| 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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|
