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 ⌀ |
|---|---|---|---|---|---|---|
d_of {G : Type u} {n : ℕ} (c : Fin (n + 1) → G) :
d k G n (Finsupp.single c 1) =
Finset.univ.sum fun p : Fin (n + 1) =>
Finsupp.single (c ∘ p.succAbove) ((-1 : k) ^ (p : ℕ)) := by
simp [d]
variable (k G) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d_of | null |
xIso (n : ℕ) : (standardComplex k G).X n ≅ Rep.ofMulAction k G (Fin (n + 1) → G) :=
Iso.refl _ | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | xIso | The `n`th object of the standard resolution of `k` is definitionally isomorphic to `k[Gⁿ⁺¹]`
equipped with the representation induced by the diagonal action of `G`. |
x_projective (G : Type u) [Group G] (n : ℕ) :
Projective ((standardComplex k G).X n) := by
classical exact inferInstanceAs <| Projective (Rep.diagonal k G (n + 1)) | instance | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | x_projective | null |
d_eq (n : ℕ) : ((standardComplex k G).d (n + 1) n).hom =
ModuleCat.ofHom (d k G (n + 1)) := by
refine ModuleCat.hom_ext <| Finsupp.lhom_ext' fun (x : Fin (n + 2) → G) => LinearMap.ext_ring ?_
simp [Action.ofMulAction_V, standardComplex, SimplicialObject.δ,
← Int.cast_smul_eq_zsmul k ((-1) ^ _ : ℤ), SimplexCategory.δ, Fin.succAboveOrderEmb] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d_eq | Simpler expression for the differential in the standard resolution of `k` as a
`G`-representation. It sends `(g₀, ..., gₙ₊₁) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ₊₁)`. |
forget₂ToModuleCat :=
((forget₂ (Rep k G) (ModuleCat.{u} k)).mapHomologicalComplex _).obj (standardComplex k G) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | forget₂ToModuleCat | The standard resolution of `k` as a trivial representation as a complex of `k`-modules. |
compForgetAugmentedIso :
AlternatingFaceMapComplex.obj
(SimplicialObject.Augmented.drop.obj (compForgetAugmented.toModule k G)) ≅
standardComplex.forget₂ToModuleCat k G :=
eqToIso
(Functor.congr_obj (map_alternatingFaceMapComplex (forget₂ (Rep k G) (ModuleCat.{u} k))).symm
(classifyingSpaceUniversalCover G ⋙ linearization k G)) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | compForgetAugmentedIso | If we apply the free functor `Type u ⥤ ModuleCat.{u} k` to the universal cover of the
classifying space of `G` as a simplicial set, then take the alternating face map complex, the result
is isomorphic to the standard resolution of the trivial `G`-representation `k` as a complex of
`k`-modules. |
forget₂ToModuleCatHomotopyEquiv :
HomotopyEquiv (standardComplex.forget₂ToModuleCat k G)
((ChainComplex.single₀ (ModuleCat k)).obj ((forget₂ (Rep k G) _).obj <| Rep.trivial k G k)) :=
(HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).trans <|
(SimplicialObject.Augmented.ExtraDegeneracy.homotopyEquiv
(extraDegeneracyCompForgetAugmentedToModule k G)).trans
(HomotopyEquiv.ofIso <|
(ChainComplex.single₀ (ModuleCat.{u} k)).mapIso
(@Finsupp.LinearEquiv.finsuppUnique k k _ _ _ (⊤_ Type u)
Types.terminalIso.toEquiv.unique).toModuleIso) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | forget₂ToModuleCatHomotopyEquiv | As a complex of `k`-modules, the standard resolution of the trivial `G`-representation `k` is
homotopy equivalent to the complex which is `k` at 0 and 0 elsewhere. |
ε : Rep.ofMulAction k G (Fin 1 → G) ⟶ Rep.trivial k G k where
hom := ModuleCat.ofHom <| Finsupp.linearCombination _ fun _ => (1 : k)
comm _ := ModuleCat.hom_ext <| Finsupp.lhom_ext' fun _ => LinearMap.ext_ring
(by simp [ModuleCat.endRingEquiv]) | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | ε | The hom of `k`-linear `G`-representations `k[G¹] → k` sending `∑ nᵢgᵢ ↦ ∑ nᵢ`. |
forget₂ToModuleCatHomotopyEquiv_f_0_eq :
(forget₂ToModuleCatHomotopyEquiv k G).1.f 0 = (forget₂ (Rep k G) _).map (ε k G) := by
refine ModuleCat.hom_ext <| Finsupp.lhom_ext fun (x : Fin 1 → G) r => ?_
change mapDomain _ _ _ = Finsupp.linearCombination _ _ _
simp only [HomotopyEquiv.ofIso, Iso.symm_hom, compForgetAugmented, compForgetAugmentedIso,
eqToIso.inv, HomologicalComplex.eqToHom_f]
change mapDomain _ (single x r) _ = _
simp [Unique.eq_default (terminal.from _), single_apply, if_pos (Subsingleton.elim _ _)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | forget₂ToModuleCatHomotopyEquiv_f_0_eq | The homotopy equivalence of complexes of `k`-modules between the standard resolution of `k` as
a trivial `G`-representation, and the complex which is `k` at 0 and 0 everywhere else, acts as
`∑ nᵢgᵢ ↦ ∑ nᵢ : k[G¹] → k` at 0. |
d_comp_ε : (standardComplex k G).d 1 0 ≫ ε k G = 0 := by
ext : 3
have : (forget₂ToModuleCat k G).d 1 0
≫ (forget₂ (Rep k G) (ModuleCat.{u} k)).map (ε k G) = 0 := by
rw [← forget₂ToModuleCatHomotopyEquiv_f_0_eq,
← (forget₂ToModuleCatHomotopyEquiv k G).1.2 1 0 rfl]
exact comp_zero
exact LinearMap.ext_iff.1 (ModuleCat.hom_ext_iff.mp this) _ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d_comp_ε | null |
εToSingle₀ :
standardComplex k G ⟶ (ChainComplex.single₀ _).obj (Rep.trivial k G k) :=
((standardComplex k G).toSingle₀Equiv _).symm ⟨ε k G, d_comp_ε k G⟩ | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | εToSingle₀ | The chain map from the standard resolution of `k` to `k[0]` given by `∑ nᵢgᵢ ↦ ∑ nᵢ` in
degree zero. |
εToSingle₀_comp_eq :
((forget₂ _ (ModuleCat.{u} k)).mapHomologicalComplex _).map (εToSingle₀ k G) ≫
(HomologicalComplex.singleMapHomologicalComplex _ _ _).hom.app _ =
(forget₂ToModuleCatHomotopyEquiv k G).hom := by
dsimp
ext1
simpa using (forget₂ToModuleCatHomotopyEquiv_f_0_eq k G).symm | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | εToSingle₀_comp_eq | null |
quasiIso_forget₂_εToSingle₀ :
QuasiIso (((forget₂ _ (ModuleCat.{u} k)).mapHomologicalComplex _).map (εToSingle₀ k G)) := by
have h : QuasiIso (forget₂ToModuleCatHomotopyEquiv k G).hom := inferInstance
rw [← εToSingle₀_comp_eq k G] at h
exact quasiIso_of_comp_right (hφφ' := h) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | quasiIso_forget₂_εToSingle₀ | null |
