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 ⌀ |
|---|---|---|---|---|---|---|
noncomputable chainsMap₂ : ModuleCat.of k (G × G →₀ A) ⟶ ModuleCat.of k (H × H →₀ B) :=
ModuleCat.ofHom <| mapRange.linearMap φ.hom.hom ∘ₗ lmapDomain A k (Prod.map f f) | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap₂ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map sending `∑ aᵢ·(gᵢ₁, gᵢ₂) : G × G →₀ A` to
`∑ φ(aᵢ)·(f(gᵢ₁), f(gᵢ₂)) : H × H →₀ B`. |
noncomputable chainsMap₃ :
ModuleCat.of k (G × G × G →₀ A) ⟶ ModuleCat.of k (H × H × H →₀ B) :=
ModuleCat.ofHom <| mapRange.linearMap φ.hom.hom ∘ₗ lmapDomain A k (Prod.map f (Prod.map f f))
@[reassoc (attr := simp), elementwise (attr := simp)] | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap₃ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map sending `∑ aᵢ·(gᵢ₁, gᵢ₂, gᵢ₃) : G × G × G →₀ A` to
`∑ φ(aᵢ)·(f(gᵢ₁), f(gᵢ₂), f(gᵢ₃)) : H × H × H →₀ B`. |
chainsMap_f_0_comp_chainsIso₀ :
(chainsMap f φ).f 0 ≫ (chainsIso₀ B).hom = (chainsIso₀ A).hom ≫ φ.hom := by
ext
simp [chainsMap_f, Unique.eq_default (α := Fin 0 → G), Unique.eq_default (α := Fin 0 → H),
chainsIso₀]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_f_0_comp_chainsIso₀ | null |
chainsMap_f_1_comp_chainsIso₁ :
(chainsMap f φ).f 1 ≫ (chainsIso₁ B).hom = (chainsIso₁ A).hom ≫ chainsMap₁ f φ := by
ext x
simp [chainsMap_f, chainsIso₁]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_f_1_comp_chainsIso₁ | null |
chainsMap_f_2_comp_chainsIso₂ :
(chainsMap f φ).f 2 ≫ (chainsIso₂ B).hom = (chainsIso₂ A).hom ≫ chainsMap₂ f φ := by
ext
simp [chainsMap_f, chainsIso₂]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_f_2_comp_chainsIso₂ | null |
chainsMap_f_3_comp_chainsIso₃ :
(chainsMap f φ).f 3 ≫ (chainsIso₃ B).hom = (chainsIso₃ A).hom ≫ chainsMap₃ f φ := by
ext
simp [chainsMap_f, chainsIso₃, ← Fin.comp_tail]
open ShortComplex | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsMap_f_3_comp_chainsIso₃ | null |
@[reassoc (attr := simp), elementwise (attr := simp)]
cyclesMap_comp_cyclesIso₀_hom :
cyclesMap f φ 0 ≫ (cyclesIso₀ B).hom = (cyclesIso₀ A).hom ≫ φ.hom := by
simp [cyclesIso₀]
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesMap_comp_cyclesIso₀_hom | null |
cyclesIso₀_inv_comp_cyclesMap :
(cyclesIso₀ A).inv ≫ cyclesMap f φ 0 = φ.hom ≫ (cyclesIso₀ B).inv :=
(CommSq.vert_inv ⟨cyclesMap_comp_cyclesIso₀_hom f φ⟩).w.symm
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesIso₀_inv_comp_cyclesMap | null |
H0π_comp_map :
H0π A ≫ map f φ 0 = φ.hom ≫ H0π B := by
simp [H0π]
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | H0π_comp_map | null |
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 ≫ (coinvariantsFunctor k G).map f := by
rw [← cancel_epi (H0π A)]
ext
simp | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | map_id_comp_H0Iso_hom | null |
epi_map_0_of_epi {A B : Rep k G} (f : A ⟶ B) [Epi f] :
Epi (map (MonoidHom.id G) f 0) where
left_cancellation g h hgh := by
simp only [← cancel_epi (H0π A)] at hgh
simp_all [cancel_epi] | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | epi_map_0_of_epi | null |
@[simps]
noncomputable mapShortComplexH1 :
shortComplexH1 A ⟶ shortComplexH1 B where
τ₁ := chainsMap₂ f φ
τ₂ := chainsMap₁ f φ
τ₃ := φ.hom
comm₁₂ := by
simp only [shortComplexH1]
ext : 3
simpa [d₂₁, map_add, map_sub, ← map_inv] using congr(single _ $((hom_comm_apply φ _ _).symm))
comm₂₃ := by
simp only [shortComplexH1]
ext : 3
simpa [← map_inv, d₁₀] using (hom_comm_apply φ _ _).symm
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH1 | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map from the short complex `(G × G →₀ A) --d₂₁--> (G →₀ A) --d₁₀--> A`
to `(H × H →₀ B) --d₂₁--> (H →₀ B) --d₁₀--> B`. |
mapShortComplexH1_zero :
mapShortComplexH1 (A := A) (B := B) f 0 = 0 := by
refine ShortComplex.hom_ext _ _ ?_ ?_ rfl
all_goals
{ simp only [shortComplexH1]
ext
simp }
@[simp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH1_zero | null |
mapShortComplexH1_id : mapShortComplexH1 (MonoidHom.id G) (𝟙 A) = 𝟙 _ := by
simp only [shortComplexH1]
ext <;> simp | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH1_id | null |
mapShortComplexH1_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) :
mapShortComplexH1 (g.comp f) (φ ≫ (Action.res _ f).map ψ) =
(mapShortComplexH1 f φ) ≫ (mapShortComplexH1 g ψ) := by
refine ShortComplex.hom_ext _ _ ?_ ?_ rfl
all_goals
{ simp only [shortComplexH1]
ext
simp [Prod.map] } | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/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 :=
