| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorHomComplex |
| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionExtZero |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat |
| import Mathlib.Algebra.Category.ModuleCat.Ext.HasExt |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped ModuleCat.Algebra |
| namespace ArExplicit.ExternalTensor |
| universe u |
| variable (k R : Type u) [Field k] [Ring R] [Algebra k R] |
| attribute [local instance 2000] moduleCatBaseModule moduleCatBaseTower |
| variable {X Y : ModuleCat.{u} R} (P : ProjectiveResolution X) |
|
|
| abbrev homShort (n : ℕ) := (homCochain k R P.complex Y).sc' (n-1) n (n+1) |
|
|
| theorem homShort_outgoing (n : ℕ) : |
| (homShort k R (Y := Y) P n).g = |
| ModuleCat.ofHom (LinearMap.lcomp k Y (P.complex.d (n+1) n).hom) := |
| homCochain_d k R P.complex Y n |
|
|
| theorem homShort_incoming_succ (n : ℕ) : |
| (homShort k R (Y := Y) P (n+1)).f = |
| ModuleCat.ofHom (LinearMap.lcomp k Y (P.complex.d (n+1) n).hom) := by |
| exact homCochain_d k R P.complex Y n |
|
|
| theorem homShort_incoming_zero : (homShort k R (Y := Y) P 0).f = 0 := by |
| exact (homCochain k R P.complex Y).shape 0 0 (by simp) |
|
|
| theorem homCycle_closed (n : ℕ) (f : LinearMap.ker (homShort k R (Y := Y) P n).g.hom) : |
| P.complex.d (n+1) n ≫ ModuleCat.ofHom (X := P.complex.X n) (Y := Y) f.val = 0 := by |
| apply ModuleCat.hom_ext |
| let g : P.complex.X n →ₗ[R] Y := f.val |
| have hf : (homShort k R (Y := Y) P n).g.hom g = 0 := f.property |
| rw [homShort_outgoing] at hf |
| exact hf |
|
|
| def homCyclesToExt (n : ℕ) : |
| LinearMap.ker (homShort k R (Y := Y) P n).g.hom →ₗ[k] Ext X Y n where |
| toFun f := P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := Y) f.val) |
| (n+1) rfl (homCycle_closed k R P n f) |
| map_add' f g := (P.add_extMk |
| (ModuleCat.ofHom (X := P.complex.X n) (Y := Y) f.val) |
| (ModuleCat.ofHom (X := P.complex.X n) (Y := Y) g.val) (n+1) rfl |
| (homCycle_closed k R P n f) (homCycle_closed k R P n g)).symm |
| map_smul' a f := ResolutionHomology.extMk_smul P n |
| (ModuleCat.ofHom (X := P.complex.X n) (Y := Y) f.val) (homCycle_closed k R P n f) a |
|
|
| theorem homCyclesToExt_surjective (n : ℕ) : |
| Function.Surjective (homCyclesToExt k R (Y := Y) P n) := by |
| intro a |
| obtain ⟨f, hf, hfa⟩ := P.extMk_surjective a (n+1) rfl |
| refine ⟨⟨f.hom, ?_⟩, hfa⟩ |
| change (homShort k R (Y := Y) P n).g.hom f.hom = 0 |
| rw [homShort_outgoing] |
| exact congrArg ModuleCat.Hom.hom hf |
|
|
| theorem homCyclesToExt_kernel (n : ℕ) : |
| LinearMap.ker (homCyclesToExt k R (Y := Y) P n) = |
| LinearMap.range (homShort k R (Y := Y) P n).moduleCatToCycles := by |
| ext x |
| cases n with |
| | zero => |
| constructor |
| · intro hx |
| have hx0 : ModuleCat.ofHom x.val = 0 := |
| (ResolutionHomology.extMk_zeroDegree_eq_zero_iff P _ (homCycle_closed k R P 0 x)).mp hx |
| refine ⟨0, Subtype.ext ?_⟩ |
| change (0 : P.complex.X 0 →ₗ[R] Y) = x.val |
| exact (congrArg ModuleCat.Hom.hom hx0).symm |
| · rintro ⟨f, rfl⟩ |
| have hf0 : (homShort k R (Y := Y) P 0).moduleCatToCycles f = 0 := by |
| apply Subtype.ext |
| change (homShort k R (Y := Y) P 0).f.hom f = 0 |
| rw [homShort_incoming_zero] |
| rfl |
| rw [hf0] |
| exact map_zero _ |
| | succ n => |
| constructor |
| · intro hx |
| obtain ⟨f, hf⟩ := (P.extMk_eq_zero_iff (ModuleCat.ofHom x.val) |
| (n+2) rfl (homCycle_closed k R P (n+1) x) n rfl).mp hx |
| refine ⟨f.hom, Subtype.ext ?_⟩ |
| change (homShort k R (Y := Y) P (n+1)).f.hom f.hom = x.val |
| rw [homShort_incoming_succ] |
| exact congrArg ModuleCat.Hom.hom hf |
| · rintro ⟨f, rfl⟩ |
| apply (P.extMk_eq_zero_iff _ (n+2) rfl |
| (homCycle_closed k R P (n+1) ((homShort k R (Y := Y) P (n+1)).moduleCatToCycles f)) n rfl).mpr |
| refine ⟨ModuleCat.ofHom f, ?_⟩ |
| apply ModuleCat.hom_ext |
| change f.comp (P.complex.d (n+1) n).hom = |
| (homShort k R (Y := Y) P (n+1)).f.hom f |
| rw [homShort_incoming_succ] |
| rfl |
|
|
| def homShortExtEquiv (n : ℕ) : |
| (homShort k R (Y := Y) P n).homology ≃ₗ[k] Ext X Y n := |
| (homShort k R (Y := Y) P n).moduleCatHomologyIso.toLinearEquiv.trans |
| ((Submodule.quotEquivOfEq _ _ (homCyclesToExt_kernel k R P n).symm).trans |
| ((homCyclesToExt k R P n).quotKerEquivOfSurjective |
| (homCyclesToExt_surjective k R P n))) |
|
|
| def homCochainExtEquiv (n : ℕ) : |
| (homCochain k R P.complex Y).homology n ≃ₗ[k] Ext X Y n := |
| ((homCochain k R P.complex Y).homologyIsoSc' (n-1) n (n+1) |
| (by cases n <;> simp) (by simp)).toLinearEquiv.trans (homShortExtEquiv k R P n) |
| |
| end ArExplicit.ExternalTensor |
| |
| end |
| |
| end OAI |
| |
