfIsTwoCocycle hf) a✝ = f a✝

theorem groupCohomology.isTwoCocycle_of_mem_twoCocycles {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] (f : G × G → A) (hf : f ∈ twoCocycles (Rep.ofDistribMulAction k G A)) : IsTwoCocycle f

def groupCohomology.twoCoboundariesOfIsTwoCoboundary {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G × G → A} (hf : IsTwoCoboundary f) : ↥(twoCoboundaries (Rep.ofDistribMulAction k G A))

Given a k-module A with a compatible DistribMulAction of G, and a function f : G × G → A satisfying the 2-coboundary condition, produces a 2-coboundary for the representation on A induced by the DistribMulAction.

Equations

• groupCohomology.twoCoboundariesOfIsTwoCoboundary hf = ⟨f, ⋯⟩

@[simp]

theorem groupCohomology.twoCoboundariesOfIsTwoCoboundary_coe {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] {f : G × G → A} (hf : IsTwoCoboundary f) (a✝ : G × G) : ↑(twoCoboundariesOfIsTwoCoboundary hf) a✝ = f a✝

theorem groupCohomology.isTwoCoboundary_of_mem_twoCoboundaries {k G A : Type u} [CommRing k] [Group G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] (f : G × G → A) (hf : f ∈ twoCoboundaries (Rep.ofDistribMulAction k G A)) : IsTwoCoboundary f

The next few sections, until the section Cohomology, are a multiplicative copy of the previous few sections beginning with IsCocycle. Unfortunately @[to_additive] doesn't work with scalar actions.

def groupCohomology.IsMulOneCocycle {G : Type u_1} {M : Type u_2} [Mul G] [CommGroup M] [SMul G M] (f : G → M) : Prop

A function f : G → M satisfies the multiplicative 1-cocycle condition if f(gh) = g • f(h) * f(g) for all g, h : G.

Equations

• groupCohomology.IsMulOneCocycle f = ∀ (g h : G), f (g * h) = g • f h * f g

def groupCohomology.IsMulTwoCocycle {G : Type u_1} {M : Type u_2} [Mul G] [CommGroup M] [SMul G M] (f : G × G → M) : Prop

A function f : G × G → M satisfies the multiplicative 2-cocycle condition if f(gh, j) * f(g, h) = g • f(h, j) * f(g, hj) for all g, h : G.

Equations

• groupCohomology.IsMulTwoCocycle f = ∀ (g h j : G), f (g * h, j) * f (g, h) = g • f (h, j) * f (g, h * j)

theorem groupCohomology.map_one_of_isMulOneCocycle {G : Type u_1} {M : Type u_2} [Monoid G] [CommGroup M] [MulAction G M] {f : G → M} (hf : IsMulOneCocycle f) : f 1 = 1

theorem groupCohomology.map_one_fst_of_isMulTwoCocycle {G : Type u_1} {M : Type u_2} [Monoid G] [CommGroup M] [MulAction G M] {f : G × G → M} (hf : IsMulTwoCocycle f) (g : G) : f (1, g) = f (1, 1)

theorem groupCohomology.map_one_snd_of_isMulTwoCocycle {G : Type u_1} {M : Type u_2} [Monoid G] [CommGroup M] [MulAction G M] {f : G × G → M} (hf : IsMulTwoCocycle f) (g : G) : f (g, 1) = g • f (1, 1)

theorem groupCohomology.map_inv_of_isMulOneCocycle {G : Type u_1} {M : Type u_2} [Group G] [CommGroup M] [MulAction G M] {f : G → M} (hf : IsMulOneCocycle f) (g : G) : g • f g⁻¹ = (f g)⁻¹

theorem groupCohomology.smul_map_inv_div_map_inv_of_isMulTwoCocycle {G : Type u_1} {M : Type u_2} [Group G] [CommGroup M] [MulAction G M] {f : G × G → M} (hf : IsMulTwoCocycle f) (g : G) : g • f (g⁻¹, g) / f (g, g⁻¹) = f (1, 1) / f (g, 1)

def groupCohomology.IsMulOneCoboundary {G : Type u_1} {M : Type u_2} [CommGroup M] [SMul G M] (f : G → M) : Prop

A function f : G → M satisfies the multiplicative 1-coboundary condition if there's x : M such that g • x / x = f(g) for all g : G.

