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 ⌀ |
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noncomputable mapShortComplex₂ (i : ℕ) := X.map (functor k G i) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mapShortComplex₂ | The short complex `Hⁱ(G, X₁) ⟶ Hⁱ(G, X₂) ⟶ Hⁱ(G, X₃)` associated to a short complex of
representations `X₁ ⟶ X₂ ⟶ X₃`. |
noncomputable mapShortComplex₃ {i j : ℕ} (hij : i + 1 = j) :=
(snakeInput (map_cochainsFunctor_shortExact hX) _ _ hij).L₁' | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mapShortComplex₃ | The short complex `Hⁱ(G, X₂) ⟶ Hⁱ(G, X₃) ⟶ Hʲ(G, X₁)`. |
mapShortComplex₁_exact {i j : ℕ} (hij : i + 1 = j) :
(mapShortComplex₁ hX hij).Exact :=
(map_cochainsFunctor_shortExact hX).homology_exact₁ i j hij | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mapShortComplex₁_exact | Exactness of `Hⁱ(G, X₃) ⟶ Hʲ(G, X₁) ⟶ Hʲ(G, X₂)`. |
mapShortComplex₂_exact (i : ℕ) :
(mapShortComplex₂ X i).Exact :=
(map_cochainsFunctor_shortExact hX).homology_exact₂ i | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mapShortComplex₂_exact | Exactness of `Hⁱ(G, X₁) ⟶ Hⁱ(G, X₂) ⟶ Hⁱ(G, X₃)`. |
mapShortComplex₃_exact {i j : ℕ} (hij : i + 1 = j) :
(mapShortComplex₃ hX hij).Exact :=
(map_cochainsFunctor_shortExact hX).homology_exact₃ i j hij | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mapShortComplex₃_exact | Exactness of `Hⁱ(G, X₂) ⟶ Hⁱ(G, X₃) ⟶ Hʲ(G, X₁)`. |
noncomputable δ (i j : ℕ) (hij : i + 1 = j) :
groupCohomology X.X₃ i ⟶ groupCohomology X.X₁ j :=
(map_cochainsFunctor_shortExact hX).δ i j hij
open Limits | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | δ | The connecting homomorphism `Hⁱ(G, X₃) ⟶ Hʲ(G, X₁)` associated to an exact sequence
`0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` of representations. |
epi_δ_of_isZero (n : ℕ) (h : IsZero (groupCohomology X.X₂ (n + 1))) :
Epi (δ hX n (n + 1) rfl) := SnakeInput.epi_δ _ h | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | epi_δ_of_isZero | null |
mono_δ_of_isZero (n : ℕ) (h : IsZero (groupCohomology X.X₂ n)) :
Mono (δ hX n (n + 1) rfl) := SnakeInput.mono_δ _ h | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mono_δ_of_isZero | null |
isIso_δ_of_isZero (n : ℕ) (h : IsZero (groupCohomology X.X₂ n))
(hs : IsZero (groupCohomology X.X₂ (n + 1))) :
IsIso (δ hX n (n + 1) rfl) := SnakeInput.isIso_δ _ h hs | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | isIso_δ_of_isZero | null |
noncomputable cocyclesMkOfCompEqD {i j : ℕ} {y : (Fin i → G) → X.X₂}
{x : (Fin j → G) → X.X₁} (hx : X.f.hom ∘ x = (inhomogeneousCochains X.X₂).d i j y) :
cocycles X.X₁ j :=
cocyclesMk x <| by simpa using
((map_cochainsFunctor_shortExact hX).d_eq_zero_of_f_eq_d_apply i j y x
(by simpa using hx) (j + 1)) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | cocyclesMkOfCompEqD | Given an exact sequence of `G`-representations `0 ⟶ X₁ ⟶f X₂ ⟶g X₃ ⟶ 0`, this expresses an
`n + 1`-cochain `x : Gⁿ⁺¹ → X₁` such that `f ∘ x ∈ Bⁿ⁺¹(G, X₂)` as a cocycle.
Stated for readability of `δ_apply`. |
δ_apply {i j : ℕ} (hij : i + 1 = j)
(z : (Fin i → G) → X.X₃) (hz : (inhomogeneousCochains X.X₃).d i j z = 0)
(y : (Fin i → G) → X.X₂) (hy : (cochainsMap (MonoidHom.id G) X.g).f i y = z)
(x : (Fin j → G) → X.X₁) (hx : X.f.hom ∘ x = (inhomogeneousCochains X.X₂).d i j y) :
δ hX i j hij (π X.X₃ i <| cocyclesMk z (by subst hij; simpa using hz)) =
π X.X₁ j (cocyclesMkOfCompEqD hX hx) := by
exact (map_cochainsFunctor_shortExact hX).δ_apply i j hij z hz y hy x
(by simpa using hx) (j + 1) (by simp) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | δ_apply | null |
mem_cocycles₁_of_comp_eq_d₀₁
{y : X.X₂} {x : G → X.X₁} (hx : X.f.hom ∘ x = d₀₁ X.X₂ y) :
x ∈ cocycles₁ X.X₁ := by
apply Function.Injective.comp_left ((Rep.mono_iff_injective X.f).1 hX.2)
have := congr($((mapShortComplexH1 (MonoidHom.id G) X.f).comm₂₃.symm) x)
simp_all [shortComplexH1, LinearMap.compLeft]
@[deprecated (since := "2025-07-02")]
alias mem_oneCocycles_of_comp_eq_dZero := mem_cocycles₁_of_comp_eq_d₀₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mem_cocycles₁_of_comp_eq_d₀₁ | Stated for readability of `δ₀_apply`. |
δ₀_apply
(z : X.X₃.ρ.invariants) (y : X.X₂) (hy : X.g.hom y = z)
(x : G → X.X₁) (hx : X.f.hom ∘ x = d₀₁ X.X₂ y) :
δ hX 0 1 rfl ((H0Iso X.X₃).inv z) = H1π X.X₁ ⟨x, mem_cocycles₁_of_comp_eq_d₀₁ hX hx⟩ := by
simpa [H0Iso, H1π, ← cocyclesMk₁_eq X.X₁, ← cocyclesMk₀_eq z] using
δ_apply hX rfl ((cochainsIso₀ X.X₃).inv z.1) (by simp) ((cochainsIso₀ X.X₂).inv y)
(by ext; simp [← hy, cochainsIso₀]) ((cochainsIso₁ X.X₁).inv x) <| by
ext g
simpa [← hx] using congr_fun (congr($((CommSq.vert_inv