mapShortComplexH1_comp (MonoidHom.id G) (MonoidHom.id G) _ _ | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH1_id_comp | null |
noncomputable mapCycles₁ :
ModuleCat.of k (cycles₁ A) ⟶ ModuleCat.of k (cycles₁ B) :=
ShortComplex.cyclesMap' (mapShortComplexH1 f φ) (shortComplexH1 A).moduleCatLeftHomologyData
(shortComplexH1 B).moduleCatLeftHomologyData | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapCycles₁ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map `Z₁(G, A) ⟶ Z₁(H, B)`. |
mapCycles₁_hom :
(mapCycles₁ f φ).hom = (chainsMap₁ f φ).hom.restrict (fun x _ => by
have := congr($((mapShortComplexH1 f φ).comm₂₃) x); simp_all [cycles₁, shortComplexH1]) :=
rfl
@[reassoc, elementwise] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapCycles₁_hom | null |
mapCycles₁_comp_i :
mapCycles₁ f φ ≫ (shortComplexH1 B).moduleCatLeftHomologyData.i =
(shortComplexH1 A).moduleCatLeftHomologyData.i ≫ chainsMap₁ f φ := by
simp
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapCycles₁_comp_i | null |
coe_mapCycles₁ (x) :
(mapCycles₁ f φ x).1 = chainsMap₁ f φ x := rfl
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | coe_mapCycles₁ | null |
cyclesMap_comp_isoCycles₁_hom :
cyclesMap f φ 1 ≫ (isoCycles₁ B).hom = (isoCycles₁ A).hom ≫ mapCycles₁ f φ := by
simp [← cancel_mono (moduleCatLeftHomologyData (shortComplexH1 B)).i, mapShortComplexH1,
chainsMap_f_1_comp_chainsIso₁ f]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesMap_comp_isoCycles₁_hom | null |
H1π_comp_map :
H1π A ≫ map f φ 1 = mapCycles₁ f φ ≫ H1π B := by
simp [H1π, Iso.inv_comp_eq, ← cyclesMap_comp_isoCycles₁_hom_assoc]
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | H1π_comp_map | null |
map₁_one (φ : A ⟶ (Action.res _ (1 : G →* H)).obj B) :
map (1 : G →* H) φ 1 = 0 := by
simp only [← cancel_epi (H1π A), H1π_comp_map, Limits.comp_zero]
ext x
rw [ModuleCat.hom_comp]
refine (H1π_eq_zero_iff _).2 ?_
simpa [coe_mapCycles₁ _ φ x, mapDomain, map_finsuppSum] using
(boundaries₁ B).finsuppSum_mem k x.1 _ fun _ _ => single_one_mem_boundaries₁ (A := B) _ | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | map₁_one | null |
mapCycles₁_quotientGroupMk'_epi :
Epi (mapCycles₁ (QuotientGroup.mk' S) (resOfQuotientIso A S).inv) := by
rw [ModuleCat.epi_iff_surjective]
rintro ⟨x, hx⟩
choose! s hs using QuotientGroup.mk_surjective (s := S)
have hs₁ : QuotientGroup.mk ∘ s = id := funext hs
refine ⟨⟨mapDomain s x, ?_⟩, Subtype.ext <| by
simp [mapCycles₁_hom, ← mapDomain_comp, hs₁]⟩
simpa [mem_cycles₁_iff, ← (mem_cycles₁_iff _).1 hx, sum_mapDomain_index_inj (f := s)
(fun x y h => by rw [← hs x, ← hs y, h])]
using Finsupp.sum_congr fun a b => QuotientGroup.induction_on a fun a => by
simp [← QuotientGroup.mk_inv, apply_eq_of_coe_eq A.ρ S (s a)⁻¹ a⁻¹ (by simp [hs])] | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapCycles₁_quotientGroupMk'_epi | null |
@[simps X₁ X₂ X₃ f g]
noncomputable H1CoresCoinfOfTrivial :
ShortComplex (ModuleCat k) where
X₁ := H1 ((Action.res _ S.subtype).obj A)
X₂ := H1 A
X₃ := H1 (ofQuotient A S)
f := map S.subtype (𝟙 _) 1
g := map (QuotientGroup.mk' S) (resOfQuotientIso A S).inv 1
zero := by rw [← map_comp, congr (QuotientGroup.mk'_comp_subtype S) (map (n := 1)), map₁_one] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | H1CoresCoinfOfTrivial | Given a `G`-representation `A` on which a normal subgroup `S ≤ G` acts trivially, this is the
short complex `H₁(S, A) ⟶ H₁(G, A) ⟶ H₁(G ⧸ S, A)`. |
map_1_quotientGroupMk'_epi :
Epi (map (QuotientGroup.mk' S) (resOfQuotientIso A S).inv 1) := by
convert epi_of_epi (H1π A) _
rw [H1π_comp_map]
exact @epi_comp _ _ _ _ _ _ (mapCycles₁_quotientGroupMk'_epi A S) (H1π _) inferInstance | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | map_1_quotientGroupMk'_epi | null |
H1CoresCoinfOfTrivial_g_epi :
Epi (H1CoresCoinfOfTrivial A S).g :=
inferInstanceAs <| Epi (map _ _ 1) | instance | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | H1CoresCoinfOfTrivial_g_epi | Given a `G`-representation `A` on which a normal subgroup `S ≤ G` acts trivially, the
induced map `H₁(G, A) ⟶ H₁(G ⧸ S, A)` is an epimorphism. |
H1CoresCoinfOfTrivial_exact :
(H1CoresCoinfOfTrivial A S).Exact := by
classical
rw [ShortComplex.moduleCat_exact_iff_ker_sub_range]
intro x hx
/- Denote `C(i) : C(S, A) ⟶ C(G, A), C(π) : C(G, A) ⟶ C(G ⧸ S, A)` and let `x : Z₁(G, A)` map to
0 in `H₁(G ⧸ S, A)`. -/
induction x using H1_induction_on with | @h x =>
rcases x with ⟨x, hxc⟩
simp_all only [H1CoresCoinfOfTrivial_X₂, H1CoresCoinfOfTrivial_X₃, H1CoresCoinfOfTrivial_g,
