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Mathlib/AlgebraicGeometry Expand file tree Collapse file tree Original file line number Diff line number Diff line change @@ -652,6 +652,20 @@ theorem morphismRestrict_appLE {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (V
652652 rw [Scheme.Hom.appLE, morphismRestrict_app', Scheme.Opens.toScheme_presheaf_map,
653653 Scheme.Hom.appLE_map]
654654
655+ @[reassoc]
656+ theorem morphismRestrict_homOfLE_isoImage_ι_hom
657+ {X : Scheme.{u}} {U V : X.Opens} (e : U ≤ V) (W : Opens V) :
658+ X.homOfLE e ∣_ W ≫ (V.ι.isoImage W).hom =
659+ (U.ι.isoImage (X.homOfLE e ⁻¹ᵁ W)).hom ≫ X.homOfLE (X.ι_image_homOfLE_le_ι_image e W) := by
660+ simp [← cancel_mono (V.ι ''ᵁ W).ι]
661+
662+ @[reassoc]
663+ theorem isoImage_ι_inv_morphismRestrict_homOfLE {X : Scheme.{u}} {U V : X.Opens}
664+ (e : U ≤ V) (W : Opens V) :
665+ (U.ι.isoImage (X.homOfLE e ⁻¹ᵁ W)).inv ≫ X.homOfLE e ∣_ W =
666+ X.homOfLE (X.ι_image_homOfLE_le_ι_image e W) ≫ (V.ι.isoImage W).inv := by
667+ simp [← cancel_mono (V.ι.isoImage W).hom, morphismRestrict_homOfLE_isoImage_ι_hom]
668+
655669set_option backward.defeqAttrib.useBackward true in
656670set_option backward.isDefEq.respectTransparency false in
657671/-- Restricting a morphism onto the image of an open immersion is isomorphic to the base change
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