⟨cochainsMap_f_1_comp_cochainsIso₁ (MonoidHom.id G) X.f⟩).w) x)) g | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | δ₀_apply | null |
mem_cocycles₂_of_comp_eq_d₁₂
{y : G → X.X₂} {x : G × G → X.X₁} (hx : X.f.hom ∘ x = d₁₂ X.X₂ y) :
x ∈ cocycles₂ X.X₁ := by
apply Function.Injective.comp_left ((Rep.mono_iff_injective X.f).1 hX.2)
have := congr($((mapShortComplexH2 (MonoidHom.id G) X.f).comm₂₃.symm) x)
simp_all [shortComplexH2, LinearMap.compLeft]
@[deprecated (since := "2025-07-02")]
alias mem_twoCocycles_of_comp_eq_dOne := mem_cocycles₂_of_comp_eq_d₁₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mem_cocycles₂_of_comp_eq_d₁₂ | Stated for readability of `δ₁_apply`. |
δ₁_apply
(z : cocycles₁ X.X₃) (y : G → X.X₂) (hy : X.g.hom ∘ y = z)
(x : G × G → X.X₁) (hx : X.f.hom ∘ x = d₁₂ X.X₂ y) :
δ hX 1 2 rfl (H1π X.X₃ z) = H2π X.X₁ ⟨x, mem_cocycles₂_of_comp_eq_d₁₂ hX hx⟩ := by
simpa [H1π, H2π, ← cocyclesMk₂_eq X.X₁, ← cocyclesMk₁_eq X.X₃] using
δ_apply hX rfl ((cochainsIso₁ X.X₃).inv z) (by simp [cocycles₁.d₁₂_apply z])
((cochainsIso₁ X.X₂).inv y) (by ext; simp [cochainsIso₁, ← hy])
((cochainsIso₂ X.X₁).inv x) <| by
ext g
simpa [← hx] using congr_fun (congr($((CommSq.vert_inv
⟨cochainsMap_f_2_comp_cochainsIso₂ (MonoidHom.id G) X.f⟩).w) x)) g | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | δ₁_apply | null |
cochainsIso₀ : (inhomogeneousCochains A).X 0 ≅ A.V :=
(LinearEquiv.funUnique (Fin 0 → G) k A).toModuleIso
@[deprecated (since := "2025-06-25")] noncomputable alias zeroCochainsIso := cochainsIso₀
@[deprecated (since := "2025-05-09")] noncomputable alias zeroCochainsLequiv := zeroCochainsIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cochainsIso₀ | The 0th object in the complex of inhomogeneous cochains of `A : Rep k G` is isomorphic
to `A` as a `k`-module. |
cochainsIso₁ : (inhomogeneousCochains A).X 1 ≅ ModuleCat.of k (G → A) :=
(LinearEquiv.funCongrLeft k A (Equiv.funUnique (Fin 1) G)).toModuleIso.symm
@[deprecated (since := "2025-06-25")] noncomputable alias oneCochainsIso := cochainsIso₁
@[deprecated (since := "2025-05-09")] noncomputable alias oneCochainsLequiv := oneCochainsIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cochainsIso₁ | The 1st object in the complex of inhomogeneous cochains of `A : Rep k G` is isomorphic
to `Fun(G, A)` as a `k`-module. |
cochainsIso₂ : (inhomogeneousCochains A).X 2 ≅ ModuleCat.of k (G × G → A) :=
(LinearEquiv.funCongrLeft k A <| (piFinTwoEquiv fun _ => G)).toModuleIso.symm
@[deprecated (since := "2025-06-25")] noncomputable alias twoCochainsIso := cochainsIso₂
@[deprecated (since := "2025-05-09")] noncomputable alias twoCochainsLequiv := twoCochainsIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cochainsIso₂ | The 2nd object in the complex of inhomogeneous cochains of `A : Rep k G` is isomorphic
to `Fun(G², A)` as a `k`-module. |
cochainsIso₃ : (inhomogeneousCochains A).X 3 ≅ ModuleCat.of k (G × G × G → A) :=
(LinearEquiv.funCongrLeft k A <| ((Fin.consEquiv _).symm.trans
((Equiv.refl G).prodCongr (piFinTwoEquiv fun _ => G)))).toModuleIso.symm
@[deprecated (since := "2025-06-25")] noncomputable alias threeCochainsIso := cochainsIso₃
@[deprecated (since := "2025-05-09")] noncomputable alias threeCochainsLequiv := threeCochainsIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cochainsIso₃ | The 3rd object in the complex of inhomogeneous cochains of `A : Rep k G` is isomorphic
to `Fun(G³, A)` as a `k`-module. |
@[simps!]
d₀₁ : A.V ⟶ ModuleCat.of k (G → A) :=
ModuleCat.ofHom
{ toFun m g := A.ρ g m - m
map_add' x y := funext fun g => by simp only [map_add, add_sub_add_comm]; rfl
map_smul' r x := funext fun g => by dsimp; rw [map_smul, smul_sub] }
@[deprecated (since := "2025-06-25")] noncomputable alias dZero := d₀₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₀₁ | 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.` |
d₀₁_ker_eq_invariants : LinearMap.ker (d₀₁ A).hom = invariants A.ρ := by
ext x
simp only [LinearMap.mem_ker, mem_invariants, ← @sub_eq_zero _ _ _ x, funext_iff]
rfl
@[deprecated (since := "2025-06-25")]
noncomputable alias dZero_ker_eq_invariants := d₀₁_ker_eq_invariants
@[simp] theorem d₀₁_eq_zero [A.IsTrivial] : d₀₁ A = 0 := by
ext
rw [d₀₁_hom_apply, isTrivial_apply, sub_self]
rfl
@[deprecated (since := "2025-06-25")] alias dZero_eq_zero := d₀₁_eq_zero
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₀₁_ker_eq_invariants | null |
subtype_comp_d₀₁ : ModuleCat.ofHom (A.ρ.invariants.subtype) ≫ d₀₁ A = 0 := by
ext ⟨x, hx⟩ g
replace hx := hx g
rw [← sub_eq_zero] at hx
exact hx
@[deprecated (since := "2025-06-25")] alias subtype_comp_dZero := subtype_comp_d₀₁ | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | subtype_comp_d₀₁ | null |
@[simps!]