LinearMap.mem_ker, H1π_comp_map_apply (QuotientGroup.mk' S)]
/- Choose `y := ∑ y(σ, τ)·(σ, τ) ∈ C₂(G ⧸ S, A)` such that `C₁(π)(x) = d(y)`. -/
rcases (H1π_eq_zero_iff _).1 hx with ⟨y, hy⟩
/- Let `s : G ⧸ S → G` be a section of the quotient map. -/
choose! s hs using QuotientGroup.mk'_surjective S
have hs₁ : QuotientGroup.mk (s := S) ∘ s = id := funext hs
/- Let `z := ∑ y(σ, τ)·(s(σ), s(τ))`. -/
let z : G × G →₀ A := lmapDomain _ k (Prod.map s s) y
/- We have that `C₂(π)(z) = y`. -/
have hz : lmapDomain _ k (QuotientGroup.mk' S) (d₂₁ A z) = d₂₁ (A.ofQuotient S) y := by
have := congr($((mapShortComplexH1 (QuotientGroup.mk' S)
(resOfQuotientIso A S).inv).comm₁₂.symm) z)
simp_all [shortComplexH1, z, ← mapDomain_comp, Prod.map_comp_map]
let v := x - d₂₁ _ z
/- We have `C₁(s ∘ π)(v) = ∑ v(g)·s(π(g)) = 0`, since `C₁(π)(v) = dC₁(π)(z) - C₁(π)(dz) = 0` by
previous assumptions. -/
have hv : mapDomain (s ∘ QuotientGroup.mk) v = 0 := by
rw [mapDomain_comp]
simp_all [v, mapDomain, sum_sub_index, coe_mapCycles₁ _ _ ⟨x, hxc⟩]
let e : G → G × G := fun (g : G) => (s (g : G ⧸ S), (s (g : G ⧸ S))⁻¹ * g)
have he : e.Injective := fun x y hxy => by
obtain ⟨(h₁ : s _ = s _), (h₂ : _ * _ = _ * _)⟩ := Prod.ext_iff.1 hxy
exact (mul_right_inj _).1 (h₁ ▸ h₂)
/- Let `ve := ∑ v(g)·(s(π(g)), s(π(g))⁻¹g)`. -/
let ve : G × G →₀ A := mapDomain e v
have hS : (v + d₂₁ _ ve).support.toSet ⊆ S := by
/- We have `d(ve) = ∑ ρ(s(π(g))⁻¹)(v(g))·s(π(g))⁻¹g - ∑ v(g)·g + ∑ v(g)·s(π(g))`.
The second sum is `v`, so cancels: -/
simp only [d₂₁, ve, ModuleCat.hom_ofHom, coe_lsum, sum_mapDomain_index_inj he, sum_single,
LinearMap.add_apply, LinearMap.sub_apply, LinearMap.coe_comp, Function.comp_apply,
lsingle_apply, sum_add, sum_sub, mul_inv_cancel_left, ← add_assoc, add_sub_cancel, e]
intro w hw
· obtain (hl | hr) := Finset.mem_union.1 (support_add hw)
/- The first sum clearly has support in `S`: -/
· obtain ⟨t, _, ht⟩ := Finset.mem_biUnion.1 (support_sum hl)
apply support_single_subset at ht
simp_all [← QuotientGroup.eq]
/- The third sum is 0, by `hv`. -/
· simp_all [mapDomain]
/- Now `v + d(ve)` has support in `S` and agrees with `x` in `H₁(G, A)`: -/
use H1π _ ⟨comapDomain Subtype.val (v + d₂₁ _ ve) <|
... | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | H1CoresCoinfOfTrivial_exact | Given a `G`-representation `A` on which a normal subgroup `S ≤ G` acts trivially, the short
complex `H₁(S, A) ⟶ H₁(G, A) ⟶ H₁(G ⧸ S, A)` is exact. |
@[simps]
noncomputable mapShortComplexH2 :
shortComplexH2 A ⟶ shortComplexH2 B where
τ₁ := chainsMap₃ f φ
τ₂ := chainsMap₂ f φ
τ₃ := chainsMap₁ f φ
comm₁₂ := by
simp only [shortComplexH2]
ext : 3
simpa [d₃₂, map_add, map_sub, ← map_inv]
using congr(Finsupp.single _ $((hom_comm_apply φ _ _).symm))
comm₂₃ := by
simp only [shortComplexH2]
ext : 3
simpa [d₂₁, map_add, map_sub, ← map_inv]
using congr(Finsupp.single _ $((hom_comm_apply φ _ _).symm))
@[simp] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH2 | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map from the short complex
`(G × G × G →₀ A) --d₃₂--> (G × G →₀ A) --d₂₁--> (G →₀ A)` to
`(H × H × H →₀ B) --d₃₂--> (H × H →₀ B) --d₂₁--> (H →₀ B)`. |
mapShortComplexH2_zero :
mapShortComplexH2 (A := A) (B := B) f 0 = 0 := by
refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
all_goals
{ simp only [shortComplexH2]
ext
simp }
@[simp] | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH2_zero | null |
mapShortComplexH2_id : mapShortComplexH2 (MonoidHom.id _) (𝟙 A) = 𝟙 _ := by
refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
all_goals
{ simp only [shortComplexH2]
ext
simp } | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH2_id | null |
mapShortComplexH2_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) :
mapShortComplexH2 (g.comp f) (φ ≫ (Action.res _ f).map ψ) =
(mapShortComplexH2 f φ) ≫ (mapShortComplexH2 g ψ) := by
refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
all_goals
{ simp only [shortComplexH2]
ext
simp [Prod.map] } | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/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 :=
mapShortComplexH2_comp (MonoidHom.id G) (MonoidHom.id G) _ _ | theorem | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapShortComplexH2_id_comp | null |
noncomputable mapCycles₂ :
ModuleCat.of k (cycles₂ A) ⟶ ModuleCat.of k (cycles₂ B) :=
ShortComplex.cyclesMap' (mapShortComplexH2 f φ) (shortComplexH2 A).moduleCatLeftHomologyData
(shortComplexH2 B).moduleCatLeftHomologyData | abbrev | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapCycles₂ | Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