standardResolution : ProjectiveResolution (Rep.trivial k G k) where
complex := standardComplex k G
π := εToSingle₀ k G
@[deprecated (since := "2025-06-06")]
alias groupCohomology.projectiveResolution := Rep.standardResolution | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | standardResolution | The standard projective resolution of `k` as a trivial `k`-linear `G`-representation. |
standardResolution.extIso (V : Rep k G) (n : ℕ) :
((Ext k (Rep k G) n).obj (Opposite.op <| Rep.trivial k G k)).obj V ≅
((standardComplex k G).linearYonedaObj k V).homology n :=
(standardResolution k G).isoExt n V
@[deprecated (since := "2025-06-06")]
alias groupCohomology.extIso := Rep.standardResolution.extIso | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | standardResolution.extIso | Given a `k`-linear `G`-representation `V`, `Extⁿ(k, V)` (where `k` is a trivial `k`-linear
`G`-representation) is isomorphic to the `n`th cohomology group of `Hom(P, V)`, where `P` is the
standard resolution of `k` called `standardComplex k G`. |
d : free k G Gⁿ⁺¹ ⟶ free k G Gⁿ :=
freeLift _ fun g => single (fun i => g i.succ) (single (g 0) 1) +
Finset.univ.sum fun j : Fin (n + 1) =>
single (Fin.contractNth j (· * ·) g) (single (1 : G) ((-1 : k) ^ ((j : ℕ) + 1)))
variable {k G} in | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d | The differential from `Gⁿ⁺¹ →₀ k[G]` to `Gⁿ →₀ k[G]` in the bar resolution of `k` as a trivial
`k`-linear `G`-representation. It sends `(g₀, ..., gₙ)` to
`g₀·(g₁, ..., gₙ) + ∑ (-1)ʲ⁺¹·(g₀, ..., gⱼgⱼ₊₁, ..., gₙ) + (-1)ⁿ⁺¹·(g₀, ..., gₙ₋₁)` for
`j = 0, ..., n - 1`. |
d_single (x : Gⁿ⁺¹) :
(d k G n).hom (single x (single 1 1)) = single (fun i => x i.succ) (Finsupp.single (x 0) 1) +
Finset.univ.sum fun j : Fin (n + 1) =>
single (Fin.contractNth j (· * ·) x) (single (1 : G) ((-1 : k) ^ ((j : ℕ) + 1))) := by
simp [d] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d_single | null |
d_comp_diagonalSuccIsoFree_inv_eq :
d k G n ≫ (diagonalSuccIsoFree k G n).inv =
(diagonalSuccIsoFree k G (n + 1)).inv ≫ (standardComplex k G).d (n + 1) n :=
free_ext _ _ fun i => by
simpa [diagonalSuccIsoFree_inv_hom_single_single (k := k), d_single (k := k),
standardComplex.d_eq, standardComplex.d_of (k := k) (Fin.partialProd i), Fin.sum_univ_succ,
Fin.partialProd_contractNth] using
congr(single $(by ext j; exact (Fin.partialProd_succ' i j).symm) 1) | lemma | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d_comp_diagonalSuccIsoFree_inv_eq | null |
noncomputable barComplex : ChainComplex (Rep k G) ℕ :=
ChainComplex.of (fun n => free k G (Fin n → G)) (fun n => d k G n) fun _ => by
ext x
simp [(diagonalSuccIsoFree k G _).comp_inv_eq.1 (d_comp_diagonalSuccIsoFree_inv_eq k G _)] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | barComplex | The projective resolution of `k` as a trivial `k`-linear `G`-representation with `n`th
differential `(Gⁿ⁺¹ →₀ k[G]) → (Gⁿ →₀ k[G])` sending `(g₀, ..., gₙ)` to
`g₀·(g₁, ..., gₙ) + ∑ (-1)ʲ⁺¹·(g₀, ..., gⱼgⱼ₊₁, ..., gₙ) + (-1)ⁿ⁺¹·(g₀, ..., gₙ₋₁)` for
`j = 0, ..., n - 1`. |
@[simp]
d_def : (barComplex k G).d (n + 1) n = d k G n := ChainComplex.of_d _ _ _ _ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | d_def | null |
isoStandardComplex : barComplex k G ≅ standardComplex k G :=
HomologicalComplex.Hom.isoOfComponents (fun i => (diagonalSuccIsoFree k G i).symm) fun i j => by
rintro (rfl : j + 1 = i)
simp only [ChainComplex.of_x, Iso.symm_hom, d_def, d_comp_diagonalSuccIsoFree_inv_eq] | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | isoStandardComplex | Isomorphism between the bar resolution and standard resolution, with `n`th map
`(Gⁿ →₀ k[G]) → k[Gⁿ⁺¹]` sending `(g₁, ..., gₙ) ↦ (1, g₁, g₁g₂, ..., g₁...gₙ)`. |
@[simps complex]
barResolution : ProjectiveResolution (Rep.trivial k G k) where
complex := barComplex k G
projective n := inferInstanceAs <| Projective (free k G (Fin n → G))
π := (isoStandardComplex k G).hom ≫ standardComplex.εToSingle₀ k G | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | barResolution | The chain complex `barComplex k G` as a projective resolution of `k` as a trivial
`k`-linear `G`-representation. |
barResolution.extIso (V : Rep k G) (n : ℕ) :
((Ext k (Rep k G) n).obj (Opposite.op <| Rep.trivial k G k)).obj V ≅
((barComplex k G).linearYonedaObj k V).homology n :=
(barResolution k G).isoExt n V | def | RepresentationTheory | [
"Mathlib.Algebra.Category.ModuleCat.Projective",
"Mathlib.AlgebraicTopology.ExtraDegeneracy",
"Mathlib.CategoryTheory.Abelian.Ext",
"Mathlib.RepresentationTheory.Rep",
"Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced"
] | Mathlib/RepresentationTheory/Homological/Resolution.lean | barResolution.extIso | Given a `k`-linear `G`-representation `V`, `Extⁿ(k, V)` (where `k` is the trivial `k`-linear
`G`-representation) is isomorphic to the `n`th cohomology group of `Hom(P, V)`, where `P` is the
bar resolution of `k`. |
@[deprecated "We now use `(Rep.barComplex k G).linearYonedaObj k A instead"
(since := "2025-06-08")]
linearYonedaObjResolution (A : Rep k G) : CochainComplex (ModuleCat.{u} k) ℕ :=
(Rep.standardComplex k G).linearYonedaObj k A | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | linearYonedaObjResolution | The complex `Hom(P, A)`, where `P` is the standard resolution of `k` as a trivial `k`-linear
`G`-representation. |
@[simps! -isSimp]
d [Monoid G] (A : Rep k G) (n : ℕ) :
ModuleCat.of k ((Fin n → G) → A) ⟶ ModuleCat.of k ((Fin (n + 1) → G) → A) :=
ModuleCat.ofHom
{ toFun f g :=
A.ρ (g 0) (f fun i => g i.succ) + Finset.univ.sum fun j : Fin (n + 1) =>
(-1 : k) ^ ((j : ℕ) + 1) • f (Fin.contractNth j (· * ·) g)
map_add' f g := by
ext
simp [Finset.sum_add_distrib, add_add_add_comm]
map_smul' r f := by
ext
simp [Finset.smul_sum, ← smul_assoc, mul_comm r] }
variable [Group G] [DecidableEq G] (A : Rep k G) (n : ℕ) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | d | The differential in the complex of inhomogeneous cochains used to