d₁₂ : ModuleCat.of k (G → A) ⟶ ModuleCat.of k (G × G → A) :=
ModuleCat.ofHom
{ toFun f g := A.ρ g.1 (f g.2) - f (g.1 * g.2) + f g.1
map_add' x y := funext fun g => by dsimp; rw [map_add, add_add_add_comm, add_sub_add_comm]
map_smul' r x := funext fun g => by dsimp; rw [map_smul, smul_add, smul_sub] }
@[deprecated (since := "2025-06-25")] noncomputable alias dOne := d₁₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₁₂ | 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).` |
@[simps!]
d₂₃ : ModuleCat.of k (G × G → A) ⟶ ModuleCat.of k (G × G × G → A) :=
ModuleCat.ofHom
{ toFun 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)
map_add' x y :=
funext fun g => by
dsimp
rw [map_add, add_sub_add_comm (A.ρ _ _), add_sub_assoc, add_sub_add_comm, add_add_add_comm,
add_sub_assoc, add_sub_assoc]
map_smul' r x := funext fun g => by dsimp; simp only [map_smul, smul_add, smul_sub] }
@[deprecated (since := "2025-06-25")] noncomputable alias dTwo := d₂₃ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₂₃ | 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).` |
comp_d₀₁_eq :
(cochainsIso₀ A).hom ≫ d₀₁ A =
(inhomogeneousCochains A).d 0 1 ≫ (cochainsIso₁ A).hom := by
ext x y
change A.ρ y (x default) - x default = _ + ({0} : Finset _).sum _
simp_rw [Fin.val_eq_zero, zero_add, pow_one, neg_smul, one_smul,
Finset.sum_singleton, sub_eq_add_neg]
rcongr i <;> exact Fin.elim0 i
@[deprecated (since := "2025-06-25")] noncomputable alias comp_dZero_eq := comp_d₀₁_eq
@[deprecated (since := "2025-05-09")] noncomputable alias dZero_comp_eq := comp_dZero_eq
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | comp_d₀₁_eq | Let `C(G, A)` denote the complex of inhomogeneous cochains of `A : Rep k G`. This lemma
says `d₀₁` gives a simpler expression for the 0th differential: that is, the following
square commutes:
```
C⁰(G, A) --d 0 1--> C¹(G, A)
| |
| |
| |
v v
A ------d₀₁-----> Fun(G, A)
```
where the vertical arrows are `cochainsIso₀` and `cochainsIso₁` respectively. |
eq_d₀₁_comp_inv :
(cochainsIso₀ A).inv ≫ (inhomogeneousCochains A).d 0 1 =
d₀₁ A ≫ (cochainsIso₁ A).inv :=
(CommSq.horiz_inv ⟨comp_d₀₁_eq A⟩).w
@[deprecated (since := "2025-06-25")] alias eq_dZero_comp_inv := eq_d₀₁_comp_inv | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | eq_d₀₁_comp_inv | null |
comp_d₁₂_eq :
(cochainsIso₁ A).hom ≫ d₁₂ A =
(inhomogeneousCochains A).d 1 2 ≫ (cochainsIso₂ A).hom := by
ext x y
change A.ρ y.1 (x _) - x _ + x _ = _ + _
rw [Fin.sum_univ_two]
simp only [Fin.val_zero, zero_add, pow_one, neg_smul, one_smul, Fin.val_one,
Nat.one_add, neg_one_sq, sub_eq_add_neg, add_assoc]
rcongr i <;> rw [Subsingleton.elim i 0] <;> rfl
@[deprecated (since := "2025-06-25")] alias comp_dOne_eq := comp_d₁₂_eq
@[deprecated (since := "2025-05-09")] alias dOne_comp_eq := comp_dOne_eq
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | comp_d₁₂_eq | Let `C(G, A)` denote the complex of inhomogeneous cochains of `A : Rep k G`. This lemma
says `d₁₂` gives a simpler expression for the 1st differential: that is, the following
square commutes:
```
C¹(G, A) ---d 1 2---> C²(G, A)
| |
| |
| |
v v
Fun(G, A) --d₁₂--> Fun(G × G, A)
```
where the vertical arrows are `cochainsIso₁` and `cochainsIso₂` respectively. |
eq_d₁₂_comp_inv :
(cochainsIso₁ A).inv ≫ (inhomogeneousCochains A).d 1 2 =
d₁₂ A ≫ (cochainsIso₂ A).inv :=
(CommSq.horiz_inv ⟨comp_d₁₂_eq A⟩).w
@[deprecated (since := "2025-06-25")] alias eq_dOne_comp_inv := eq_d₁₂_comp_inv | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | eq_d₁₂_comp_inv | null |
comp_d₂₃_eq :
(cochainsIso₂ A).hom ≫ d₂₃ A =
(inhomogeneousCochains A).d 2 3 ≫ (cochainsIso₃ A).hom := by
ext x y
change A.ρ y.1 (x _) - x _ + x _ - x _ = _ + _
dsimp
rw [Fin.sum_univ_three]
simp only [sub_eq_add_neg, add_assoc, Fin.val_zero, zero_add, pow_one, neg_smul,
one_smul, Fin.val_one, Fin.val_two, pow_succ' (-1 : k) 2, neg_sq, Nat.one_add, one_pow, mul_one]
rcongr i <;> fin_cases i <;> rfl
@[deprecated (since := "2025-06-25")] alias comp_dTwo_eq := comp_d₂₃_eq
@[deprecated (since := "2025-05-09")] alias dTwo_comp_eq := comp_dTwo_eq
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | comp_d₂₃_eq | Let `C(G, A)` denote the complex of inhomogeneous cochains of `A : Rep k G`. This lemma
says `d₂₃` gives a simpler expression for the 2nd differential: that is, the following
square commutes:
```
C²(G, A) ----d 2 3----> C³(G, A)
| |
| |
| |
v v
Fun(G × G, A) --d₂₃--> Fun(G × G × G, A)
```
where the vertical arrows are `cochainsIso₂` and `cochainsIso₃` respectively. |
eq_d₂₃_comp_inv :
(cochainsIso₂ A).inv ≫ (inhomogeneousCochains A).d 2 3 =
d₂₃ A ≫ (cochainsIso₃ A).inv :=
(CommSq.horiz_inv ⟨comp_d₂₃_eq A⟩).w
@[deprecated (since := "2025-06-25")] alias eq_dTwo_comp_inv := eq_d₂₃_comp_inv
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | eq_d₂₃_comp_inv | null |
d₀₁_comp_d₁₂ : d₀₁ A ≫ d₁₂ A = 0 := by
ext
simp [Pi.zero_apply (M := fun _ => A)]
@[deprecated (since := "2025-06-25")] alias dZero_comp_dOne := d₀₁_comp_d₁₂
@[deprecated (since := "2025-05-14")] alias dOne_comp_dZero := dZero_comp_dOne
