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 ⌀ |
|---|---|---|---|---|---|---|
@[simps]
coboundariesOfIsMulCoboundary₁ {f : G → M} (hf : IsMulCoboundary₁ f) :
coboundaries₁ (Rep.ofMulDistribMulAction G M) :=
⟨f, hf.choose, funext hf.choose_spec⟩
@[deprecated (since := "2025-06-25")]
alias oneCoboundariesOfIsMulOneCoboundary := coboundariesOfIsMulCoboundary₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesOfIsMulCoboundary₁ | Given an abelian group `M` with a `MulDistribMulAction` of `G`, and a function
`f : G → M` satisfying the multiplicative 1-coboundary condition, produces a
1-coboundary for the representation on `Additive M` induced by the `MulDistribMulAction`. |
isMulCoboundary₁_of_mem_coboundaries₁
(f : G → M) (hf : f ∈ coboundaries₁ (Rep.ofMulDistribMulAction G M)) :
IsMulCoboundary₁ (M := M) (Additive.ofMul ∘ f) := by
rcases hf with ⟨x, rfl⟩
exact ⟨x, fun _ => rfl⟩
@[deprecated (since := "2025-07-02")]
alias isMulOneCoboundary_of_mem_oneCoboundaries := isMulCoboundary₁_of_mem_coboundaries₁ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isMulCoboundary₁_of_mem_coboundaries₁ | null |
@[simps]
cocyclesOfIsMulCocycle₂ {f : G × G → M} (hf : IsMulCocycle₂ f) :
cocycles₂ (Rep.ofMulDistribMulAction G M) :=
⟨Additive.ofMul ∘ f, (mem_cocycles₂_iff (A := Rep.ofMulDistribMulAction G M) f).2 hf⟩
@[deprecated (since := "2025-06-25")] alias twoCocyclesOfIsMulTwoCocycle := cocyclesOfIsMulCocycle₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesOfIsMulCocycle₂ | Given an abelian group `M` with a `MulDistribMulAction` of `G`, and a function
`f : G × G → M` satisfying the multiplicative 2-cocycle condition, produces a 2-cocycle for the
representation on `Additive M` induced by the `MulDistribMulAction`. |
isMulCocycle₂_of_mem_cocycles₂
(f : G × G → M) (hf : f ∈ cocycles₂ (Rep.ofMulDistribMulAction G M)) :
IsMulCocycle₂ (Additive.toMul ∘ f) :=
(mem_cocycles₂_iff (A := Rep.ofMulDistribMulAction G M) f).1 hf
@[deprecated (since := "2025-07-02")]
alias isMulTwoCocycle_of_mem_twoCocycles := isMulCocycle₂_of_mem_cocycles₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isMulCocycle₂_of_mem_cocycles₂ | null |
coboundariesOfIsMulCoboundary₂ {f : G × G → M} (hf : IsMulCoboundary₂ f) :
coboundaries₂ (Rep.ofMulDistribMulAction G M) :=
⟨f, hf.choose, funext fun g ↦ hf.choose_spec g.1 g.2⟩
@[deprecated (since := "2025-06-25")]
alias twoCoboundariesOfIsMulTwoCoboundary := coboundariesOfIsMulCoboundary₂ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | coboundariesOfIsMulCoboundary₂ | Given an abelian group `M` with a `MulDistribMulAction` of `G`, and a function
`f : G × G → M` satisfying the multiplicative 2-coboundary condition, produces a
2-coboundary for the representation on `M` induced by the `MulDistribMulAction`. |
isMulCoboundary₂_of_mem_coboundaries₂
(f : G × G → M) (hf : f ∈ coboundaries₂ (Rep.ofMulDistribMulAction G M)) :
IsMulCoboundary₂ (M := M) (Additive.toMul ∘ f) := by
rcases hf with ⟨x, rfl⟩
exact ⟨x, fun _ _ => rfl⟩
@[deprecated (since := "2025-07-02")]
alias isMulTwoCoboundary_of_mem_twoCoboundaries := isMulCoboundary₂_of_mem_coboundaries₂ | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isMulCoboundary₂_of_mem_coboundaries₂ | null |
shortComplexH0_exact : (shortComplexH0 A).Exact := by
rw [ShortComplex.moduleCat_exact_iff]
intro (x : A) (hx : d₀₁ _ x = 0)
refine ⟨⟨x, fun g => ?_⟩, rfl⟩
rw [← sub_eq_zero]
exact congr_fun hx g | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | shortComplexH0_exact | null |
@[simps! hom_left hom_right inv_left inv_right]
dArrowIso₀₁ :
Arrow.mk ((inhomogeneousCochains A).d 0 1) ≅ Arrow.mk (d₀₁ A) :=
Arrow.isoMk (cochainsIso₀ A) (cochainsIso₁ A) (comp_d₀₁_eq A)
@[deprecated (since := "2025-06-25")] noncomputable alias dZeroArrowIso := dArrowIso₀₁ | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | dArrowIso₀₁ | The arrow `A --d₀₁--> Fun(G, A)` is isomorphic to the differential
`(inhomogeneousCochains A).d 0 1` of the complex of inhomogeneous cochains of `A`. |
cocyclesIso₀ : cocycles A 0 ≅ ModuleCat.of k A.ρ.invariants :=
KernelFork.mapIsoOfIsLimit
((inhomogeneousCochains A).cyclesIsKernel 0 1 (by simp)) (shortComplexH0_exact A).fIsKernel
(dArrowIso₀₁ A)
@[deprecated (since := "2025-07-02")] noncomputable alias zeroCocyclesIso := cocyclesIso₀
@[deprecated (since := "2025-06-12")]
noncomputable alias isoZeroCocycles := zeroCocyclesIso
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesIso₀ | The 0-cocycles of the complex of inhomogeneous cochains of `A` are isomorphic to
`A.ρ.invariants`, which is a simpler type. |
cocyclesIso₀_hom_comp_f :
(cocyclesIso₀ A).hom ≫ (shortComplexH0 A).f = iCocycles A 0 ≫ (cochainsIso₀ A).hom := by
dsimp [cocyclesIso₀]
apply KernelFork.mapOfIsLimit_ι
@[deprecated (since := "2025-07-02")]