this is the induced map `Z₂(G, A) ⟶ Z₂(H, B)`. |
mapCycles₂_hom :
(mapCycles₂ f φ).hom = (chainsMap₂ f φ).hom.restrict (fun x _ => by
have := congr($((mapShortComplexH2 f φ).comm₂₃) x); simp_all [cycles₂, shortComplexH2]) :=
rfl
@[reassoc, elementwise] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapCycles₂_hom | null |
mapCycles₂_comp_i :
mapCycles₂ f φ ≫ (shortComplexH2 B).moduleCatLeftHomologyData.i =
(shortComplexH2 A).moduleCatLeftHomologyData.i ≫ chainsMap₂ f φ := by
simp
@[simp] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | mapCycles₂_comp_i | null |
coe_mapCycles₂ (x) :
(mapCycles₂ f φ x).1 = chainsMap₂ f φ x := rfl
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | coe_mapCycles₂ | null |
cyclesMap_comp_isoCycles₂_hom :
cyclesMap f φ 2 ≫ (isoCycles₂ B).hom = (isoCycles₂ A).hom ≫ mapCycles₂ f φ := by
simp [← cancel_mono (moduleCatLeftHomologyData (shortComplexH2 B)).i, mapShortComplexH2,
chainsMap_f_2_comp_chainsIso₂ f]
@[reassoc (attr := simp), elementwise (attr := simp)] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | cyclesMap_comp_isoCycles₂_hom | null |
H2π_comp_map :
H2π A ≫ map f φ 2 = mapCycles₂ f φ ≫ H2π B := by
simp [H2π, Iso.inv_comp_eq, ← cyclesMap_comp_isoCycles₂_hom_assoc] | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | H2π_comp_map | null |
@[simps]
noncomputable chainsFunctor :
Rep k G ⥤ ChainComplex (ModuleCat k) ℕ where
obj A := inhomogeneousChains A
map f := chainsMap (MonoidHom.id _) f
map_id _ := chainsMap_id
map_comp φ ψ := chainsMap_comp (MonoidHom.id G) (MonoidHom.id G) φ ψ | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | chainsFunctor | The functor sending a representation to its complex of inhomogeneous chains. |
@[simps]
noncomputable functor (n : ℕ) : Rep k G ⥤ ModuleCat k where
obj A := groupHomology A n
map {A B} φ := map (MonoidHom.id _) φ n
map_id A := by simp [map, groupHomology]
map_comp f g := by
simp only [← HomologicalComplex.homologyMap_comp, ← chainsMap_comp]
rfl | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | functor | The functor sending a `G`-representation `A` to `Hₙ(G, A)`. |
@[simps]
noncomputable coresNatTrans (n : ℕ) :
Action.res (ModuleCat k) f ⋙ functor k G n ⟶ functor k H n where
app X := map f (𝟙 _) n
naturality {X Y} φ := by simp [← cancel_epi (groupHomology.π _ n),
← HomologicalComplex.cyclesMap_comp_assoc, ← chainsMap_comp, congr (MonoidHom.id_comp _)
chainsMap, congr (MonoidHom.comp_id _) chainsMap, Category.id_comp
(X := (Action.res _ _).obj _)] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | coresNatTrans | Given a group homomorphism `f : G →* H`, this is a natural transformation between the functors
sending `A : Rep k H` to `Hₙ(G, Res(f)(A))` and to `Hₙ(H, A)`. |
@[simps]
noncomputable coinfNatTrans (S : Subgroup G) [S.Normal] (n : ℕ) :
functor k G n ⟶ quotientToCoinvariantsFunctor k S ⋙ functor k (G ⧸ S) n where
app A := map (QuotientGroup.mk' S) (mkQ _ _ <| Coinvariants.le_comap_ker A.ρ S) n
naturality {X Y} φ := by
simp only [Functor.comp_map, functor_map, ← cancel_epi (groupHomology.π _ n),
HomologicalComplex.homologyπ_naturality_assoc, HomologicalComplex.homologyπ_naturality,
← HomologicalComplex.cyclesMap_comp_assoc, ← chainsMap_comp]
congr 1 | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean | coinfNatTrans | Given a normal subgroup `S ≤ G`, this is a natural transformation between the functors
sending `A : Rep k G` to `Hₙ(G, A)` and to `Hₙ(G ⧸ S, A_S)`. |
map_chainsFunctor_shortExact :
ShortExact (X.map (chainsFunctor k G)) :=
letI := hX.mono_f
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, ker_mapRange, range_mapRange_linearMap X.f.hom.hom
(LinearMap.ker_eq_bot.2 <| (ModuleCat.mono_iff_injective _).1 _), this]
mono_f := chainsMap_id_f_map_mono X.f i
epi_g := letI := hX.epi_g; chainsMap_id_f_map_epi X.g i }
open HomologicalComplex.HomologySequence | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | map_chainsFunctor_shortExact | null |
noncomputable mapShortComplex₁ {i j : ℕ} (hij : j + 1 = i) :=
(snakeInput (map_chainsFunctor_shortExact hX) _ _ hij).L₂'
variable (X) in | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/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`. |
noncomputable mapShortComplex₂ (i : ℕ) := X.map (functor k G i) | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | mapShortComplex₂ | The short complex `Hᵢ(G, X₁) ⟶ Hᵢ(G, X₂) ⟶ Hᵢ(G, X₃)` associated to a short complex of
representations `X₁ ⟶ X₂ ⟶ X₃`. |