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₀₁_comp_d₁₂ | null |
d₁₂_comp_d₂₃ : d₁₂ A ≫ d₂₃ A = 0 := by
ext f g
simp [mul_assoc, Pi.zero_apply (M := fun _ => A)]
abel
@[deprecated (since := "2025-06-25")] alias dOne_comp_dTwo := d₁₂_comp_d₂₃
@[deprecated (since := "2025-05-14")] alias dTwo_comp_dOne := dOne_comp_dTwo
open ShortComplex | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₁₂_comp_d₂₃ | null |
@[simps! -isSimp f g]
shortComplexH0 : ShortComplex (ModuleCat k) :=
mk _ _ (subtype_comp_d₀₁ A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | shortComplexH0 | The (exact) short complex `A.ρ.invariants ⟶ A ⟶ (G → A)`. |
@[simps! -isSimp f g]
shortComplexH1 : ShortComplex (ModuleCat k) :=
mk (d₀₁ A) (d₁₂ A) (d₀₁_comp_d₁₂ A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | shortComplexH1 | The short complex `A --d₀₁--> Fun(G, A) --d₁₂--> Fun(G × G, A)`. |
@[simps! -isSimp f g]
shortComplexH2 : ShortComplex (ModuleCat k) :=
mk (d₁₂ A) (d₂₃ A) (d₁₂_comp_d₂₃ A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | shortComplexH2 | The short complex `Fun(G, A) --d₁₂--> Fun(G × G, A) --d₂₃--> Fun(G × G × G, A)`. |
cocycles₁ : Submodule k (G → A) := LinearMap.ker (d₁₂ A).hom
@[deprecated (since := "2025-06-25")] noncomputable alias oneCocycles := cocycles₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₁ | 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).` |
cocycles₂ : Submodule k (G × G → A) := LinearMap.ker (d₂₃ A).hom
@[deprecated (since := "2025-06-25")] noncomputable alias twoCocycles := cocycles₂
variable {A} | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂ | 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).` |
@[simp]
cocycles₁.coe_mk (f : G → A) (hf) : ((⟨f, hf⟩ : cocycles₁ A) : G → A) = f := rfl
@[deprecated (since := "2025-06-25")] alias oneCocycles.coe_mk := cocycles₁.coe_mk
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₁.coe_mk | null |
cocycles₁.val_eq_coe (f : cocycles₁ A) : f.1 = f := rfl
@[deprecated (since := "2025-06-25")] alias oneCocycles.val_eq_coe := cocycles₁.val_eq_coe
@[ext] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₁.val_eq_coe | null |
cocycles₁_ext {f₁ f₂ : cocycles₁ A} (h : ∀ g : G, f₁ g = f₂ g) : f₁ = f₂ :=
DFunLike.ext f₁ f₂ h
@[deprecated (since := "2025-06-25")] alias oneCocycles_ext := cocycles₁_ext | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₁_ext | null |
mem_cocycles₁_def (f : G → A) :
f ∈ cocycles₁ A ↔ ∀ g h : G, A.ρ g (f h) - f (g * h) + f g = 0 :=
LinearMap.mem_ker.trans <| by
simp_rw [funext_iff, d₁₂_hom_apply, Prod.forall]
rfl
@[deprecated (since := "2025-06-25")] alias mem_oneCocycles_def := mem_cocycles₁_def | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | mem_cocycles₁_def | null |
mem_cocycles₁_iff (f : G → A) :
f ∈ cocycles₁ A ↔ ∀ g h : G, f (g * h) = A.ρ g (f h) + f g := by
simp_rw [mem_cocycles₁_def, sub_add_eq_add_sub, sub_eq_zero, eq_comm]
@[deprecated (since := "2025-06-25")] alias mem_oneCocycles_iff := mem_cocycles₁_iff
@[simp] theorem cocycles₁_map_one (f : cocycles₁ A) : f 1 = 0 := by
have := (mem_cocycles₁_def f).1 f.2 1 1
simpa only [map_one, Module.End.one_apply, mul_one, sub_self, zero_add] using this
@[deprecated (since := "2025-06-25")] alias oneCocycles_map_one := cocycles₁_map_one
@[simp] theorem cocycles₁_map_inv (f : cocycles₁ A) (g : G) :
A.ρ g (f g⁻¹) = - f g := by
rw [← add_eq_zero_iff_eq_neg, ← cocycles₁_map_one f, ← mul_inv_cancel g,
(mem_cocycles₁_iff f).1 f.2 g g⁻¹]
@[deprecated (since := "2025-06-25")] alias oneCocycles_map_inv := cocycles₁_map_inv | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | mem_cocycles₁_iff | null |
d₀₁_apply_mem_cocycles₁ (x : A) :
d₀₁ A x ∈ cocycles₁ A :=
d₀₁_comp_d₁₂_apply _ _
@[deprecated (since := "2025-06-25")] alias dZero_apply_mem_oneCocycles := d₀₁_apply_mem_cocycles₁
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₀₁_apply_mem_cocycles₁ | null |
cocycles₁.d₁₂_apply (x : cocycles₁ A) :
d₁₂ A x = 0 := x.2
@[deprecated (since := "2025-06-25")] alias oneCocycles.dOne_apply := cocycles₁.d₁₂_apply | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₁.d₁₂_apply | null |
cocycles₁_map_mul_of_isTrivial [A.IsTrivial] (f : cocycles₁ A) (g h : G) :
f (g * h) = f g + f h := by
rw [(mem_cocycles₁_iff f).1 f.2, isTrivial_apply A.ρ g (f h), add_comm]
@[deprecated (since := "2025-06-25")]
alias oneCocycles_map_mul_of_isTrivial := cocycles₁_map_mul_of_isTrivial | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₁_map_mul_of_isTrivial | null |
mem_cocycles₁_of_addMonoidHom [A.IsTrivial] (f : Additive G →+ A) :
f ∘ Additive.ofMul ∈ cocycles₁ A :=
(mem_cocycles₁_iff _).2 fun g h => by
simp only [Function.comp_apply, ofMul_mul, map_add,
isTrivial_apply A.ρ g (f (Additive.ofMul h)), add_comm (f (Additive.ofMul g))]
@[deprecated (since := "2025-06-25")]
alias mem_oneCocycles_of_addMonoidHom := mem_cocycles₁_of_addMonoidHom
variable (A) in | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | mem_cocycles₁_of_addMonoidHom | null |
@[simps!]