Equations

• groupCohomology.IsMulOneCoboundary f = ∃ (x : M), ∀ (g : G), g • x / x = f g

def groupCohomology.IsMulTwoCoboundary {G : Type u_1} {M : Type u_2} [Mul G] [CommGroup M] [SMul G M] (f : G × G → M) : Prop

A function f : G × G → M satisfies the 2-coboundary condition if there's x : G → M such that g • x(h) / x(gh) * x(g) = f(g, h) for all g, h : G.

Equations

• groupCohomology.IsMulTwoCoboundary f = ∃ (x : G → M), ∀ (g h : G), g • x h / x (g * h) * x g = f (g, h)

def groupCohomology.oneCocyclesOfIsMulOneCocycle {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] {f : G → M} (hf : IsMulOneCocycle f) : ↥(oneCocycles (Rep.ofMulDistribMulAction G M))

Given an abelian group M with a MulDistribMulAction of G, and a function f : G → M satisfying the multiplicative 1-cocycle condition, produces a 1-cocycle for the representation on Additive M induced by the MulDistribMulAction.

Equations

• groupCohomology.oneCocyclesOfIsMulOneCocycle hf = ⟨⇑Additive.ofMul ∘ f, ⋯⟩

@[simp]

theorem groupCohomology.oneCocyclesOfIsMulOneCocycle_coe {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] {f : G → M} (hf : IsMulOneCocycle f) (a✝ : G) : ↑(oneCocyclesOfIsMulOneCocycle hf) a✝ = (⇑Additive.ofMul ∘ f) a✝

theorem groupCohomology.isMulOneCocycle_of_mem_oneCocycles {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] (f : G → M) (hf : f ∈ oneCocycles (Rep.ofMulDistribMulAction G M)) : IsMulOneCocycle (⇑Additive.toMul ∘ f)

def groupCohomology.oneCoboundariesOfIsMulOneCoboundary {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] {f : G → M} (hf : IsMulOneCoboundary f) : ↥(oneCoboundaries (Rep.ofMulDistribMulAction G M))

Given an abelian group M with a MulDistribMulAction of G, and a function f : G → M satisfying the multiplicative 1-coboundary condition, produces a 1-coboundary for the representation on Additive M induced by the MulDistribMulAction.

Equations

• groupCohomology.oneCoboundariesOfIsMulOneCoboundary hf = ⟨f, ⋯⟩

@[simp]

theorem groupCohomology.oneCoboundariesOfIsMulOneCoboundary_coe {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] {f : G → M} (hf : IsMulOneCoboundary f) (a✝ : G) : ↑(oneCoboundariesOfIsMulOneCoboundary hf) a✝ = f a✝

theorem groupCohomology.isMulOneCoboundary_of_mem_oneCoboundaries {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] (f : G → M) (hf : f ∈ oneCoboundaries (Rep.ofMulDistribMulAction G M)) : IsMulOneCoboundary (⇑Additive.ofMul ∘ f)

def groupCohomology.twoCocyclesOfIsMulTwoCocycle {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] {f : G × G → M} (hf : IsMulTwoCocycle f) : ↥(twoCocycles (Rep.ofMulDistribMulAction G M))

Given an abelian group M with a MulDistribMulAction of G, and a function f : G × G → M satisfying the multiplicative 2-cocycle condition, produces a 2-cocycle for the representation on Additive M induced by the MulDistribMulAction.

Equations

• groupCohomology.twoCocyclesOfIsMulTwoCocycle hf = ⟨⇑Additive.ofMul ∘ f, ⋯⟩

@[simp]

theorem groupCohomology.twoCocyclesOfIsMulTwoCocycle_coe {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] {f : G × G → M} (hf : IsMulTwoCocycle f) (a✝ : G × G) : ↑(twoCocyclesOfIsMulTwoCocycle hf) a✝ = (⇑Additive.ofMul ∘ f) a✝

theorem groupCohomology.isMulTwoCocycle_of_mem_twoCocycles {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] (f : G × G → M) (hf : f ∈ twoCocycles (Rep.ofMulDistribMulAction G M)) : IsMulTwoCocycle (⇑Additive.toMul ∘ f)

def groupCohomology.twoCoboundariesOfIsMulTwoCoboundary {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] {f : G × G → M} (hf : IsMulTwoCoboundary f) : ↥(twoCoboundaries (Rep.ofMulDistribMulAction G M))

Given an abelian group M with a MulDistribMulAction of G, and a function f : G × G → M satisfying the multiplicative 2-coboundary condition, produces a 2-coboundary for the representation on M induced by the MulDistribMulAction.

