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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 | 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
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