calculate group cohomology. |
d_eq :
d A n =
(freeLiftLEquiv (Fin n → G) A).toModuleIso.inv ≫
((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫
(freeLiftLEquiv (Fin (n + 1) → G) A).toModuleIso.hom := by
ext
simp [d_hom_apply, map_add, barComplex.d_single (k := k)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | d_eq | null |
noncomputable inhomogeneousCochains : CochainComplex (ModuleCat k) ℕ :=
CochainComplex.of (fun n => ModuleCat.of k ((Fin n → G) → A))
(fun n => inhomogeneousCochains.d A n) fun n => by
classical
simp only [d_eq]
slice_lhs 3 4 => { rw [Iso.hom_inv_id] }
slice_lhs 2 4 => { rw [Category.id_comp, ((barComplex k G).linearYonedaObj k A).d_comp_d] }
simp
variable {A n} in
@[ext] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | inhomogeneousCochains | Given a `k`-linear `G`-representation `A`, this is the complex of inhomogeneous cochains
$$0 \to \mathrm{Fun}(G^0, A) \to \mathrm{Fun}(G^1, A) \to \mathrm{Fun}(G^2, A) \to \dots$$
which calculates the group cohomology of `A`. |
inhomogeneousCochains.ext {x y : (inhomogeneousCochains A).X n} (h : ∀ g, x g = y g) :
x = y := funext h | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | inhomogeneousCochains.ext | null |
inhomogeneousCochains.d_def (n : ℕ) :
(inhomogeneousCochains A).d n (n + 1) = d A n := by
simp | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | inhomogeneousCochains.d_def | null |
inhomogeneousCochains.d_comp_d :
d A n ≫ d A (n + 1) = 0 := by
simpa [CochainComplex.of] using (inhomogeneousCochains A).d_comp_d n (n + 1) (n + 2) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | inhomogeneousCochains.d_comp_d | null |
inhomogeneousCochainsIso [DecidableEq G] :
inhomogeneousCochains A ≅ (barComplex k G).linearYonedaObj k A := by
refine HomologicalComplex.Hom.isoOfComponents
(fun i => (Rep.freeLiftLEquiv (Fin i → G) A).toModuleIso.symm) ?_
rintro i j (h : i + 1 = j)
subst h
simp [d_eq, -LinearEquiv.toModuleIso_hom, -LinearEquiv.toModuleIso_inv] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | inhomogeneousCochainsIso | Given a `k`-linear `G`-representation `A`, the complex of inhomogeneous cochains is isomorphic
to `Hom(P, A)`, where `P` is the bar resolution of `k` as a trivial `G`-representation. |
cocycles (n : ℕ) : ModuleCat k := (inhomogeneousCochains A).cycles n
variable {A} in | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | cocycles | The `n`-cocycles `Zⁿ(G, A)` of a `k`-linear `G`-representation `A`, i.e. the kernel of the
`n`th differential in the complex of inhomogeneous cochains. |
cocyclesMk {n : ℕ} (f : (Fin n → G) → A) (h : inhomogeneousCochains.d A n f = 0) :
cocycles A n :=
(inhomogeneousCochains A).cyclesMk f (n + 1) (by simp) (by simp [h]) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | cocyclesMk | Make an `n`-cocycle out of an element of the kernel of the `n`th differential. |
iCocycles (n : ℕ) : cocycles A n ⟶ (inhomogeneousCochains A).X n :=
(inhomogeneousCochains A).iCycles n | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | iCocycles | The natural inclusion of the `n`-cocycles `Zⁿ(G, A)` into the `n`-cochains `Cⁿ(G, A).` |
toCocycles (i j : ℕ) : (inhomogeneousCochains A).X i ⟶ cocycles A j :=
(inhomogeneousCochains A).toCycles i j
variable {A} in | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | toCocycles | This is the map from `i`-cochains to `j`-cocycles induced by the differential in the complex of
inhomogeneous cochains. |
iCocycles_mk {n : ℕ} (f : (Fin n → G) → A) (h : inhomogeneousCochains.d A n f = 0) :
iCocycles A n (cocyclesMk f h) = f := by
exact (inhomogeneousCochains A).i_cyclesMk (i := n) f (n + 1) (by simp) (by simp [h]) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | iCocycles_mk | null |
groupCohomology [Group G] (A : Rep k G) (n : ℕ) : ModuleCat k :=
(inhomogeneousCochains A).homology n | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | groupCohomology | The group cohomology of a `k`-linear `G`-representation `A`, as the cohomology of its complex
of inhomogeneous cochains. |
groupCohomology.π [Group G] (A : Rep k G) (n : ℕ) :
groupCohomology.cocycles A n ⟶ groupCohomology A n :=
(inhomogeneousCochains A).homologyπ n
@[deprecated (since := "2025-06-11")]
noncomputable alias groupCohomologyπ := groupCohomology.π
@[elab_as_elim] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | groupCohomology.π | The natural map from `n`-cocycles to `n`th group cohomology for a `k`-linear
`G`-representation `A`. |
groupCohomology_induction_on [Group G] {A : Rep k G} {n : ℕ}
{C : groupCohomology A n → Prop} (x : groupCohomology A n)
(h : ∀ x : cocycles A n, C (π A n x)) : C x := by
rcases (ModuleCat.epi_iff_surjective (π A n)).1 inferInstance x with ⟨y, rfl⟩
exact h y | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | groupCohomology_induction_on | null |
groupCohomologyIsoExt [Group G] [DecidableEq G] (A : Rep k G) (n : ℕ) :
groupCohomology A n ≅ ((Ext k (Rep k G) n).obj (Opposite.op <| Rep.trivial k G k)).obj A :=
isoOfQuasiIsoAt (HomotopyEquiv.ofIso (inhomogeneousCochainsIso A)).hom n ≪≫
(Rep.barResolution.extIso k G A n).symm | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | groupCohomologyIsoExt | The `n`th group cohomology of a `k`-linear `G`-representation `A` is isomorphic to
`Extⁿ(k, A)` (taken in `Rep k G`), where `k` is a trivial `k`-linear `G`-representation. |
groupCohomologyIso [Group G] [DecidableEq G] (A : Rep k G) (n : ℕ)
(P : ProjectiveResolution (Rep.trivial k G k)) :
groupCohomology A n ≅ (P.complex.linearYonedaObj k A).homology n :=
groupCohomologyIsoExt A n ≪≫ P.isoExt _ _ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | groupCohomologyIso | The `n`th group cohomology of a `k`-linear `G`-representation `A` is isomorphic to
`Hⁿ(Hom(P, A))`, where `P` is any projective resolution of `k` as a trivial `k`-linear
`G`-representation. |
isZero_groupCohomology_succ_of_subsingleton
