fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
d₂₁_apply_mem_cycles₁ (x : G × G →₀ A) :
d₂₁ A x ∈ cycles₁ A :=
congr($(d₂₁_comp_d₁₀ A) x)
variable (A) in | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_apply_mem_cycles₁ | null |
cycles₁_eq_top_of_isTrivial [A.IsTrivial] : cycles₁ A = ⊤ := by
rw [cycles₁, d₁₀_eq_zero_of_isTrivial, ModuleCat.hom_zero, LinearMap.ker_zero]
variable (A) in | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cycles₁_eq_top_of_isTrivial | null |
cycles₁IsoOfIsTrivial [A.IsTrivial] :
ModuleCat.of k (cycles₁ A) ≅ ModuleCat.of k (G →₀ A) :=
(LinearEquiv.ofTop _ (cycles₁_eq_top_of_isTrivial A)).toModuleIso
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cycles₁IsoOfIsTrivial | The natural inclusion `Z₁(G, A) ⟶ C₁(G, A)` is an isomorphism when the representation
on `A` is trivial. |
cycles₁IsoOfIsTrivial_hom_apply [A.IsTrivial] (x : cycles₁ A) :
(cycles₁IsoOfIsTrivial A).hom x = x.1 := rfl
@[simp] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cycles₁IsoOfIsTrivial_hom_apply | null |
cycles₁IsoOfIsTrivial_inv_apply [A.IsTrivial] (x : G →₀ A) :
((cycles₁IsoOfIsTrivial A).inv x).1 = x := rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cycles₁IsoOfIsTrivial_inv_apply | null |
mem_cycles₂_iff (x : G × G →₀ A) :
x ∈ cycles₂ A ↔ x.sum (fun g a => single g.2 (A.ρ g.1⁻¹ a) + single g.1 a) =
x.sum (fun g a => single (g.1 * g.2) a) := by
change x.sum (fun g a => _) = 0 ↔ _
simp [sub_add_eq_add_sub, sub_eq_zero] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | mem_cycles₂_iff | null |
single_mem_cycles₂_iff_inv (g : G × G) (a : A) :
single g a ∈ cycles₂ A ↔ single g.2 (A.ρ g.1⁻¹ a) + single g.1 a = single (g.1 * g.2) a := by
simp [mem_cycles₂_iff] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_mem_cycles₂_iff_inv | null |
single_mem_cycles₂_iff (g : G × G) (a : A) :
single g a ∈ cycles₂ A ↔
single (g.1 * g.2) (A.ρ g.1 a) = single g.2 a + single g.1 (A.ρ g.1 a) := by
rw [← (mapRange_injective (α := G) _ (map_zero _) (A.ρ.apply_bijective g.1⁻¹).1).eq_iff]
simp [mem_cycles₂_iff, mapRange_add, eq_comm] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_mem_cycles₂_iff | null |
d₃₂_apply_mem_cycles₂ (x : G × G × G →₀ A) :
d₃₂ A x ∈ cycles₂ A :=
congr($(d₃₂_comp_d₂₁ A) x) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₃₂_apply_mem_cycles₂ | null |
boundaries₁ : Submodule k (G →₀ A) :=
LinearMap.range (d₂₁ A).hom | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundaries₁ | The 1-boundaries `B₁(G, A)` of `A : Rep k G`, defined as the image of the map
`(G² →₀ A) → (G →₀ A)` defined by `a·(g₁, g₂) ↦ ρ_A(g₁⁻¹)(a)·g₂ - a·g₁g₂ + a·g₁`. |
boundaries₂ : Submodule k (G × G →₀ A) :=
LinearMap.range (d₃₂ A).hom
variable {A} | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundaries₂ | The 2-boundaries `B₂(G, A)` of `A : Rep k G`, defined as the image of the map
`(G³ →₀ A) → (G² →₀ A)` defined by
`a·(g₁, g₂, g₃) ↦ ρ_A(g₁⁻¹)(a)·(g₂, g₃) - a·(g₁g₂, g₃) + a·(g₁, g₂g₃) - a·(g₁, g₂)`. |
mem_cycles₁_of_mem_boundaries₁ (f : G →₀ A) (h : f ∈ boundaries₁ A) :
f ∈ cycles₁ A := by
rcases h with ⟨x, rfl⟩
exact d₂₁_apply_mem_cycles₁ x
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | mem_cycles₁_of_mem_boundaries₁ | null |
boundaries₁_le_cycles₁ : boundaries₁ A ≤ cycles₁ A :=
mem_cycles₁_of_mem_boundaries₁
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundaries₁_le_cycles₁ | null |
boundariesToCycles₁ : boundaries₁ A →ₗ[k] cycles₁ A :=
Submodule.inclusion (boundaries₁_le_cycles₁ A)
@[simp] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundariesToCycles₁ | The natural inclusion `B₁(G, A) →ₗ[k] Z₁(G, A)`. |
boundariesToCycles₁_apply (x : boundaries₁ A) :
(boundariesToCycles₁ A x).1 = x.1 := rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundariesToCycles₁_apply | null |
single_one_mem_boundaries₁ (a : A) :
single 1 a ∈ boundaries₁ A := by
use single (1, 1) a
simp [d₂₁] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_one_mem_boundaries₁ | null |
single_ρ_self_add_single_inv_mem_boundaries₁ (g : G) (a : A) :