noncomputable alias zeroCocyclesIso_hom_comp_f := cocyclesIso₀_hom_comp_f
@[deprecated (since := "2025-06-12")]
alias isoZeroCocycles_hom_comp_subtype := zeroCocyclesIso_hom_comp_f
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesIso₀_hom_comp_f | null |
cocyclesIso₀_inv_comp_iCocycles :
(cocyclesIso₀ A).inv ≫ iCocycles A 0 =
(shortComplexH0 A).f ≫ (cochainsIso₀ A).inv := by
rw [Iso.inv_comp_eq, ← Category.assoc, Iso.eq_comp_inv, cocyclesIso₀_hom_comp_f]
@[deprecated (since := "2025-07-02")]
noncomputable alias zeroCocyclesIso_inv_comp_iCocycles := cocyclesIso₀_inv_comp_iCocycles
@[deprecated (since := "2025-06-12")]
alias isoZeroCocycles_inv_comp_iCocycles := zeroCocyclesIso_inv_comp_iCocycles
variable {A} in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesIso₀_inv_comp_iCocycles | null |
cocyclesMk₀_eq (x : A.ρ.invariants) :
cocyclesMk ((cochainsIso₀ A).inv x.1) (by ext g; simp [cochainsIso₀, x.2 (g 0),
inhomogeneousCochains.d, Pi.zero_apply (M := fun _ => A)]) = (cocyclesIso₀ A).inv x :=
(ModuleCat.mono_iff_injective <| iCocycles A 0).1 inferInstance <| by
rw [iCocycles_mk]
exact (cocyclesIso₀_inv_comp_iCocycles_apply A x).symm
@[deprecated (since := "2025-07-02")] alias cocyclesMk_0_eq := cocyclesMk₀_eq | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesMk₀_eq | null |
@[simps! hom inv]
isoShortComplexH1 : (inhomogeneousCochains A).sc 1 ≅ shortComplexH1 A :=
(inhomogeneousCochains A).isoSc' 0 1 2 (by simp) (by simp) ≪≫
isoMk (cochainsIso₀ A) (cochainsIso₁ A) (cochainsIso₂ A)
(comp_d₀₁_eq A) (comp_d₁₂_eq A)
@[deprecated (since := "2025-07-11")] noncomputable alias shortComplexH1Iso := isoShortComplexH1 | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoShortComplexH1 | The short complex `A --d₀₁--> Fun(G, A) --d₁₂--> Fun(G × G, A)` is isomorphic to the 1st
short complex associated to the complex of inhomogeneous cochains of `A`. |
isoCocycles₁ : cocycles A 1 ≅ ModuleCat.of k (cocycles₁ A) :=
cyclesMapIso' (isoShortComplexH1 A) _ (shortComplexH1 A).moduleCatLeftHomologyData
@[deprecated (since := "2025-06-25")] noncomputable alias isoOneCocycles := isoCocycles₁
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoCocycles₁ | The 1-cocycles of the complex of inhomogeneous cochains of `A` are isomorphic to
`cocycles₁ A`, which is a simpler type. |
isoCocycles₁_hom_comp_i :
(isoCocycles₁ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.i =
iCocycles A 1 ≫ (cochainsIso₁ A).hom := by
simp [isoCocycles₁, iCocycles, HomologicalComplex.iCycles, iCycles]
@[deprecated (since := "2025-06-25")] alias isoOneCocycles_hom_comp_i := isoCocycles₁_hom_comp_i
@[deprecated (since := "2025-05-09")]
alias isoOneCocycles_hom_comp_subtype := isoOneCocycles_hom_comp_i
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoCocycles₁_hom_comp_i | null |
isoCocycles₁_inv_comp_iCocycles :
(isoCocycles₁ A).inv ≫ iCocycles A 1 =
(shortComplexH1 A).moduleCatLeftHomologyData.i ≫ (cochainsIso₁ A).inv :=
(CommSq.horiz_inv ⟨isoCocycles₁_hom_comp_i A⟩).w
@[deprecated (since := "2025-06-25")]
alias isoOneCocycles_inv_comp_iCocycles := isoCocycles₁_inv_comp_iCocycles
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoCocycles₁_inv_comp_iCocycles | null |
toCocycles_comp_isoCocycles₁_hom :
toCocycles A 0 1 ≫ (isoCocycles₁ A).hom =
(cochainsIso₀ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.f' := by
simp [← cancel_mono (shortComplexH1 A).moduleCatLeftHomologyData.i, comp_d₀₁_eq,
shortComplexH1_f]
@[deprecated (since := "2025-06-25")]
alias toCocycles_comp_isoOneCocycles_hom := toCocycles_comp_isoCocycles₁_hom | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | toCocycles_comp_isoCocycles₁_hom | null |
cocyclesMk₁_eq (x : cocycles₁ A) :
cocyclesMk ((cochainsIso₁ A).inv x) (by
simp [← inhomogeneousCochains.d_def, cocycles₁.d₁₂_apply x]) =
(isoCocycles₁ A).inv x := by
apply_fun (forget₂ _ Ab).map ((inhomogeneousCochains A).iCycles 1) using
(AddCommGrp.mono_iff_injective _).1 <| (forget₂ _ _).map_mono _
simpa only [HomologicalComplex.i_cyclesMk] using
(isoCocycles₁_inv_comp_iCocycles_apply _ x).symm
@[deprecated (since := "2025-07-02")] alias cocyclesMk_1_eq := cocyclesMk₁_eq | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesMk₁_eq | null |
@[simps! hom inv]
isoShortComplexH2 :
(inhomogeneousCochains A).sc 2 ≅ shortComplexH2 A :=
(inhomogeneousCochains A).isoSc' 1 2 3 (by simp) (by simp) ≪≫
isoMk (cochainsIso₁ A) (cochainsIso₂ A) (cochainsIso₃ A)
(comp_d₁₂_eq A) (comp_d₂₃_eq A)
@[deprecated (since := "2025-07-11")] noncomputable alias shortComplexH2Iso := isoShortComplexH2 | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoShortComplexH2 | The short complex `Fun(G, A) --d₁₂--> Fun(G × G, A) --dTwo--> Fun(G × G × G, A)` is
isomorphic to the 2nd short complex associated to the complex of inhomogeneous cochains of `A`. |