noncomputable mapShortComplex₃ {i j : ℕ} (hij : j + 1 = i) :=
(snakeInput (map_chainsFunctor_shortExact hX) _ _ hij).L₁' | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | mapShortComplex₃ | The short complex `Hᵢ(G, X₂) ⟶ Hᵢ(G, X₃) ⟶ Hⱼ(G, X₁)` associated to an exact sequence of
representations `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0`. |
mapShortComplex₁_exact {i j : ℕ} (hij : j + 1 = i) :
(mapShortComplex₁ hX hij).Exact :=
(map_chainsFunctor_shortExact hX).homology_exact₁ i j hij | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | mapShortComplex₁_exact | Exactness of `Hᵢ(G, X₃) ⟶ Hⱼ(G, X₁) ⟶ Hⱼ(G, X₂)`. |
mapShortComplex₂_exact (i : ℕ) :
(mapShortComplex₂ X i).Exact :=
(map_chainsFunctor_shortExact hX).homology_exact₂ i | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | mapShortComplex₂_exact | Exactness of `Hᵢ(G, X₁) ⟶ Hᵢ(G, X₂) ⟶ Hᵢ(G, X₃)`. |
mapShortComplex₃_exact {i j : ℕ} (hij : j + 1 = i) :
(mapShortComplex₃ hX hij).Exact :=
(map_chainsFunctor_shortExact hX).homology_exact₃ i j hij | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | mapShortComplex₃_exact | Exactness of `Hᵢ(G, X₂) ⟶ Hᵢ(G, X₃) ⟶ Hⱼ(G, X₁)`. |
noncomputable δ (i j : ℕ) (hij : j + 1 = i) :
groupHomology X.X₃ i ⟶ groupHomology X.X₁ j :=
(map_chainsFunctor_shortExact hX).δ i j hij
open Limits | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | δ | The connecting homomorphism `Hᵢ(G, X₃) ⟶ Hⱼ(G, X₁)` associated to an exact sequence
`0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` of representations. |
epi_δ_of_isZero (n : ℕ) (h : IsZero (groupHomology X.X₂ n)) :
Epi (δ hX (n + 1) n rfl) := SnakeInput.epi_δ _ h | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | epi_δ_of_isZero | null |
mono_δ_of_isZero (n : ℕ) (h : IsZero (groupHomology X.X₂ (n + 1))) :
Mono (δ hX (n + 1) n rfl) := SnakeInput.mono_δ _ h | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | mono_δ_of_isZero | null |
isIso_δ_of_isZero (n : ℕ) (hs : IsZero (groupHomology X.X₂ (n + 1)))
(h : IsZero (groupHomology X.X₂ n)) :
IsIso (δ hX (n + 1) n rfl) := SnakeInput.isIso_δ _ hs h | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | isIso_δ_of_isZero | null |
noncomputable cyclesMkOfCompEqD {i j : ℕ} {y : (Fin i → G) →₀ X.X₂}
{x : (Fin j → G) →₀ X.X₁}
(hx : mapRange.linearMap X.f.hom.hom x = (inhomogeneousChains X.X₂).d i j y) :
cycles X.X₁ j :=
cyclesMk j _ rfl x <| by
simpa using (map_chainsFunctor_shortExact hX).d_eq_zero_of_f_eq_d_apply i j y x
(by simpa using hx) _ | abbrev | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | cyclesMkOfCompEqD | Given an exact sequence of `G`-representations `0 ⟶ X₁ ⟶f X₂ ⟶g X₃ ⟶ 0`, this expresses an
`n`-chain `x : Gⁿ →₀ X₁` such that `f ∘ x ∈ Bₙ(G, X₂)` as a cycle. Stated for readability of
`δ_apply`. |
δ_apply {i j : ℕ} (hij : j + 1 = i)
(z : (Fin i → G) →₀ X.X₃) (hz : (inhomogeneousChains X.X₃).d i j z = 0)
(y : (Fin i → G) →₀ X.X₂) (hy : (chainsMap (MonoidHom.id G) X.g).f i y = z)
(x : (Fin j → G) →₀ X.X₁)
(hx : mapRange.linearMap X.f.hom.hom x = (inhomogeneousChains X.X₂).d i j y) :
δ hX i j hij (π X.X₃ i <| cyclesMk i j (by simp [← hij]) z (by simpa using hz)) =
π X.X₁ j (cyclesMkOfCompEqD hX hx) := by
exact (map_chainsFunctor_shortExact hX).δ_apply i j hij z hz y hy x (by simpa using hx) _ rfl | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | δ_apply | null |
δ₀_apply
(z : cycles₁ X.X₃) (y : G →₀ X.X₂) (hy : mapRange.linearMap X.g.hom.hom y = z.1)
(x : X.X₁) (hx : X.f.hom x = d₁₀ X.X₂ y) :
δ hX 1 0 rfl (H1π X.X₃ z) = H0π X.X₁ x := by
simpa only [H1π, ModuleCat.hom_comp, LinearMap.coe_comp, Function.comp_apply, H0π,
← cyclesMk₀_eq X.X₁, ← cyclesMk₁_eq X.X₃]
using δ_apply hX (i := 1) (j := 0) rfl ((chainsIso₁ X.X₃).inv z.1) (by simp)
((chainsIso₁ X.X₂).inv y) (Finsupp.ext fun _ => by simp [chainsIso₁, ← hy])
((chainsIso₀ X.X₁).inv x) (Finsupp.ext fun _ => by simp [chainsIso₀, ← hx]) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | δ₀_apply | null |
mem_cycles₁_of_comp_eq_d₂₁
{y : G × G →₀ X.X₂} {x : G →₀ X.X₁} (hx : mapRange.linearMap X.f.hom.hom x = d₂₁ X.X₂ y) :
x ∈ cycles₁ X.X₁ := LinearMap.mem_ker.2 <| (Rep.mono_iff_injective X.f).1 hX.2 <| by
have := congr($((mapShortComplexH1 (MonoidHom.id G) X.f).comm₂₃.symm) x)
simp_all [shortComplexH1] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | mem_cycles₁_of_comp_eq_d₂₁ | Stated for readability of `δ₁_apply`. |
δ₁_apply
(z : cycles₂ X.X₃) (y : G × G →₀ X.X₂) (hy : mapRange.linearMap X.g.hom.hom y = z.1)
(x : G →₀ X.X₁) (hx : mapRange.linearMap X.f.hom.hom x = d₂₁ X.X₂ y) :