single g (A.ρ g a) + single g⁻¹ a ∈ boundaries₁ A := by
rw [← d₂₁_single_ρ_add_single_inv_mul g 1]
exact Set.mem_range_self _ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_ρ_self_add_single_inv_mem_boundaries₁ | null |
single_inv_ρ_self_add_single_mem_boundaries₁ (g : G) (a : A) :
single g⁻¹ (A.ρ g⁻¹ a) + single g a ∈ boundaries₁ A := by
rw [← d₂₁_single_inv_mul_ρ_add_single g 1]
exact Set.mem_range_self _ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_inv_ρ_self_add_single_mem_boundaries₁ | null |
mem_cycles₂_of_mem_boundaries₂ (x : G × G →₀ A) (h : x ∈ boundaries₂ A) :
x ∈ cycles₂ A := by
rcases h with ⟨x, rfl⟩
exact d₃₂_apply_mem_cycles₂ x
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | mem_cycles₂_of_mem_boundaries₂ | null |
boundaries₂_le_cycles₂ : boundaries₂ A ≤ cycles₂ A :=
mem_cycles₂_of_mem_boundaries₂
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundaries₂_le_cycles₂ | null |
boundariesToCycles₂ : boundaries₂ A →ₗ[k] cycles₂ A :=
Submodule.inclusion (boundaries₂_le_cycles₂ A)
@[simp] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundariesToCycles₂ | The natural inclusion `B₂(G, A) →ₗ[k] Z₂(G, A)`. |
boundariesToCycles₂_apply (x : boundaries₂ A) :
(boundariesToCycles₂ A x).1 = x.1 := rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundariesToCycles₂_apply | null |
single_one_fst_sub_single_one_fst_mem_boundaries₂ (g h : G) (a : A) :
single (1, g * h) a - single (1, g) a ∈ boundaries₂ A := by
use single (1, g, h) a
simp [d₃₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_one_fst_sub_single_one_fst_mem_boundaries₂ | null |
single_one_fst_sub_single_one_snd_mem_boundaries₂ (g h : G) (a : A) :
single (1, h) (A.ρ g⁻¹ a) - single (g, 1) a ∈ boundaries₂ A := by
use single (g, 1, h) a
simp [d₃₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_one_fst_sub_single_one_snd_mem_boundaries₂ | null |
single_one_snd_sub_single_one_fst_mem_boundaries₂ (g h : G) (a : A) :
single (g, 1) (A.ρ g a) - single (1, h) a ∈ boundaries₂ A := by
use single (g, 1, h) (A.ρ g (-a))
simp [d₃₂_single (G := G)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_one_snd_sub_single_one_fst_mem_boundaries₂ | null |
single_one_snd_sub_single_one_snd_mem_boundaries₂ (g h : G) (a : A) :
single (h, 1) (A.ρ g⁻¹ a) - single (g * h, 1) a ∈ boundaries₂ A := by
use single (g, h, 1) a
simp [d₃₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_one_snd_sub_single_one_snd_mem_boundaries₂ | null |
IsCycle₁ (x : G →₀ A) : Prop := x.sum (fun g a => g⁻¹ • a) = x.sum (fun _ a => a) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | IsCycle₁ | A finsupp `∑ aᵢ·gᵢ : G →₀ A` satisfies the 1-cycle condition if `∑ gᵢ⁻¹ • aᵢ = ∑ aᵢ`. |
IsCycle₂ (x : G × G →₀ A) : Prop :=
x.sum (fun g a => single g.2 (g.1⁻¹ • a) + single g.1 a) =
x.sum (fun g a => single (g.1 * g.2) a) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | IsCycle₂ | A finsupp `∑ aᵢ·(gᵢ, hᵢ) : G × G →₀ A` satisfies the 2-cycle condition if
`∑ (gᵢ⁻¹ • aᵢ)·hᵢ + aᵢ·gᵢ = ∑ aᵢ·gᵢhᵢ`. |
@[simp]
single_isCycle₁_iff (g : G) (a : A) :
IsCycle₁ (single g a) ↔ g • a = a := by
rw [← (MulAction.bijective g⁻¹).1.eq_iff]
simp [IsCycle₁, eq_comm] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_isCycle₁_iff | null |
single_isCycle₁_of_mem_fixedPoints
(g : G) (a : A) (ha : a ∈ MulAction.fixedPoints G A) :
IsCycle₁ (single g a) := by
simp_all [IsCycle₁] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_isCycle₁_of_mem_fixedPoints | null |
single_isCycle₂_iff_inv (g : G × G) (a : A) :
IsCycle₂ (single g a) ↔
single g.2 (g.1⁻¹ • a) + single g.1 a = single (g.1 * g.2) a := by
simp [IsCycle₂]
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_isCycle₂_iff_inv | null |
single_isCycle₂_iff (g : G × G) (a : A) :
IsCycle₂ (single g a) ↔
single g.2 a + single g.1 (g.1 • a) = single (g.1 * g.2) (g.1 • a) := by
rw [← (Finsupp.mapRange_injective (α := G) _ (smul_zero _) (MulAction.bijective g.1⁻¹).1).eq_iff]
simp [mapRange_add, IsCycle₂] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_isCycle₂_iff | null |