isoCocycles₂ : cocycles A 2 ≅ ModuleCat.of k (cocycles₂ A) :=
cyclesMapIso' (isoShortComplexH2 A) _ (shortComplexH2 A).moduleCatLeftHomologyData
@[deprecated (since := "2025-06-25")] noncomputable alias isoTwoCocycles := isoCocycles₂
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoCocycles₂ | The 2-cocycles of the complex of inhomogeneous cochains of `A` are isomorphic to
`cocycles₂ A`, which is a simpler type. |
isoCocycles₂_hom_comp_i :
(isoCocycles₂ A).hom ≫ (shortComplexH2 A).moduleCatLeftHomologyData.i =
iCocycles A 2 ≫ (cochainsIso₂ A).hom := by
simp [isoCocycles₂, iCocycles, HomologicalComplex.iCycles, iCycles]
@[deprecated (since := "2025-06-25")] alias isoTwoCocycles_hom_comp_i := isoCocycles₂_hom_comp_i
@[deprecated (since := "2025-05-09")]
alias isoTwoCocycles_hom_comp_subtype := isoTwoCocycles_hom_comp_i
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoCocycles₂_hom_comp_i | null |
isoCocycles₂_inv_comp_iCocycles :
(isoCocycles₂ A).inv ≫ iCocycles A 2 =
(shortComplexH2 A).moduleCatLeftHomologyData.i ≫ (cochainsIso₂ A).inv :=
(CommSq.horiz_inv ⟨isoCocycles₂_hom_comp_i A⟩).w
@[deprecated (since := "2025-06-25")]
alias isoTwoCocycles_inv_comp_iCocycles := isoCocycles₂_inv_comp_iCocycles
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | isoCocycles₂_inv_comp_iCocycles | null |
toCocycles_comp_isoCocycles₂_hom :
toCocycles A 1 2 ≫ (isoCocycles₂ A).hom =
(cochainsIso₁ A).hom ≫ (shortComplexH2 A).moduleCatLeftHomologyData.f' := by
simp [← cancel_mono (shortComplexH2 A).moduleCatLeftHomologyData.i, comp_d₁₂_eq,
shortComplexH2_f]
@[deprecated (since := "2025-06-25")]
alias toCocycles_comp_isoTwoCocycles_hom := toCocycles_comp_isoCocycles₂_hom | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | toCocycles_comp_isoCocycles₂_hom | null |
cocyclesMk₂_eq (x : cocycles₂ A) :
cocyclesMk ((cochainsIso₂ A).inv x) (by
simp [← inhomogeneousCochains.d_def, cocycles₂.d₂₃_apply x]) =
(isoCocycles₂ A).inv x := by
apply_fun (forget₂ _ Ab).map ((inhomogeneousCochains A).iCycles 2) using
(AddCommGrp.mono_iff_injective _).1 <| (forget₂ _ _).map_mono _
simpa only [HomologicalComplex.i_cyclesMk] using
(isoCocycles₂_inv_comp_iCocycles_apply _ x).symm
@[deprecated (since := "2025-07-02")] alias cocyclesMk_2_eq := cocyclesMk₂_eq | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | cocyclesMk₂_eq | null |
H0 := groupCohomology A 0 | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H0 | Shorthand for the 0th group cohomology of a `k`-linear `G`-representation `A`, `H⁰(G, A)`,
defined as the 0th cohomology of the complex of inhomogeneous cochains of `A`. |
H0Iso : H0 A ≅ ModuleCat.of k A.ρ.invariants :=
(CochainComplex.isoHomologyπ₀ _).symm ≪≫ cocyclesIso₀ A
@[deprecated (since := "2025-06-11")]
noncomputable alias isoH0 := H0Iso
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H0Iso | The 0th group cohomology of `A`, defined as the 0th cohomology of the complex of inhomogeneous
cochains, is isomorphic to the invariants of the representation on `A`. |
π_comp_H0Iso_hom :
π A 0 ≫ (H0Iso A).hom = (cocyclesIso₀ A).hom := by
simp [H0Iso]
@[deprecated (since := "2025-06-12")]
alias groupCohomologyπ_comp_isoH0_hom := π_comp_H0Iso_hom
@[elab_as_elim] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | π_comp_H0Iso_hom | null |
H0_induction_on {C : H0 A → Prop} (x : H0 A)
(h : ∀ x : A.ρ.invariants, C ((H0Iso A).inv x)) : C x := by
simpa using h ((H0Iso A).hom x) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H0_induction_on | null |
H0IsoOfIsTrivial :
H0 A ≅ A.V := H0Iso A ≪≫ (LinearEquiv.ofTop _ (invariants_eq_top A.ρ)).toModuleIso
@[deprecated (since := "2025-05-09")]
noncomputable alias H0LequivOfIsTrivial := H0IsoOfIsTrivial
@[simp, elementwise] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H0IsoOfIsTrivial | When the representation on `A` is trivial, then `H⁰(G, A)` is all of `A.` |
H0IsoOfIsTrivial_hom :
(H0IsoOfIsTrivial A).hom = (H0Iso A).hom ≫ (shortComplexH0 A).f := rfl
@[deprecated (since := "2025-06-11")]
alias H0LequivOfIsTrivial_eq_subtype := H0IsoOfIsTrivial_hom
@[deprecated (since := "2025-05-09")]
alias H0LequivOfIsTrivial_apply := H0IsoOfIsTrivial_hom_apply
@[reassoc, elementwise] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H0IsoOfIsTrivial_hom | null |
π_comp_H0IsoOfIsTrivial_hom :
π A 0 ≫ (H0IsoOfIsTrivial A).hom = iCocycles A 0 ≫ (cochainsIso₀ A).hom := by
simp
variable {A} in
@[simp] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | π_comp_H0IsoOfIsTrivial_hom | null |
H0IsoOfIsTrivial_inv_apply (x : A) :
(H0IsoOfIsTrivial A).inv x = (H0Iso A).inv ⟨x, by simp⟩ := rfl
@[deprecated (since := "2025-05-09")]
alias H0LequivOfIsTrivial_symm_apply := H0IsoOfIsTrivial_inv_apply | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H0IsoOfIsTrivial_inv_apply | null |
