Theorems · Theorem · algebraic geometry

AlgebraicGeometry.IsZariskiLocalAtTarget.of_forall_source_exists_preimage

∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsZariskiLocalAtTarget P]
  {X Y : AlgebraicGeometry.Scheme} [P.RespectsRight AlgebraicGeometry.IsOpenImmersion]
  [P.HasOfPostcompProperty AlgebraicGeometry.IsOpenImmersion] (f : X ⟶ Y),
  (∀ (x : ↥X), ∃ U, f x ∈ U ∧ P (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι f)) →
    P f

Defined in: Mathlib.AlgebraicGeometry.Morphisms.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 151 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.IsZariskiLocalAtTarget CategoryTheory.MorphismProperty.RespectsRight CategoryTheory.MorphismProperty.HasOfPostcompProperty

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Cites33

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
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
Set.rangeproof · cited by 4,705
CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
TopCat.carrierstatement and proof · cited by 3,184
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
ContinuousMapstatement · cited by 2,491
CommRingCatstatement · cited by 2,333
CategoryTheory.MorphismPropertystatement and proof · cited by 2,179
TopologicalSpace.Opensstatement · cited by 2,040

Cited by1

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Scheme.IsQuasiAffine.of_forall_exists_mem_basicOpenproof · cited by 2