IsBoundary₀ (a : A) : Prop :=
∃ (x : G →₀ A), x.sum (fun g z => g⁻¹ • z - z) = a | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | IsBoundary₀ | A term `x : A` satisfies the 0-boundary condition if there exists a finsupp
`∑ aᵢ·gᵢ : G →₀ A` such that `∑ gᵢ⁻¹ • aᵢ - aᵢ = x`. |
IsBoundary₁ (x : G →₀ A) : Prop :=
∃ y : G × G →₀ A, y.sum
(fun g a => single g.2 (g.1⁻¹ • a) - single (g.1 * g.2) a + single g.1 a) = x | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | IsBoundary₁ | A finsupp `x : G →₀ A` satisfies the 1-boundary condition if there's a finsupp
`∑ aᵢ·(gᵢ, hᵢ) : G × G →₀ A` such that `∑ (gᵢ⁻¹ • aᵢ)·hᵢ - aᵢ·gᵢhᵢ + aᵢ·gᵢ = x`. |
IsBoundary₂ (x : G × G →₀ A) : Prop :=
∃ y : G × G × G →₀ A, y.sum (fun g a => single (g.2.1, g.2.2) (g.1⁻¹ • a) -
single (g.1 * g.2.1, g.2.2) a + single (g.1, g.2.1 * g.2.2) a - single (g.1, g.2.1) a) = x | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | IsBoundary₂ | A finsupp `x : G × G →₀ A` satisfies the 2-boundary condition if there's a finsupp
`∑ aᵢ·(gᵢ, hᵢ, jᵢ) : G × G × G →₀ A` such that
`∑ (gᵢ⁻¹ • aᵢ)·(hᵢ, jᵢ) - aᵢ·(gᵢhᵢ, jᵢ) + aᵢ·(gᵢ, hᵢjᵢ) - aᵢ·(gᵢ, hᵢ) = x.` |
isBoundary₀_iff (a : A) :
IsBoundary₀ G a ↔ ∃ x : G →₀ A, x.sum (fun g z => g • z - z) = a := by
constructor
· rintro ⟨x, hx⟩
use x.sum (fun g a => single g (- (g⁻¹ • a)))
simp_all [sum_neg_index, sum_sum_index, neg_add_eq_sub]
· rintro ⟨x, hx⟩
use x.sum (fun g a => single g (- (g • a)))
simp_all [sum_neg_index, sum_sum_index, neg_add_eq_sub] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isBoundary₀_iff | null |
isBoundary₁_iff (x : G →₀ A) :
IsBoundary₁ x ↔ ∃ y : G × G →₀ A, y.sum
(fun g a => single g.2 a - single (g.1 * g.2) (g.1 • a) + single g.1 (g.1 • a)) = x := by
constructor
· rintro ⟨y, hy⟩
use y.sum (fun g a => single g (g.1⁻¹ • a))
simp_all [sum_sum_index]
· rintro ⟨x, hx⟩
use x.sum (fun g a => single g (g.1 • a))
simp_all [sum_sum_index] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isBoundary₁_iff | null |
isBoundary₂_iff (x : G × G →₀ A) :
IsBoundary₂ x ↔ ∃ y : G × G × G →₀ A, y.sum
(fun g a => single (g.2.1, g.2.2) a - single (g.1 * g.2.1, g.2.2) (g.1 • a) +
single (g.1, g.2.1 * g.2.2) (g.1 • a) - single (g.1, g.2.1) (g.1 • a)) = x := by
constructor
· rintro ⟨y, hy⟩
use y.sum (fun g a => single g (g.1⁻¹ • a))
simp_all [sum_sum_index]
· rintro ⟨x, hx⟩
use x.sum (fun g a => single g (g.1 • a))
simp_all [sum_sum_index] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isBoundary₂_iff | null |
@[simps]
coinvariantsKerOfIsBoundary₀ (x : A) (hx : IsBoundary₀ G x) :
Coinvariants.ker (Representation.ofDistribMulAction k G A) :=
⟨x, by
rcases (isBoundary₀_iff G x).1 hx with ⟨y, rfl⟩
exact Submodule.finsuppSum_mem _ _ _ _ fun g _ => Coinvariants.mem_ker_of_eq g (y g) _ rfl⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | coinvariantsKerOfIsBoundary₀ | Given a `k`-module `A` with a compatible `DistribMulAction` of `G`, and a term
`x : A` satisfying the 0-boundary condition, this produces an element of the kernel of the quotient
map `A → A_G` for the representation on `A` induced by the `DistribMulAction`. |
isBoundary₀_of_mem_coinvariantsKer
(x : A) (hx : x ∈ Coinvariants.ker (Representation.ofDistribMulAction k G A)) :
IsBoundary₀ G x :=
Submodule.span_induction (fun _ ⟨g, hg⟩ => ⟨single g.1⁻¹ g.2, by simp_all⟩) ⟨0, by simp⟩
(fun _ _ _ _ ⟨X, hX⟩ ⟨Y, hY⟩ => ⟨X + Y, by simp_all [sum_add_index', add_sub_add_comm]⟩)
(fun r _ _ ⟨X, hX⟩ => ⟨r • X, by simp [← hX, sum_smul_index', smul_comm, smul_sub, smul_sum]⟩)
hx | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isBoundary₀_of_mem_coinvariantsKer | null |
@[simps]
cyclesOfIsCycle₁ (x : G →₀ A) (hx : IsCycle₁ x) :
cycles₁ (Rep.ofDistribMulAction k G A) :=
⟨x, (mem_cycles₁_iff (A := Rep.ofDistribMulAction k G A) x).2 hx⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesOfIsCycle₁ | Given a `k`-module `A` with a compatible `DistribMulAction` of `G`, and a finsupp
`x : G →₀ A` satisfying the 1-cycle condition, produces a 1-cycle for the representation on
`A` induced by the `DistribMulAction`. |
