Theorems · Theorem · algebraic geometry

AlgebraicGeometry.HasRingHomProperty.of_stalkMap

∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme}
  {Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop}
  [AlgebraicGeometry.HasRingHomProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},
  (RingHom.OfLocalizationPrime fun {R S} [CommRing R] [CommRing S] => Q) →
    (∀ (x : ↥X), Q (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap f x))) → P f

If Q is a property of ring maps that can be checked on prime ideals, the associated property of scheme morphisms can be checked on stalks.

Defined in

Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Cited by

1 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AlgebraicGeometry.HasRingHomProperty