Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_iff_isOpen_singleton_asFiber

∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [AlgebraicGeometry.LocallyOfFiniteType f] {x : ↥X},
  AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt f x ↔ IsOpen {AlgebraicGeometry.Scheme.Hom.asFiber f x}

Defined in: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite
Cited by: 2 results in Mathlib
Foundations: Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.LocallyOfFiniteType

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites81

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 and proof · cited by 53,352
Quiver.Homstatement and proof · cited by 32,603
TopologicalSpaceproof · cited by 24,529
CategoryTheory.CategoryStruct.compproof · cited by 17,999
Algebraproof · cited by 11,388
Set.Elemproof · cited by 7,166
Set.imageproof · cited by 5,609
Set.preimageproof · cited by 4,946
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

Cited by2

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.isClopen_singleton_asFiberproof · cited by 1
AlgebraicGeometry.exists_isFinite_morphismRestrict_of_finite_preimage_singletonproof · cited by 1