cocycles₁IsoOfIsTrivial [hA : A.IsTrivial] :
ModuleCat.of k (cocycles₁ A) ≅ ModuleCat.of k (Additive G →+ A) :=
LinearEquiv.toModuleIso
{ toFun f :=
{ toFun := f ∘ Additive.toMul
map_zero' := cocycles₁_map_one f
map_add' := cocycles₁_map_mul_of_isTrivial f }
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f :=
{ val := f
property := mem_cocycles₁_of_addMonoidHom f } }
@[deprecated (since := "2025-06-25")]
noncomputable alias oneCocyclesIsoOfIsTrivial := cocycles₁IsoOfIsTrivial
@[deprecated (since := "2025-05-09")]
noncomputable alias oneCocyclesLequivOfIsTrivial := oneCocyclesIsoOfIsTrivial | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₁IsoOfIsTrivial | When `A : Rep k G` is a trivial representation of `G`, `Z¹(G, A)` is isomorphic to the
group homs `G → A`. |
@[simp]
cocycles₂.coe_mk (f : G × G → A) (hf) : ((⟨f, hf⟩ : cocycles₂ A) : G × G → A) = f := rfl
@[deprecated (since := "2025-06-25")] alias twoCocycles.coe_mk := cocycles₂.coe_mk
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂.coe_mk | null |
cocycles₂.val_eq_coe (f : cocycles₂ A) : f.1 = f := rfl
@[deprecated (since := "2025-06-25")] alias twoCocycles.val_eq_coe := cocycles₂.val_eq_coe
@[ext] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂.val_eq_coe | null |
cocycles₂_ext {f₁ f₂ : cocycles₂ A} (h : ∀ g h : G, f₁ (g, h) = f₂ (g, h)) : f₁ = f₂ :=
DFunLike.ext f₁ f₂ (Prod.forall.mpr h)
@[deprecated (since := "2025-06-25")] alias twoCocycles_ext := cocycles₂_ext | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂_ext | null |
mem_cocycles₂_def (f : G × G → A) :
f ∈ cocycles₂ A ↔ ∀ g h j : G,
A.ρ g (f (h, j)) - f (g * h, j) + f (g, h * j) - f (g, h) = 0 :=
LinearMap.mem_ker.trans <| by
simp_rw [funext_iff, d₂₃_hom_apply, Prod.forall]
rfl
@[deprecated (since := "2025-06-25")] alias mem_twoCocycles_def := mem_cocycles₂_def | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | mem_cocycles₂_def | null |
mem_cocycles₂_iff (f : G × G → A) :
f ∈ cocycles₂ A ↔ ∀ g h j : G,
f (g * h, j) + f (g, h) =
A.ρ g (f (h, j)) + f (g, h * j) := by
simp_rw [mem_cocycles₂_def, sub_eq_zero, sub_add_eq_add_sub, sub_eq_iff_eq_add, eq_comm,
add_comm (f (_ * _, _))]
@[deprecated (since := "2025-06-25")] alias mem_twoCocycles_iff := mem_cocycles₂_iff | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | mem_cocycles₂_iff | null |
cocycles₂_map_one_fst (f : cocycles₂ A) (g : G) :
f (1, g) = f (1, 1) := by
have := ((mem_cocycles₂_iff f).1 f.2 1 1 g).symm
simpa only [map_one, Module.End.one_apply, one_mul, add_right_inj, this]
@[deprecated (since := "2025-06-25")] alias twoCocycles_map_one_fst := cocycles₂_map_one_fst | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂_map_one_fst | null |
cocycles₂_map_one_snd (f : cocycles₂ A) (g : G) :
f (g, 1) = A.ρ g (f (1, 1)) := by
have := (mem_cocycles₂_iff f).1 f.2 g 1 1
simpa only [mul_one, add_left_inj, this]
@[deprecated (since := "2025-06-25")] alias twoCocycles_map_one_snd := cocycles₂_map_one_snd | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂_map_one_snd | null |
cocycles₂_ρ_map_inv_sub_map_inv (f : cocycles₂ A) (g : G) :
A.ρ g (f (g⁻¹, g)) - f (g, g⁻¹)
= f (1, 1) - f (g, 1) := by
have := (mem_cocycles₂_iff f).1 f.2 g g⁻¹ g
simp only [mul_inv_cancel, inv_mul_cancel, cocycles₂_map_one_fst _ g]
at this
exact sub_eq_sub_iff_add_eq_add.2 this.symm
@[deprecated (since := "2025-06-25")]
alias twoCocycles_ρ_map_inv_sub_map_inv := cocycles₂_ρ_map_inv_sub_map_inv | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂_ρ_map_inv_sub_map_inv | null |
d₁₂_apply_mem_cocycles₂ (x : G → A) :
d₁₂ A x ∈ cocycles₂ A :=
d₁₂_comp_d₂₃_apply _ _
@[deprecated (since := "2025-06-25")] alias dOne_apply_mem_twoCocycles := d₁₂_apply_mem_cocycles₂
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | d₁₂_apply_mem_cocycles₂ | null |
cocycles₂.d₂₃_apply (x : cocycles₂ A) :
d₂₃ A x = 0 := x.2
@[deprecated (since := "2025-06-25")] alias twoCocycles.dTwo_apply := cocycles₂.d₂₃_apply | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocycles₂.d₂₃_apply | null |
coboundaries₁ : Submodule k (G → A) :=
LinearMap.range (d₀₁ A).hom
@[deprecated (since := "2025-06-25")] noncomputable alias oneCoboundaries := coboundaries₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₁ | 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.` |
coboundaries₂ : Submodule k (G × G → A) :=
LinearMap.range (d₁₂ A).hom
@[deprecated (since := "2025-06-25")] noncomputable alias twoCoboundaries := coboundaries₂
variable {A} | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₂ | 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).` |
@[simp]
coboundaries₁.coe_mk (f : G → A) (hf) :
((⟨f, hf⟩ : coboundaries₁ A) : G → A) = f := rfl