[Group G] [Subsingleton G] (A : Rep k G) (n : ℕ) :
Limits.IsZero (groupCohomology A (n + 1)) :=
(isZero_Ext_succ_of_projective (Rep.trivial k G k) A n).of_iso <| groupCohomologyIsoExt _ _ | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.Opposite",
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean | isZero_groupCohomology_succ_of_subsingleton | null |
congr {f₁ f₂ : G →* H} (h : f₁ = f₂) {φ : (Action.res _ f₁).obj A ⟶ B} {T : Type*}
(F : (f : G →* H) → (φ : (Action.res _ f).obj A ⟶ B) → T) :
F f₁ φ = F f₂ (h ▸ φ) := by
subst h
rfl | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | congr | null |
@[simps! -isSimp f f_hom]
noncomputable cochainsMap :
inhomogeneousCochains A ⟶ inhomogeneousCochains B where
f i := ModuleCat.ofHom <|
φ.hom.hom.compLeft (Fin i → G) ∘ₗ LinearMap.funLeft k A (fun x : Fin i → G => (f ∘ x))
comm' i j (hij : _ = _) := by
subst hij
ext
simpa [inhomogeneousCochains.d_hom_apply, Fin.comp_contractNth]
using (hom_comm_apply φ _ _).symm
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the chain map sending `x : Hⁿ → A` to `(g : Gⁿ) ↦ φ (x (f ∘ g))`. |
cochainsMap_id :
cochainsMap (MonoidHom.id _) (𝟙 A) = 𝟙 (inhomogeneousCochains A) := by
rfl
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_id | null |
cochainsMap_id_f_hom_eq_compLeft {A B : Rep k G} (f : A ⟶ B) (i : ℕ) :
((cochainsMap (MonoidHom.id G) f).f i).hom = f.hom.hom.compLeft _ := by
ext
rfl
@[deprecated (since := "2025-06-11")]
alias cochainsMap_id_f_eq_compLeft := cochainsMap_id_f_hom_eq_compLeft
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_id_f_hom_eq_compLeft | null |
cochainsMap_comp {G H K : Type u} [Group G] [Group H]
[Group K] {A : Rep k K} {B : Rep k H} {C : Rep k G} (f : H →* K) (g : G →* H)
(φ : (Action.res _ f).obj A ⟶ B) (ψ : (Action.res _ g).obj B ⟶ C) :
cochainsMap (f.comp g) ((Action.res _ g).map φ ≫ ψ) =
cochainsMap f φ ≫ cochainsMap g ψ := by
rfl
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_comp | null |
cochainsMap_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) :
cochainsMap (MonoidHom.id G) (φ ≫ ψ) =
cochainsMap (MonoidHom.id G) φ ≫ cochainsMap (MonoidHom.id G) ψ := by
rfl
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_id_comp | null |
cochainsMap_zero : cochainsMap (A := A) (B := B) f 0 = 0 := by rfl | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_zero | null |
cochainsMap_f_map_mono (hf : Function.Surjective f) [Mono φ] (i : ℕ) :
Mono ((cochainsMap f φ).f i) := by
simpa [ModuleCat.mono_iff_injective] using
((Rep.mono_iff_injective φ).1 inferInstance).comp_left.comp <|
LinearMap.funLeft_injective_of_surjective k A _ hf.comp_left | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_f_map_mono | null |
cochainsMap_id_f_map_mono {A B : Rep k G} (φ : A ⟶ B) [Mono φ] (i : ℕ) :
Mono ((cochainsMap (MonoidHom.id G) φ).f i) :=
cochainsMap_f_map_mono (MonoidHom.id G) φ (fun x => ⟨x, rfl⟩) i | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_id_f_map_mono | null |
cochainsMap_f_map_epi (hf : Function.Injective f) [Epi φ] (i : ℕ) :
Epi ((cochainsMap f φ).f i) := by
simpa [ModuleCat.epi_iff_surjective] using
((Rep.epi_iff_surjective φ).1 inferInstance).comp_left.comp <|
LinearMap.funLeft_surjective_of_injective k A _ hf.comp_left | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_f_map_epi | null |
cochainsMap_id_f_map_epi {A B : Rep k G} (φ : A ⟶ B) [Epi φ] (i : ℕ) :
Epi ((cochainsMap (MonoidHom.id G) φ).f i) :=
cochainsMap_f_map_epi (MonoidHom.id G) φ (fun _ _ h => h) i | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_id_f_map_epi | null |
noncomputable cocyclesMap (n : ℕ) :
groupCohomology.cocycles A n ⟶ groupCohomology.cocycles B n :=
HomologicalComplex.cyclesMap (cochainsMap f φ) n
@[simp] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cocyclesMap | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the induced map `Zⁿ(H, A) ⟶ Zⁿ(G, B)` sending `x : Hⁿ → A` to
`(g : Gⁿ) ↦ φ (x (f ∘ g))`. |
cocyclesMap_id : cocyclesMap (MonoidHom.id G) (𝟙 B) n = 𝟙 _ :=
HomologicalComplex.cyclesMap_id _ _
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cocyclesMap_id | null |
cocyclesMap_comp {G H K : Type u} [Group G] [Group H]
[Group K] {A : Rep k K} {B : Rep k H} {C : Rep k G} (f : H →* K) (g : G →* H)
(φ : (Action.res _ f).obj A ⟶ B) (ψ : (Action.res _ g).obj B ⟶ C) (n : ℕ) :
cocyclesMap (f.comp g) ((Action.res _ g).map φ ≫ ψ) n =
cocyclesMap f φ n ≫ cocyclesMap g ψ n := by
simp [cocyclesMap, ← HomologicalComplex.cyclesMap_comp, ← cochainsMap_comp]
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cocyclesMap_comp | null |
cocyclesMap_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) (n : ℕ) :
cocyclesMap (MonoidHom.id G) (φ ≫ ψ) n =
cocyclesMap (MonoidHom.id G) φ n ≫ cocyclesMap (MonoidHom.id G) ψ n := by
simp [cocyclesMap, cochainsMap_id_comp, HomologicalComplex.cyclesMap_comp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cocyclesMap_id_comp | null |
noncomputable map (n : ℕ) :
groupCohomology A n ⟶ groupCohomology B n :=
HomologicalComplex.homologyMap (cochainsMap f φ) n
@[reassoc, elementwise] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | map | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the induced map `Hⁿ(H, A) ⟶ Hⁿ(G, B)` sending `x : Hⁿ → A` to
`(g : Gⁿ) ↦ φ (x (f ∘ g))`. |
π_map (n : ℕ) :
π A n ≫ map f φ n = cocyclesMap f φ n ≫ π B n := by
simp [map, cocyclesMap]
@[simp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | π_map | null |
map_id : map (MonoidHom.id G) (𝟙 B) n = 𝟙 _ := HomologicalComplex.homologyMap_id _ _
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | map_id | null |
map_comp {G H K : Type u} [Group G] [Group H]
[Group K] {A : Rep k K} {B : Rep k H} {C : Rep k G} (f : H →* K) (g : G →* H)
(φ : (Action.res _ f).obj A ⟶ B) (ψ : (Action.res _ g).obj B ⟶ C) (n : ℕ) :
map (f.comp g) ((Action.res _ g).map φ ≫ ψ) n = map f φ n ≫ map g ψ n := by
simp [map, ← HomologicalComplex.homologyMap_comp, ← cochainsMap_comp]