H1 := groupCohomology A 1 | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1 | Shorthand for the 1st group cohomology of a `k`-linear `G`-representation `A`, `H¹(G, A)`,
defined as the 1st cohomology of the complex of inhomogeneous cochains of `A`. |
H1π : ModuleCat.of k (cocycles₁ A) ⟶ H1 A :=
(isoCocycles₁ A).inv ≫ π A 1 | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1π | The quotient map from the 1-cocycles of `A`, as a submodule of `G → A`, to `H¹(G, A)`. |
H1π_eq_zero_iff (x : cocycles₁ A) : H1π A x = 0 ↔ ⇑x ∈ coboundaries₁ A := by
have h := leftHomologyπ_naturality'_assoc (isoShortComplexH1 A).inv
(shortComplexH1 A).moduleCatLeftHomologyData (leftHomologyData _)
((inhomogeneousCochains A).sc 1).leftHomologyIso.hom
simp only [H1π, isoCocycles₁, π, HomologicalComplex.homologyπ, homologyπ,
cyclesMapIso'_inv, leftHomologyπ, ← h, ← leftHomologyMapIso'_inv, ModuleCat.hom_comp,
LinearMap.coe_comp, Function.comp_apply, map_eq_zero_iff _
((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
simp [LinearMap.range_codRestrict, coboundaries₁, shortComplexH1, cocycles₁] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1π_eq_zero_iff | null |
H1π_eq_iff (x y : cocycles₁ A) :
H1π A x = H1π A y ↔ ⇑x - ⇑y ∈ coboundaries₁ A := by
rw [← sub_eq_zero, ← map_sub, H1π_eq_zero_iff]
rfl
@[elab_as_elim] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1π_eq_iff | null |
H1_induction_on {C : H1 A → Prop} (x : H1 A) (h : ∀ x : cocycles₁ A, C (H1π A x)) :
C x :=
groupCohomology_induction_on x fun y => by simpa [H1π] using h ((isoCocycles₁ A).hom y)
variable (A) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1_induction_on | null |
H1Iso : H1 A ≅ (shortComplexH1 A).moduleCatLeftHomologyData.H :=
(leftHomologyIso _).symm ≪≫ (leftHomologyMapIso' (isoShortComplexH1 A) _ _)
@[deprecated (since := "2025-06-11")]
noncomputable alias isoH1 := H1Iso
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1Iso | The 1st group cohomology of `A`, defined as the 1st cohomology of the complex of inhomogeneous
cochains, is isomorphic to `cocycles₁ A ⧸ coboundaries₁ A`, which is a simpler type. |
π_comp_H1Iso_hom :
π A 1 ≫ (H1Iso A).hom = (isoCocycles₁ A).hom ≫
(shortComplexH1 A).moduleCatLeftHomologyData.π := by
simp [H1Iso, isoCocycles₁, π, HomologicalComplex.homologyπ, leftHomologyπ]
@[deprecated (since := "2025-06-12")]
alias groupCohomologyπ_comp_isoH1_hom := π_comp_H1Iso_hom | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | π_comp_H1Iso_hom | null |
H1IsoOfIsTrivial :
H1 A ≅ ModuleCat.of k (Additive G →+ A) :=
(HomologicalComplex.isoHomologyπ _ 0 1 (CochainComplex.prev_nat_succ 0) <| by
ext; simp [inhomogeneousCochains.d_def, inhomogeneousCochains.d,
Unique.eq_default (α := Fin 0 → G), Pi.zero_apply (M := fun _ => A)]).symm ≪≫
isoCocycles₁ A ≪≫ cocycles₁IsoOfIsTrivial A
@[deprecated (since := "2025-05-09")]
noncomputable alias H1LequivOfIsTrivial := H1IsoOfIsTrivial
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1IsoOfIsTrivial | When `A : Rep k G` is a trivial representation of `G`, `H¹(G, A)` is isomorphic to the
group homs `G → A`. |
H1π_comp_H1IsoOfIsTrivial_hom :
H1π A ≫ (H1IsoOfIsTrivial A).hom = (cocycles₁IsoOfIsTrivial A).hom := by
simp [H1IsoOfIsTrivial, H1π]
@[deprecated (since := "2025-05-09")]
alias H1LequivOfIsTrivial_comp_H1π := H1π_comp_H1IsoOfIsTrivial_hom
variable {A} | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1π_comp_H1IsoOfIsTrivial_hom | null |
H1IsoOfIsTrivial_H1π_apply_apply
(f : cocycles₁ A) (x : Additive G) :
(H1IsoOfIsTrivial A).hom (H1π A f) x = f x.toMul := by simp
@[deprecated (since := "2025-05-09")]
alias H1LequivOfIsTrivial_comp_H1_π_apply_apply := H1IsoOfIsTrivial_H1π_apply_apply | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1IsoOfIsTrivial_H1π_apply_apply | null |
H1IsoOfIsTrivial_inv_apply (f : Additive G →+ A) :
(H1IsoOfIsTrivial A).inv f = H1π A ((cocycles₁IsoOfIsTrivial A).inv f) := rfl
@[deprecated (since := "2025-05-09")]
alias H1LequivOfIsTrivial_symm_apply := H1IsoOfIsTrivial_inv_apply | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H1IsoOfIsTrivial_inv_apply | null |
H2 := groupCohomology A 2 | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H2 | Shorthand for the 2nd group cohomology of a `k`-linear `G`-representation `A`, `H²(G, A)`,
defined as the 2nd cohomology of the complex of inhomogeneous cochains of `A`. |
H2π : ModuleCat.of k (cocycles₂ A) ⟶ H2 A :=
(isoCocycles₂ A).inv ≫ π A 2 | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H2π | The quotient map from the 2-cocycles of `A`, as a submodule of `G × G → A`, to `H²(G, A)`. |
H2π_eq_zero_iff (x : cocycles₂ A) : H2π A x = 0 ↔ ⇑x ∈ coboundaries₂ A := by
have h := leftHomologyπ_naturality'_assoc (isoShortComplexH2 A).inv
(shortComplexH2 A).moduleCatLeftHomologyData (leftHomologyData _)