isCycle₁_of_mem_cycles₁
(x : G →₀ A) (hx : x ∈ cycles₁ (Rep.ofDistribMulAction k G A)) :
IsCycle₁ x := by
simpa using (mem_cycles₁_iff (A := Rep.ofDistribMulAction k G A) x).1 hx | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isCycle₁_of_mem_cycles₁ | null |
@[simps]
boundariesOfIsBoundary₁ (x : G →₀ A) (hx : IsBoundary₁ x) :
boundaries₁ (Rep.ofDistribMulAction k G A) :=
⟨x, hx⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundariesOfIsBoundary₁ | Given a `k`-module `A` with a compatible `DistribMulAction` of `G`, and a finsupp
`x : G →₀ A` satisfying the 1-boundary condition, produces a 1-boundary for the representation
on `A` induced by the `DistribMulAction`. |
isBoundary₁_of_mem_boundaries₁
(x : G →₀ A) (hx : x ∈ boundaries₁ (Rep.ofDistribMulAction k G A)) :
IsBoundary₁ x := hx | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isBoundary₁_of_mem_boundaries₁ | null |
@[simps]
cyclesOfIsCycle₂ (x : G × G →₀ A) (hx : IsCycle₂ x) :
cycles₂ (Rep.ofDistribMulAction k G A) :=
⟨x, (mem_cycles₂_iff (A := Rep.ofDistribMulAction k G A) x).2 hx⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesOfIsCycle₂ | Given a `k`-module `A` with a compatible `DistribMulAction` of `G`, and a finsupp
`x : G × G →₀ A` satisfying the 2-cycle condition, produces a 2-cycle for the representation on
`A` induced by the `DistribMulAction`. |
isCycle₂_of_mem_cycles₂
(x : G × G →₀ A) (hx : x ∈ cycles₂ (Rep.ofDistribMulAction k G A)) :
IsCycle₂ x := (mem_cycles₂_iff (A := Rep.ofDistribMulAction k G A) x).1 hx | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isCycle₂_of_mem_cycles₂ | null |
@[simps]
boundariesOfIsBoundary₂ (x : G × G →₀ A) (hx : IsBoundary₂ x) :
boundaries₂ (Rep.ofDistribMulAction k G A) :=
⟨x, hx⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | boundariesOfIsBoundary₂ | Given a `k`-module `A` with a compatible `DistribMulAction` of `G`, and a finsupp
`x : G × G →₀ A` satisfying the 2-boundary condition, produces a 2-boundary for the
representation on `A` induced by the `DistribMulAction`. |
isBoundary₂_of_mem_boundaries₂
(x : G × G →₀ A) (hx : x ∈ boundaries₂ (Rep.ofDistribMulAction k G A)) :
IsBoundary₂ x := hx | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isBoundary₂_of_mem_boundaries₂ | null |
shortComplexH0_exact : (shortComplexH0 A).Exact := by
rw [ShortComplex.moduleCat_exact_iff]
intro x (hx : Coinvariants.mk _ _ = 0)
rw [Coinvariants.mk_eq_zero, ← range_d₁₀_eq_coinvariantsKer] at hx
rcases hx with ⟨x, hx, rfl⟩
use x
rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | shortComplexH0_exact | null |
cyclesIso₀ : cycles A 0 ≅ A.V :=
(inhomogeneousChains A).iCyclesIso _ 0 (by aesop) (by aesop) ≪≫ chainsIso₀ A
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesIso₀ | The 0-cycles of the complex of inhomogeneous chains of `A` are isomorphic to `A`. |
cyclesIso₀_inv_comp_iCycles :
(cyclesIso₀ A).inv ≫ iCycles A 0 = (chainsIso₀ A).inv := by
simp [cyclesIso₀] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesIso₀_inv_comp_iCycles | null |
@[simps! hom_left hom_right inv_left inv_right]
d₁₀ArrowIso :
Arrow.mk ((inhomogeneousChains A).d 1 0) ≅ Arrow.mk (d₁₀ A) :=
Arrow.isoMk (chainsIso₁ A) (chainsIso₀ A) (comp_d₁₀_eq A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₁₀ArrowIso | The arrow `(G →₀ A) --d₁₀--> A` is isomorphic to the differential
`(inhomogeneousChains A).d 1 0` of the complex of inhomogeneous chains of `A`. |
opcyclesIso₀ : (inhomogeneousChains A).opcycles 0 ≅ (coinvariantsFunctor k G).obj A :=
CokernelCofork.mapIsoOfIsColimit
((inhomogeneousChains A).opcyclesIsCokernel 1 0 (by simp)) (shortComplexH0_exact A).gIsCokernel
(d₁₀ArrowIso A)
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | opcyclesIso₀ | The 0-cycles of the complex of inhomogeneous chains of `A` are isomorphic to
`A.ρ.coinvariants`, which is a simpler type. |
pOpcycles_comp_opcyclesIso_hom :
(inhomogeneousChains A).pOpcycles 0 ≫ (opcyclesIso₀ A).hom =
(chainsIso₀ A).hom ≫ (coinvariantsMk k G).app A :=
