Theorems · Theorem · algebraic geometry

AlgebraicGeometry.IsOpenImmersion.isPullback

∀ {U V X Y : AlgebraicGeometry.Scheme} (g : U ⟶ V) (iU : U ⟶ X) (iV : V ⟶ Y) (f : X ⟶ Y)
  [inst : AlgebraicGeometry.IsOpenImmersion iU] [inst_1 : AlgebraicGeometry.IsOpenImmersion iV],
  CategoryTheory.CategoryStruct.comp iU f = CategoryTheory.CategoryStruct.comp g iV →
    (TopologicalSpace.Opens.map f.base).obj (AlgebraicGeometry.Scheme.Hom.opensRange iV) =
        AlgebraicGeometry.Scheme.Hom.opensRange iU →
      CategoryTheory.IsPullback g iU iV f

Defined in: Mathlib.AlgebraicGeometry.OpenImmersion
Cited by: 4 results in Mathlib
Foundations: Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.IsOpenImmersion AlgebraicGeometry.IsOpenImmersion

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Cites44

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
CategoryTheory.Categoryproof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.Functor.objstatement and proof · cited by 19,642
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.Iso.invproof · cited by 6,514
CategoryTheory.Category.assocproof · cited by 6,433
CategoryTheory.CategoryStruct.idproof · cited by 6,235
Set.preimageproof · cited by 4,946
Set.rangeproof · cited by 4,705
CategoryTheory.ConcreteCategory.homproof · cited by 4,022
CategoryTheory.Isoproof · cited by 3,963

Cited by4

Results whose statement or proof uses this declaration.

AlgebraicGeometry.isPullback_morphismRestrictproof · cited by 10
AlgebraicGeometry.isPullback_inl_inl_coprodMapproof · cited by 2
AlgebraicGeometry.Scheme.Hom.isPullback_resLEproof · cited by 1
AlgebraicGeometry.diagonal_isAffine_iff_forall_isAffineOpen_infproof · cited by 0