((inhomogeneousCochains A).sc 2).leftHomologyIso.hom
simp only [H2π, isoCocycles₂, π, HomologicalComplex.homologyπ, homologyπ,
cyclesMapIso'_inv, leftHomologyπ, ← h, ← leftHomologyMapIso'_inv, ModuleCat.hom_comp,
LinearMap.coe_comp, Function.comp_apply, map_eq_zero_iff _
((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
simp [LinearMap.range_codRestrict, coboundaries₂, shortComplexH2, cocycles₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H2π_eq_zero_iff | null |
H2π_eq_iff (x y : cocycles₂ A) :
H2π A x = H2π A y ↔ ⇑x - ⇑y ∈ coboundaries₂ A := by
rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl
@[elab_as_elim] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H2π_eq_iff | null |
H2_induction_on {C : H2 A → Prop} (x : H2 A) (h : ∀ x : cocycles₂ A, C (H2π A x)) :
C x :=
groupCohomology_induction_on x fun y => by simpa [H2π] using h ((isoCocycles₂ A).hom y)
variable (A) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H2_induction_on | null |
H2Iso : H2 A ≅ (shortComplexH2 A).moduleCatLeftHomologyData.H :=
(leftHomologyIso _).symm ≪≫ (leftHomologyMapIso' (isoShortComplexH2 A) _ _)
@[deprecated (since := "2025-06-11")]
noncomputable alias isoH2 := H2Iso
@[reassoc (attr := simp), elementwise (attr := simp)] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | H2Iso | The 2nd group cohomology of `A`, defined as the 2nd cohomology of the complex of inhomogeneous
cochains, is isomorphic to `cocycles₂ A ⧸ coboundaries₂ A`, which is a simpler type. |
π_comp_H2Iso_hom :
π A 2 ≫ (H2Iso A).hom = (isoCocycles₂ A).hom ≫
(shortComplexH2 A).moduleCatLeftHomologyData.π := by
simp [H2Iso, isoCocycles₂, π, HomologicalComplex.homologyπ, leftHomologyπ]
@[deprecated (since := "2025-06-12")]
alias groupCohomologyπ_comp_isoH2_hom := π_comp_H2Iso_hom | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean | π_comp_H2Iso_hom | null |
noncomputable linearYonedaObjResProjectiveResolutionIso
(P : ProjectiveResolution (trivial k G k)) (A : Rep k S) :
((Action.res _ S.subtype).mapProjectiveResolution P).complex.linearYonedaObj k A ≅
P.complex.linearYonedaObj k (coind S.subtype A) :=
HomologicalComplex.Hom.isoOfComponents
(fun _ => (resCoindHomEquiv _ _ _).toModuleIso) fun _ _ _ =>
ModuleCat.hom_ext (LinearMap.ext fun f => Action.Hom.ext <| by ext; simp [hom_comm_apply]) | def | RepresentationTheory | [
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Shapiro.lean | linearYonedaObjResProjectiveResolutionIso | Given a projective resolution `P` of `k` as a `k`-linear `G`-representation, a subgroup
`S ≤ G`, and a `k`-linear `S`-representation `A`, this is an isomorphism of complexes
`Hom(Res(S)(P), A) ≅ Hom(P, Coind_S^G(A)).` |
noncomputable coindIso [DecidableEq G] (A : Rep k S) (n : ℕ) :
groupCohomology (coind S.subtype A) n ≅ groupCohomology A n :=
(HomologicalComplex.homologyFunctor _ _ _).mapIso
(inhomogeneousCochainsIso (coind S.subtype A) ≪≫
(linearYonedaObjResProjectiveResolutionIso (barResolution k G) A).symm) ≪≫
(groupCohomologyIso A n ((Action.res _ _).mapProjectiveResolution <| barResolution k G)).symm | def | RepresentationTheory | [
"Mathlib.CategoryTheory.Preadditive.Projective.Resolution",
"Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic",
"Mathlib.RepresentationTheory.Coinduced",
"Mathlib.RepresentationTheory.Induced"
] | Mathlib/RepresentationTheory/Homological/GroupCohomology/Shapiro.lean | coindIso | Shapiro's lemma: given a subgroup `S ≤ G` and an `S`-representation `A`, we have
`Hⁿ(G, Coind_S^G(A)) ≅ Hⁿ(S, A).` |
defined by `g • a := A.ρ g⁻¹ a`, but currently mathlib's `TensorProduct` is only defined
for commutative rings. | structure | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | defined | null |
HomologicalComplex.coinvariantsTensorObj {α : Type*} [AddRightCancelSemigroup α] [One α]
(A : Rep k G) (P : ChainComplex (Rep k G) α) :
ChainComplex (ModuleCat k) α :=
(((Rep.coinvariantsTensor k G).obj A).mapHomologicalComplex _).obj P | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | HomologicalComplex.coinvariantsTensorObj | Given `A : Rep k G` and a chain complex `P` in `Rep k G`, this is the chain complex whose
`n`th object is `(A ⊗ Pₙ)_G`. |
@[simps]
Tor (n : ℕ) : Rep k G ⥤ Rep k G ⥤ ModuleCat k where
obj X := Functor.leftDerived ((coinvariantsTensor k G).obj X) n
map f := NatTrans.leftDerived ((coinvariantsTensor k G).map f) n
variable {k G} (A : Rep k G) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | Tor | The left-derived functors given by deriving the second argument of `A, B ↦ (A ⊗[k] B)_G`. |
torIso (A : Rep k G) {B : Rep k G} (P : ProjectiveResolution B) (n : ℕ) :
((Rep.Tor k G n).obj A).obj B ≅ (P.complex.coinvariantsTensorObj A).homology n :=
P.isoLeftDerivedObj _ n | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | torIso | `Tor` can be computed using a projective resolution. |
