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Mathlib/AlgebraicGeometry/Modules/Tilde.lean

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@@ -398,19 +398,6 @@ set_option backward.isDefEq.respectTransparency false in
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instance : (tilde M).IsQuasicoherent :=
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(presentationTilde.{u} _ .univ (by simp) _ (Submodule.span_eq _)).isQuasicoherent
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set_option backward.isDefEq.respectTransparency false in
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lemma isIso_fromTildeΓ_of_presentation (M : (Spec R).Modules) (P : M.Presentation) :
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IsIso M.fromTildeΓ := by
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rw [isIso_fromTildeΓ_iff]
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let g := (tilde.functor _).preimage <| (tildeFinsupp _).hom ≫ P.relations.π ≫ kernel.ι _ ≫
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(tildeFinsupp _).inv
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let iso : cokernel ((tilde.functor R).map g) ≅ cokernel (P.relations.π ≫ kernel.ι _) := by
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refine cokernel.mapIso _ _ (tildeFinsupp _) (tildeFinsupp _) ?_
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simp only [g, (tilde.functor R).map_preimage]
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simp
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exact ⟨cokernel g, ⟨PreservesCokernel.iso (tilde.functor R) g ≪≫ iso ≪≫
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IsColimit.coconePointUniqueUpToIso (colimit.isColimit _) P.isColimit⟩⟩
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end IsQuasicoherent
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end AlgebraicGeometry

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