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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Algebra.AuslanderReiten.Ext.ExtKunnethMap
import OAI.Algebra.AuslanderReiten.HomComplex.HomCochainNativeClass
import OAI.Algebra.AuslanderReiten.Resolutions.MapResolutionExtConjugate
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian HomologicalComplex
open scoped ModuleCat.Algebra
namespace ArExplicit.ExternalTensor
universe u
variable (k R : Type u) [Field k] [Ring R] [Algebra k R]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable {X : ModuleCat.{u} R} (P : ProjectiveResolution X)
variable (Y : Type u) [AddCommGroup Y] [Module R Y] [Module k Y] [IsScalarTower k R Y]
variable (H : homCochain k R P.complex Y ⟶ homCochain k R P.complex Y)
theorem nativeHomEndExt_extMk (n : ℕ) (f g : P.complex.X n →ₗ[R] Y)
(hf : f.comp (P.complex.d (n+1) n).hom = 0)
(hg : g.comp (P.complex.d (n+1) n).hom = 0)
(hfg : (H.f n).hom f = g) :
nativeHomEndExt k R P Y H n
(P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) f)
(n+1) rfl (ModuleCat.hom_ext hf)) =
P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) g)
(n+1) rfl (ModuleCat.hom_ext hg) := by
rw [← homCochainNativeExtEquiv_literal k R P Y n f hf,
← homCochainNativeExtEquiv_literal k R P Y n g hg]
change (homCochainNativeExtEquiv k R P Y n)
((homologyMap H n).hom ((homCochainNativeExtEquiv k R P Y n).symm
((homCochainNativeExtEquiv k R P Y n)
(VectorCochain.literalClass k (homCochain k R P.complex Y) n _)))) = _
rw [LinearEquiv.symm_apply_apply]
exact congrArg (homCochainNativeExtEquiv k R P Y n)
(VectorCochain.literalClass_map k H n _ _ hfg)
theorem nativeHomEnd_closed (n : ℕ) (f : P.complex.X n →ₗ[R] Y)
(hf : f.comp (P.complex.d (n+1) n).hom = 0) :
(show P.complex.X n →ₗ[R] Y from (H.f n).hom f).comp
(P.complex.d (n+1) n).hom = 0 := by
have hh := LinearMap.congr_fun (congrArg ModuleCat.Hom.hom (H.comm n (n+1))) f
change ((homCochain k R P.complex Y).d n (n+1)).hom ((H.f n).hom f) =
(H.f (n+1)).hom (((homCochain k R P.complex Y).d n (n+1)).hom f) at hh
rw [homCochain_d] at hh
change (show P.complex.X n →ₗ[R] Y from (H.f n).hom f).comp
(P.complex.d (n+1) n).hom =
(H.f (n+1)).hom (f.comp (P.complex.d (n+1) n).hom) at hh
rw [hf] at hh
exact hh.trans (H.f (n+1)).hom.map_zero
end ArExplicit.ExternalTensor
namespace ArExplicit.ExternalTensor
universe u
variable (k R : Type u) [Field k] [Ring R] [Algebra k R]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable (Y : Type u) [addY : AddCommGroup Y] [modY : Module R Y]
[baseY : Module k Y] [towerY : IsScalarTower k R Y]
variable (P : ProjectiveResolution (ModuleCat.of R Y))
variable (H : homCochain k R P.complex Y ⟶ homCochain k R P.complex Y)
variable (F : ModuleCat.{u} R ⥤ ModuleCat.{u} R) [F.Additive]
[CategoryTheory.Limits.PreservesFiniteLimits F]
[CategoryTheory.Limits.PreservesFiniteColimits F] [F.PreservesProjectiveObjects]
variable (e : ModuleCat.of R Y ≅ F.obj (ModuleCat.of R Y))
variable (φ : ProjectiveResolution.Hom P (ResolutionProducts.mapResolution F P) e.hom)
theorem nativeHomEndExt_eq_exactFunctor
(hφ : ∀ n (a : P.complex.X n ⟶ ModuleCat.of R Y),
ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) ((H.f n).hom a.hom) =
(φ.hom.f n ≫ F.map a) ≫ e.inv)
(n : ℕ) (x : Ext (ModuleCat.of R Y) (ModuleCat.of R Y) n) :
nativeHomEndExt k R P Y H n x =
((Ext.mk₀ e.hom).comp (x.mapExactFunctor F) (zero_add n)).comp
(Ext.mk₀ e.inv) (add_zero n) := by
obtain ⟨a, ha, rfl⟩ := P.extMk_surjective x (n+1) rfl
let b : P.complex.X n →ₗ[R] Y := (H.f n).hom a.hom
have ha' : a.hom.comp (P.complex.d (n+1) n).hom = 0 :=
congrArg ModuleCat.Hom.hom ha
have hb := nativeHomEnd_closed k R P Y H n a.hom ha'
have ht := nativeHomEndExt_extMk k R P Y H n a.hom b ha' hb rfl
refine ht.trans ?_
exact (ResolutionProducts.map_extMk_conjugate F P e.hom e.inv φ a ha
(ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) b)
(ModuleCat.hom_ext hb) (hφ n a)).symm
end ArExplicit.ExternalTensor
end
end OAI
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