Equations

• groupCohomology.twoCoboundariesOfIsMulTwoCoboundary hf = ⟨f, ⋯⟩

theorem groupCohomology.isMulTwoCoboundary_of_mem_twoCoboundaries {G M : Type} [Group G] [CommGroup M] [MulDistribMulAction G M] (f : G × G → M) (hf : f ∈ twoCoboundaries (Rep.ofMulDistribMulAction G M)) : IsMulTwoCoboundary (⇑Additive.toMul ∘ f)

@[reducible, inline]

abbrev groupCohomology.H0 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ModuleCat k

We define the 0th group cohomology of a k-linear G-representation A, H⁰(G, A), to be the invariants of the representation, Aᴳ.

Equations

• groupCohomology.H0 A = ModuleCat.of k ↥A.ρ.invariants

@[reducible, inline]

abbrev groupCohomology.H1 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ModuleCat k

We define the 1st group cohomology of a k-linear G-representation A, H¹(G, A), to be 1-cocycles (i.e. Z¹(G, A) := Ker(d¹ : Fun(G, A) → Fun(G², A)) modulo 1-coboundaries (i.e. B¹(G, A) := Im(d⁰: A → Fun(G, A))).

Equations

• groupCohomology.H1 A = (groupCohomology.shortComplexH1 A).moduleCatHomology

@[reducible, inline]

abbrev groupCohomology.H1π {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ModuleCat.of k ↥(oneCocycles A) ⟶ H1 A

The quotient map Z¹(G, A) → H¹(G, A).

Equations

• groupCohomology.H1π A = (groupCohomology.shortComplexH1 A).moduleCatHomologyπ

theorem groupCohomology.H1π_eq_zero_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (x : ↥(oneCocycles A)) : (CategoryTheory.ConcreteCategory.hom (H1π A)) x = 0 ↔ ⇑x ∈ oneCoboundaries A

@[reducible, inline]

abbrev groupCohomology.H2 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ModuleCat k

We define the 2nd group cohomology of a k-linear G-representation A, H²(G, A), to be 2-cocycles (i.e. Z²(G, A) := Ker(d² : Fun(G², A) → Fun(G³, A)) modulo 2-coboundaries (i.e. B²(G, A) := Im(d¹: Fun(G, A) → Fun(G², A))).

Equations

• groupCohomology.H2 A = (groupCohomology.shortComplexH2 A).moduleCatHomology

@[reducible, inline]

abbrev groupCohomology.H2π {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : ModuleCat.of k ↥(twoCocycles A) ⟶ H2 A

The quotient map Z²(G, A) → H²(G, A).

Equations

• groupCohomology.H2π A = (groupCohomology.shortComplexH2 A).moduleCatHomologyπ

theorem groupCohomology.H2π_eq_zero_iff {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (x : ↥(twoCocycles A)) : (CategoryTheory.ConcreteCategory.hom (H2π A)) x = 0 ↔ ⇑x ∈ twoCoboundaries A

def groupCohomology.H0LequivOfIsTrivial {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] : ↑(H0 A) ≃ₗ[k] ↑A.V

When the representation on A is trivial, then H⁰(G, A) is all of A.

Equations

• groupCohomology.H0LequivOfIsTrivial A = LinearEquiv.ofTop A.ρ.invariants ⋯

@[simp]

theorem groupCohomology.H0LequivOfIsTrivial_eq_subtype {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] : ↑(H0LequivOfIsTrivial A) = A.ρ.invariants.subtype

theorem groupCohomology.H0LequivOfIsTrivial_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] (x : ↑(H0 A)) : (H0LequivOfIsTrivial A) x = ↑x

@[simp]

theorem groupCohomology.H0LequivOfIsTrivial_symm_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] (x : ↑A.V) : ↑((H0LequivOfIsTrivial A).symm x) = x

def groupCohomology.H1LequivOfIsTrivial {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] : ↑(H1 A) ≃ₗ[k] Additive G →+ ↑A.V

When A : Rep k G is a trivial representation of G, H¹(G, A) is isomorphic to the group homs G → A.

