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Theorems · Definition · algebraic geometry

AlgebraicGeometry.Scheme.Hom.isoOpensRange

{X Y : AlgebraicGeometry.Scheme} →
  (f : X ⟶ Y) → [inst : AlgebraicGeometry.IsOpenImmersion f] → X ≅ ↑(AlgebraicGeometry.Scheme.Hom.opensRange f)

If f : X ⟶ Y is an open immersion, then X is isomorphic to its image in Y.

Defined in
Mathlib.AlgebraicGeometry.Restrict
Cited by
17 results in Mathlib
Foundations
Depth 141 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
AlgebraicGeometry.IsOpenImmersion
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

AlgebraicGeometry.Scheme.Hom.isoOpensRange_hom_ι · cited by 6Hom.isoOpensRange_hom_ιAlgebraicGeometry.Scheme.Hom.partialIso · cited by 5Hom.partialIsoAlgebraicGeometry.Scheme.IsLocallyDirected.tAux · cited by 4IsLocallyDirected.tAuxAlgebraicGeometry.Scheme.Hom.isoOpensRange_inv_comp · cited by 3Hom.isoOpensRange_inv_compAlgebraicGeometry.Scheme.Hom.isoOpensRange_hom_ι_assoc · cited by 2Hom.isoOpensRange_hom_ι_a…AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux · cited by 2IsLocallyDirected.homOfLE…AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAt · cited by 1AlgebraicGeometry.exists_…AlgebraicGeometry.Scheme.Hom.partialIso_iso · cited by 1Hom.partialIso_isoAlgebraicGeometry.sourceLocalClosure.iff_forall_exists · cited by 1sourceLocalClosure.iff_fo…AlgebraicGeometry.Scheme.IsLocallyDirected.glueDataι_naturality · cited by 1IsLocallyDirected.glueDat…AlgebraicGeometry.Scheme.Hom.isLocallyConstructible_image · cited by 1Hom.isLocallyConstructibl…AlgebraicGeometry.Scheme.Hom.exists_mem_and_isIso_morphismRestrict_toNormalization · cited by 1Hom.exists_mem_and_isIso_…AlgebraicGeometry.Scheme.Hom.isoOpensRange_inv_comp_assoc · cited by 0Hom.isoOpensRange_inv_com…AlgebraicGeometry.Scheme.Hom.birationalOver · cited by 0Hom.birationalOverAlgebraicGeometry.Scheme.IsLocallyDirected.exists_of_pullback_V_V · cited by 0IsLocallyDirected.exists_…Quiver.Hom · cited by 32603Quiver.HomCategoryTheory.Iso · cited by 3963CategoryTheory.IsoAlgebraicGeometry.Scheme · cited by 2540AlgebraicGeometry.SchemeAlgebraicGeometry.IsOpenImmersion · cited by 476AlgebraicGeometry.IsOpenI…AlgebraicGeometry.Scheme.Opens.toScheme · cited by 433Opens.toSchemeAlgebraicGeometry.Scheme.Opens.ι · cited by 275Opens.ιAlgebraicGeometry.Scheme.Hom.opensRange · cited by 113Hom.opensRangeAlgebraicGeometry.IsOpenImmersion.isoOfRangeEq · cited by 16IsOpenImmersion.isoOfRang…Hom.isoOpensRangeCITED BYCITES

Cites8

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

Cited by19

Results whose statement or proof uses this declaration.