isZero_Tor_succ_of_projective (X Y : Rep k G) [Projective Y] (n : ℕ) :
IsZero (((Tor k G (n + 1)).obj X).obj Y) :=
Functor.isZero_leftDerived_obj_projective_succ .. | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | isZero_Tor_succ_of_projective | The higher `Tor` groups for `X` and `Y` are zero if `Y` is projective. |
@[deprecated "Use `(barComplex k G).coinvariantsTensorObj A` instead." (since := "2025-06-17")]
coinvariantsTensorBarResolution [DecidableEq G] :=
(((coinvariantsTensor k G).obj A).mapHomologicalComplex _).obj (barComplex k G) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | coinvariantsTensorBarResolution | Given a `k`-linear `G`-representation `A`, this is the chain complex `(A ⊗[k] P)_G`, where
`P` is the bar resolution of `k` as a trivial representation. |
d : ModuleCat.of k ((Fin (n + 1) → G) →₀ A) ⟶ ModuleCat.of k ((Fin n → G) →₀ A) :=
ModuleCat.ofHom <| lsum (R := k) k fun g => lsingle (fun i => g i.succ) ∘ₗ A.ρ (g 0)⁻¹ +
Finset.univ.sum fun j : Fin (n + 1) =>
(-1 : k) ^ ((j : ℕ) + 1) • lsingle (Fin.contractNth j (· * ·) g)
variable {A n} in
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | d | The differential in the complex of inhomogeneous chains used to calculate group homology. |
d_single (n : ℕ) (g : Fin (n + 1) → G) (a : A) :
d A n (single g a) = single (fun i => g i.succ) (A.ρ (g 0)⁻¹ a) +
Finset.univ.sum fun j : Fin (n + 1) =>
(-1 : k) ^ ((j : ℕ) + 1) • single (Fin.contractNth j (· * ·) g) a := by
simp [d]
open ModuleCat.MonoidalCategory | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | d_single | null |
d_eq [DecidableEq G] :
d A n = (coinvariantsTensorFreeLEquiv A (Fin (n + 1) → G)).toModuleIso.inv ≫
((barComplex k G).coinvariantsTensorObj A).d (n + 1) n ≫
(coinvariantsTensorFreeLEquiv A (Fin n → G)).toModuleIso.hom := by
ext : 3
simp [d_single (k := k), ModuleCat.MonoidalCategory.tensorObj,
ModuleCat.MonoidalCategory.whiskerLeft, tensorObj_def, whiskerLeft_def, TensorProduct.tmul_add,
TensorProduct.tmul_sum, barComplex.d_single (k := k)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | d_eq | null |
noncomputable inhomogeneousChains :
ChainComplex (ModuleCat k) ℕ :=
ChainComplex.of (fun n => ModuleCat.of k ((Fin n → G) →₀ A))
(fun n => inhomogeneousChains.d A n) fun n => by
classical
simp only [inhomogeneousChains.d_eq]
slice_lhs 3 4 => { rw [Iso.hom_inv_id] }
slice_lhs 2 4 => { rw [Category.id_comp, ((barComplex k G).coinvariantsTensorObj A).d_comp_d] }
simp
open inhomogeneousChains
variable {A n} in
@[ext] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | inhomogeneousChains | Given a `k`-linear `G`-representation `A`, this is the complex of inhomogeneous chains
$$\dots \to \bigoplus_{G^1} A \to \bigoplus_{G^0} A \to 0$$
which calculates the group homology of `A`. |
inhomogeneousChains.ext {M : ModuleCat k} {x y : (inhomogeneousChains A).X n ⟶ M}
(h : ∀ g, ModuleCat.ofHom (lsingle g) ≫ x = ModuleCat.ofHom (lsingle g) ≫ y) :
x = y := ModuleCat.hom_ext <| lhom_ext' fun g => ModuleCat.hom_ext_iff.1 (h g) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | inhomogeneousChains.ext | null |
inhomogeneousChains.d_def (n : ℕ) :
(inhomogeneousChains A).d (n + 1) n = d A n := by
simp [inhomogeneousChains] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | inhomogeneousChains.d_def | null |
inhomogeneousChains.d_comp_d :
d A (n + 1) ≫ d A n = 0 := by
simpa [ChainComplex.of] using ((inhomogeneousChains A).d_comp_d (n + 2) (n + 1) n) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | inhomogeneousChains.d_comp_d | null |
inhomogeneousChainsIso [DecidableEq G] :
inhomogeneousChains A ≅ (barComplex k G).coinvariantsTensorObj A := by
refine HomologicalComplex.Hom.isoOfComponents ?_ ?_
· intro i
apply (coinvariantsTensorFreeLEquiv A (Fin i → G)).toModuleIso.symm
rintro i j rfl
simp [d_eq, -LinearEquiv.toModuleIso_hom, -LinearEquiv.toModuleIso_inv] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | inhomogeneousChainsIso | Given a `k`-linear `G`-representation `A`, the complex of inhomogeneous chains is isomorphic
to `(A ⊗[k] P)_G`, where `P` is the bar resolution of `k` as a trivial `G`-representation. |
cycles (n : ℕ) : ModuleCat k := (inhomogeneousChains A).cycles n
open HomologicalComplex
variable {A} in | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | cycles | The `n`-cycles `Zₙ(G, A)` of a `k`-linear `G`-representation `A`, i.e. the kernel of the
differential `Cₙ(G, A) ⟶ Cₙ₋₁(G, A)` in the complex of inhomogeneous chains. |
cyclesMk (m n : ℕ) (h : (ComplexShape.down ℕ).next m = n) (f : (Fin m → G) →₀ A)
(hf : (inhomogeneousChains A).d m n f = 0) : cycles A m :=
(inhomogeneousChains A).cyclesMk f n h hf | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | cyclesMk | When `m = 0` this makes a term of `cycles A 0` from any element of `A` (or more precisely