δ hX 2 1 rfl (H2π X.X₃ z) = H1π X.X₁ ⟨x, mem_cycles₁_of_comp_eq_d₂₁ hX hx⟩ := by
simpa only [H2π, ModuleCat.hom_comp, LinearMap.coe_comp, Function.comp_apply, H1π,
← cyclesMk₂_eq X.X₃, ← cyclesMk₁_eq X.X₁]
using δ_apply hX (i := 2) (j := 1) rfl ((chainsIso₂ X.X₃).inv z.1) (by simp)
((chainsIso₂ X.X₂).inv y) (Finsupp.ext fun _ => by simp [chainsIso₂, ← hy])
((chainsIso₁ X.X₁).inv x) (Finsupp.ext fun _ => by simp [chainsIso₁, ← hx]) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ConcreteCategory",
"Mathlib.Algebra.Homology.HomologicalComplexAbelian",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.lean | δ₁_apply | null |
chainsIso₀ : (inhomogeneousChains A).X 0 ≅ A.V :=
(LinearEquiv.finsuppUnique _ _ _).toModuleIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | chainsIso₀ | The 0th object in the complex of inhomogeneous chains of `A : Rep k G` is isomorphic
to `A` as a `k`-module. |
chainsIso₁ : (inhomogeneousChains A).X 1 ≅ ModuleCat.of k (G →₀ A) :=
(Finsupp.domLCongr (Equiv.funUnique (Fin 1) G)).toModuleIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | chainsIso₁ | The 1st object in the complex of inhomogeneous chains of `A : Rep k G` is isomorphic
to `G →₀ A` as a `k`-module. |
chainsIso₂ : (inhomogeneousChains A).X 2 ≅ ModuleCat.of k (G × G →₀ A) :=
(Finsupp.domLCongr (piFinTwoEquiv fun _ => G)).toModuleIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | chainsIso₂ | The 2nd object in the complex of inhomogeneous chains of `A : Rep k G` is isomorphic
to `G² →₀ A` as a `k`-module. |
chainsIso₃ : (inhomogeneousChains A).X 3 ≅ ModuleCat.of k (G × G × G →₀ A) :=
(Finsupp.domLCongr ((Fin.consEquiv _).symm.trans
((Equiv.refl G).prodCongr (piFinTwoEquiv fun _ => G)))).toModuleIso | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | chainsIso₃ | The 3rd object in the complex of inhomogeneous chains of `A : Rep k G` is isomorphic
to `G³ → A` as a `k`-module. |
d₁₀ : ModuleCat.of k (G →₀ A) ⟶ A.V :=
ModuleCat.ofHom <| lsum k fun g => A.ρ g⁻¹ - LinearMap.id
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₁₀ | The 0th differential in the complex of inhomogeneous chains of `A : Rep k G`, as a
`k`-linear map `(G →₀ A) → A`. It is defined by `single g a ↦ ρ_A(g⁻¹)(a) - a.` |
d₁₀_single (g : G) (a : A) : d₁₀ A (single g a) = A.ρ g⁻¹ a - a := by
simp [d₁₀] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₁₀_single | null |
d₁₀_single_one (a : A) : d₁₀ A (single 1 a) = 0 := by
simp [d₁₀] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₁₀_single_one | null |
d₁₀_single_inv (g : G) (a : A) :
d₁₀ A (single g⁻¹ a) = - d₁₀ A (single g (A.ρ g a)) := by
simp [d₁₀] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₁₀_single_inv | null |
range_d₁₀_eq_coinvariantsKer :
LinearMap.range (d₁₀ A).hom = Coinvariants.ker A.ρ := by
symm
apply Submodule.span_eq_of_le
· rintro _ ⟨x, rfl⟩
use single x.1⁻¹ x.2
simp [d₁₀]
· rintro x ⟨y, hy⟩
induction y using Finsupp.induction generalizing x with
| zero => simp [← hy]
| single_add _ _ _ _ _ h =>
simpa [← hy, add_sub_add_comm, sum_add_index, d₁₀_single (G := G)]
using Submodule.add_mem _ (Coinvariants.mem_ker_of_eq _ _ _ rfl) (h rfl)
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | range_d₁₀_eq_coinvariantsKer | null |
d₁₀_comp_coinvariantsMk : d₁₀ A ≫ (coinvariantsMk k G).app A = 0 := by
ext
simp [d₁₀] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₁₀_comp_coinvariantsMk | null |
chains₁ToCoinvariantsKer :
ModuleCat.of k (G →₀ A) ⟶ ModuleCat.of k (Coinvariants.ker A.ρ) :=
ModuleCat.ofHom <| (d₁₀ A).hom.codRestrict _ <|
range_d₁₀_eq_coinvariantsKer A ▸ LinearMap.mem_range_self _
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | chains₁ToCoinvariantsKer | The 0th differential in the complex of inhomogeneous chains of a `G`-representation `A` as a
linear map into the `k`-submodule of `A` spanned by elements of the form
`ρ(g)(x) - x, g ∈ G, x ∈ A`. |
d₁₀_eq_zero_of_isTrivial [A.IsTrivial] : d₁₀ A = 0 := by
ext
simp [d₁₀] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₁₀_eq_zero_of_isTrivial | null |
d₂₁ : ModuleCat.of k (G × G →₀ A) ⟶ ModuleCat.of k (G →₀ A) :=
ModuleCat.ofHom <| lsum k fun g => lsingle g.2 ∘ₗ A.ρ g.1⁻¹ - lsingle (g.1 * g.2) + lsingle g.1
variable {A}
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁ | The 1st differential in the complex of inhomogeneous chains of `A : Rep k G`, as a
`k`-linear map `(G² →₀ A) → (G →₀ A)`. It is defined by
`a·(g₁, g₂) ↦ ρ_A(g₁⁻¹)(a)·g₂ - a·g₁g₂ + a·g₁`. |
d₂₁_single (g : G × G) (a : A) :