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | map_comp | null |
map_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) (n : ℕ) :
map (MonoidHom.id G) (φ ≫ ψ) n =
map (MonoidHom.id G) φ n ≫ map (MonoidHom.id G) ψ n := by
rw [map, cochainsMap_id_comp, HomologicalComplex.homologyMap_comp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | map_id_comp | null |
noncomputable cochainsMap₁ :
ModuleCat.of k (H → A) ⟶ ModuleCat.of k (G → B) :=
ModuleCat.ofHom <| φ.hom.hom.compLeft G ∘ₗ LinearMap.funLeft k A f
@[deprecated (since := "2025-07-12")] alias f₁ := cochainsMap₁
@[deprecated (since := "2025-06-25")] noncomputable alias fOne := f₁ | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap₁ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the induced map sending `x : H → A` to `(g : G) ↦ φ (x (f g))`. |
noncomputable cochainsMap₂ :
ModuleCat.of k (H × H → A) ⟶ ModuleCat.of k (G × G → B) :=
ModuleCat.ofHom <| φ.hom.hom.compLeft (G × G) ∘ₗ LinearMap.funLeft k A (Prod.map f f)
@[deprecated (since := "2025-07-12")] alias f₂ := cochainsMap₂
@[deprecated (since := "2025-06-25")] noncomputable alias fTwo := f₂ | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap₂ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the induced map sending `x : H × H → A` to `(g₁, g₂ : G × G) ↦ φ (x (f g₁, f g₂))`. |
noncomputable cochainsMap₃ :
ModuleCat.of k (H × H × H → A) ⟶ ModuleCat.of k (G × G × G → B) :=
ModuleCat.ofHom <|
φ.hom.hom.compLeft (G × G × G) ∘ₗ LinearMap.funLeft k A (Prod.map f (Prod.map f f))
@[deprecated (since := "2025-07-12")] alias f₃ := cochainsMap₃
@[deprecated (since := "2025-06-25")] noncomputable alias fThree := f₃
@[reassoc (attr := simp), elementwise (attr := simp)] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap₃ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the induced map sending `x : H × H × H → A` to
`(g₁, g₂, g₃ : G × G × G) ↦ φ (x (f g₁, f g₂, f g₃))`. |
cochainsMap_f_0_comp_cochainsIso₀ :
(cochainsMap f φ).f 0 ≫ (cochainsIso₀ B).hom = (cochainsIso₀ A).hom ≫ φ.hom := by
ext x
simp only [cochainsMap_f, Unique.eq_default (f ∘ _)]
rfl
@[deprecated (since := "2025-06-25")]
alias cochainsMap_f_0_comp_zeroCochainsIso := cochainsMap_f_0_comp_cochainsIso₀
@[deprecated (since := "2025-05-09")]
alias cochainsMap_f_0_comp_zeroCochainsLequiv := cochainsMap_f_0_comp_cochainsIso₀
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_f_0_comp_cochainsIso₀ | null |
cochainsMap_f_1_comp_cochainsIso₁ :
(cochainsMap f φ).f 1 ≫ (cochainsIso₁ B).hom = (cochainsIso₁ A).hom ≫ cochainsMap₁ f φ := rfl
@[deprecated (since := "2025-06-25")]
alias cochainsMap_f_1_comp_oneCochainsIso := cochainsMap_f_1_comp_cochainsIso₁
@[deprecated (since := "2025-05-09")]
alias cochainsMap_f_1_comp_oneCochainsLequiv := cochainsMap_f_1_comp_oneCochainsIso
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_f_1_comp_cochainsIso₁ | null |
cochainsMap_f_2_comp_cochainsIso₂ :
(cochainsMap f φ).f 2 ≫ (cochainsIso₂ B).hom = (cochainsIso₂ A).hom ≫ cochainsMap₂ f φ := by
ext x g
change φ.hom (x _) = φ.hom (x _)
rcongr x
fin_cases x <;> rfl
@[deprecated (since := "2025-06-25")]
alias cochainsMap_f_2_comp_twoCochainsIso := cochainsMap_f_2_comp_cochainsIso₂
@[deprecated (since := "2025-05-09")]
alias cochainsMap_f_2_comp_twoCochainsLequiv := cochainsMap_f_2_comp_twoCochainsIso
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_f_2_comp_cochainsIso₂ | null |
cochainsMap_f_3_comp_cochainsIso₃ :
(cochainsMap f φ).f 3 ≫ (cochainsIso₃ B).hom = (cochainsIso₃ A).hom ≫ cochainsMap₃ f φ := by
ext x g
change φ.hom (x _) = φ.hom (x _)
rcongr x
fin_cases x <;> rfl
@[deprecated (since := "2025-06-25")]
alias cochainsMap_f_3_comp_threeCochainsIso := cochainsMap_f_3_comp_cochainsIso₃
@[deprecated (since := "2025-05-09")]
alias cochainsMap_f_3_comp_threeCochainsLequiv := cochainsMap_f_3_comp_threeCochainsIso | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsMap_f_3_comp_cochainsIso₃ | null |
@[simps]
noncomputable mapShortComplexH1 :
shortComplexH1 A ⟶ shortComplexH1 B where
τ₁ := φ.hom
τ₂ := cochainsMap₁ f φ
τ₃ := cochainsMap₂ f φ
comm₁₂ := by
ext x
funext g
simpa [shortComplexH1, d₀₁, cochainsMap₁] using (hom_comm_apply φ g x).symm
comm₂₃ := by
ext x
funext g
simpa [shortComplexH1, d₁₂, cochainsMap₁, cochainsMap₂] using (hom_comm_apply φ _ _).symm
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH1 | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is induced map `Aᴴ ⟶ Bᴳ`. -/
@[deprecated (since := "2025-06-09")]
alias H0Map := map
@[deprecated (since := "2025-06-09")]
alias H0Map_id := map_id
@[deprecated (since := "2025-06-09")]
alias H0Map_comp := map_comp
@[deprecated (since := "2025-06-09")]
alias H0Map_id_comp := map_id_comp
@[reassoc (attr := simp), elementwise (attr := simp)]
theorem map_H0Iso_hom_f :
map f φ 0 ≫ (H0Iso B).hom ≫ (shortComplexH0 B).f =
(H0Iso A).hom ≫ (shortComplexH0 A).f ≫ φ.hom := by
simp [← cancel_epi (π _ _)]
@[deprecated (since := "2025-06-09")]
alias H0Map_comp_f := map_H0Iso_hom_f
@[reassoc (attr := simp), elementwise (attr := simp)]
theorem map_id_comp_H0Iso_hom {A B : Rep k G} (f : A ⟶ B) :
map (MonoidHom.id G) f 0 ≫ (H0Iso B).hom = (H0Iso A).hom ≫ (invariantsFunctor k G).map f := by
simp only [← cancel_mono (shortComplexH0 B).f, Category.assoc, map_H0Iso_hom_f]
rfl
@[deprecated (since := "2025-06-09")]
alias H0Map_id_eq_invariantsFunctor_map := map_id_comp_H0Iso_hom
instance mono_map_0_of_mono {A B : Rep k G} (f : A ⟶ B) [Mono f] :
Mono (map (MonoidHom.id G) f 0) where
right_cancellation g h hgh := by
simp only [← cancel_mono (H0Iso B).hom, Category.assoc, map_id_comp_H0Iso_hom] at hgh
simp_all [cancel_mono]
@[deprecated (since := "2025-06-09")]
alias mono_H0Map_of_mono := mono_map_0_of_mono
@[reassoc, elementwise]
theorem cocyclesMap_cocyclesIso₀_hom_f :
cocyclesMap f φ 0 ≫ (cocyclesIso₀ B).hom ≫ (shortComplexH0 B).f =
(cocyclesIso₀ A).hom ≫ (shortComplexH0 A).f ≫ φ.hom := by
simp
@[deprecated (since := "2025-07-02")]