CokernelCofork.π_mapOfIsColimit (φ := (d₁₀ArrowIso A).hom) _ _
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | pOpcycles_comp_opcyclesIso_hom | null |
coinvariantsMk_comp_opcyclesIso₀_inv :
(coinvariantsMk k G).app A ≫ (opcyclesIso₀ A).inv =
(chainsIso₀ A).inv ≫ (inhomogeneousChains A).pOpcycles 0 :=
(CommSq.vert_inv ⟨pOpcycles_comp_opcyclesIso_hom A⟩).w | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | coinvariantsMk_comp_opcyclesIso₀_inv | null |
cyclesMk₀_eq (x : A) :
cyclesMk 0 0 (by simp) ((chainsIso₀ A).inv x) (by simp) = (cyclesIso₀ A).inv x :=
(ModuleCat.mono_iff_injective <| iCycles A 0).1 inferInstance <| by rw [iCycles_mk]; simp | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesMk₀_eq | null |
@[simps! hom inv]
isoShortComplexH1 : (inhomogeneousChains A).sc 1 ≅ shortComplexH1 A :=
(inhomogeneousChains A).isoSc' 2 1 0 (by simp) (by simp) ≪≫
isoMk (chainsIso₂ A) (chainsIso₁ A) (chainsIso₀ A) (comp_d₂₁_eq A) (comp_d₁₀_eq A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoShortComplexH1 | The short complex `(G² →₀ A) --d₂₁--> (G →₀ A) --d₁₀--> A` is isomorphic to the 1st
short complex associated to the complex of inhomogeneous chains of `A`. |
isoCycles₁ : cycles A 1 ≅ ModuleCat.of k (cycles₁ A) :=
cyclesMapIso' (isoShortComplexH1 A) ((inhomogeneousChains A).sc 1).leftHomologyData
(shortComplexH1 A).moduleCatLeftHomologyData
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoCycles₁ | The 1-cycles of the complex of inhomogeneous chains of `A` are isomorphic to
`cycles₁ A`, which is a simpler type. |
isoCycles₁_hom_comp_i :
(isoCycles₁ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.i =
iCycles A 1 ≫ (chainsIso₁ A).hom := by
simp [isoCycles₁, iCycles, HomologicalComplex.iCycles, ShortComplex.iCycles]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoCycles₁_hom_comp_i | null |
isoCycles₁_inv_comp_iCycles :
(isoCycles₁ A).inv ≫ iCycles A 1 =
(shortComplexH1 A).moduleCatLeftHomologyData.i ≫ (chainsIso₁ A).inv :=
(CommSq.horiz_inv ⟨isoCycles₁_hom_comp_i A⟩).w
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoCycles₁_inv_comp_iCycles | null |
toCycles_comp_isoCycles₁_hom :
toCycles A 2 1 ≫ (isoCycles₁ A).hom =
(chainsIso₂ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.f' := by
simp [← cancel_mono (shortComplexH1 A).moduleCatLeftHomologyData.i, comp_d₂₁_eq,
shortComplexH1_f] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | toCycles_comp_isoCycles₁_hom | null |
cyclesMk₁_eq (x : cycles₁ A) :
cyclesMk 1 0 (by simp) ((chainsIso₁ A).inv x) (by simp) = (isoCycles₁ A).inv x :=
(ModuleCat.mono_iff_injective <| iCycles A 1).1 inferInstance <| by
rw [iCycles_mk]
simp only [ChainComplex.of_x, isoCycles₁_inv_comp_iCycles_apply]
rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesMk₁_eq | null |
@[simps! hom inv]
isoShortComplexH2 : (inhomogeneousChains A).sc 2 ≅ shortComplexH2 A :=
(inhomogeneousChains A).isoSc' 3 2 1 (by simp) (by simp) ≪≫
isoMk (chainsIso₃ A) (chainsIso₂ A) (chainsIso₁ A) (comp_d₃₂_eq A) (comp_d₂₁_eq A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoShortComplexH2 | The short complex `(G³ →₀ A) --d₃₂--> (G² →₀ A) --d₂₁--> (G →₀ A)` is isomorphic to the 2nd
short complex associated to the complex of inhomogeneous chains of `A`. |
isoCycles₂ : cycles A 2 ≅ ModuleCat.of k (cycles₂ A) :=
cyclesMapIso' (isoShortComplexH2 A) ((inhomogeneousChains A).sc 2).leftHomologyData
(shortComplexH2 A).moduleCatLeftHomologyData
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoCycles₂ | The 2-cycles of the complex of inhomogeneous chains of `A` are isomorphic to
`cycles₂ A`, which is a simpler type. |
isoCycles₂_hom_comp_i :
(isoCycles₂ A).hom ≫ (shortComplexH2 A).moduleCatLeftHomologyData.i =
iCycles A 2 ≫ (chainsIso₂ A).hom := by
simp [isoCycles₂, iCycles, HomologicalComplex.iCycles, ShortComplex.iCycles]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoCycles₂_hom_comp_i | null |
isoCycles₂_inv_comp_iCycles :
(isoCycles₂ A).inv ≫ iCycles A 2 =
(shortComplexH2 A).moduleCatLeftHomologyData.i ≫ (chainsIso₂ A).inv :=