Equations

• groupCohomology.H1LequivOfIsTrivial A = ((LinearMap.range (groupCohomology.shortComplexH1 A).moduleCatToCycles).quotEquivOfEqBot ⋯).trans (groupCohomology.oneCocyclesLequivOfIsTrivial A)

theorem groupCohomology.H1LequivOfIsTrivial_comp_H1π {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] : ↑(H1LequivOfIsTrivial A) ∘ₗ ModuleCat.Hom.hom (H1π A) = ↑(oneCocyclesLequivOfIsTrivial A)

@[simp]

theorem groupCohomology.H1LequivOfIsTrivial_H1_π_apply_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] (f : ↥(oneCocycles A)) (x : Additive G) : ((H1LequivOfIsTrivial A) (Submodule.Quotient.mk f)) x = f (Additive.toMul x)

@[simp]

theorem groupCohomology.H1LequivOfIsTrivial_symm_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial] (f : Additive G →+ ↑A.V) : (H1LequivOfIsTrivial A).symm f = (CategoryTheory.ConcreteCategory.hom (H1π A)) ((oneCocyclesLequivOfIsTrivial A).symm f)

instance groupCohomology.instMonoModuleCatFShortComplexH0 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.Mono (shortComplexH0 A).f

theorem groupCohomology.shortComplexH0_exact {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (shortComplexH0 A).Exact

def groupCohomology.dZeroArrowIso {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.Arrow.mk ((inhomogeneousCochains A).d 0 1) ≅ CategoryTheory.Arrow.mk (ModuleCat.ofHom (dZero A))

The arrow A --dZero--> Fun(G, A) is isomorphic to the differential (inhomogeneousCochains A).d 0 1 of the complex of inhomogeneous cochains of A.

Equations

• groupCohomology.dZeroArrowIso A = CategoryTheory.Arrow.isoMk (groupCohomology.zeroCochainsLequiv A).toModuleIso (groupCohomology.oneCochainsLequiv A).toModuleIso ⋯

@[simp]

theorem groupCohomology.dZeroArrowIso_inv_right {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (dZeroArrowIso A).inv.right = ModuleCat.ofHom ↑(oneCochainsLequiv A).symm

@[simp]

theorem groupCohomology.dZeroArrowIso_hom_left {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (dZeroArrowIso A).hom.left = ModuleCat.ofHom ↑(zeroCochainsLequiv A)

@[simp]

theorem groupCohomology.dZeroArrowIso_hom_right {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (dZeroArrowIso A).hom.right = ModuleCat.ofHom ↑(oneCochainsLequiv A)

@[simp]

theorem groupCohomology.dZeroArrowIso_inv_left {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (dZeroArrowIso A).inv.left = ModuleCat.ofHom ↑(zeroCochainsLequiv A).symm

def groupCohomology.isoZeroCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : cocycles A 0 ≅ H0 A

The 0-cocycles of the complex of inhomogeneous cochains of A are isomorphic to A.ρ.invariants, which is a simpler type.

Equations

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

@[simp]

theorem groupCohomology.isoZeroCocycles_hom_comp_subtype {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (isoZeroCocycles A).hom (ModuleCat.ofHom A.ρ.invariants.subtype) = CategoryTheory.CategoryStruct.comp (iCocycles A 0) (zeroCochainsLequiv A).toModuleIso.hom

@[simp]

theorem groupCohomology.isoZeroCocycles_hom_comp_subtype_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : ModuleCat.of k ↑A.V ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoZeroCocycles A).hom (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom A.ρ.invariants.subtype) h) = CategoryTheory.CategoryStruct.comp (iCocycles A 0) (CategoryTheory.CategoryStruct.comp (zeroCochainsLequiv A).toModuleIso.hom h)

@[simp]

theorem groupCohomology.isoZeroCocycles_hom_comp_subtype_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (cocycles A 0)) : ↑((CategoryTheory.ConcreteCategory.hom (isoZeroCocycles A).hom) x) = (zeroCochainsLequiv A) ((CategoryTheory.ConcreteCategory.hom (iCocycles A 0)) x)

