File size: 3,405 Bytes
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 | import OAI.Algebra.AuslanderReiten.HomComplex.HomKunnethClass
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]
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)
include hf hg in
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]
end ArExplicit.ExternalTensor
end
end OAI
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