d₂₁ A (single g a) = single g.2 (A.ρ g.1⁻¹ a) - single (g.1 * g.2) a + single g.1 a := by
simp [d₂₁] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_single | null |
d₂₁_single_one_fst (g : G) (a : A) :
d₂₁ A (single (1, g) a) = single 1 a := by
simp [d₂₁] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_single_one_fst | null |
d₂₁_single_one_snd (g : G) (a : A) :
d₂₁ A (single (g, 1) a) = single 1 (A.ρ g⁻¹ a) := by
simp [d₂₁] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_single_one_snd | null |
d₂₁_single_inv_self_ρ_sub_self_inv (g : G) (a : A) :
d₂₁ A (single (g⁻¹, g) (A.ρ g⁻¹ a) - single (g, g⁻¹) a) =
single 1 a - single 1 (A.ρ g⁻¹ a) := by
simp only [map_sub, d₂₁_single (G := G), inv_inv, self_inv_apply, inv_mul_cancel,
mul_inv_cancel]
abel | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_single_inv_self_ρ_sub_self_inv | null |
d₂₁_single_self_inv_ρ_sub_inv_self (g : G) (a : A) :
d₂₁ A (single (g, g⁻¹) (A.ρ g a) - single (g⁻¹, g) a) =
single 1 a - single 1 (A.ρ g a) := by
simp only [map_sub, d₂₁_single (G := G), inv_self_apply, mul_inv_cancel, inv_inv,
inv_mul_cancel]
abel | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_single_self_inv_ρ_sub_inv_self | null |
d₂₁_single_ρ_add_single_inv_mul (g h : G) (a : A) :
d₂₁ A (single (g, h) (A.ρ g a) + single (g⁻¹, g * h) a) =
single g (A.ρ g a) + single g⁻¹ a := by
simp only [map_add, d₂₁_single (G := G), inv_self_apply, inv_inv, inv_mul_cancel_left]
abel | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_single_ρ_add_single_inv_mul | null |
d₂₁_single_inv_mul_ρ_add_single (g h : G) (a : A) :
d₂₁ A (single (g⁻¹, g * h) (A.ρ g⁻¹ a) + single (g, h) a) =
single g⁻¹ (A.ρ g⁻¹ a) + single g a := by
simp only [map_add, d₂₁_single (G := G), inv_inv, self_inv_apply, inv_mul_cancel_left]
abel
variable (A) in | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_single_inv_mul_ρ_add_single | null |
d₃₂ : ModuleCat.of k (G × G × G →₀ A) ⟶ ModuleCat.of k (G × G →₀ A) :=
ModuleCat.ofHom <| lsum k fun g =>
lsingle (g.2.1, g.2.2) ∘ₗ A.ρ g.1⁻¹ - lsingle (g.1 * g.2.1, g.2.2) +
lsingle (g.1, g.2.1 * g.2.2) - lsingle (g.1, g.2.1)
@[simp] | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₃₂ | The 2nd differential in the complex of inhomogeneous chains of `A : Rep k G`, as a
`k`-linear map `(G³ →₀ A) → (G² →₀ A)`. It is defined by
`a·(g₁, g₂, g₃) ↦ ρ_A(g₁⁻¹)(a)·(g₂, g₃) - a·(g₁g₂, g₃) + a·(g₁, g₂g₃) - a·(g₁, g₂)`. |
d₃₂_single (g : G × G × G) (a : A) :
d₃₂ A (single g a) = single (g.2.1, g.2.2) (A.ρ g.1⁻¹ a) - single (g.1 * g.2.1, g.2.2) a +
single (g.1, g.2.1 * g.2.2) a - single (g.1, g.2.1) a := by
simp [d₃₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₃₂_single | null |
d₃₂_single_one_fst (g h : G) (a : A) :
d₃₂ A (single (1, g, h) a) = single (1, g * h) a - single (1, g) a := by
simp [d₃₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₃₂_single_one_fst | null |
d₃₂_single_one_snd (g h : G) (a : A) :
d₃₂ A (single (g, 1, h) a) = single (1, h) (A.ρ g⁻¹ a) - single (g, 1) a := by
simp [d₃₂] | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₃₂_single_one_snd | null |
d₃₂_single_one_thd (g h : G) (a : A) :
d₃₂ A (single (g, h, 1) a) = single (h, 1) (A.ρ g⁻¹ a) - single (g * h, 1) a := by
simp [d₃₂]
variable (A) | lemma | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₃₂_single_one_thd | null |
comp_d₁₀_eq :
(chainsIso₁ A).hom ≫ d₁₀ A = (inhomogeneousChains A).d 1 0 ≫ (chainsIso₀ A).hom :=
ModuleCat.hom_ext <| lhom_ext fun _ _ => by
simp [inhomogeneousChains.d_def, chainsIso₀, chainsIso₁, d₁₀_single (G := G),
Unique.eq_default (α := Fin 0 → G), sub_eq_add_neg, inhomogeneousChains.d_single (G := G)]
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | comp_d₁₀_eq | Let `C(G, A)` denote the complex of inhomogeneous chains of `A : Rep k G`. This lemma
says `d₁₀` gives a simpler expression for the 0th differential: that is, the following
square commutes:
```
C₁(G, A) --d 1 0--> C₀(G, A)
| |
| |
| |
v v
(G →₀ A) ----d₁₀----> A
```
where the vertical arrows are `chainsIso₁` and `chainsIso₀` respectively. |
eq_d₁₀_comp_inv :
(chainsIso₁ A).inv ≫ (inhomogeneousChains A).d 1 0 = d₁₀ A ≫ (chainsIso₀ A).inv :=
(CommSq.horiz_inv ⟨comp_d₁₀_eq A⟩).w | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | eq_d₁₀_comp_inv | null |
comp_d₂₁_eq :
(chainsIso₂ A).hom ≫ d₂₁ A = (inhomogeneousChains A).d 2 1 ≫ (chainsIso₁ A).hom :=
ModuleCat.hom_ext <| lhom_ext fun _ _ => by
simp [inhomogeneousChains.d_def, chainsIso₁, add_assoc, chainsIso₂, d₂₁_single (G := G),
-Finsupp.domLCongr_apply, domLCongr_single, sub_eq_add_neg, Fin.contractNth,
