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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Algebra.AuslanderReiten.Ext.ExtKunneth
import OAI.Algebra.AuslanderReiten.HomComplex.HomCochainNaturality
import OAI.Algebra.AuslanderReiten.Homology.VectorCochainKunnethMap
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian HomologicalComplex
open scoped TensorProduct ModuleCat.Algebra
namespace ArExplicit.ExternalTensor
universe u
variable (k : Type u) [Field k]
private theorem tensor_conjugate {E F G H : Type u}
[AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [AddCommGroup H]
[Module k E] [Module k F] [Module k G] [Module k H]
(e : E ≃ₗ[k] F) (d : G ≃ₗ[k] H) (f : E →ₗ[k] E) (g : G →ₗ[k] G)
(x : E ⊗[k] G) :
TensorProduct.map e.toLinearMap d.toLinearMap (TensorProduct.map f g x) =
TensorProduct.map (e.toLinearMap.comp (f.comp e.symm.toLinearMap))
(d.toLinearMap.comp (g.comp d.symm.toLinearMap))
(TensorProduct.map e.toLinearMap d.toLinearMap x) := by
induction x using TensorProduct.inductionOn with
| add x y hx hy => simp only [map_add, hx, hy]
| tmul x y => simp only [TensorProduct.map_tmul, LinearMap.comp_apply,
LinearEquiv.coe_coe, LinearEquiv.symm_apply_apply]
private theorem homology_conjugate {K L : CochainComplex (ModuleCat.{u} k) ℕ}
(e : K ≅ L) (f : K ⟶ K) (n : ℕ) (x : L.homology n) :
((homologyFunctor (ModuleCat.{u} k) (.up ℕ) n).mapIso e).inv.hom
((homologyMap (e.inv ≫ f ≫ e.hom) n).hom x) =
(homologyMap f n).hom
(((homologyFunctor (ModuleCat.{u} k) (.up ℕ) n).mapIso e).inv.hom x) := by
have h : homologyMap (e.inv ≫ f ≫ e.hom) n ≫ homologyMap e.inv n =
homologyMap e.inv n ≫ homologyMap f n := by
rw [← homologyMap_comp, Category.assoc, Category.assoc, e.hom_inv_id,
Category.comp_id, homologyMap_comp]
exact congrArg (fun a => a.hom x) h
variable (R : Type u) [Ring R] [Algebra k R]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable {X : ModuleCat.{u} R} (P : ProjectiveResolution X)
variable (U : Type u) [AddCommGroup U] [Module R U] [Module k U] [IsScalarTower k R U]
def nativeHomEndExt (f : homCochain k R P.complex U ⟶ homCochain k R P.complex U)
(n : ℕ) : Ext X (ModuleCat.of R U) n →ₗ[k] Ext X (ModuleCat.of R U) n :=
(homCochainNativeExtEquiv k R P U n).toLinearMap.comp
((homologyMap f n).hom.comp (homCochainNativeExtEquiv k R P U n).symm.toLinearMap)
variable (A B : Type u) [Ring A] [Ring B] [Algebra k A] [Algebra k B]
variable [Module.Finite k A] [Module.Finite k B]
variable {M : ModuleCat.{u} A} {N : ModuleCat.{u} B}
variable (P : ProjectiveResolution M) (Q : ProjectiveResolution N)
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)
variable (f : homCochain k A P.complex U ⟶ homCochain k A P.complex U)
variable (g : homCochain k B Q.complex V ⟶ homCochain k B Q.complex V)
local notation "T" => MonoidalCategory.curriedTensor (ModuleCat k)
def homTensorMap :
homCochain k (A ⊗[k] B) (complex k A B P.complex Q.complex) (moduleObject k A B U V) ⟶
homCochain k (A ⊗[k] B) (complex k A B P.complex Q.complex) (moduleObject k A B U V) :=
(homCochainTotalIso k A B P.complex Q.complex U V).inv ≫