(CommSq.horiz_inv ⟨isoCycles₂_hom_comp_i A⟩).w
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | isoCycles₂_inv_comp_iCycles | null |
toCycles_comp_isoCycles₂_hom :
toCycles A 3 2 ≫ (isoCycles₂ A).hom =
(chainsIso₃ A).hom ≫ (shortComplexH2 A).moduleCatLeftHomologyData.f' := by
simp [← cancel_mono (shortComplexH2 A).moduleCatLeftHomologyData.i, comp_d₃₂_eq,
shortComplexH2_f] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | toCycles_comp_isoCycles₂_hom | null |
cyclesMk₂_eq (x : cycles₂ A) :
cyclesMk 2 1 (by simp) ((chainsIso₂ A).inv x) (by simp) = (isoCycles₂ A).inv x :=
(ModuleCat.mono_iff_injective <| iCycles A 2).1 inferInstance <| by
rw [iCycles_mk]
simp only [ChainComplex.of_x, isoCycles₂_inv_comp_iCycles_apply]
rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesMk₂_eq | null |
H0 := groupHomology A 0 | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H0 | Shorthand for the 0th group homology of a `k`-linear `G`-representation `A`, `H₀(G, A)`,
defined as the 0th homology of the complex of inhomogeneous chains of `A`. |
H0Iso : H0 A ≅ (coinvariantsFunctor k G).obj A :=
(ChainComplex.isoHomologyι₀ _) ≪≫ opcyclesIso₀ A | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H0Iso | The 0th group homology of `A`, defined as the 0th homology of the complex of inhomogeneous
chains, is isomorphic to the invariants of the representation on `A`. |
H0π : A.V ⟶ H0 A := (cyclesIso₀ A).inv ≫ π A 0 | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H0π | The quotient map from `A` to `H₀(G, A)`. |
@[reassoc (attr := simp), elementwise (attr := simp)]
π_comp_H0Iso_hom :
π A 0 ≫ (H0Iso A).hom = (cyclesIso₀ A).hom ≫ (coinvariantsMk k G).app A := by
simp [H0Iso, cyclesIso₀]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | π_comp_H0Iso_hom | null |
coinvariantsMk_comp_H0Iso_inv :
(coinvariantsMk k G).app A ≫ (H0Iso A).inv = H0π A :=
(CommSq.vert_inv ⟨π_comp_H0Iso_hom A⟩).w
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | coinvariantsMk_comp_H0Iso_inv | null |
H0π_comp_H0Iso_hom :
H0π A ≫ (H0Iso A).hom = (coinvariantsMk k G).app A := by
simp [H0π]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H0π_comp_H0Iso_hom | null |
cyclesIso₀_comp_H0π :
(cyclesIso₀ A).hom ≫ H0π A = π A 0 := by
simp [H0π]
@[elab_as_elim] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cyclesIso₀_comp_H0π | null |
H0_induction_on {C : H0 A → Prop} (x : H0 A)
(h : ∀ x : A, C (H0π A x)) : C x :=
groupHomology_induction_on x fun y => by simpa using h ((cyclesIso₀ A).hom y) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H0_induction_on | null |
H0IsoOfIsTrivial :
H0 A ≅ A.V :=
((inhomogeneousChains A).isoHomologyπ 1 0 (by simp) <| by
ext; simp [inhomogeneousChains.d_def, inhomogeneousChains.d_single (G := G),
Unique.eq_default (α := Fin 0 → G)]).symm ≪≫ cyclesIso₀ A
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H0IsoOfIsTrivial | When the representation on `A` is trivial, then `H₀(G, A)` is all of `A.` |
H0IsoOfIsTrivial_inv_eq_π :
(H0IsoOfIsTrivial A).inv = H0π A := rfl
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H0IsoOfIsTrivial_inv_eq_π | null |
π_comp_H0IsoOfIsTrivial_hom :
π A 0 ≫ (H0IsoOfIsTrivial A).hom = (cyclesIso₀ A).hom := by
simp [H0IsoOfIsTrivial] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | π_comp_H0IsoOfIsTrivial_hom | null |
H1 := groupHomology A 1 | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1 | Shorthand for the 1st group homology of a `k`-linear `G`-representation `A`, `H₁(G, A)`,
defined as the 1st homology of the complex of inhomogeneous chains of `A`. |
H1π : ModuleCat.of k (cycles₁ A) ⟶ H1 A :=
(isoCycles₁ A).inv ≫ π A 1 | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1π | The quotient map from the 1-cycles of `A`, as a submodule of `G →₀ A`, to `H₁(G, A)`. |
H1π_eq_zero_iff (x : cycles₁ A) : H1π A x = 0 ↔ x.1 ∈ boundaries₁ A := by
have h := leftHomologyπ_naturality'_assoc (isoShortComplexH1 A).inv
(shortComplexH1 A).moduleCatLeftHomologyData (leftHomologyData _)
((inhomogeneousChains A).sc 1).leftHomologyIso.hom