inhomogeneousChains.d_single (G := G)]
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | comp_d₂₁_eq | Let `C(G, A)` denote the complex of inhomogeneous chains of `A : Rep k G`. This lemma
says `d₂₁` gives a simpler expression for the 1st differential: that is, the following
square commutes:
```
C₂(G, A) --d 2 1--> C₁(G, A)
| |
| |
| |
v v
(G² →₀ A) --d₂₁--> (G →₀ A)
```
where the vertical arrows are `chainsIso₂` and `chainsIso₁` respectively. |
eq_d₂₁_comp_inv :
(chainsIso₂ A).inv ≫ (inhomogeneousChains A).d 2 1 = d₂₁ A ≫ (chainsIso₁ A).inv :=
(CommSq.horiz_inv ⟨comp_d₂₁_eq A⟩).w | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | eq_d₂₁_comp_inv | null |
comp_d₃₂_eq :
(chainsIso₃ A).hom ≫ d₃₂ A = (inhomogeneousChains A).d 3 2 ≫ (chainsIso₂ A).hom :=
ModuleCat.hom_ext <| lhom_ext fun _ _ => by
simp [inhomogeneousChains.d_def, chainsIso₂, pow_succ, chainsIso₃,
-domLCongr_apply, domLCongr_single, d₃₂, Fin.sum_univ_three,
Fin.contractNth, Fin.tail_def, sub_eq_add_neg, add_assoc,
inhomogeneousChains.d_single (G := G), add_rotate' (-(single (_ * _, _) _)),
add_left_comm (single (_, _ * _) _)]
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | comp_d₃₂_eq | Let `C(G, A)` denote the complex of inhomogeneous chains of `A : Rep k G`. This lemma
says `d₃₂` gives a simpler expression for the 2nd differential: that is, the following
square commutes:
```
C₃(G, A) --d 3 2--> C₂(G, A)
| |
| |
| |
v v
(G³ →₀ A) --d₃₂--> (G² →₀ A)
```
where the vertical arrows are `chainsIso₃` and `chainsIso₂` respectively. |
eq_d₃₂_comp_inv :
(chainsIso₃ A).inv ≫ (inhomogeneousChains A).d 3 2 = d₃₂ A ≫ (chainsIso₂ A).inv :=
(CommSq.horiz_inv ⟨comp_d₃₂_eq A⟩).w
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | eq_d₃₂_comp_inv | null |
d₂₁_comp_d₁₀ : d₂₁ A ≫ d₁₀ A = 0 := by
ext x g
simp [d₁₀, d₂₁, sum_add_index', sum_sub_index, sub_sub_sub_comm, add_sub_add_comm]
@[reassoc (attr := simp), elementwise (attr := simp)] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₂₁_comp_d₁₀ | null |
d₃₂_comp_d₂₁ : d₃₂ A ≫ d₂₁ A = 0 := by
simp [← cancel_mono (chainsIso₁ A).inv, ← eq_d₂₁_comp_inv, ← eq_d₃₂_comp_inv_assoc]
open ShortComplex | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | d₃₂_comp_d₂₁ | null |
@[simps! -isSimp f g]
shortComplexH0 : ShortComplex (ModuleCat k) :=
mk _ _ (d₁₀_comp_coinvariantsMk A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | shortComplexH0 | The (exact) short complex `(G →₀ A) ⟶ A ⟶ A.ρ.coinvariants`. |
@[simps! -isSimp f g]
shortComplexH1 : ShortComplex (ModuleCat k) :=
mk _ _ (d₂₁_comp_d₁₀ A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | shortComplexH1 | The short complex `(G² →₀ A) --d₂₁--> (G →₀ A) --d₁₀--> A`. |
@[simps! -isSimp f g]
shortComplexH2 : ShortComplex (ModuleCat k) :=
mk _ _ (d₃₂_comp_d₂₁ A) | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | shortComplexH2 | The short complex `(G³ →₀ A) --d₃₂--> (G² →₀ A) --d₂₁--> (G →₀ A)`. |
cycles₁ : Submodule k (G →₀ A) := LinearMap.ker (d₁₀ A).hom | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cycles₁ | The 1-cycles `Z₁(G, A)` of `A : Rep k G`, defined as the kernel of the map
`(G →₀ A) → A` defined by `single g a ↦ ρ_A(g⁻¹)(a) - a`. |
cycles₂ : Submodule k (G × G →₀ A) := LinearMap.ker (d₂₁ A).hom
variable {A} | def | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | cycles₂ | The 2-cycles `Z₂(G, A)` of `A : Rep k G`, defined as the kernel of the map
`(G² →₀ A) → (G →₀ A)` defined by `a·(g₁, g₂) ↦ ρ_A(g₁⁻¹)(a)·g₂ - a·g₁g₂ + a·g₁`. |
mem_cycles₁_iff (x : G →₀ A) :
x ∈ cycles₁ A ↔ x.sum (fun g a => A.ρ g⁻¹ a) = x.sum (fun _ a => a) := by
change x.sum (fun g a => A.ρ g⁻¹ a - a) = 0 ↔ _
rw [sum_sub, sub_eq_zero] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | mem_cycles₁_iff | null |
single_mem_cycles₁_iff (g : G) (a : A) :
single g a ∈ cycles₁ A ↔ A.ρ g a = a := by
simp [mem_cycles₁_iff, ← (A.ρ.apply_bijective g).1.eq_iff (a := A.ρ g⁻¹ a), eq_comm] | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_mem_cycles₁_iff | null |
single_mem_cycles₁_of_mem_invariants (g : G) (a : A) (ha : a ∈ A.ρ.invariants) :
single g a ∈ cycles₁ A :=
(single_mem_cycles₁_iff g a).2 (ha g) | theorem | RepresentationTheory | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.GroupTheory.Abelianization.Defs",
"Mathlib.RepresentationTheory.Homological.GroupHomology.Basic",
"Mathlib.RepresentationTheory.Invariants"
] | Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean | single_mem_cycles₁_of_mem_invariants | null |
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