@[simp]

theorem groupCohomology.isoZeroCocycles_inv_comp_iCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (isoZeroCocycles A).inv (iCocycles A 0) = CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom A.ρ.invariants.subtype) (zeroCochainsLequiv A).toModuleIso.inv

@[simp]

theorem groupCohomology.isoZeroCocycles_inv_comp_iCocycles_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : ModuleCat.of k ((Fin 0 → G) → ↑A.V) ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoZeroCocycles A).inv (CategoryTheory.CategoryStruct.comp (iCocycles A 0) h) = CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom A.ρ.invariants.subtype) (CategoryTheory.CategoryStruct.comp (zeroCochainsLequiv A).toModuleIso.inv h)

@[simp]

theorem groupCohomology.isoZeroCocycles_inv_comp_iCocycles_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (H0 A)) : (CategoryTheory.ConcreteCategory.hom (iCocycles A 0)) ((CategoryTheory.ConcreteCategory.hom (isoZeroCocycles A).inv) x) = (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom A.ρ.invariants.subtype) (ModuleCat.ofHom ↑(zeroCochainsLequiv A).symm)) x

def groupCohomology.isoH0 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : groupCohomology A 0 ≅ H0 A

The 0th group cohomology of A, defined as the 0th cohomology of the complex of inhomogeneous cochains, is isomorphic to the invariants of the representation on A.

Equations

• groupCohomology.isoH0 A = (groupCohomology.inhomogeneousCochains A).isoHomologyπ₀.symm ≪≫ groupCohomology.isoZeroCocycles A

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH0_hom {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (groupCohomologyπ A 0) (isoH0 A).hom = (isoZeroCocycles A).hom

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH0_hom_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : H0 A ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupCohomologyπ A 0) (CategoryTheory.CategoryStruct.comp (isoH0 A).hom h) = CategoryTheory.CategoryStruct.comp (isoZeroCocycles A).hom h

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH0_hom_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (cocycles A 0)) : (CategoryTheory.ConcreteCategory.hom (isoH0 A).hom) ((CategoryTheory.ConcreteCategory.hom (groupCohomologyπ A 0)) x) = (CategoryTheory.ConcreteCategory.hom (isoZeroCocycles A).hom) x

def groupCohomology.shortComplexH1Iso {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : HomologicalComplex.sc' (inhomogeneousCochains A) 0 1 2 ≅ shortComplexH1 A

The short complex A --dZero--> Fun(G, A) --dOne--> Fun(G × G, A) is isomorphic to the 1st short complex associated to the complex of inhomogeneous cochains of A.

Equations

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

@[simp]

theorem groupCohomology.shortComplexH1Iso_hom {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (shortComplexH1Iso A).hom = { τ₁ := ModuleCat.ofHom ↑(zeroCochainsLequiv A), τ₂ := ModuleCat.ofHom ↑(oneCochainsLequiv A), τ₃ := ModuleCat.ofHom ↑(twoCochainsLequiv A), comm₁₂ := ⋯, comm₂₃ := ⋯ }

@[simp]

theorem groupCohomology.shortComplexH1Iso_inv {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (shortComplexH1Iso A).inv = CategoryTheory.ShortComplex.homMk (ModuleCat.ofHom ↑(zeroCochainsLequiv A).symm) (ModuleCat.ofHom ↑(oneCochainsLequiv A).symm) (ModuleCat.ofHom ↑(twoCochainsLequiv A).symm) ⋯ ⋯

def groupCohomology.isoOneCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : cocycles A 1 ≅ ModuleCat.of k ↥(oneCocycles A)

The 1-cocycles of the complex of inhomogeneous cochains of A are isomorphic to oneCocycles A, which is a simpler type.