alias cocyclesMap_zeroIsoCocycles_hom_f := cocyclesMap_cocyclesIso₀_hom_f
@[deprecated (since := "2025-06-12")]
alias cocyclesMap_comp_isoZeroCocycles_hom := cocyclesMap_zeroIsoCocycles_hom_f
end H0
section H1
/-- Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the induced map from the short complex `A --d₀₁--> Fun(H, A) --d₁₂--> Fun(H × H, A)`
to `B --d₀₁--> Fun(G, B) --d₁₂--> Fun(G × G, B)`. |
mapShortComplexH1_zero :
mapShortComplexH1 (A := A) (B := B) f 0 = 0 := by
rfl
@[simp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH1_zero | null |
mapShortComplexH1_id :
mapShortComplexH1 (MonoidHom.id _) (𝟙 A) = 𝟙 _ := by
rfl
@[reassoc] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH1_id | null |
mapShortComplexH1_comp {G H K : Type u} [Group G] [Group H] [Group K]
{A : Rep k K} {B : Rep k H} {C : Rep k G} (f : H →* K) (g : G →* H)
(φ : (Action.res _ f).obj A ⟶ B) (ψ : (Action.res _ g).obj B ⟶ C) :
mapShortComplexH1 (f.comp g) ((Action.res _ g).map φ ≫ ψ) =
mapShortComplexH1 f φ ≫ mapShortComplexH1 g ψ := rfl
@[reassoc] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH1_comp | null |
mapShortComplexH1_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) :
mapShortComplexH1 (MonoidHom.id G) (φ ≫ ψ) =
mapShortComplexH1 (MonoidHom.id G) φ ≫ mapShortComplexH1 (MonoidHom.id G) ψ := rfl | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH1_id_comp | null |
noncomputable mapCocycles₁ :
ModuleCat.of k (cocycles₁ A) ⟶ ModuleCat.of k (cocycles₁ B) :=
ShortComplex.cyclesMap' (mapShortComplexH1 f φ) (shortComplexH1 A).moduleCatLeftHomologyData
(shortComplexH1 B).moduleCatLeftHomologyData
@[deprecated (since := "2025-06-25")] alias mapOneCocycles := mapCocycles₁
@[reassoc, elementwise] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapCocycles₁ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is induced map `Z¹(H, A) ⟶ Z¹(G, B)`. |
mapCocycles₁_comp_i :
mapCocycles₁ f φ ≫ (shortComplexH1 B).moduleCatLeftHomologyData.i =
(shortComplexH1 A).moduleCatLeftHomologyData.i ≫ cochainsMap₁ f φ := by
simp
@[deprecated (since := "2025-06-25")] alias mapOneCocycles_comp_i := mapCocycles₁_comp_i
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapCocycles₁_comp_i | null |
coe_mapCocycles₁ (x) :
⇑(mapCocycles₁ f φ x) = cochainsMap₁ f φ x := rfl
@[deprecated (since := "2025-06-25")] alias coe_mapOneCocycles := coe_mapCocycles₁
@[deprecated (since := "2025-05-09")]
alias mapOneCocycles_comp_subtype := mapOneCocycles_comp_i
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | coe_mapCocycles₁ | null |
cocyclesMap_comp_isoCocycles₁_hom :
cocyclesMap f φ 1 ≫ (isoCocycles₁ B).hom = (isoCocycles₁ A).hom ≫ mapCocycles₁ f φ := by
simp [← cancel_mono (moduleCatLeftHomologyData (shortComplexH1 B)).i, mapShortComplexH1,
cochainsMap_f_1_comp_cochainsIso₁ f]
@[deprecated (since := "2025-06-25")]
alias cocyclesMap_comp_isoOneCocycles_hom := cocyclesMap_comp_isoCocycles₁_hom
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cocyclesMap_comp_isoCocycles₁_hom | null |
mapCocycles₁_one (φ : (Action.res _ 1).obj A ⟶ B) :
mapCocycles₁ 1 φ = 0 := by
rw [← cancel_mono (moduleCatLeftHomologyData (shortComplexH1 B)).i, cyclesMap'_i]
refine ModuleCat.hom_ext (LinearMap.ext fun _ ↦ funext fun y => ?_)
simp [mapShortComplexH1, shortComplexH1, Pi.zero_apply y]
@[deprecated (since := "2025-06-25")] alias mapOneCocycles_one := mapCocycles₁_one | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapCocycles₁_one | null |
@[simps X₁ X₂ X₃ f g]
noncomputable H1InfRes :
ShortComplex (ModuleCat k) where
X₁ := groupCohomology (A.quotientToInvariants S) 1
X₂ := groupCohomology A 1
X₃ := groupCohomology ((Action.res _ S.subtype).obj A) 1
f := map (QuotientGroup.mk' S) (subtype _ _ <| le_comap_invariants A.ρ S) 1
g := map S.subtype (𝟙 _) 1
zero := by rw [← map_comp, Category.comp_id, congr (QuotientGroup.mk'_comp_subtype S)
(fun f φ => map f φ 1), map₁_one] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | H1InfRes | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is induced map `H¹(H, A) ⟶ H¹(G, B)`. -/
@[deprecated (since := "2025-06-09")]
alias H1Map := map
@[deprecated (since := "2025-6-09")]
alias H1Map_id := map_id
@[deprecated (since := "2025-06-09")]
alias H1Map_comp := map_comp
@[deprecated (since := "2025-06-09")]
alias H1Map_id_comp := map_id_comp
@[reassoc (attr := simp), elementwise (attr := simp)]
lemma H1π_comp_map :
H1π A ≫ map f φ 1 = mapCocycles₁ f φ ≫ H1π B := by
simp [H1π, Iso.inv_comp_eq, ← cocyclesMap_comp_isoCocycles₁_hom_assoc]
@[deprecated (since := "2025-06-12")]
alias H1π_comp_H1Map := H1π_comp_map
@[simp]
theorem map₁_one (φ : (Action.res _ 1).obj A ⟶ B) :
map 1 φ 1 = 0 := by
simp [← cancel_epi (H1π _)]
@[deprecated (since := "2025-07-31")]
alias map_1_one := map₁_one
@[deprecated (since := "2025-06-09")]
alias H1Map_one := map_1_one
section InfRes
variable (A : Rep k G) (S : Subgroup G) [S.Normal]
/-- The short complex `H¹(G ⧸ S, A^S) ⟶ H¹(G, A) ⟶ H¹(S, A)`. |
H1InfRes_exact : (H1InfRes A S).Exact := by
rw [moduleCat_exact_iff_ker_sub_range]
intro x hx
induction x using H1_induction_on with | @h x =>
simp_all only [H1InfRes_X₂, H1InfRes_X₃, H1InfRes_g, H1InfRes_X₁, LinearMap.mem_ker,
H1π_comp_map_apply S.subtype, H1InfRes_f]
rcases (H1π_eq_zero_iff _).1 hx with ⟨(y : A), hy⟩
have h1 := (mem_cocycles₁_iff x).1 x.2
have h2 : ∀ s ∈ S, x s = A.ρ s y - y :=
fun s hs => funext_iff.1 hy.symm ⟨s, hs⟩
refine ⟨H1π _ ⟨fun g => Quotient.liftOn' g (fun g => ⟨x.1 g - A.ρ g y + y, ?_⟩) ?_, ?_⟩, ?_⟩
· intro s
calc
_ = x (s * g) - x s - A.ρ s (A.ρ g y) + (x s + y) := by
simp [add_eq_of_eq_sub (h2 s s.2), sub_eq_of_eq_add (h1 s g)]
_ = x (g * (g⁻¹ * s * g)) - A.ρ g (A.ρ (g⁻¹ * s * g) y - y) - A.ρ g y + y := by
simp only [mul_assoc, mul_inv_cancel_left, map_mul, Module.End.mul_apply, map_sub,
Representation.self_inv_apply]
abel
_ = x g - A.ρ g y + y := by
simp [eq_sub_of_add_eq' (h1 g (g⁻¹ * s * g)).symm,