any element in the kernel of `d₀,₀ = 0`). When `m` is positive, this makes a term of `cycles A m`
from any element of the kernel of `dₘ,ₘ₋₁`. |
iCycles (n : ℕ) : cycles A n ⟶ (inhomogeneousChains A).X n :=
(inhomogeneousChains A).iCycles n
variable {A} in | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | iCycles | The natural inclusion of the `n`-cycles `Zₙ(G, A)` into the `n`-chains `Cₙ(G, A).` |
iCycles_mk {m n : ℕ} (h : (ComplexShape.down ℕ).next m = n) (f : (Fin m → G) →₀ A)
(hf : (inhomogeneousChains A).d m n f = 0) :
iCycles A m (cyclesMk m n h f hf) = f := by
exact (inhomogeneousChains A).i_cyclesMk f n h hf | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | iCycles_mk | null |
toCycles (i j : ℕ) : (inhomogeneousChains A).X i ⟶ cycles A j :=
(inhomogeneousChains A).toCycles i j | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | toCycles | This is the map from `i`-chains to `j`-cycles induced by the differential in the complex of
inhomogeneous chains. |
groupHomology (n : ℕ) : ModuleCat k :=
(inhomogeneousChains A).homology n | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | groupHomology | The group homology of a `k`-linear `G`-representation `A`, as the homology of its complex
of inhomogeneous chains. |
groupHomology.π (n : ℕ) :
cycles A n ⟶ groupHomology A n :=
(inhomogeneousChains A).homologyπ n
variable {A} in
@[elab_as_elim] | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | groupHomology.π | The natural map from `n`-cycles to `n`th group homology for a `k`-linear
`G`-representation `A`. |
groupHomology_induction_on {n : ℕ}
{C : groupHomology A n → Prop} (x : groupHomology A n)
(h : ∀ x : cycles 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.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | groupHomology_induction_on | null |
groupHomologyIsoTor [DecidableEq G] (n : ℕ) :
groupHomology A n ≅ ((Tor k G n).obj A).obj (Rep.trivial k G k) :=
isoOfQuasiIsoAt (HomotopyEquiv.ofIso (inhomogeneousChainsIso A)).hom n ≪≫
(torIso A (barResolution k G) n).symm | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | groupHomologyIsoTor | The `n`th group homology of a `k`-linear `G`-representation `A` is isomorphic to
`Torₙ(A, k)` (taken in `Rep k G`), where `k` is a trivial `k`-linear `G`-representation. |
groupHomologyIso [DecidableEq G] (A : Rep k G) (n : ℕ)
(P : ProjectiveResolution (Rep.trivial k G k)) :
groupHomology A n ≅ (P.complex.coinvariantsTensorObj A).homology n :=
groupHomologyIsoTor A n ≪≫ torIso A P n | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | groupHomologyIso | The `n`th group homology of a `k`-linear `G`-representation `A` is isomorphic to
`Hₙ((A ⊗ P)_G)`, where `P` is any projective resolution of `k` as a trivial `k`-linear
`G`-representation. |
isZero_groupHomology_succ_of_subsingleton [Subsingleton G] (n : ℕ) :
Limits.IsZero (groupHomology A (n + 1)) :=
(isZero_Tor_succ_of_projective A (Rep.trivial k G k) n).of_iso <| groupHomologyIsoTor _ _ | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.RepresentationTheory.Coinvariants",
"Mathlib.RepresentationTheory.Homological.Resolution",
"Mathlib.Tactic.CategoryTheory.Slice",
"Mathlib.CategoryTheory.Abelian.LeftDerived"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean | isZero_groupHomology_succ_of_subsingleton | null |
congr {f₁ f₂ : G →* H} (h : f₁ = f₂) {φ : A ⟶ (Action.res _ f₁).obj B} {T : Type*}
(F : (f : G →* H) → (φ : A ⟶ (Action.res _ f).obj B) → T) :
F f₁ φ = F f₂ (h ▸ φ) := by
subst h
rfl | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | congr | null |
@[simps! -isSimp f f_hom]
noncomputable chainsMap :
inhomogeneousChains A ⟶ inhomogeneousChains B where
f i := ModuleCat.ofHom <| mapRange.linearMap φ.hom.hom ∘ₗ lmapDomain A k (f ∘ ·)
comm' i j (hij : _ = _) := by
subst hij
ext
simpa [Fin.comp_contractNth, map_add, inhomogeneousChains.d]
using congr(single _ $((hom_comm_apply φ (_)⁻¹ _).symm))
@[reassoc (attr := simp)] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the chain map sending `∑ aᵢ·gᵢ : Gⁿ →₀ A` to `∑ φ(aᵢ)·(f ∘ gᵢ) : Hⁿ →₀ B`. |
lsingle_comp_chainsMap_f (n : ℕ) (x : Fin n → G) :
ModuleCat.ofHom (lsingle x) ≫ (chainsMap f φ).f n =
φ.hom ≫ ModuleCat.ofHom (lsingle (f ∘ x)) := by
ext
simp [chainsMap_f] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | lsingle_comp_chainsMap_f | null |
chainsMap_f_single (n : ℕ) (x : Fin n → G) (a : A) :
(chainsMap f φ).f n (single x a) = single (f ∘ x) (φ.hom a) := by
simp [chainsMap_f]
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_f_single | null |
chainsMap_id :
chainsMap (MonoidHom.id G) (𝟙 A) = 𝟙 (inhomogeneousChains A) :=
HomologicalComplex.hom_ext _ _ fun _ => ModuleCat.hom_ext <| lhom_ext' fun _ =>
ModuleCat.hom_ext_iff.1 <| lsingle_comp_chainsMap_f (k := k) (MonoidHom.id G) ..