@[deprecated (since := "2025-06-25")] alias oneCoboundaries.coe_mk := coboundaries₁.coe_mk
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₁.coe_mk | null |
coboundaries₁.val_eq_coe (f : coboundaries₁ A) : f.1 = f := rfl
@[deprecated (since := "2025-06-25")] alias oneCoboundaries.val_eq_coe := coboundaries₁.val_eq_coe
@[ext] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₁.val_eq_coe | null |
coboundaries₁_ext {f₁ f₂ : coboundaries₁ A} (h : ∀ g : G, f₁ g = f₂ g) : f₁ = f₂ :=
DFunLike.ext f₁ f₂ h
@[deprecated (since := "2025-06-25")] alias oneCoboundaries_ext := coboundaries₁_ext
variable (A) in | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₁_ext | null |
coboundaries₁_le_cocycles₁ : coboundaries₁ A ≤ cocycles₁ A := by
rintro _ ⟨x, rfl⟩
exact d₀₁_apply_mem_cocycles₁ x
@[deprecated (since := "2025-06-25")]
alias oneCoboundaries_le_oneCocycles := coboundaries₁_le_cocycles₁
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₁_le_cocycles₁ | null |
coboundariesToCocycles₁ : coboundaries₁ A →ₗ[k] cocycles₁ A :=
Submodule.inclusion (coboundaries₁_le_cocycles₁ A)
@[deprecated (since := "2025-06-25")]
noncomputable alias oneCoboundariesToOneCocycles := coboundariesToCocycles₁
@[simp] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesToCocycles₁ | Natural inclusion `B¹(G, A) →ₗ[k] Z¹(G, A)`. |
coboundariesToCocycles₁_apply (x : coboundaries₁ A) :
coboundariesToCocycles₁ A x = x.1 := rfl
@[deprecated (since := "2025-06-25")]
alias oneCoboundariesToOneCocycles_apply := coboundariesToCocycles₁_apply | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesToCocycles₁_apply | null |
coboundaries₁_eq_bot_of_isTrivial (A : Rep k G) [A.IsTrivial] :
coboundaries₁ A = ⊥ := by
simp_rw [coboundaries₁, d₀₁_eq_zero]
exact LinearMap.range_eq_bot.2 rfl
@[deprecated (since := "2025-06-25")]
alias oneCoboundaries_eq_bot_of_isTrivial := coboundaries₁_eq_bot_of_isTrivial | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₁_eq_bot_of_isTrivial | null |
@[simp]
coboundaries₂.coe_mk (f : G × G → A) (hf) :
((⟨f, hf⟩ : coboundaries₂ A) : G × G → A) = f := rfl
@[deprecated (since := "2025-06-25")] alias twoCoboundaries.coe_mk := coboundaries₂.coe_mk
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₂.coe_mk | null |
coboundaries₂.val_eq_coe (f : coboundaries₂ A) : f.1 = f := rfl
@[deprecated (since := "2025-06-25")] alias twoCoboundaries.val_eq_coe := coboundaries₂.val_eq_coe
@[ext] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₂.val_eq_coe | null |
coboundaries₂_ext {f₁ f₂ : coboundaries₂ A} (h : ∀ g h : G, f₁ (g, h) = f₂ (g, h)) :
f₁ = f₂ :=
DFunLike.ext f₁ f₂ (Prod.forall.mpr h)
@[deprecated (since := "2025-06-25")] alias twoCoboundaries_ext := coboundaries₂_ext
variable (A) in | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₂_ext | null |
coboundaries₂_le_cocycles₂ : coboundaries₂ A ≤ cocycles₂ A := by
rintro _ ⟨x, rfl⟩
exact d₁₂_apply_mem_cocycles₂ x
@[deprecated (since := "2025-06-25")]
alias twoCoboundaries_le_twoCocycles := coboundaries₂_le_cocycles₂
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundaries₂_le_cocycles₂ | null |
coboundariesToCocycles₂ : coboundaries₂ A →ₗ[k] cocycles₂ A :=
Submodule.inclusion (coboundaries₂_le_cocycles₂ A)
@[deprecated (since := "2025-06-25")]
noncomputable alias twoCoboundariesToTwoCocycles := coboundariesToCocycles₂
@[simp] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesToCocycles₂ | Natural inclusion `B²(G, A) →ₗ[k] Z²(G, A)`. |
coboundariesToCocycles₂_apply (x : coboundaries₂ A) :
coboundariesToCocycles₂ A x = x.1 := rfl
@[deprecated (since := "2025-06-25")]
alias twoCoboundariesToTwoCocycles_apply := coboundariesToCocycles₂_apply | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesToCocycles₂_apply | null |
IsCocycle₁ (f : G → A) : Prop := ∀ g h : G, f (g * h) = g • f h + f g
@[deprecated (since := "2025-06-25")] alias IsOneCocycle := IsCocycle₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsCocycle₁ | A function `f : G → A` satisfies the 1-cocycle condition if
`f(gh) = g • f(h) + f(g)` for all `g, h : G`. |
IsCocycle₂ (f : G × G → A) : Prop :=
∀ g h j : G, f (g * h, j) + f (g, h) = g • (f (h, j)) + f (g, h * j)
@[deprecated (since := "2025-06-25")] alias IsTwoCocycle := IsCocycle₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsCocycle₂ | 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`. |
map_one_of_isCocycle₁ {f : G → A} (hf : IsCocycle₁ f) :
f 1 = 0 := by
simpa only [mul_one, one_smul, left_eq_add] using hf 1 1
@[deprecated (since := "2025-06-25")] alias map_one_of_isOneCocycle := map_one_of_isCocycle₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_one_of_isCocycle₁ | null |
map_one_fst_of_isCocycle₂ {f : G × G → A} (hf : IsCocycle₂ f) (g : G) :
