| import OAI.Algebra.AuslanderReiten.HomComplex.HomCochainExt |
|
|
| namespace OAI |
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|
| 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) |
|
|
| def homCochainQuotientEquiv (n : ℕ) : |
| (homCochain k R P.complex Y).homology n ≃ₗ[k] |
| (LinearMap.ker (homShort k R (Y := Y) P n).g.hom ⧸ |
| LinearMap.range (homShort k R (Y := Y) P n).moduleCatToCycles) := |
| ((homCochain k R P.complex Y).homologyIsoSc' (n-1) n (n+1) |
| (by cases n <;> simp) (by simp)).toLinearEquiv.trans |
| (homShort k R (Y := Y) P n).moduleCatHomologyIso.toLinearEquiv |
|
|
| def homCochainCycleClass (n : ℕ) |
| (f : LinearMap.ker (homShort k R (Y := Y) P n).g.hom) : |
| (homCochain k R P.complex Y).homology n := |
| (homCochainQuotientEquiv k R P n).symm (Submodule.Quotient.mk f) |
|
|
| theorem homCochainExtEquiv_cycleClass (n : ℕ) |
| (f : LinearMap.ker (homShort k R (Y := Y) P n).g.hom) : |
| homCochainExtEquiv k R P n (homCochainCycleClass k R P n f) = |
| P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := Y) f.val) |
| (n+1) rfl (homCycle_closed k R P n f) := by |
| change ((Submodule.quotEquivOfEq _ _ (homCyclesToExt_kernel k R P n).symm).trans |
| ((homCyclesToExt k R P n).quotKerEquivOfSurjective |
| (homCyclesToExt_surjective k R P n))) |
| ((homCochainQuotientEquiv k R P n) |
| ((homCochainQuotientEquiv k R P n).symm (Submodule.Quotient.mk f))) = _ |
| rw [LinearEquiv.apply_symm_apply] |
| rfl |
|
|
| end ArExplicit.ExternalTensor |
|
|
| end |
|
|
| end OAI |
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|