h2 (g⁻¹ * s * g) (Subgroup.Normal.conj_mem' ‹_› _ s.2 _)]
· intro g h hgh
have := congr(A.ρ g $(h2 (g⁻¹ * h) <| QuotientGroup.leftRel_apply.1 hgh))
simp_all [← sub_eq_add_neg, sub_eq_sub_iff_sub_eq_sub]
· rw [mem_cocycles₁_iff]
intro g h
induction g using QuotientGroup.induction_on with | @H g =>
induction h using QuotientGroup.induction_on with | @H h =>
apply Subtype.ext
simp [← QuotientGroup.mk_mul, h1 g h, sub_add_eq_add_sub, add_assoc]
· symm
simp only [H1π_comp_map_apply, H1π_eq_iff (A := A)]
use y
ext g
simp [coe_mapCocycles₁ (QuotientGroup.mk' S),
cocycles₁.coe_mk (A := A.quotientToInvariants S), ← sub_sub] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | H1InfRes_exact | The inflation map `H¹(G ⧸ S, A^S) ⟶ H¹(G, A)` is a monomorphism. -/
instance : Mono (H1InfRes A S).f := by
rw [ModuleCat.mono_iff_injective, injective_iff_map_eq_zero]
intro x hx
induction x using H1_induction_on with | @h x =>
simp_all only [H1InfRes_X₂, H1InfRes_X₁, H1InfRes_f, H1π_comp_map_apply (QuotientGroup.mk' S)]
rcases (H1π_eq_zero_iff _).1 hx with ⟨y, hy⟩
refine (H1π_eq_zero_iff _).2 ⟨⟨y, fun s => ?_⟩, funext fun g => QuotientGroup.induction_on g
fun g => Subtype.ext <| by simpa [-SetLike.coe_eq_coe] using congr_fun hy g⟩
simpa [coe_mapCocycles₁ (x := x), sub_eq_zero, (QuotientGroup.eq_one_iff s.1).2 s.2] using
congr_fun hy s.1
/-- Given a `G`-representation `A` and a normal subgroup `S ≤ G`, the short complex
`H¹(G ⧸ S, A^S) ⟶ H¹(G, A) ⟶ H¹(S, A)` is exact. |
@[simps]
noncomputable mapShortComplexH2 :
shortComplexH2 A ⟶ shortComplexH2 B where
τ₁ := cochainsMap₁ f φ
τ₂ := cochainsMap₂ f φ
τ₃ := cochainsMap₃ f φ
comm₁₂ := by
ext x
funext g
simpa [shortComplexH2, d₁₂, cochainsMap₁, cochainsMap₂] using (hom_comm_apply φ _ _).symm
comm₂₃ := by
ext x
funext g
simpa [shortComplexH2, d₂₃, cochainsMap₂, cochainsMap₃] using (hom_comm_apply φ _ _).symm
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH2 | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is the induced map from the short complex
`Fun(H, A) --d₁₂--> Fun(H × H, A) --d₂₃--> Fun(H × H × H, A)` to
`Fun(G, B) --d₁₂--> Fun(G × G, B) --d₂₃--> Fun(G × G × G, B)`. |
mapShortComplexH2_zero :
mapShortComplexH2 (A := A) (B := B) f 0 = 0 := rfl
@[simp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH2_zero | null |
mapShortComplexH2_id :
mapShortComplexH2 (MonoidHom.id _) (𝟙 A) = 𝟙 _ := by
rfl
@[reassoc] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH2_id | null |
mapShortComplexH2_comp {G H K : Type u} [Group G] [Group H] [Group K]
{A : Rep k K} {B : Rep k H} {C : Rep k G} (f : H →* K) (g : G →* H)
(φ : (Action.res _ f).obj A ⟶ B) (ψ : (Action.res _ g).obj B ⟶ C) :
mapShortComplexH2 (f.comp g) ((Action.res _ g).map φ ≫ ψ) =
mapShortComplexH2 f φ ≫ mapShortComplexH2 g ψ := rfl
@[reassoc] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH2_comp | null |
mapShortComplexH2_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) :
mapShortComplexH2 (MonoidHom.id G) (φ ≫ ψ) =
mapShortComplexH2 (MonoidHom.id G) φ ≫ mapShortComplexH2 (MonoidHom.id G) ψ := rfl | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapShortComplexH2_id_comp | null |
noncomputable mapCocycles₂ :
ModuleCat.of k (cocycles₂ A) ⟶ ModuleCat.of k (cocycles₂ B) :=
ShortComplex.cyclesMap' (mapShortComplexH2 f φ) (shortComplexH2 A).moduleCatLeftHomologyData
(shortComplexH2 B).moduleCatLeftHomologyData
@[deprecated (since := "2025-06-25")] alias mapTwoCocycles := mapCocycles₂
@[reassoc, elementwise] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapCocycles₂ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is induced map `Z²(H, A) ⟶ Z²(G, B)`. |
mapCocycles₂_comp_i :
mapCocycles₂ f φ ≫ (shortComplexH2 B).moduleCatLeftHomologyData.i =
(shortComplexH2 A).moduleCatLeftHomologyData.i ≫ cochainsMap₂ f φ := by
simp
@[deprecated (since := "2025-06-25")] alias mapTwoCocycles_comp_i := mapCocycles₂_comp_i
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | mapCocycles₂_comp_i | null |
coe_mapCocycles₂ (x) :
⇑(mapCocycles₂ f φ x) = cochainsMap₂ f φ x := rfl
@[deprecated (since := "2025-06-25")] alias coe_mapTwoCocycles := coe_mapCocycles₂
@[deprecated (since := "2025-05-09")]
alias mapTwoCocycles_comp_subtype := mapTwoCocycles_comp_i
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | coe_mapCocycles₂ | null |
cocyclesMap_comp_isoCocycles₂_hom :
cocyclesMap f φ 2 ≫ (isoCocycles₂ B).hom = (isoCocycles₂ A).hom ≫ mapCocycles₂ f φ := by
simp [← cancel_mono (moduleCatLeftHomologyData (shortComplexH2 B)).i, mapShortComplexH2,
cochainsMap_f_2_comp_cochainsIso₂ f]
@[deprecated (since := "2025-06-25")]
alias cocyclesMap_comp_isoTwoCocycles_hom := cocyclesMap_comp_isoCocycles₂_hom | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cocyclesMap_comp_isoCocycles₂_hom | null |
@[simps]
noncomputable cochainsFunctor : Rep k G ⥤ CochainComplex (ModuleCat k) ℕ where
obj A := inhomogeneousCochains A
map f := cochainsMap (MonoidHom.id _) f
map_id _ := cochainsMap_id
map_comp φ ψ := cochainsMap_comp (MonoidHom.id G) (MonoidHom.id G) φ ψ | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | cochainsFunctor | Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
this is induced map `H²(H, A) ⟶ H²(G, B)`. -/
@[deprecated (since := "2025-06-09")]
alias H2Map := map
@[deprecated (since := "2025-06-09")]
alias H2Map_id := map_id
@[deprecated (since := "2025-06-09")]
alias H2Map_comp := map_comp
@[deprecated (since := "2025-06-09")]
alias H2Map_id_comp := map_id_comp
@[reassoc (attr := simp), elementwise (attr := simp)]
lemma H2π_comp_map :
H2π A ≫ map f φ 2 = mapCocycles₂ f φ ≫ H2π B := by
simp [H2π, Iso.inv_comp_eq, ← cocyclesMap_comp_isoCocycles₂_hom_assoc]
@[deprecated (since := "2025-06-12")]
alias H2π_comp_H2Map := H2π_comp_map
end H2
variable (k G)
/-- The functor sending a representation to its complex of inhomogeneous cochains. |
@[simps]
noncomputable functor (n : ℕ) : Rep k G ⥤ ModuleCat k where
obj A := groupCohomology A n
map φ := map (MonoidHom.id _) φ n