Equations

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

@[simp]

theorem groupCohomology.isoOneCocycles_hom_comp_subtype {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (isoOneCocycles A).hom (ModuleCat.ofHom (oneCocycles A).subtype) = CategoryTheory.CategoryStruct.comp (iCocycles A 1) (oneCochainsLequiv A).toModuleIso.hom

@[simp]

theorem groupCohomology.isoOneCocycles_hom_comp_subtype_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : ModuleCat.of k (G → ↑A.V) ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoOneCocycles A).hom (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (oneCocycles A).subtype) h) = CategoryTheory.CategoryStruct.comp (iCocycles A 1) (CategoryTheory.CategoryStruct.comp (oneCochainsLequiv A).toModuleIso.hom h)

@[simp]

theorem groupCohomology.isoOneCocycles_hom_comp_subtype_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (cocycles A 1)) : ⇑((CategoryTheory.ConcreteCategory.hom (isoOneCocycles A).hom) x) = (oneCochainsLequiv A) ((CategoryTheory.ConcreteCategory.hom (iCocycles A 1)) x)

@[simp]

theorem groupCohomology.toCocycles_comp_isoOneCocycles_hom {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (toCocycles A 0 1) (isoOneCocycles A).hom = CategoryTheory.CategoryStruct.comp (zeroCochainsLequiv A).toModuleIso.hom (ModuleCat.ofHom (shortComplexH1 A).moduleCatToCycles)

@[simp]

theorem groupCohomology.toCocycles_comp_isoOneCocycles_hom_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : ModuleCat.of k ↥(oneCocycles A) ⟶ Z) : CategoryTheory.CategoryStruct.comp (toCocycles A 0 1) (CategoryTheory.CategoryStruct.comp (isoOneCocycles A).hom h) = CategoryTheory.CategoryStruct.comp (zeroCochainsLequiv A).toModuleIso.hom (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (shortComplexH1 A).moduleCatToCycles) h)

@[simp]

theorem groupCohomology.toCocycles_comp_isoOneCocycles_hom_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (ModuleCat.of k ((Fin 0 → G) → ↑A.V))) : (CategoryTheory.ConcreteCategory.hom (isoOneCocycles A).hom) ((CategoryTheory.ConcreteCategory.hom (toCocycles A 0 1)) x) = (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom ↑(zeroCochainsLequiv A)) (ModuleCat.ofHom (shortComplexH1 A).moduleCatToCycles)) x

def groupCohomology.isoH1 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : groupCohomology A 1 ≅ H1 A

The 1st group cohomology of A, defined as the 1st cohomology of the complex of inhomogeneous cochains, is isomorphic to oneCocycles A ⧸ oneCoboundaries A, which is a simpler type.

Equations

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

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH1_hom {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (groupCohomologyπ A 1) (isoH1 A).hom = CategoryTheory.CategoryStruct.comp (isoOneCocycles A).hom (H1π A)

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH1_hom_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : H1 A ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupCohomologyπ A 1) (CategoryTheory.CategoryStruct.comp (isoH1 A).hom h) = CategoryTheory.CategoryStruct.comp (isoOneCocycles A).hom (CategoryTheory.CategoryStruct.comp (H1π A) h)

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH1_hom_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (cocycles A 1)) : (CategoryTheory.ConcreteCategory.hom (isoH1 A).hom) ((CategoryTheory.ConcreteCategory.hom (groupCohomologyπ A 1)) x) = Submodule.Quotient.mk ((CategoryTheory.ConcreteCategory.hom (isoOneCocycles A).hom) x)

def groupCohomology.shortComplexH2Iso {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : HomologicalComplex.sc' (inhomogeneousCochains A) 1 2 3 ≅ shortComplexH2 A

The short complex Fun(G, A) --dOne--> Fun(G × G, A) --dTwo--> Fun(G × G × G, A) is isomorphic to the 2nd short complex associated to the complex of inhomogeneous cochains of A.

Equations

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

@[simp]

theorem groupCohomology.shortComplexH2Iso_inv {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (shortComplexH2Iso A).inv = CategoryTheory.ShortComplex.homMk (ModuleCat.ofHom ↑(oneCochainsLequiv A).symm) (ModuleCat.ofHom ↑(twoCochainsLequiv A).symm) (ModuleCat.ofHom ↑(threeCochainsLequiv A).symm) ⋯ ⋯

@[simp]

theorem groupCohomology.shortComplexH2Iso_hom {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : (shortComplexH2Iso A).hom = { τ₁ := ModuleCat.ofHom ↑(oneCochainsLequiv A), τ₂ := ModuleCat.ofHom ↑(twoCochainsLequiv A), τ₃ := ModuleCat.ofHom ↑(threeCochainsLequiv A), comm₁₂ := ⋯, comm₂₃ := ⋯ }

def groupCohomology.isoTwoCocycles {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : cocycles A 2 ≅ ModuleCat.of k ↥(twoCocycles A)

The 2-cocycles of the complex of inhomogeneous cochains of A are isomorphic to twoCocycles A, which is a simpler type.

