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2c7d975 | 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 | import OAI.Algebra.AuslanderReiten.Ext.ExtKunneth
import OAI.Algebra.AuslanderReiten.HomComplex.HomCochainNaturality
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian HomologicalComplex
open scoped ModuleCat.Algebra
namespace ArExplicit.ExternalTensor
universe u
variable (k R : Type u) [Field k] [Ring R] [Algebra k R]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable {X : ModuleCat.{u} R} (P : ProjectiveResolution X)
variable (Y : Type u) [AddCommGroup Y] [Module R Y] [Module k Y] [IsScalarTower k R Y]
theorem homCochain_mem_cycles (n : ℕ) (f : P.complex.X n →ₗ[R] Y)
(hf : f.comp (P.complex.d (n+1) n).hom=0) :
f ∈ LinearMap.ker ((homCochain k R P.complex Y).d n (n+1)).hom := by
change ((homCochain k R P.complex Y).d n (n+1)).hom f=0
rw [homCochain_d]
exact hf
theorem homCochainNativeExtEquiv_literal (n : ℕ)
(f : P.complex.X n →ₗ[R] Y) (hf : f.comp (P.complex.d (n+1) n).hom=0) :
homCochainNativeExtEquiv k R P Y n
(VectorCochain.literalClass k (homCochain k R P.complex Y) n ⟨f,homCochain_mem_cycles k R P Y n f hf⟩) =
P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) f)
(n+1) rfl (ModuleCat.hom_ext hf) := by
let C := canonicalHomCochain k R P.complex (ModuleCat.of R Y)
let z : LinearMap.ker (C.d n (n+1)).hom := ⟨f,by
change (C.d n (n+1)).hom f=0
dsimp only [C, canonicalHomCochain]
rw [homCochain_d]
exact hf⟩
have hz := VectorCochain.literalClass_map k (homCochainScalarIso k R P.complex Y).hom n
(⟨f,homCochain_mem_cycles k R P Y n f hf⟩ : LinearMap.ker ((homCochain k R P.complex Y).d n (n+1)).hom) z rfl
change homCochainExtEquiv k R (Y := ModuleCat.of R Y) P n
((homologyMap (homCochainScalarIso k R P.complex Y).hom n).hom
(VectorCochain.literalClass k (homCochain k R P.complex Y) n ⟨f,homCochain_mem_cycles k R P Y n f hf⟩)) = _
rw [hz]
have he := homCochainExtEquiv_cycleClass k R (Y := ModuleCat.of R Y) P n z
erw [homCochainCycleClass_eq_literal] at he
exact he
theorem homCochainExtEquiv_literal (N : ModuleCat.{u} R) (n : ℕ)
(f : P.complex.X n →ₗ[R] N) (hf : f.comp (P.complex.d (n+1) n).hom=0) :
homCochainExtEquiv k R (Y := N) P n
(VectorCochain.literalClass k (homCochain k R P.complex N) n
⟨f,homCochain_mem_cycles k R P N n f hf⟩) =
P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := N) f)
(n+1) rfl (ModuleCat.hom_ext hf) := by
let z : LinearMap.ker (homShort k R (Y := N) P n).g.hom :=
⟨f,homCochain_mem_cycles k R P N n f hf⟩
have he := homCochainExtEquiv_cycleClass k R (Y := N) P n z
rw [homCochainCycleClass_eq_literal] at he
exact he
end ArExplicit.ExternalTensor
end
end OAI
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