| import Mathlib.CategoryTheory.Abelian.Projective.Resolution | |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat | |
| import Mathlib.Algebra.Category.ModuleCat.Projective | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit | |
| open CategoryTheory CategoryTheory.Limits | |
| universe u | |
| variable {R : Type u} [Ring R] {M : ModuleCat.{u} R} | |
| variable (P : ProjectiveResolution M) | |
| theorem resolution_range_eq_ker : | |
| LinearMap.range (P.complex.d 1 0).hom = LinearMap.ker (P.π.f 0).hom := by | |
| have h := P.exact₀ | |
| rw [ShortComplex.moduleCat_exact_iff_range_eq_ker] at h | |
| exact h | |
| def resolutionCokernelEquiv : | |
| (P.complex.X 0 ⧸ LinearMap.range (P.complex.d 1 0).hom) ≃ₗ[R] M := | |
| (Submodule.quotEquivOfEq _ _ (resolution_range_eq_ker P)).trans | |
| ((P.π.f 0).hom.quotKerEquivOfSurjective | |
| ((ModuleCat.epi_iff_surjective (P.π.f 0)).mp inferInstance)) | |
| theorem resolutionCokernelEquiv_mk (x : P.complex.X 0) : | |
| resolutionCokernelEquiv P (Submodule.Quotient.mk x) = (P.π.f 0).hom x := rfl | |
| end ArExplicit | |
| end | |
| end OAI | |