simp only [H1π, isoCycles₁, π, HomologicalComplex.homologyπ, homologyπ,
cyclesMapIso'_inv, leftHomologyπ, ← h, ← leftHomologyMapIso'_inv, ModuleCat.hom_comp,
LinearMap.coe_comp, Function.comp_apply, map_eq_zero_iff _
((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
simp [LinearMap.range_codRestrict, boundaries₁, shortComplexH1, cycles₁] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1π_eq_zero_iff | null |
H1π_eq_iff (x y : cycles₁ A) :
H1π A x = H1π A y ↔ x.1 - y.1 ∈ boundaries₁ A := by
rw [← sub_eq_zero, ← map_sub, H1π_eq_zero_iff]
rfl
@[elab_as_elim] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1π_eq_iff | null |
H1_induction_on {C : H1 A → Prop} (x : H1 A) (h : ∀ x : cycles₁ A, C (H1π A x)) :
C x :=
groupHomology_induction_on x fun y => by simpa [H1π] using h ((isoCycles₁ A).hom y)
variable (A) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1_induction_on | null |
H1Iso : H1 A ≅ (shortComplexH1 A).moduleCatLeftHomologyData.H :=
(leftHomologyIso _).symm ≪≫ (leftHomologyMapIso' (isoShortComplexH1 A) _ _)
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1Iso | The 1st group homology of `A`, defined as the 1st homology of the complex of inhomogeneous
chains, is isomorphic to `cycles₁ A ⧸ boundaries₁ A`, which is a simpler type. |
π_comp_H1Iso_hom :
π A 1 ≫ (H1Iso A).hom = (isoCycles₁ A).hom ≫
(shortComplexH1 A).moduleCatLeftHomologyData.π := by
simp [H1Iso, isoCycles₁, π, HomologicalComplex.homologyπ, leftHomologyπ]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | π_comp_H1Iso_hom | null |
π_comp_H1Iso_inv :
(shortComplexH1 A).moduleCatLeftHomologyData.π ≫ (H1Iso A).inv = H1π A :=
(CommSq.vert_inv ⟨π_comp_H1Iso_hom A⟩).w | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | π_comp_H1Iso_inv | null |
mkH1OfIsTrivial : Additive (Abelianization G) →ₗ[ℤ] A →ₗ[ℤ] H1 A :=
AddMonoidHom.toIntLinearMap <| AddMonoidHom.toMultiplicative'.symm <| Abelianization.lift {
toFun g := Multiplicative.ofAdd (AddMonoidHom.toIntLinearMap (AddMonoidHomClass.toAddMonoidHom
((H1π A).hom ∘ₗ (cycles₁IsoOfIsTrivial A).inv.hom ∘ₗ lsingle g)))
map_one' := Multiplicative.toAdd.injective <|
LinearMap.ext fun _ => (H1π_eq_zero_iff _).2 <| single_one_mem_boundaries₁ _
map_mul' g h := Multiplicative.toAdd.injective <| LinearMap.ext fun a => by
simpa [← map_add] using ((H1π_eq_iff _ _).2 ⟨single (g, h) a, by
simp [cycles₁IsoOfIsTrivial, sub_add_eq_add_sub, add_comm (single h a),
d₂₁_single (A := A)]⟩).symm }
variable {A} in
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | mkH1OfIsTrivial | If a `G`-representation on `A` is trivial, this is the natural map `Gᵃᵇ → A → H₁(G, A)`
sending `⟦g⟧, a` to `⟦single g a⟧`. |
mkH1OfIsTrivial_apply (g : G) (a : A) :
mkH1OfIsTrivial A (Additive.ofMul (Abelianization.of g)) a =
H1π A ((cycles₁IsoOfIsTrivial A).inv (single g a)) := rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | mkH1OfIsTrivial_apply | null |
H1ToTensorOfIsTrivial : H1 A →ₗ[ℤ] (Additive <| Abelianization G) ⊗[ℤ] A :=
((QuotientAddGroup.lift _ ((Finsupp.liftAddHom fun g => AddMonoidHomClass.toAddMonoidHom
(TensorProduct.mk ℤ _ _ (Additive.ofMul (Abelianization.of g)))).comp
(cycles₁ A).toAddSubgroup.subtype) fun ⟨y, hy⟩ ⟨z, hz⟩ => AddMonoidHom.mem_ker.2 <| by
simp [← hz, d₂₁, sum_sum_index, sum_add_index', tmul_add, sum_sub_index, tmul_sub,
shortComplexH1]).comp <| AddMonoidHomClass.toAddMonoidHom (H1Iso A).hom.hom).toIntLinearMap
variable {A} in
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1ToTensorOfIsTrivial | If a `G`-representation on `A` is trivial, this is the natural map `H₁(G, A) → Gᵃᵇ ⊗[ℤ] A`
sending `⟦single g a⟧` to `⟦g⟧ ⊗ₜ a`. |
H1ToTensorOfIsTrivial_H1π_single (g : G) (a : A) :
H1ToTensorOfIsTrivial A (H1π A <| (cycles₁IsoOfIsTrivial A).inv (single g a)) =
Additive.ofMul (Abelianization.of g) ⊗ₜ[ℤ] a := by
simp only [H1ToTensorOfIsTrivial, H1π, AddMonoidHom.coe_toIntLinearMap, AddMonoidHom.coe_comp]
change QuotientAddGroup.lift _ _ _ ((H1Iso A).hom _) = _
simp [π_comp_H1Iso_hom_apply, Submodule.Quotient.mk, QuotientAddGroup.lift, AddCon.lift,