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_id | null |
chainsMap_id_f_hom_eq_mapRange {A B : Rep k G} (i : ℕ) (φ : A ⟶ B) :
((chainsMap (MonoidHom.id G) φ).f i).hom = mapRange.linearMap φ.hom.hom := by
refine lhom_ext fun _ _ => ?_
simp [chainsMap_f, MonoidHom.coe_id] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_id_f_hom_eq_mapRange | null |
chainsMap_comp {G H K : Type u} [Group G] [Group H] [Group K]
{A : Rep k G} {B : Rep k H} {C : Rep k K}
(f : G →* H) (g : H →* K) (φ : A ⟶ (Action.res _ f).obj B) (ψ : B ⟶ (Action.res _ g).obj C) :
chainsMap (g.comp f) (φ ≫ (Action.res _ f).map ψ) = chainsMap f φ ≫ chainsMap g ψ := by
ext
simp [chainsMap_f, Function.comp_assoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_comp | null |
chainsMap_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) :
chainsMap (MonoidHom.id G) (φ ≫ ψ) =
chainsMap (MonoidHom.id G) φ ≫ chainsMap (MonoidHom.id G) ψ :=
chainsMap_comp (MonoidHom.id G) (MonoidHom.id G) _ _
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_id_comp | null |
chainsMap_zero : chainsMap f (0 : A ⟶ (Action.res _ f).obj B) = 0 := by
ext; simp [chainsMap_f, LinearMap.zero_apply (M₂ := B)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_zero | null |
chainsMap_f_map_mono (hf : Function.Injective f) [Mono φ] (i : ℕ) :
Mono ((chainsMap f φ).f i) := by
simpa [ModuleCat.mono_iff_injective] using
(mapRange_injective φ.hom (map_zero _) <| (Rep.mono_iff_injective φ).1
inferInstance).comp (mapDomain_injective hf.comp_left) | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_f_map_mono | null |
chainsMap_id_f_map_mono {A B : Rep k G} (φ : A ⟶ B) [Mono φ] (i : ℕ) :
Mono ((chainsMap (MonoidHom.id G) φ).f i) :=
chainsMap_f_map_mono (MonoidHom.id G) φ (fun _ _ h => h) _ | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_id_f_map_mono | null |
chainsMap_f_map_epi (hf : Function.Surjective f) [Epi φ] (i : ℕ) :
Epi ((chainsMap f φ).f i) := by
simpa [ModuleCat.epi_iff_surjective] using
(mapRange_surjective φ.hom (map_zero _) ((Rep.epi_iff_surjective φ).1 inferInstance)).comp
(mapDomain_surjective hf.comp_left) | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_f_map_epi | null |
chainsMap_id_f_map_epi {A B : Rep k G} (φ : A ⟶ B) [Epi φ] (i : ℕ) :
Epi ((chainsMap (MonoidHom.id G) φ).f i) :=
chainsMap_f_map_epi _ _ (fun x => ⟨x, rfl⟩) _ | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_id_f_map_epi | null |
noncomputable cyclesMap (n : ℕ) :
groupHomology.cycles A n ⟶ groupHomology.cycles B n :=
HomologicalComplex.cyclesMap (chainsMap f φ) n
@[simp] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesMap | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map `Zₙ(G, A) ⟶ Zₙ(H, B)` sending `∑ aᵢ·gᵢ : Gⁿ →₀ A` to
`∑ φ(aᵢ)·(f ∘ gᵢ) : Hⁿ →₀ B`. |
cyclesMap_id : cyclesMap (MonoidHom.id G) (𝟙 A) n = 𝟙 _ := by
simp [cyclesMap]
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesMap_id | null |
cyclesMap_comp {G H K : Type u} [Group G] [Group H] [Group K]
{A : Rep k G} {B : Rep k H} {C : Rep k K} (f : G →* H) (g : H →* K)
(φ : A ⟶ (Action.res _ f).obj B) (ψ : B ⟶ (Action.res _ g).obj C) (n : ℕ) :
cyclesMap (g.comp f) (φ ≫ (Action.res _ f).map ψ) n = cyclesMap f φ n ≫ cyclesMap g ψ n := by
simp [cyclesMap, ← HomologicalComplex.cyclesMap_comp, ← chainsMap_comp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesMap_comp | null |
cyclesMap_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) (n : ℕ) :
cyclesMap (MonoidHom.id G) (φ ≫ ψ) n =
cyclesMap (MonoidHom.id G) φ n ≫ cyclesMap (MonoidHom.id G) ψ n := by
simp [cyclesMap, chainsMap_id_comp, HomologicalComplex.cyclesMap_comp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesMap_id_comp | null |
noncomputable map (n : ℕ) :
groupHomology A n ⟶ groupHomology B n :=
HomologicalComplex.homologyMap (chainsMap f φ) n
@[reassoc, elementwise] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | map | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map `Hₙ(G, A) ⟶ Hₙ(H, B)` sending `∑ aᵢ·gᵢ : Gⁿ →₀ A` to
`∑ φ(aᵢ)·(f ∘ gᵢ) : Hⁿ →₀ B`. |
π_map (n : ℕ) :
π A n ≫ map f φ n = cyclesMap f φ n ≫ π B n := by
simp [map, cyclesMap]
@[simp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | π_map | null |
map_id : map (MonoidHom.id G) (𝟙 A) n = 𝟙 _ := by
simp [map, groupHomology]
@[reassoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | map_id | null |
map_comp {G H K : Type u} [Group G] [Group H] [Group K]
{A : Rep k G} {B : Rep k H} {C : Rep k K} (f : G →* H) (g : H →* K)
(φ : A ⟶ (Action.res _ f).obj B) (ψ : B ⟶ (Action.res _ g).obj C) (n : ℕ) :
map (g.comp f) (φ ≫ (Action.res _ f).map ψ) n = map f φ n ≫ map g ψ n := by
simp [map, ← HomologicalComplex.homologyMap_comp, ← chainsMap_comp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/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, chainsMap_id_comp, HomologicalComplex.homologyMap_comp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | map_id_comp | null |
noncomputable chainsMap₁ : ModuleCat.of k (G →₀ A) ⟶ ModuleCat.of k (H →₀ B) :=
ModuleCat.ofHom <| mapRange.linearMap φ.hom.hom ∘ₗ lmapDomain A k f | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap₁ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map sending `∑ aᵢ·gᵢ : G →₀ A` to `∑ φ(aᵢ)·f(gᵢ) : H →₀ B`. |
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