Equations

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

@[simp]

theorem groupCohomology.isoTwoCocycles_hom_comp_subtype {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (isoTwoCocycles A).hom (ModuleCat.ofHom (twoCocycles A).subtype) = CategoryTheory.CategoryStruct.comp (iCocycles A 2) (twoCochainsLequiv A).toModuleIso.hom

@[simp]

theorem groupCohomology.isoTwoCocycles_hom_comp_subtype_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : ModuleCat.of k (G × G → ↑A.V) ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoTwoCocycles A).hom (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (twoCocycles A).subtype) h) = CategoryTheory.CategoryStruct.comp (iCocycles A 2) (CategoryTheory.CategoryStruct.comp (twoCochainsLequiv A).toModuleIso.hom h)

@[simp]

theorem groupCohomology.isoTwoCocycles_hom_comp_subtype_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (cocycles A 2)) : ⇑((CategoryTheory.ConcreteCategory.hom (isoTwoCocycles A).hom) x) = (twoCochainsLequiv A) ((CategoryTheory.ConcreteCategory.hom (iCocycles A 2)) x)

@[simp]

theorem groupCohomology.toCocycles_comp_isoTwoCocycles_hom {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (toCocycles A 1 2) (isoTwoCocycles A).hom = CategoryTheory.CategoryStruct.comp (oneCochainsLequiv A).toModuleIso.hom (ModuleCat.ofHom (shortComplexH2 A).moduleCatToCycles)

@[simp]

theorem groupCohomology.toCocycles_comp_isoTwoCocycles_hom_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : ModuleCat.of k ↥(twoCocycles A) ⟶ Z) : CategoryTheory.CategoryStruct.comp (toCocycles A 1 2) (CategoryTheory.CategoryStruct.comp (isoTwoCocycles A).hom h) = CategoryTheory.CategoryStruct.comp (oneCochainsLequiv A).toModuleIso.hom (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (shortComplexH2 A).moduleCatToCycles) h)

@[simp]

theorem groupCohomology.toCocycles_comp_isoTwoCocycles_hom_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (ModuleCat.of k ((Fin 1 → G) → ↑A.V))) : (CategoryTheory.ConcreteCategory.hom (isoTwoCocycles A).hom) ((CategoryTheory.ConcreteCategory.hom (toCocycles A 1 2)) x) = (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom ↑(oneCochainsLequiv A)) (ModuleCat.ofHom (shortComplexH2 A).moduleCatToCycles)) x

def groupCohomology.isoH2 {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : groupCohomology A 2 ≅ H2 A

The 2nd group cohomology of A, defined as the 2nd cohomology of the complex of inhomogeneous cochains, is isomorphic to twoCocycles A ⧸ twoCoboundaries A, which is a simpler type.

Equations

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

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH2_hom {k G : Type u} [CommRing k] [Group G] (A : Rep k G) : CategoryTheory.CategoryStruct.comp (groupCohomologyπ A 2) (isoH2 A).hom = CategoryTheory.CategoryStruct.comp (isoTwoCocycles A).hom (H2π A)

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH2_hom_assoc {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {Z : ModuleCat k} (h : H2 A ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupCohomologyπ A 2) (CategoryTheory.CategoryStruct.comp (isoH2 A).hom h) = CategoryTheory.CategoryStruct.comp (isoTwoCocycles A).hom (CategoryTheory.CategoryStruct.comp (H2π A) h)

@[simp]

theorem groupCohomology.groupCohomologyπ_comp_isoH2_hom_apply {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (x : CategoryTheory.ToType (cocycles A 2)) : (CategoryTheory.ConcreteCategory.hom (isoH2 A).hom) ((CategoryTheory.ConcreteCategory.hom (groupCohomologyπ A 2)) x) = Submodule.Quotient.mk ((CategoryTheory.ConcreteCategory.hom (isoTwoCocycles A).hom) x)