AddCon.liftOn, AddSubgroup.subtype, cycles₁IsoOfIsTrivial] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1ToTensorOfIsTrivial_H1π_single | null |
@[simps! -isSimp]
H1AddEquivOfIsTrivial :
H1 A ≃+ (Additive <| Abelianization G) ⊗[ℤ] A :=
LinearEquiv.toAddEquiv <| LinearEquiv.ofLinear
(H1ToTensorOfIsTrivial A) (lift <| mkH1OfIsTrivial A)
(ext <| LinearMap.toAddMonoidHom_injective <| by
ext g a
simp [TensorProduct.mk_apply, TensorProduct.lift.tmul, mkH1OfIsTrivial_apply,
H1ToTensorOfIsTrivial_H1π_single g a])
(LinearMap.toAddMonoidHom_injective <|
(H1Iso A).symm.toLinearEquiv.toAddEquiv.comp_left_injective <|
QuotientAddGroup.addMonoidHom_ext _ <|
(cycles₁IsoOfIsTrivial A).symm.toLinearEquiv.toAddEquiv.comp_left_injective <| by
ext
simp only [H1ToTensorOfIsTrivial, Iso.toLinearEquiv, AddMonoidHom.coe_comp,
LinearMap.toAddMonoidHom_coe, LinearMap.coe_comp, AddMonoidHom.coe_toIntLinearMap]
change TensorProduct.lift _ (QuotientAddGroup.lift _ _ _ ((H1Iso A).hom _)) = _
simpa [AddSubgroup.subtype, cycles₁IsoOfIsTrivial_inv_apply (A := A),
-π_comp_H1Iso_inv_apply] using (π_comp_H1Iso_inv_apply A _).symm)
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1AddEquivOfIsTrivial | If a `G`-representation on `A` is trivial, this is the group isomorphism between
`H₁(G, A) ≃+ Gᵃᵇ ⊗[ℤ] A` defined by `⟦single g a⟧ ↦ ⟦g⟧ ⊗ a`. |
H1AddEquivOfIsTrivial_single (g : G) (a : A) :
H1AddEquivOfIsTrivial A (H1π A <| (cycles₁IsoOfIsTrivial A).inv (single g a)) =
Additive.ofMul (Abelianization.of g) ⊗ₜ[ℤ] a := by
rw [H1AddEquivOfIsTrivial_apply, H1ToTensorOfIsTrivial_H1π_single g a]
@[simp] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1AddEquivOfIsTrivial_single | null |
H1AddEquivOfIsTrivial_symm_tmul (g : G) (a : A) :
(H1AddEquivOfIsTrivial A).symm (Additive.ofMul (Abelianization.of g) ⊗ₜ[ℤ] a) =
H1π A ((cycles₁IsoOfIsTrivial A).inv <| single g a) := by
rfl | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H1AddEquivOfIsTrivial_symm_tmul | null |
H2 := groupHomology A 2 | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H2 | Shorthand for the 2nd group homology of a `k`-linear `G`-representation `A`, `H₂(G, A)`,
defined as the 2nd homology of the complex of inhomogeneous chains of `A`. |
H2π : ModuleCat.of k (cycles₂ A) ⟶ H2 A :=
(isoCycles₂ A).inv ≫ π A 2 | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H2π | The quotient map from the 2-cycles of `A`, as a submodule of `G × G →₀ A`, to `H₂(G, A)`. |
H2π_eq_zero_iff (x : cycles₂ A) : H2π A x = 0 ↔ x.1 ∈ boundaries₂ A := by
have h := leftHomologyπ_naturality'_assoc (isoShortComplexH2 A).inv
(shortComplexH2 A).moduleCatLeftHomologyData (leftHomologyData _)
((inhomogeneousChains A).sc 2).leftHomologyIso.hom
simp only [H2π, isoCycles₂, π, HomologicalComplex.homologyπ, homologyπ,
cyclesMapIso'_inv, leftHomologyπ, ← h, ← leftHomologyMapIso'_inv, ModuleCat.hom_comp,
LinearMap.coe_comp, Function.comp_apply, map_eq_zero_iff _
((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
simp [LinearMap.range_codRestrict, boundaries₂, shortComplexH2, cycles₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H2π_eq_zero_iff | null |
H2π_eq_iff (x y : cycles₂ A) :
H2π A x = H2π A y ↔ x.1 - y.1 ∈ boundaries₂ A := by
rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl
@[elab_as_elim] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H2π_eq_iff | null |
H2_induction_on {C : H2 A → Prop} (x : H2 A) (h : ∀ x : cycles₂ A, C (H2π A x)) :
C x :=
groupHomology_induction_on x (fun y => by simpa [H2π] using h ((isoCycles₂ A).hom y))
variable (A) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H2_induction_on | null |
H2Iso : H2 A ≅ (shortComplexH2 A).moduleCatLeftHomologyData.H :=
(leftHomologyIso _).symm ≪≫ (leftHomologyMapIso' (isoShortComplexH2 A) _ _)
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | H2Iso | The 2nd group homology of `A`, defined as the 2nd homology of the complex of inhomogeneous
chains, is isomorphic to `cycles₂ A ⧸ boundaries₂ A`, which is a simpler type. |
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