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d7f9a33 | 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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Algebra.AuslanderReiten.Ext.ExtKunnethClass
import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorCubicExt
import OAI.Algebra.AuslanderReiten.Ext.TensorExtCoefficients
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open scoped TensorProduct ModuleCat.Algebra
namespace ArExplicit.ExternalTensor
universe u
variable (k A B : Type u) [Field k] [Ring A] [Ring B] [Algebra k A] [Algebra k B]
variable [CharP k 2] [Module.Finite k A] [Module.Finite k B]
variable [HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)]
variable [HasDerivedCategory (ModuleCat.{u} (A ⊗[k] B))]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable {X : ModuleCat.{u} A} {Y : ModuleCat.{u} B}
variable (P : ProjectiveResolution X) (Q : ProjectiveResolution Y)
variable [∀ n, Module.Finite A (P.complex.X n)] [∀ n, Module.Finite B (Q.complex.X n)]
variable (U : ModuleCat.{u} A) (V : ModuleCat.{u} B)
private theorem extMk_of_eq {R : Type u} [Ring R] {M N : ModuleCat.{u} R}
(L : ProjectiveResolution M) {n : ℕ} (f g : L.complex.X n ⟶ N)
(hf : L.complex.d (n+1) n ≫ f=0) (hg : L.complex.d (n+1) n ≫ g=0) (hfg : f=g) :
L.extMk f (n+1) rfl hf = L.extMk g (n+1) rfl hg := by
subst g
rfl
def elementaryExt (i j n : ℕ) (h : i+j=n)
(f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V)
(hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) :
Ext (moduleObject k A B X Y) (moduleObject k A B U V) n :=
(projectiveResolution k A B P Q).extMk
(ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X n)
(Y := moduleObject k A B U V) (elementaryHom k A B P.complex Q.complex U V i j n h f g))
(n+1) rfl (ModuleCat.hom_ext (elementary_closed k A B P.complex Q.complex U V i j n h f g hf hg))
omit [CharP k 2] [Module.Finite k A] [Module.Finite k B]
[HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)]
[HasDerivedCategory (ModuleCat.{u} (A ⊗[k] B))] in
theorem elementaryExt_kunneth (i j n : ℕ) (h : i+j=n)
(f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V)
(hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) :
extKunnethEquiv k A B P Q U V n (elementaryExt k A B P Q U V i j n h f g hf hg) =
(Pi.single ⟨(i,j),h⟩
(P.extMk (ModuleCat.ofHom (X := P.complex.X i) (Y := U) f) (i+1) rfl (ModuleCat.hom_ext hf) ⊗ₜ[k]
Q.extMk (ModuleCat.ofHom (X := Q.complex.X j) (Y := V) g) (j+1) rfl (ModuleCat.hom_ext hg)) :
extCoefficients k A B X U Y V n) :=
extKunnethEquiv_elementary k A B P Q U V i j n h f g hf hg
variable (p : ∀ n, P.complex.X (n+3) ⟶ P.complex.X n)
variable (hp : ∀ n, p (n+1) ≫ P.complex.d (n+1) n = P.complex.d (n+4) (n+3) ≫ p n)
omit [HasDerivedCategory (ModuleCat.{u} A)] [∀ n, Module.Finite A (P.complex.X n)] in
include hp in
theorem firstFactor_closed (i : ℕ) (f : P.complex.X i →ₗ[A] U)
(hf : f.comp (P.complex.d (i+1) i).hom=0) :
(f.comp (p i).hom).comp (P.complex.d (i+4) (i+3)).hom=0 :=
congrArg ModuleCat.Hom.hom (ResolutionProducts.cubicPrecomp_closed P p hp
(ModuleCat.ofHom (X := P.complex.X i) (Y := U) f) (ModuleCat.hom_ext hf))
omit [Module.Finite k A] [Module.Finite k B]
[HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)] in
theorem firstCubicExt_elementary (i j n : ℕ) (h : i+j=n)
(f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V)
(hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) :
(firstCubicExt k A B P Q p hp).comp
(elementaryExt k A B P Q U V i j n h f g hf hg) (by omega : 3+n=n+3) =
elementaryExt k A B P Q U V (i+3) j (n+3) (by omega) (f.comp (p i).hom) g
(firstFactor_closed A P U p hp i f hf) hg := by
unfold elementaryExt
erw [firstCubicExt_action]
have he : firstComparison k A B P Q p n ≫
ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X n)
(Y := moduleObject k A B U V) (elementaryHom k A B P.complex Q.complex U V i j n h f g) =
ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X (n+3))
(Y := moduleObject k A B U V)
(elementaryHom k A B P.complex Q.complex U V (i+3) j (n+3) (by omega) (f.comp (p i).hom) g) :=
ModuleCat.hom_ext (elementary_firstCubic k A B P.complex Q.complex U V p i j n h f g)
exact extMk_of_eq (projectiveResolution k A B P Q) _ _ _ _ he
variable (r : ∀ n, Q.complex.X (n+3) ⟶ Q.complex.X n)
variable (hr : ∀ n, r (n+1) ≫ Q.complex.d (n+1) n = Q.complex.d (n+4) (n+3) ≫ r n)
omit [HasDerivedCategory (ModuleCat.{u} B)] [∀ n, Module.Finite B (Q.complex.X n)] in
include hr in
theorem secondFactor_closed (j : ℕ) (g : Q.complex.X j →ₗ[B] V)
(hg : g.comp (Q.complex.d (j+1) j).hom=0) :
(g.comp (r j).hom).comp (Q.complex.d (j+4) (j+3)).hom=0 :=
congrArg ModuleCat.Hom.hom (ResolutionProducts.cubicPrecomp_closed Q r hr
(ModuleCat.ofHom (X := Q.complex.X j) (Y := V) g) (ModuleCat.hom_ext hg))
omit [Module.Finite k A] [Module.Finite k B]
[HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)] in
theorem secondCubicExt_elementary (i j n : ℕ) (h : i+j=n)
(f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V)
(hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) :
(secondCubicExt k A B P Q r hr).comp
(elementaryExt k A B P Q U V i j n h f g hf hg) (by omega : 3+n=n+3) =
elementaryExt k A B P Q U V i (j+3) (n+3) (by omega) f (g.comp (r j).hom)
hf (secondFactor_closed B Q V r hr j g hg) := by
unfold elementaryExt
erw [secondCubicExt_action]
have he : secondComparison k A B P Q r n ≫
ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X n)
(Y := moduleObject k A B U V) (elementaryHom k A B P.complex Q.complex U V i j n h f g) =
ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X (n+3))
(Y := moduleObject k A B U V)
(elementaryHom k A B P.complex Q.complex U V i (j+3) (n+3) (by omega) f (g.comp (r j).hom)) :=
ModuleCat.hom_ext (elementary_secondCubic k A B P.complex Q.complex U V r i j n h f g)
exact extMk_of_eq (projectiveResolution k A B P Q) _ _ _ _ he
end ArExplicit.ExternalTensor
end
end OAI
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