Theorems · Theorem · algebraic geometry

AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso

∀ {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}
  {Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop},
  (RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => Q') →
    (∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S),
        Q f → ∀ (J : Ideal S) (x : J.IsPrime), Q' (Localization.localRingHom (Ideal.comap f J) J f ⋯)) →
      P f → ∀ (x : ↥X), Q' (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap f x))

Let Q be a property of ring maps that implies Q' on stalks. Then if the associated property of scheme morphisms holds for f, Q' holds on all stalks.

Defined in

Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Cited by

2 results in Mathlib

Foundations

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

Assumes

AlgebraicGeometry.HasRingHomProperty