mapBifunctorMap f g T (.up ℕ) ≫
(homCochainTotalIso k A B P.complex Q.complex U V).hom
def homKunnethCoordinates (n : ℕ) :
(homCochain k (A ⊗[k] B) (complex k A B P.complex Q.complex)
(moduleObject k A B U V)).homology n ≃ₗ[k]
(∀ p : Diagonal n, (homCochain k A P.complex U).homology p.val.1 ⊗[k]
(homCochain k B Q.complex V).homology p.val.2) :=
(homKunnethEquiv k A B P.complex Q.complex U V n).trans
DFinsupp.linearEquivFunOnFintype
omit [Module.Finite k A] [Module.Finite k B] in
theorem homKunnethCoordinates_natural (n : ℕ)
(x : (homCochain k (A ⊗[k] B) (complex k A B P.complex Q.complex)
(moduleObject k A B U V)).homology n) :
homKunnethCoordinates k A B P Q U V n
((homologyMap (homTensorMap k A B P Q U V f g) n).hom x) =
VectorCochain.diagonalTermMap k (VectorCochain.gradedMap k f)
(VectorCochain.gradedMap k g) n (homKunnethCoordinates k A B P Q U V n x) := by
let e := homCochainTotalIso k A B P.complex Q.complex U V
change VectorCochain.homologyCoordinates k n
(((homologyFunctor (ModuleCat.{u} k) (.up ℕ) n).mapIso e).inv.hom
((homologyMap (e.inv ≫ mapBifunctorMap f g T (.up ℕ) ≫ e.hom) n).hom x)) = _
rw [homology_conjugate]
exact VectorCochain.homologyCoordinates_natural k f g n _
def tensorHomEndExt (n : ℕ) :
Ext (moduleObject k A B M N) (moduleObject k A B U V) n →ₗ[k]
Ext (moduleObject k A B M N) (moduleObject k A B U V) n :=
nativeHomEndExt k (A ⊗[k] B) (projectiveResolution k A B P Q)
(moduleObject k A B U V) (homTensorMap k A B P Q U V f g) n
omit [Module.Finite k A] [Module.Finite k B] in
theorem extKunnethEquiv_apply (n : ℕ)
(x : Ext (moduleObject k A B M N) (moduleObject k A B U V) n) (p : Diagonal n) :
extKunnethEquiv k A B P Q U V n x p =
TensorProduct.map (homCochainExtEquiv k A (Y := U) P p.val.1).toLinearMap
(homCochainExtEquiv k B (Y := V) Q p.val.2).toLinearMap
(homKunnethCoordinates k A B P Q U V n
((homCochainNativeExtEquiv k (A ⊗[k] B) (projectiveResolution k A B P Q)
(moduleObject k A B U V) n).symm x) p) := rfl
omit [Module.Finite k A] [Module.Finite k B] in
theorem extKunnethEquiv_tensorHomEndExt (n : ℕ)
(x : Ext (moduleObject k A B M N) (moduleObject k A B U V) n) (p : Diagonal n) :
extKunnethEquiv k A B P Q U V n (tensorHomEndExt k A B P Q U V f g n x) p =
TensorProduct.map (homCochainEndExt k A P f p.val.1)
(homCochainEndExt k B Q g p.val.2)
(extKunnethEquiv k A B P Q U V n x p) := by
rw [extKunnethEquiv_apply, extKunnethEquiv_apply]
let e := homCochainNativeExtEquiv k (A ⊗[k] B) (projectiveResolution k A B P Q)
(moduleObject k A B U V) n
change TensorProduct.map (homCochainExtEquiv k A (Y := U) P p.val.1).toLinearMap
(homCochainExtEquiv k B (Y := V) Q p.val.2).toLinearMap
(homKunnethCoordinates k A B P Q U V n
(e.symm (e ((homologyMap (homTensorMap k A B P Q U V f g) n).hom (e.symm x)))) p) = _
erw [LinearEquiv.symm_apply_apply, homKunnethCoordinates_natural]
exact tensor_conjugate k (homCochainExtEquiv k A (Y := U) P p.val.1)
(homCochainExtEquiv k B (Y := V) Q p.val.2) (homologyMap f p.val.1).hom
(homologyMap g p.val.2).hom _
end ArExplicit.ExternalTensor
end
end OAI
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