f (1, g) = f (1, 1) := by
simpa only [one_smul, one_mul, mul_one, add_right_inj] using (hf 1 1 g).symm
@[deprecated (since := "2025-06-25")] alias map_one_fst_of_isTwoCocycle := map_one_fst_of_isCocycle₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_one_fst_of_isCocycle₂ | null |
map_one_snd_of_isCocycle₂ {f : G × G → A} (hf : IsCocycle₂ f) (g : G) :
f (g, 1) = g • f (1, 1) := by
simpa only [mul_one, add_left_inj] using hf g 1 1
@[deprecated (since := "2025-06-25")] alias map_one_snd_of_isTwoCocycle := map_one_snd_of_isCocycle₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_one_snd_of_isCocycle₂ | null |
@[scoped simp] map_inv_of_isCocycle₁ {f : G → A} (hf : IsCocycle₁ f) (g : G) :
g • f g⁻¹ = - f g := by
rw [← add_eq_zero_iff_eq_neg, ← map_one_of_isCocycle₁ hf, ← mul_inv_cancel g, hf g g⁻¹]
@[deprecated (since := "2025-06-25")] alias map_inv_of_isOneCocycle := map_inv_of_isCocycle₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_inv_of_isCocycle₁ | null |
smul_map_inv_sub_map_inv_of_isCocycle₂ {f : G × G → A} (hf : IsCocycle₂ f) (g : G) :
g • f (g⁻¹, g) - f (g, g⁻¹) = f (1, 1) - f (g, 1) := by
have := hf g g⁻¹ g
simp only [mul_inv_cancel, inv_mul_cancel, map_one_fst_of_isCocycle₂ hf g] at this
exact sub_eq_sub_iff_add_eq_add.2 this.symm
@[deprecated (since := "2025-06-25")]
alias smul_map_inv_sub_map_inv_of_isTwoCocycle := smul_map_inv_sub_map_inv_of_isCocycle₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | smul_map_inv_sub_map_inv_of_isCocycle₂ | null |
IsCoboundary₁ (f : G → A) : Prop := ∃ x : A, ∀ g : G, g • x - x = f g
@[deprecated (since := "2025-06-25")] alias IsOneCoboundary := IsCoboundary₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsCoboundary₁ | 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`. |
IsCoboundary₂ (f : G × G → A) : Prop :=
∃ x : G → A, ∀ g h : G, g • x h - x (g * h) + x g = f (g, h)
@[deprecated (since := "2025-06-25")] alias IsTwoCoboundary := IsCoboundary₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsCoboundary₂ | 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`. |
@[simps]
cocyclesOfIsCocycle₁ {f : G → A} (hf : IsCocycle₁ f) :
cocycles₁ (Rep.ofDistribMulAction k G A) :=
⟨f, (mem_cocycles₁_iff (A := Rep.ofDistribMulAction k G A) f).2 hf⟩
@[deprecated (since := "2025-06-25")] alias oneCocyclesOfIsOneCocycle := cocyclesOfIsCocycle₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesOfIsCocycle₁ | 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`. |
isCocycle₁_of_mem_cocycles₁
(f : G → A) (hf : f ∈ cocycles₁ (Rep.ofDistribMulAction k G A)) :
IsCocycle₁ f :=
fun _ _ => (mem_cocycles₁_iff (A := Rep.ofDistribMulAction k G A) f).1 hf _ _
@[deprecated (since := "2025-07-02")]
alias isOneCocycle_of_mem_oneCocycles := isCocycle₁_of_mem_cocycles₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isCocycle₁_of_mem_cocycles₁ | null |
@[simps]
coboundariesOfIsCoboundary₁ {f : G → A} (hf : IsCoboundary₁ f) :
coboundaries₁ (Rep.ofDistribMulAction k G A) :=
⟨f, hf.choose, funext hf.choose_spec⟩
@[deprecated (since := "2025-06-25")]
alias oneCoboundariesOfIsOneCoboundary := coboundariesOfIsCoboundary₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesOfIsCoboundary₁ | 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`. |
isCoboundary₁_of_mem_coboundaries₁
(f : G → A) (hf : f ∈ coboundaries₁ (Rep.ofDistribMulAction k G A)) :
IsCoboundary₁ f := by
rcases hf with ⟨a, rfl⟩
exact ⟨a, fun _ => rfl⟩
@[deprecated (since := "2025-07-02")]
alias isOneCoboundary_of_mem_oneCoboundaries := isCoboundary₁_of_mem_coboundaries₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isCoboundary₁_of_mem_coboundaries₁ | null |
@[simps]
cocyclesOfIsCocycle₂ {f : G × G → A} (hf : IsCocycle₂ f) :
cocycles₂ (Rep.ofDistribMulAction k G A) :=
⟨f, (mem_cocycles₂_iff (A := Rep.ofDistribMulAction k G A) f).2 hf⟩
@[deprecated (since := "2025-06-25")] alias twoCocyclesOfIsTwoCocycle := cocyclesOfIsCocycle₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesOfIsCocycle₂ | 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`. |
isCocycle₂_of_mem_cocycles₂
(f : G × G → A) (hf : f ∈ cocycles₂ (Rep.ofDistribMulAction k G A)) :
IsCocycle₂ f := (mem_cocycles₂_iff (A := Rep.ofDistribMulAction k G A) f).1 hf
@[deprecated (since := "2025-07-02")]
alias isTwoCocycle_of_mem_twoCocycles := isCocycle₂_of_mem_cocycles₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isCocycle₂_of_mem_cocycles₂ | null |
@[simps]
coboundariesOfIsCoboundary₂ {f : G × G → A} (hf : IsCoboundary₂ f) :
coboundaries₂ (Rep.ofDistribMulAction k G A) :=
⟨f, hf.choose,funext fun g ↦ hf.choose_spec g.1 g.2⟩
@[deprecated (since := "2025-06-25")]
