| import OAI.Algebra.AuslanderReiten.HomComplex.HomKunnethClass |
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| namespace OAI |
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| 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] |
| 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) |
| variable (i j n : ℕ) (h : i+j=n) |
| variable (f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V) |
| variable (hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) |
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| include hf hg in |
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| theorem extKunnethEquiv_elementary : |
| extKunnethEquiv k A B P Q 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))) = |
| (Pi.single ⟨(i,j),h⟩ |
| (P.extMk (ModuleCat.ofHom f) (i+1) rfl (ModuleCat.hom_ext hf) ⊗ₜ[k] |
| Q.extMk (ModuleCat.ofHom g) (j+1) rfl (ModuleCat.hom_ext hg)) : |
| ∀ p : Diagonal n, Ext X U p.val.1 ⊗[k] Ext Y V p.val.2) := by |
| let e := elementaryHom k A B P.complex Q.complex U V i j n h f g |
| have he := elementary_closed k A B P.complex Q.complex U V i j n h f g hf hg |
| let z := (⟨e,homCochain_mem_cycles k (A ⊗[k] B) (projectiveResolution k A B P Q) |
| (moduleObject k A B U V) n e he⟩ : |
| LinearMap.ker ((homCochain k (A ⊗[k] B) (complex k A B P.complex Q.complex) |
| (moduleObject k A B U V)).d n (n+1)).hom) |
| let xf := (⟨f,homCochain_mem_cycles k A P U i f hf⟩ : |
| LinearMap.ker ((homCochain k A P.complex U).d i (i+1)).hom) |
| let xg := (⟨g,homCochain_mem_cycles k B Q V j g hg⟩ : |
| LinearMap.ker ((homCochain k B Q.complex V).d j (j+1)).hom) |
| have hext := homCochainNativeExtEquiv_literal k (A ⊗[k] B) |
| (projectiveResolution k A B P Q) (moduleObject k A B U V) n e he |
| change extKunnethEquiv k A B P Q 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) e) (n+1) rfl (ModuleCat.hom_ext he)) = _ |
| refine (congrArg (extKunnethEquiv k A B P Q U V n) hext.symm).trans ?_ |
| unfold extKunnethEquiv |
| erw [LinearEquiv.trans_apply, LinearEquiv.trans_apply, LinearEquiv.trans_apply, LinearEquiv.symm_apply_apply] |
| have hk := homKunnethEquiv_elementary k A B P.complex Q.complex U V i j n h f g hf hg |
| xf xg rfl rfl z rfl |
| erw [hk] |
| funext a |
| by_cases ha : a=⟨(i,j),h⟩ |
| · subst a |
| simp only [DFinsupp.linearEquivFunOnFintype_apply, DFinsupp.single_eq_same, LinearEquiv.piCongrRight_apply, Pi.single_eq_same, TensorProduct.congr_tmul] |
| rw [homCochainExtEquiv_literal k A P U i f hf, homCochainExtEquiv_literal k B Q V j g hg] |
| · simp only [DFinsupp.linearEquivFunOnFintype_apply, DFinsupp.single_eq_of_ne ha, LinearEquiv.piCongrRight_apply, Pi.single_eq_of_ne ha, map_zero] |
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| end ArExplicit.ExternalTensor |
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| end |
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| end OAI |
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