map_id _ := HomologicalComplex.homologyMap_id _ _
map_comp _ _ := by
simp only [← HomologicalComplex.homologyMap_comp]
rfl | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | functor | The functor sending a `G`-representation `A` to `Hⁿ(G, A)`. |
@[simps]
noncomputable resNatTrans (n : ℕ) :
functor k H n ⟶ Action.res (ModuleCat k) f ⋙ functor k G n where
app X := map f (𝟙 _) n
naturality {X Y} φ := by simp [← cancel_epi (groupCohomology.π _ n),
← HomologicalComplex.cyclesMap_comp_assoc, ← cochainsMap_comp, congr (MonoidHom.id_comp _)
cochainsMap, congr (MonoidHom.comp_id _) cochainsMap, Category.id_comp
(X := (Action.res _ _).obj _)] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | resNatTrans | Given a group homomorphism `f : G →* H`, this is a natural transformation between the functors
sending `A : Rep k H` to `Hⁿ(H, A)` and to `Hⁿ(G, Res(f)(A))`. |
@[simps]
noncomputable infNatTrans (S : Subgroup G) [S.Normal] (n : ℕ) :
quotientToInvariantsFunctor k S ⋙ functor k (G ⧸ S) n ⟶ functor k G n where
app A := map (QuotientGroup.mk' S) (subtype _ _ <| le_comap_invariants A.ρ S) n
naturality {X Y} φ := by
simp only [Functor.comp_map, functor_map, ← cancel_epi (groupCohomology.π _ n),
HomologicalComplex.homologyπ_naturality_assoc, HomologicalComplex.homologyπ_naturality,
← HomologicalComplex.cyclesMap_comp_assoc, ← cochainsMap_comp]
congr 1 | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean | infNatTrans | Given a normal subgroup `S ≤ G`, this is a natural transformation between the functors
sending `A : Rep k G` to `Hⁿ(G ⧸ S, A^S)` and to `Hⁿ(G, A)`. |
noncomputable aux (f : (L ≃ₐ[K] L) → Lˣ) : L → L :=
Finsupp.linearCombination L (fun φ : L ≃ₐ[K] L ↦ (φ : L → L))
(Finsupp.equivFunOnFinite.symm (fun φ => (f φ : L))) | def | RepresentationTheory | [
"Mathlib.FieldTheory.Fixed",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Hilbert90.lean | aux | Given `f : Aut_K(L) → Lˣ`, the sum `∑ f(φ) • φ` for `φ ∈ Aut_K(L)`, as a function `L → L`. |
aux_ne_zero (f : (L ≃ₐ[K] L) → Lˣ) : aux f ≠ 0 :=
/- the set `Aut_K(L)` is linearly independent in the `L`-vector space `L → L`, by Dedekind's
linear independence of characters -/
have : LinearIndependent L (fun (f : L ≃ₐ[K] L) => (f : L → L)) :=
LinearIndependent.comp (ι' := L ≃ₐ[K] L)
(linearIndependent_monoidHom L L) (fun f => f)
(fun x y h => by ext; exact DFunLike.ext_iff.1 h _)
have h := linearIndependent_iff.1 this
(Finsupp.equivFunOnFinite.symm (fun φ => (f φ : L)))
fun H => Units.ne_zero (f 1) (DFunLike.ext_iff.1 (h H) 1) | theorem | RepresentationTheory | [
"Mathlib.FieldTheory.Fixed",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Hilbert90.lean | aux_ne_zero | null |
isMulCoboundary₁_of_isMulCocycle₁_of_aut_to_units
(f : (L ≃ₐ[K] L) → Lˣ) (hf : IsMulCocycle₁ f) :
IsMulCoboundary₁ f := by
/- Let `z : L` be such that `∑ f(h) * h(z) ≠ 0`, for `h ∈ Aut_K(L)` -/
obtain ⟨z, hz⟩ : ∃ z, aux f z ≠ 0 :=
not_forall.1 (fun H => aux_ne_zero f <| funext <| fun x => H x)
have : aux f z = ∑ h, f h * h z := by simp [aux, Finsupp.linearCombination, Finsupp.sum_fintype]
/- Let `β = (∑ f(h) * h(z))⁻¹.` -/
use (Units.mk0 (aux f z) hz)⁻¹
intro g
/- Then the equality follows from the hypothesis that `f` is a 1-cocycle. -/
simp only [IsMulCocycle₁, AlgEquiv.smul_units_def,
map_inv, div_inv_eq_mul, inv_mul_eq_iff_eq_mul, Units.ext_iff, this,
Units.val_mul, Units.coe_map, Units.val_mk0, MonoidHom.coe_coe] at hf ⊢
simp_rw [map_sum, map_mul, Finset.sum_mul, mul_assoc, mul_comm _ (f _ : L), ← mul_assoc, ← hf g]
exact eq_comm.1 (Fintype.sum_bijective (fun i => g * i)
(Group.mulLeft_bijective g) _ _ (fun i => rfl))
@[deprecated (since := "2025-06-26")]
alias isMulOneCoboundary_of_isMulOneCocycle_of_aut_to_units :=
isMulCoboundary₁_of_isMulCocycle₁_of_aut_to_units | theorem | RepresentationTheory | [
"Mathlib.FieldTheory.Fixed",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Hilbert90.lean | isMulCoboundary₁_of_isMulCocycle₁_of_aut_to_units | Noether's generalization of Hilbert's Theorem 90: given a finite extension of fields and a
function `f : Aut_K(L) → Lˣ` satisfying `f(gh) = g(f(h)) * f(g)` for all `g, h : Aut_K(L)`, there
exists `β : Lˣ` such that `g(β)/β = f(g)` for all `g : Aut_K(L).` |
noncomputable H1ofAutOnUnitsUnique : Unique (H1 (Rep.ofAlgebraAutOnUnits K L)) where
default := 0
uniq := fun a => H1_induction_on a fun x => (H1π_eq_zero_iff _).2 <| by
refine (coboundariesOfIsMulCoboundary₁ ?_).2
rcases isMulCoboundary₁_of_isMulCocycle₁_of_aut_to_units x.1
(isMulCocycle₁_of_mem_cocycles₁ _ x.2) with ⟨β, hβ⟩
use β | instance | RepresentationTheory | [
"Mathlib.FieldTheory.Fixed",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Hilbert90.lean | H1ofAutOnUnitsUnique | Noether's generalization of Hilbert's Theorem 90: given a finite extension of fields `L/K`, the
first group cohomology `H¹(Aut_K(L), Lˣ)` is trivial. |
map_cochainsFunctor_shortExact :
ShortExact (X.map (cochainsFunctor k G)) :=
HomologicalComplex.shortExact_of_degreewise_shortExact _ fun i => {
exact := by
have : LinearMap.range X.f.hom.hom = LinearMap.ker X.g.hom.hom :=
(hX.exact.map (forget₂ (Rep k G) (ModuleCat k))).moduleCat_range_eq_ker
simp [moduleCat_exact_iff_range_eq_ker, LinearMap.range_compLeft,
LinearMap.ker_compLeft, this]
mono_f := letI := hX.mono_f; cochainsMap_id_f_map_mono X.f i
epi_g := letI := hX.epi_g; cochainsMap_id_f_map_epi X.g i }
open HomologicalComplex.HomologySequence | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | map_cochainsFunctor_shortExact | null |
noncomputable mapShortComplex₁ {i j : ℕ} (hij : i + 1 = j) :=
(snakeInput (map_cochainsFunctor_shortExact hX) _ _ hij).L₂'
variable (X) in | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean | mapShortComplex₁ | The short complex `Hⁱ(G, X₃) ⟶ Hʲ(G, X₁) ⟶ Hʲ(G, X₂)` associated to an exact
sequence of representations `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0`. |
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