alias twoCoboundariesOfIsTwoCoboundary := coboundariesOfIsCoboundary₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesOfIsCoboundary₂ | 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`. |
isCoboundary₂_of_mem_coboundaries₂
(f : G × G → A) (hf : f ∈ coboundaries₂ (Rep.ofDistribMulAction k G A)) :
IsCoboundary₂ f := by
rcases hf with ⟨a, rfl⟩
exact ⟨a, fun _ _ => rfl⟩
@[deprecated (since := "2025-07-02")]
alias isTwoCoboundary_of_mem_twoCoboundaries := isCoboundary₂_of_mem_coboundaries₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isCoboundary₂_of_mem_coboundaries₂ | null |
IsMulCocycle₁ (f : G → M) : Prop := ∀ g h : G, f (g * h) = g • f h * f g
@[deprecated (since := "2025-06-25")] alias IsMulOneCocycle := IsMulCocycle₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsMulCocycle₁ | A function `f : G → M` satisfies the multiplicative 1-cocycle condition if
`f(gh) = g • f(h) * f(g)` for all `g, h : G`. |
IsMulCocycle₂ (f : G × G → M) : Prop :=
∀ g h j : G, f (g * h, j) * f (g, h) = g • (f (h, j)) * f (g, h * j)
@[deprecated (since := "2025-06-25")] alias IsMulTwoCocycle := IsMulCocycle₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsMulCocycle₂ | 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`. |
map_one_of_isMulCocycle₁ {f : G → M} (hf : IsMulCocycle₁ f) :
f 1 = 1 := by
simpa only [mul_one, one_smul, left_eq_mul] using hf 1 1
@[deprecated (since := "2025-06-25")] alias map_one_of_isMulOneCocycle := map_one_of_isMulCocycle₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_one_of_isMulCocycle₁ | null |
map_one_fst_of_isMulCocycle₂ {f : G × G → M} (hf : IsMulCocycle₂ f) (g : G) :
f (1, g) = f (1, 1) := by
simpa only [one_smul, one_mul, mul_one, mul_right_inj] using (hf 1 1 g).symm
@[deprecated (since := "2025-06-25")]
alias map_one_fst_of_isMulTwoCocycle := map_one_fst_of_isMulCocycle₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_one_fst_of_isMulCocycle₂ | null |
map_one_snd_of_isMulCocycle₂ {f : G × G → M} (hf : IsMulCocycle₂ f) (g : G) :
f (g, 1) = g • f (1, 1) := by
simpa only [mul_one, mul_left_inj] using hf g 1 1
@[deprecated (since := "2025-06-25")]
alias map_one_snd_of_isMulTwoCocycle := map_one_snd_of_isMulCocycle₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_one_snd_of_isMulCocycle₂ | null |
@[scoped simp] map_inv_of_isMulCocycle₁ {f : G → M} (hf : IsMulCocycle₁ f) (g : G) :
g • f g⁻¹ = (f g)⁻¹ := by
rw [← mul_eq_one_iff_eq_inv, ← map_one_of_isMulCocycle₁ hf, ← mul_inv_cancel g, hf g g⁻¹]
@[deprecated (since := "2025-06-25")] alias map_inv_of_isMulOneCocycle := map_inv_of_isMulCocycle₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | map_inv_of_isMulCocycle₁ | null |
smul_map_inv_div_map_inv_of_isMulCocycle₂
{f : G × G → M} (hf : IsMulCocycle₂ f) (g : G) :
g • f (g⁻¹, g) / f (g, g⁻¹) = f (1, 1) / f (g, 1) := by
have := hf g g⁻¹ g
simp only [mul_inv_cancel, inv_mul_cancel, map_one_fst_of_isMulCocycle₂ hf g] at this
exact div_eq_div_iff_mul_eq_mul.2 this.symm
@[deprecated (since := "2025-07-02")]
alias smul_map_inv_div_map_inv_of_isMulTwoCocycle := smul_map_inv_div_map_inv_of_isMulCocycle₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | smul_map_inv_div_map_inv_of_isMulCocycle₂ | null |
IsMulCoboundary₁ (f : G → M) : Prop := ∃ x : M, ∀ g : G, g • x / x = f g
@[deprecated (since := "2025-06-25")] alias IsMulOneCoboundary := IsMulCoboundary₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsMulCoboundary₁ | 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`. |
IsMulCoboundary₂ (f : G × G → M) : Prop :=
∃ x : G → M, ∀ g h : G, g • x h / x (g * h) * x g = f (g, h)
@[deprecated (since := "2025-06-25")] alias IsMulTwoCoboundary := IsMulCoboundary₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | IsMulCoboundary₂ | 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`. |
@[simps]
cocyclesOfIsMulCocycle₁ {f : G → M} (hf : IsMulCocycle₁ f) :
cocycles₁ (Rep.ofMulDistribMulAction G M) :=
⟨Additive.ofMul ∘ f, (mem_cocycles₁_iff (A := Rep.ofMulDistribMulAction G M) f).2 hf⟩
@[deprecated (since := "2025-06-25")] alias oneCocyclesOfIsMulOneCocycle := cocyclesOfIsMulCocycle₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesOfIsMulCocycle₁ | 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`. |
isMulCocycle₁_of_mem_cocycles₁
(f : G → M) (hf : f ∈ cocycles₁ (Rep.ofMulDistribMulAction G M)) :
IsMulCocycle₁ (Additive.toMul ∘ f) :=
(mem_cocycles₁_iff (A := Rep.ofMulDistribMulAction G M) f).1 hf
@[deprecated (since := "2025-07-02")]
alias isMulOneCocycle_of_mem_oneCocycles := isMulCocycle₁_of_mem_cocycles₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isMulCocycle₁_of_mem_cocycles₁ | null |
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