| import OAI.Algebra.AuslanderReiten.Complete.CompleteComparison | |
| import Mathlib.Algebra.Module.Injective | |
| import Mathlib.LinearAlgebra.Isomorphisms | |
| namespace OAI | |
| namespace ArExplicit | |
| universe u | |
| variable {R : Type u} [Ring R] [Module.Injective R R] | |
| variable {K L M : Type u} | |
| variable [AddCommGroup K] [Module R K] [AddCommGroup L] [Module R L] | |
| variable [AddCommGroup M] [Module R M] | |
| theorem homExactAt_of_injective (e : K →ₗ[R] L) (d : L →ₗ[R] M) | |
| (hex : LinearMap.range e = LinearMap.ker d) : HomExactAt e d := by | |
| intro f hf | |
| have hk : LinearMap.ker d ≤ LinearMap.ker f := by | |
| rw [← hex] | |
| rintro _ ⟨x, rfl⟩ | |
| exact DFunLike.congr_fun hf x | |
| let u : LinearMap.range d →ₗ[R] R := | |
| ((LinearMap.ker d).liftQ f hk).comp d.quotKerEquivRange.symm.toLinearMap | |
| obtain ⟨g, hg⟩ := Module.Injective.extension_property R R (LinearMap.range d) M | |
| (LinearMap.range d).subtype Subtype.val_injective u | |
| refine ⟨g, ?_⟩ | |
| apply LinearMap.ext | |
| intro x | |
| have h := DFunLike.congr_fun hg (d.rangeRestrict x) | |
| change g (d x) = f x | |
| calc | |
| g (d x) = u (d.rangeRestrict x) := h | |
| _ = f x := by | |
| exact congrArg ((LinearMap.ker d).liftQ f hk) | |
| (d.quotKerEquivRange_symm_apply_image x (LinearMap.mem_range_self d x)) | |
| theorem ExactProjectiveComplex.totallyAcyclic | |
| {P : ChainComplex (ModuleCat.{u} R) ℤ} (hP : ExactProjectiveComplex P) : | |
| TotallyAcyclicComplex P where | |
| toExactProjectiveComplex := hP | |
| dualExact n := homExactAt_of_injective _ _ (hP.exactAt n) | |
| end ArExplicit | |
| end OAI | |