Theorems · Definition · algebraic geometry
AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq
{X Y Z : AlgebraicGeometry.Scheme} →
(f : X ⟶ Z) →
(g : Y ⟶ Z) →
[H : AlgebraicGeometry.IsOpenImmersion f] →
[AlgebraicGeometry.IsOpenImmersion g] → Set.range ⇑f = Set.range ⇑g → (X ≅ Y)Two open immersions with equal range are isomorphic.
- Defined in
- Mathlib.AlgebraicGeometry.OpenImmersion
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- Quiver.Homstatement and proof · cited by 32,603
- Set.rangestatement and proof · cited by 4,705
- CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
- CategoryTheory.Isostatement · cited by 3,963
- TopCat.carrierstatement · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- ContinuousMapstatement · cited by 2,491
- CommRingCatstatement · cited by 2,333
- AlgebraicGeometry.PresheafedSpace.carrierstatement · cited by 2,020
- AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement · cited by 1,988
Cited by27
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.isoOfEqproof · cited by 33
- AlgebraicGeometry.Scheme.Hom.isoImageproof · cited by 32
- AlgebraicGeometry.Scheme.Hom.isoOpensRangeproof · cited by 17
- AlgebraicGeometry.isAffineOpen_opensRangeproof · cited by 13
- AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_facstatement · cited by 13
- AlgebraicGeometry.Proj.pullbackAwayιIsoproof · cited by 12
- AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_inv_facstatement · cited by 12
- AlgebraicGeometry.pullbackRestrictIsoRestrictproof · cited by 10
- AlgebraicGeometry.Scheme.Opens.isoOfLEproof · cited by 9
- AlgebraicGeometry.basicOpenIsoSpecAwayproof · cited by 7
- AlgebraicGeometry.morphismRestrictOpensRangeproof · cited by 7
- AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersionproof · cited by 7